research article calculation of the quantum-mechanical...
TRANSCRIPT
Research ArticleCalculation of the Quantum-MechanicalTunneling in Bound Potentials
Sophya Garashchuk Bing Gu and James Mazzuca
Department of Chemistry and Biochemistry University of South Carolina Columbia SC 29208 USA
Correspondence should be addressed to Sophya Garashchuk sgarashcmailchemscedu
Received 28 January 2014 Accepted 1 April 2014 Published 24 April 2014
Academic Editor Anton Kokalj
Copyright copy 2014 Sophya Garashchuk et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The quantum-mechanical tunneling is often important in low-energy reactions which involve motion of light nuclei occurringin condensed phase The potential energy profile for such processes is typically represented as a double-well potential along thereaction coordinate In a potential of this type defining reaction probabilities rigorously formulated only for unbound potentials interms of the scattering states with incomingoutgoing scattering boundary conditions becomes ambiguous Based on the analysisof a rectangular double-well potential a modified expression for the reaction probabilities and rate constants suitable for arbitrarydouble- (or multiple-) well potentials is developed with the goal of quantifying tunneling The proposed definition involves energyeigenstates of the bound potential and exact quantum-mechanical transmission probability through the barrier region of thecorresponding scattering potential Applications are given for several model systems including proton transfer in a HOndashHndashCH
3
model and the differences between the quantum-mechanical and quasiclassical tunneling probabilities are examined
1 Introduction
There is great interest in understanding the role of quantum-mechanical (QM) effects on reactivity at low temperature incomplex systems such as liquids [1] proteins [2] at surfaces[3] and so forth One example which motivated many theo-retical investigations [4] is the unusually high experimentalkinetic isotope effect (KIE = 81) of the transferring hydro-gendeuterium substitution on the reaction rate constant insoybean lipoxygenase-1 [5] attributed to QM tunneling asthe dominant reaction mechanism A few other exampleswhere tunneling defines the reaction rate are isomerization ofmethylhydroxycarbene [6] and reaction of hydroxyl radicalwith methanol [7] To understand the role of tunneling for aspecific system a typical approach is to construct an electronicpotential energy profile of a double-well character along one-dimensional reaction coordinate and to compute tunnel-ing probabilities using the quasiclassical Wentzel-Kramers-Brillouin (WKB) approximationThis was done for examplein recent studies of the barrier shape effect on the classicaland QM contributions to reactivity in enzyme catalysis [8ndash10] and in a study of the tunneling control of reactivity in
carbenes [6] Our goal here is to define and analyze QMreaction probabilities and thermal reaction rate constants inthe bound potentials in a more rigorous way than the WKBtheory There are two QM effects that should be considered(i) decay of the wavefunction inside a barrier for a range ofenergies treated approximately in the WKB approach and(ii) quantization of the energy levels due to the bound char-acter of a potential (The latter is not included in the WKBexcept via the harmonic zero-point energy of the reactantwell)
The ultimate information about a reactive process in gasphase is contained in the scatteringmatrix defined for a givencollision energy 119864 Its elements 119878
119903119901(119864) subscripts 119903 and 119901
label specific internal states of reactants and products aredefined in terms of the energy eigenstates with incom-ingoutgoing boundary conditions in the reaction coordinatefor a potential which vanishes at large reactant-product sep-aration [11] From the scattering matrix the cumulative reac-tion probability119873(119864) may be computed as
119873(119864) = sum
119903119901
10038161003816100381610038161003816
119878119903119901(119864)
10038161003816100381610038161003816
2
(1)
Hindawi Publishing CorporationJournal of eoretical ChemistryVolume 2014 Article ID 240491 11 pageshttpdxdoiorg1011552014240491
2 Journal of Theoretical Chemistry
Furthermore 119873(119864) determines the thermal reaction rateconstant as the Boltzmann average over the collision energy[12]
119896 (120573) =
1
2120587
int
infin
minusinfin119873(119864) 119890
minus120573119864119889119864
int
infin
minusinfin119890
minus120573119864119889119864
(2)
where 120573 is the inverse temperature 120573 = (119896119861119879)
minus1 (The atomicunits ℎ = 1 are used throughout) The cumulative reactionprobability can also be equivalently determined from expres-sions involving correlation functions of the flux operatoreigenstates evolved in time [12]These flux-based approachesnaturally connect to quantum and classical transition statetheories (beyond the scope of this paper)
Reactions in condensed phase are typically described bythe potential energy profile along the reaction coordinate of adouble-well character Therefore definitions of probabilitiesor rate constants related to scattering eigenstates are inappli-cable The flux-based approaches involving time-correlationfunctions are not formally guaranteed to reach a plateau (anddo not reach it in one-dimensional models) at long timeswithout inclusion of other degrees of freedom In the full-dimensional treatment the other ldquobathrdquo degrees of freedomare responsible for the wavefunction decoherence and forthe energy dissipation from the reactive degree of freedomThe plateau behavior of correlation functions is required fora rigorous definition of thermal reaction rate constants Ithas been argued though not formally proven that alreadyin two degrees of freedom oscillatory behavior of correlationfunctions is sufficiently damped with time to make the flux-based approaches work [13] Of course high-dimensionalmodels give more realistic representation of reactions incondensed phase but are often impractical If the couplingof the reaction coordinate to the bath degrees of freedom isweak or the bath degrees of freedom are not involved in 119905rearrangement of atoms then a reliable estimate of quantumtunneling probability as a characteristic of the barrier and ofthe potential energy profile along the reaction coordinate isimportant in itself Given that significance of QM tunnelingis the highest in the regime of just a few reactantproducteigenstates contributing to reactivity a simple expression forthe QM tunneling probabilities which can be implementedby solving the time-independent Schrodinger equation isuseful In the time domain treating a chemical reaction asan event proceeding from reactants to products rather thanthe transition state flux can be useful as well especially if thepotential depends on time for example mimicking the effectof substrate motion in reactions on surfaces [14]
For bound potentials the QM reaction probabilities andrate constants are usually estimated using the transition statetheory [15] or using the quasiclassicalWKB expression for thetunneling probability evaluated at the zero-point energy ofthe reactant well 119864 = 119864
0[16]
119879
QC(119864) = exp(minus2int
1199092
1199091
radic2119898 (119881 (119909) minus 119864)119889119909)
119881 (1199091) = 119881 (119909
2) = 119864
(3)
The transition state estimate is accurate for a single barrier atenergies comparable to its height while the WKB expressionis accurate at low energies when transmission from productsto reactants is justifiably neglected However the product re-gion and a possibility of a reverse reaction should be incor-porated for a more rigorous treatment the energy level split-tings measured spectroscopically are due to tunneling anddepend on all regions of the potential
Even though the approximate treatments of tunneling canbe surprisingly accurate the search for more conceptuallyrigorous expressions led to the development of less traditionalapproaches such as the stationary state decomposition forquadratic potentials [17] via the quantum Hamilton-Jacobiequation adapted for energy eigenstates [18 19] the exte-rior complex scaling method [20] and more recently thequantum instanton theory [21] ring polymer dynamics [22]and perturbative ldquobeyond instantonsrdquo correction to theWKBapproximation [23] The quantum tunneling is known to bevery sensitive to the barrier shape and energy as shownfor example in the context of enzyme catalysis by Hayand coworkers [9] Thus to assess the importance of QMtunneling we propose an adaptation of definitions of the QMtunneling probability and reaction rate constant based onexact energy levels of the bound system and exact QM tun-neling through the barrier of the corresponding scatteringsystem The expressions are obtained from the analysis ofa rectangular double-well potential (Section 2) and its unam-biguous mapping onto a scattering problem (For generalpotentials there is some ambiguity associated with the def-inition of the reactant and product regions) After modelapplications to rectangular and piecewise quadratic potentialwe use the approach to compute KIE on the reaction rateconstant of protondeuteron transfer in a model HOndashHndashCH3along a linear reaction path with realistic potential
(Section 3) Section 4 concludes
2 Formalism
The modified expression for the reaction rate constants isbased on the common functional form of the energy eigen-states for the open scattering system and the double-wellsystem (or any sequence of wells and barriers) consistingof rectangular potential steps Let us consider a one-dimen-sional open scattering system with a potential being constantin the asymptotic regions as 119909 rarr plusmninfin and arbitrary in theinteraction region referred to as the ldquobarrierrdquo region
119881 (119909) =
0 119909 lt 1199091reactant region
119881119887 (119909) 119909
1le 119909 le 119909
2barrier region
1198810
119909 gt 1199092product region
(4)
Describing scattering in the energy domain the energy eigen-states with appropriate boundary conditions such as a wave-function incoming from the left in the direction from
Journal of Theoretical Chemistry 3
reactants to products have the following asymptotic form for119864 gt 119881
0(119898 is the particle mass)
120595119864(119909)
=
exp (120580119896119903119909)
+119903 (119864) exp (minus120580119896119903119909) 119909 lt 1199091 119896119903=radic2119898119864
119905 (119864) exp (120580119896119901119909) 119909 gt 119909
2 119896119901= radic2119898 (119864 minus 119881
0)
(5)
The wavefunction in the barrier region
120595119864(119909) = 119891
119864(119909) 119909
1le 119909 le 119909
2 (6)
is an unspecified solution to the one-dimensional Schro-dinger equation which smoothly connects solutions in thetwo asymptotic regions and does not explicitly enter the ex-pression for the transmission probability 119879(119864)
119879 (119864) equiv 119873 (119864) = |119905 (119864)|
2119896119901
119896119903
(7)
For the double-well potential the energy eigenstateswith scat-tering boundary conditions [11] are undefined making (5)and (7) inapplicable Weiner [17 24] extended the conceptof scattering states to a piecewise quadratic double-wellpotential defining the incoming and outgoing boundary con-ditions in the parabolic wells These asymptotically divergingsolutions were determined from the quantum Hamilton-Jac-obi equation [25] resulting in impractical implementationbesides being limited by the functional form of the potential
Dynamics the probability of finding a particle initiallylocalized in the reactant well on the product side oscillatesin time without converging to a well-defined value unlessenergy dissipation and interaction of the reactive systemswith the environment are fully included
We will consider the rectangular double-well potentialwhich is easy to analyze in terms of the plane-wave solutionsFor such a potential 119881 shown in Figure 1
119881 (119909) =
infin 119909 le 1199090 119909 ge 119909
3walls
0 1199090lt 119909 lt 119909
1reactant region
119881119887 1199091le 119909 le 119909
2barrier region
1198810 1199092lt 119909 lt 119909
3product region
(8)
the energy eigenstates and the corresponding transmissionprobabilities for the open scattering system defined by (4)and for the full double-well potential of (8) are the sameat the energy levels 119864
119899of the full potential in the region
of finite 119881 Another feature of this bound potential is thatthe reactant and product regions are uniquely defined Thusthe expression for the thermal reaction rates of (2) can bemodified (i) using the transmission probability in the opensystem having the same barrier region as the double-wellpotential and (ii) using the energy eigenstates of the double-well potential
119896 (120573) =
1
2120587
sum119899120588119899119879 (119864119899) 119890
minus120573119864119899
sum119899120588119899119890
minus120573119864119899
(9)
minus4 minus2 0 2 4
Coordinate x [a0]
Ener
gyE
[har
tree]
4
2
0
x0 x1 x2 x3
n = 0dwn = 1dwn = 2dw
n = 0 rwn = 1 rw
Ψ1
Ψ0
Ψ2
Figure 1 The asymmetric rectangular double-well potential ofSection 31 Its three lowest eigenfunctions are shown as dashesThehorizontal lines mark the corresponding three energy levels (labeled119899 = 0 1 2 dw) and also two lowest energy levels of the reactant wellwith infinitely high walls (labeled 1198991015840 = 0 1 rw)
The summation goes over the energy eigenstates 120601119899of the
bound system
119867120601119899= 119864119899120601119899 (10)
and 120588119899is the probability of finding a particle on the reactant
side for each state 119899
120588119899= ⟨120601119899
10038161003816100381610038161003816
119875119903
10038161003816100381610038161003816
120601119899⟩ = int
119909119903
minusinfin
1003816100381610038161003816
120601119899(119909)
1003816100381610038161003816
2119889119909 (11)
Operator 119875119903projects a wavefunction onto the reactant region
defined as 119909 lt 119909119903
In case of the rectangular double well of (8) the projec-tions of the eigenstates on the reactant well are unambiguous(119909119903= 1199091)
120588119899= int
1199091
1199090
1003816100381610038161003816
120601119899 (119909)
1003816100381610038161003816
2119889119909 (12)
Generally a definition of 119875119903requires specification of the
reactant region Some possibilities are as follows (i) 119909119903is a
position of the barrier top for any 119899 and (ii) 119909119903is a classical
turning point that is 119909(119899)119903
is a root of 119881(119909(119899)119903) = 119864
119899in the
reactant wellThe first option is reasonable for simple barrierswith a singlemaximum while the second one is more generaland consistent with the WKB tunneling expression
To summarize the formalism for general potentials withreactant and product well regions the transmission probabil-ity is defined by the properties of the open scattering systemthat is by the barrier with the 119881(119909) set to constants beyondthe reactantproduct well minima 119909
0and 119909
3 respectively
4 Journal of Theoretical Chemistry
minus2 minus1 0 1 2
0
05
x1 xL xR x2
Vb
V0
Pote
ntia
l [Eh
]
Coordinate [a0]
Figure 2 A double-well potential (solid line) and the correspondingto it scattering potential (dash) Coordinates 119909
1and 119909
2are the
positions of the minima in the reactant and product regionsrespectively and also define the barrier region 119909
119871and 119909
119877are
inflection points of the potential given by (20)
This is shown in Figure 2 for a piecewise quadratic continuouspotential The double-well potential simply specifies whichenergies out of the continuum of the scattering energy statesto include into the Boltzmann averaging in (9) For anasymmetric potential the reactant well may have eigenstatesbelow the asymptote119881
0of the product wellThose eigenstates
should be included into the Boltzmann averaging in (9)with the zero transmission probability The same argumentsand expressions are applicable to more complicated boundpotentials as long as the reactant and product regions arespecified
3 Examples and Discussion
In this section we examine reaction probabilities and rateconstants computed from (9) for several model systems withthe emphasis on comparison of the QM andWKB tunnelingprobabilities We also consider better-than-WKB approxi-mations relevant to each model a ldquominimalisticrdquo two-statedescription for a symmetric rectangular well (Section 31)a mixed QCQM calculation for an asymmetric well (Sec-tion 32) and a parabolic barrier approximation for a piece-wise quadratic potential (Section 33) Accuracy of transmis-sion probabilities and the energy level splittings followingfrom the quasiclassical WKB expression is examined Exam-ple of the KIE calculation for a model proton transfer systemis described in Section 34
31 Comparison of the QMWKB and Two-State Model Prob-abilities for a Symmetric Double Well Let us apply (9) toa symmetric rectangular double well and compare reactionprobabilities computed using the exact QM and approximateenergy levels and transmission probabilities For a general
rectangular barrier given by (8) without the infinitely highwalls the transmission probability is
119879 (119864)
=
4119896119903119896
2
119887119896119901
(119896119903+ 119896119901)
2
119896
2
119887+[119896
2
119903119896
2
119901+119896
2
119887(119896
2
119887minus 119896
2
119903minus 119896
2
119901)] sin2 (119896
119887119871)
for 119864 gt 119881119887
4119896119903119896
2
119887119896119901
(119896119903+ 119896119901)
2
119896
2
119887+[119896
2
119903119896
2
119901+119896
2
119887(119896
2
119887+ 119896
2
119903+ 119896
2
119901)] sinh2 (119896
119887119871)
for 1198810lt 119864 lt 119881
119887
(13)
In the expression above 119896119887= radic2119898|119881
119887minus 119864| and 119871 = 119909
2minus 1199091
is the barrier width 119896119903and 119896
119901are given in (5) Analytical
expression for QM transmission probability 119879(119864) allowshigh precision calculations performed with Maple [26] Theparticle mass is119898 = 1 a u
We are considering a symmetric barrier of fixed height119881119887
and variablewidthThewidth of thewells119889119889 = 1199091minus1199090= 1199093minus
1199092 is fixed at 119889 = 2119886
0 The barrier height 119881
119887is set to 4119864
ℎ The
barrier width 119871 is in the range of 119871 = [025 80]1198860 In the limit
of infinitely wide barrier the reactant and product wells havetwo bound states The transmission probabilities 119879(119864) andthe energy level splitting Δ associated with both asymptoticenergy levels are computed by considering two lowest pairsof energy eigenstates of the finite-width potentials ExactQM transmission probabilities are compared with thoseobtained using the quasiclassical WKB approach (labeledQC) and the two-state description (labeled 2119904) of this systemThe WKB tunneling probabilities are used in comparisoneven though the WKB method is inaccurate for potentialswith discontinuous derivatives because of its routine use intunneling calculations for chemical systems
In a symmetric potential the ground and first excitedstates have evenodd symmetry so the half-sumhalf-difference of the two states represents a particle located on thereactantproduct side Therefore for consistent comparisonof exact and approximate expressions the QM transmission119879
QM is defined as
119879
QM=
1205880119879 (1198640) + 1205881119879 (1198641)
1205880+ 1205881
(14)
to account for localization of the ldquoinitialrdquo wavefunction inthe reactant well The difference between (9) and (14) isthat the latter neglects the effect of temperature on theoccupation of 119899 = 0 and 119899 = 1 states This omission isaccurate in the limit of small energy level splittings Equation(14) is also consistent with the two-state representation of awavefunction in the basis formed by the ground eigenstatesof reactant and product wells 120601
119903and 120601
119901 respectively These
functions are the lowest energy eigenfunctions of the reactantand product wells in the limit of an infinitely wide barrier(The eigenfunctions of the isolated wells with infinitely highwalls cannot be used for a rectangular potential becausesuch eigenfunctions will have zero overlap) In the two-state
Journal of Theoretical Chemistry 5
6 80
0
05
1
15
2
25
42Barrier width L [a0]
TQCTQM
T2sTQM
Figure 3 Tunneling in the symmetric rectangular double-wellpotential of Section 31 Ratios of approximate transmission prob-abilities that is the quasiclassical WKB (squaresdash) and the two-state representation (circlessolid lines) to the exact QM probabilityare given as functions of the barrier width
representation the energies of the ground and excited statesare the generalized eigenvalues of the 2-by-2 HamiltonianmatrixHwith the overlapmatrix S For a symmetric potentialthe matrix elements are
ℎ11= ℎ22= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119903⟩ ℎ12= ℎ21= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119901⟩ (15)
11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)
H1205952119904119894= 119864
2119904
119894S1205952119904119894119894 = 0 1 (17)
The energy level splitting is Δ2119904 = 11986421199041minus 119864
2119904
0
Following [24] we use the quasiclassical relation betweenthe energy level splitting and the tunneling probability
1198641minus 1198640= Δ =
21198640
120587
radic119879 (1198640) (18)
with the ground state energy used instead of the frequency ofthe harmonic reactant well In the two-state representationthe energy level splitting Δ2119904 following from (17) yields anestimate for the tunneling 1198792119904 The same (18) gives estimatesof the quasiclassical energy level splitting ΔQC once thequasiclassical WKB tunneling probability 119879QC is computedfrom (3) Two pairs of states of the full potential 119899 = 0 119899 = 1and 119899 = 2 119899 = 3 are analyzed the higher energy states 119899 = 2and 119899 = 3 are described in the basis of the first excited statesof the reactant and product wells
Table 1 lists the ground and first excited energy levels ob-tained exactly using quasiclassical WKB approximation andwithin the two-state approximation labeled QM QC and2119904 respectively are listed in the upper half of Table 1The samequantities for the next highest pair of levels (Δ = 119864
3minus1198642) are
given in the lower half of Table 1 In the quasiclassical WKBtreatment the energy levels are defined by the width of the
1 2 3 4
Energy [Eh]
00001
1e minus 08
Tunn
elin
gT
(E)
L = 2 QML = 2 QC
E0
E1
Figure 4 Transmission through the symmetric rectangular barrierof Section 31 of the width 119871 = 2119886
0 Exact QM and QC WKB
probabilities are shown as a solid line and a dash respectively Theenergies of the ground and first excited states for the well width119889 = 2119886
0are indicated with dot-dashes
reactant well and therefore do not depend on the barrierwidth The ground state energy in the two-state approx-imation is remarkably accurate even for narrow barrierswhile the accuracy of 119899 = 2 level deteriorates for smallerbarrier widths The ratios of the quasiclassical and two-stateestimates of tunneling probability to the exact QM values asa function of the barrier width are plotted in Figure 3 for thelowest pair of states The agreement between 1198792119904 and 119879QM isexcellent for the barrier width 119871 gt 10 119886
0 The quasiclassical
WKB results underestimate the tunneling probability by afactor of 2This discrepancy can be understood by comparingquasiclassical WKB and QM tunneling at the energies ofthe asymptotic eigenstates shown in Figure 4 for the barrierwidth 119871 = 2119886
0 The transmission probabilities at these ener-
gies differ by a factor of 2 For a wider well of the width119889 = 4119886
0(not shown here) the ground state energy happens to
be near the intersection of the QM and quasiclassical WKBtunneling curves at 119864 = 02599119864
ℎ which coincidentally
yields much better agreement than the results shown inFigure 3 Therefore the accuracy of the quasiclassical resultsdepends on the reactant well width even if the accuracy ofthe quasiclassical tunneling probability does not At very lowenergies the discrepancy between the QM and QC tunnelingis in orders of magnitude but fortunately the energy regimebelow the zero-point energy of reactants does not contributeto the tunneling in a double-well system
To summarize the two-state tunneling probabilities forthe lowest pair of eigenstates are remarkably accurate exceptfor very narrow barriers and may provide useful estimatesif the rest of the states are of much higher energy Thebarrier can be considered sufficiently wide when the overlapof the reactant and product well eigenstates ⟨120601
119901|120601119903⟩ is less
than a few percent (5 percent in the current example)The QC expression (18) relating the energy level splitting
6 Journal of Theoretical Chemistry
Table1Energylevelssplittin
gsand
tunn
elingp
robabilityfor
the119899=01and119899=23pairofstatesofthesym
metric
rectangu
lard
ouble-wellpotentia
lofSectio
n31Th
ereactantp
rodu
ctstates
overlap
ofthetwo-stated
escriptio
nisgivenin
thelastcolum
nTh
eenergylevelsandlevelsplittings
areg
iven
in119864ℎN
umbersin
parenthesesa
rethep
owerso
f10
119871[1198860]
119864
QM0
Δ
QM
119879
QM
119864
QC0
Δ
QC
119879
QC
119864
2s 0Δ
2119904
119879
2119904
⟨120601119901|120601119903⟩
Forthe119899=01pairof
energy
states
808800
1919
(minus9)
1205
(minus17)
08800
1174
(minus9)
439
(minus18)
08800
1913
(minus9)
117(minus17)
3554(minus9)
408800
4192(minus5)
5752(minus9)
08800
2564(minus5)
2095(minus9)
08800
4210(minus5)
564
6(minus9)
440
8(minus5)
208770
6196(minus3)
1257
(minus4)
08800
3790(minus3)
4577(minus5)
08769
6224(minus3)
1243
(minus4)
4035(minus3)
108422
00753
00185
08800
4608(minus3)
6765(minus3)
08425
7564
(minus2)
1989
(minus2)
3398(minus2)
05
07433
02618
02105
08800
01607
8225(minus2)
07483
02637
03064
00922
025
060
9804881
05900
08800
03000
02868
06289
04906
1502
01476
Forthe119899=23pairof
energy
states
833053
1099
(minus4)
1481
(minus8)
33053
1690
(minus4)
6450(minus9)
33053
0974(minus4)
2141(minus9)
4712(minus4)
432993
1227
(minus2)
1848
(minus4)
33053
1886
(minus2)
8031(minus5)
32999
1087
(minus2)
2676(minus5)
2790(minus2)
232477
01334
2144(minus2)
33053
01992
8962(minus3)
32472
0114
63071(minus3)
01644
131337
04856
02267
33053
064
7400947
30860
03487
00315
03225
05
300
6810
078
06166
33053
11672
03077
28262
05329
00877
03905
025
29086
14864
08529
33053
15672
05547
25604
06101
01401
03961
Journal of Theoretical Chemistry 7
Table 2 Thermal rate constants 119896(119879) for the asymmetric rectangular double well of Section 32 The first column lists the temperature thatis 119896119861119879 in units of the barrier height119881
119887The thermal rate constants obtained using (9) using exact energy levels and QM and QC transmission
probabilities are given in the second and third columns respectively The last column contains the thermal rate constants obtained from QCtunneling probability and the energy levels of the reactant potential with infinitely high walls
119896119861119879119881119887
119879
QM119864
QM119899
119879
QC119864
QM119899
119879
QC119864
QC119899
001 691 (minus13) 265 (minus13) 738 (minus15)002 101 (minus8) 380 (minus9) 778 (minus10)004 123 (minus6) 464 (minus7) 252 (minus7)01 399 (minus5) 267 (minus5) 196 (minus5)02 131 (minus3) 197 (minus3) 241 (minus3)04 195 (minus2) 266 (minus2) 364 (minus2)10 0134 0159 019820 0286 0320 037140 0450 0483 0533
and tunneling is inaccurate for the higher energy pairs ofldquosplitrdquo levelsThe accuracy of the QC approximation dependson both the accuracy of QC tunneling probability anddiscrepancy between the energy levels of the full potentialand of the asymptotic reactant well For one-dimensionalsystems the use of exact energy eigenstates of the full potentialrather than those of an isolated reactant well in (9) ismore appropriate even with approximate calculation of thetunneling probability
32 Asymmetric Rectangular Barrier Now let us consider anasymmetric rectangular double-well potential given by (8)and sketched in Figure 1 There are 7 states with energiesbelow the barrier top The ground and first excited statesare localized on the reactant and product sides respectivelyThe eigenstates of the full potential with even quantum num-bers correlate with the eigenstates of the reactant well thedifference in energies of the full and asymptotic potentialsincreases with the quantum number The parameter valuesfor the potential are 119881
119887= 40119864
ℎ 1198810= 04119864
ℎ and 119871 = 10119886
0
The well width is 119889 = 401198860 The particle mass is119898 = 1 a u
Exact QM energy levels and eigenfunctions needed tocompute projections on the reactant well are obtained aseigenvalues and eigenvectors of the Hamiltonian matrix inthe Colbert-Miller discrete variable representation [27]
Exact QM transmission probability is given by (13) Qua-siclassical transmission probability is defined by (3) and set to(1) for energies above the barrier top The exact and approxi-mate rate constants computed from (9) are listed inTable 2 fortemperatures measured in units of the barrier top 119881
119887 Results
of the fully QM fully QC and mixed QCQM (QC trans-mission evaluated at exact QM energy levels) descriptions areshown in Figure 5 and in Table 2 At the lowest temperatureequivalent to 1 of the barrier height the QC rate constantis 93 times lower than the QM result while the mixedQCQM estimate is only 2 times lower At high temperatureswhen the high energy eigenstates significantly contribute tothe reaction rate constant the discrepancy between exactand approximate rate constants becomes smaller with themixed QCQM estimate being closer to the exact result at allenergies One concludes that generally the accuracy of 119896(119879)
0 005 01 015 02
Rate
cons
tant
Exact QM QM levelsQC T(E)Reactant levelsQC T(E)
k(T
)
0
02
04
06
08
Temperature [Vb]
0 10 20 30
00001
1e minus 08
1e minus 12
1e minus 16
Figure 5 Transmission through the asymmetric rectangular barrierof Section 32 The thermal reaction rate constant is given as afunction of temperature measured in units of the barrier height[119896119861119879119881119887] where119881
119887= 4119864ℎThe rate constants are obtained using the
exact QM transmission probability and energy levels (circlessolidline) the QC transmission probability evaluated at the exact QMenergy levels (trianglesdot-dash) and at the QC energy levels of theisolated reactant well (squaresdash) The insert shows 119896(119879) on thelinear scale for wide range of temperatures
depends both on the accuracy ofQC transmission probability119879
QC and on the difference between exact and asymptotic-wellenergy levels Using exact QM energy levels and functions inconjunction with quasiclassical transmission probabilities ismore accurate than the fully QC description
Note that for the rectangular potential considered here119879
QC(119864) = 1 for 119864 gt 119881
119887is inaccurate because QC approxi-
mation does not describe above-the-barrier reflection leadingto oscillations of 119879QM For general potentials the reactionprobability may be computed using for example the time-dependent QM wave packet approach [28] used in the next
8 Journal of Theoretical Chemistry
example yielding reaction probabilities for a range of ener-gies from a single calculation or with a time-independentmethod for a few energy values
33 Piecewise Quadratic Potential Next we examine a piece-wise quadratic potential sketched in Figure 2 The reactantwell and barrier are quadratic functions in 119909 The reactionrate constants are obtained using the QM and quasiclassicalWKB expressions and using the analytical QM transmission
119879
QMpar (119864) = (1 + exp(minus
2120587119864
119908119887
))
minus1
119908119887=radic
119896119887
119898
(19)
through a parabolic barrier Equation (19) is more accuratethan the QC expression near the barrier top [16] A continu-ous piecewise quadratic potential is defined as follows
119881 (119909)
=
119896(119909 minus 1199091)
2
2
119909 lt 119909119871reactant region
minus
119896119887119909
2
2
+ 119881119887
119909119871le 119909 le 119909
119877barrier top region
119896(119909 minus 1199092)
2
2
+ 1198810119909 gt 119909
119877product region
(20)
with the proper choice of parametersmdash119881119887 1198810 119909119871 and 119909
119877mdash
the potential and its first derivative is continuous function at119909 = 119909
119871and 119909 = 119909
119877 Here a symmetric double well 119881
0= 0
is considered the remaining parameter values are 1199092= minus1199091
= 11198860 119896 = 119896
119887= 80119864
ℎ119886
minus2
0 119881119887= 20119864
ℎ and 119909
119877= minus119909119871= 05119886
0
The particle mass is 119898 = 1 a u (If the Hamiltonian werescaled by the mass of the proton 119881
119887would be equivalent to
03 eV or 7 kcalmol)The barrier parameters are chosen suchthat there are only few energy levels under the barrier topwhich is typical for a chemical reaction at low temperature
The time-dependent wave packet correlation approach[28] has been used to calculate QM transmission probability119879(119864) = |119878
119903119901(119864)|
2 entering (9) since we examine dependenceof the reaction rate on the accuracy of the energy levels usedin (9) The scattering matrix element 119878
119903119901(119864) describes trans-
mission from reactants to products
119878119903119901(119864) =
(2120587)
minus1
120578
lowast
119901(119864) 120578119903 (
119864)
int
infin
minusinfin
⟨120601
minus
119901
100381610038161003816100381610038161003816
119890
minus120580119905100381610038161003816100381610038161003816
120601
+
119903⟩ 119890
120580119864119905119889119905 (21)
The subscripts 119903119901 refer to reactantproduct reaction chan-nels and plusmn refers to the asymptotically incomingoutgoingwave relative to the barrier An incoming wave packet |120601+
119903⟩
taken as a Gaussian function in the left asymptotic regionof 119881 is evolved in time in the ldquounfoldedrdquo potential shownin Figure 2 with a dash Another Gaussian function |120601minus
119901⟩
is placed in the product region of the potential The time-dependent overlap of evolving |120601+
119903⟩ with the stationary ⟨120601minus
119901|
or correlation function 119862(119905) = ⟨120601
minus
119901(119909 0) | 120601
+
119903(119909 119905)⟩ is
computed and Fourier-transformed into the energy domain
The denominator in (21) accounts for the distributions ofenergy eigenstates in the reactant and product wave packetsat time 119905 = 0 these distributions depend on the wave packetlocalization and kinetic energy
120578119903119901(119864) = radic
119898
2120587119901
int
infin
minusinfin
119890
minus120580119901119909120601
plusmn
119903119901(119909) 119889119909 119901 =
radic2119898119864
(22)
A single wave packet propagation accomplished on a gridusing the split-operator method [29 30] gives transmissionprobability for a range of energies represented in the reac-tantproduct wave packets which is convenient if multipleeigenstates contribute to the rate constant At low energiesit is hard to converge transmission probability using time-dependent dynamics methods but the energies below thezero-point energy of the bound potential do not contributeto 119896(119879)The exact QM energy levels and projections of eigen-functions on the reactant well are performed as in Section 32There are five energy levels below the barrier top but manymore states contribute to 119896(119879) at higher temperatures
The energy-resolved exactQM parabolic barrier approxi-mation and QC transmission probabilities are shown in Fig-ure 6(a) Tunneling probabilities are underestimated in theQC approximation while the parabolic barrier expressiongives accurate results for energies above 119881
1198875 = 4119864
ℎ(the
barrier is in fact parabolic in this energy range) The discrep-ancy grows at energy below 4119864
ℎ This energy range however
is below the zero-point energy and does not contributeto the thermal rate constants shown in Figure 6(b) Theapproximate rate constants differ from exact QM results byat most 25ndash30
34 Proton Transfer As a chemically relevant model we con-sider the proton transfer in the HOndashHndashCH
3system for a
constrained collinear OndashHndashC geometry In this model sys-tem the proton is transferred from a donor carbon to ac-ceptor oxygenThe potential energy surface is obtained fromdensity function theory (DFT) electronic structure calcula-tions in particular at the B3LYP6-31G(dp) level of theory[31] The energies were compared to those obtained at theCCSD(T)aug-cc-pVDZ theory level for the same geome-tries and were found in excellent agreement In this sys-tem collinear donor-proton-acceptor arrangement is a goodapproximation to the fully optimized reaction path as deter-mined by a set of constrained geometry optimizations Inthese optimizations119877OH was fixed to a value in the range 08ndash20 A the average deviation of angCHO from linearity was lessthan 2∘ and the maximum deviation was 10∘
One-dimensional potential energy surfaces of 40 pointswere generated as functions of 119877OH for a fixed 119877CO distanceProton transfer on three surfaces for119877CO = 27 28 and 29 Ais analyzed below The surfaces were parametrized as sixthdegree polynomials in 119877OH The potential energy curvesshown in Figure 7(a) have characteristic asymmetric double-well shape
We examine the effect of donor-acceptor distance 119877COon the tunneling rate constants using QM and QC transmis-sion probabilities and exact and approximate energy levels
Journal of Theoretical Chemistry 9
00001
001
1
QMQCParabolic
0 10 20 30
1
1e minus 08
1e minus 06
Energy [Eh]
T(E
)
(a)
0 05 1 15 20
05
Rate
cons
tant
0 01 02
00001
001
Temperature [Vb]
(b)
Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896
119861119879119881119887] where 119881
119887= 20119864
ℎ
0
10
20
30
40
50
08 1 12 14 16 18 2
272829
Ener
gy (m
E h)
ROH (A)
(a)
0 500 1000 1500
103
102
101
100
KIE
Temperature (K)
(b)
Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO
distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896
119867119896119863(KIE) using exact QM formulation
in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top
The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896
119867119896119863should be largest at low
temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Carbohydrate Chemistry
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Chemistry
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Advances in
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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
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Chromatography Research International
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Analytical ChemistryInternational Journal of
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Quantum Chemistry
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ElectrochemistryInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
2 Journal of Theoretical Chemistry
Furthermore 119873(119864) determines the thermal reaction rateconstant as the Boltzmann average over the collision energy[12]
119896 (120573) =
1
2120587
int
infin
minusinfin119873(119864) 119890
minus120573119864119889119864
int
infin
minusinfin119890
minus120573119864119889119864
(2)
where 120573 is the inverse temperature 120573 = (119896119861119879)
minus1 (The atomicunits ℎ = 1 are used throughout) The cumulative reactionprobability can also be equivalently determined from expres-sions involving correlation functions of the flux operatoreigenstates evolved in time [12]These flux-based approachesnaturally connect to quantum and classical transition statetheories (beyond the scope of this paper)
Reactions in condensed phase are typically described bythe potential energy profile along the reaction coordinate of adouble-well character Therefore definitions of probabilitiesor rate constants related to scattering eigenstates are inappli-cable The flux-based approaches involving time-correlationfunctions are not formally guaranteed to reach a plateau (anddo not reach it in one-dimensional models) at long timeswithout inclusion of other degrees of freedom In the full-dimensional treatment the other ldquobathrdquo degrees of freedomare responsible for the wavefunction decoherence and forthe energy dissipation from the reactive degree of freedomThe plateau behavior of correlation functions is required fora rigorous definition of thermal reaction rate constants Ithas been argued though not formally proven that alreadyin two degrees of freedom oscillatory behavior of correlationfunctions is sufficiently damped with time to make the flux-based approaches work [13] Of course high-dimensionalmodels give more realistic representation of reactions incondensed phase but are often impractical If the couplingof the reaction coordinate to the bath degrees of freedom isweak or the bath degrees of freedom are not involved in 119905rearrangement of atoms then a reliable estimate of quantumtunneling probability as a characteristic of the barrier and ofthe potential energy profile along the reaction coordinate isimportant in itself Given that significance of QM tunnelingis the highest in the regime of just a few reactantproducteigenstates contributing to reactivity a simple expression forthe QM tunneling probabilities which can be implementedby solving the time-independent Schrodinger equation isuseful In the time domain treating a chemical reaction asan event proceeding from reactants to products rather thanthe transition state flux can be useful as well especially if thepotential depends on time for example mimicking the effectof substrate motion in reactions on surfaces [14]
For bound potentials the QM reaction probabilities andrate constants are usually estimated using the transition statetheory [15] or using the quasiclassicalWKB expression for thetunneling probability evaluated at the zero-point energy ofthe reactant well 119864 = 119864
0[16]
119879
QC(119864) = exp(minus2int
1199092
1199091
radic2119898 (119881 (119909) minus 119864)119889119909)
119881 (1199091) = 119881 (119909
2) = 119864
(3)
The transition state estimate is accurate for a single barrier atenergies comparable to its height while the WKB expressionis accurate at low energies when transmission from productsto reactants is justifiably neglected However the product re-gion and a possibility of a reverse reaction should be incor-porated for a more rigorous treatment the energy level split-tings measured spectroscopically are due to tunneling anddepend on all regions of the potential
Even though the approximate treatments of tunneling canbe surprisingly accurate the search for more conceptuallyrigorous expressions led to the development of less traditionalapproaches such as the stationary state decomposition forquadratic potentials [17] via the quantum Hamilton-Jacobiequation adapted for energy eigenstates [18 19] the exte-rior complex scaling method [20] and more recently thequantum instanton theory [21] ring polymer dynamics [22]and perturbative ldquobeyond instantonsrdquo correction to theWKBapproximation [23] The quantum tunneling is known to bevery sensitive to the barrier shape and energy as shownfor example in the context of enzyme catalysis by Hayand coworkers [9] Thus to assess the importance of QMtunneling we propose an adaptation of definitions of the QMtunneling probability and reaction rate constant based onexact energy levels of the bound system and exact QM tun-neling through the barrier of the corresponding scatteringsystem The expressions are obtained from the analysis ofa rectangular double-well potential (Section 2) and its unam-biguous mapping onto a scattering problem (For generalpotentials there is some ambiguity associated with the def-inition of the reactant and product regions) After modelapplications to rectangular and piecewise quadratic potentialwe use the approach to compute KIE on the reaction rateconstant of protondeuteron transfer in a model HOndashHndashCH3along a linear reaction path with realistic potential
(Section 3) Section 4 concludes
2 Formalism
The modified expression for the reaction rate constants isbased on the common functional form of the energy eigen-states for the open scattering system and the double-wellsystem (or any sequence of wells and barriers) consistingof rectangular potential steps Let us consider a one-dimen-sional open scattering system with a potential being constantin the asymptotic regions as 119909 rarr plusmninfin and arbitrary in theinteraction region referred to as the ldquobarrierrdquo region
119881 (119909) =
0 119909 lt 1199091reactant region
119881119887 (119909) 119909
1le 119909 le 119909
2barrier region
1198810
119909 gt 1199092product region
(4)
Describing scattering in the energy domain the energy eigen-states with appropriate boundary conditions such as a wave-function incoming from the left in the direction from
Journal of Theoretical Chemistry 3
reactants to products have the following asymptotic form for119864 gt 119881
0(119898 is the particle mass)
120595119864(119909)
=
exp (120580119896119903119909)
+119903 (119864) exp (minus120580119896119903119909) 119909 lt 1199091 119896119903=radic2119898119864
119905 (119864) exp (120580119896119901119909) 119909 gt 119909
2 119896119901= radic2119898 (119864 minus 119881
0)
(5)
The wavefunction in the barrier region
120595119864(119909) = 119891
119864(119909) 119909
1le 119909 le 119909
2 (6)
is an unspecified solution to the one-dimensional Schro-dinger equation which smoothly connects solutions in thetwo asymptotic regions and does not explicitly enter the ex-pression for the transmission probability 119879(119864)
119879 (119864) equiv 119873 (119864) = |119905 (119864)|
2119896119901
119896119903
(7)
For the double-well potential the energy eigenstateswith scat-tering boundary conditions [11] are undefined making (5)and (7) inapplicable Weiner [17 24] extended the conceptof scattering states to a piecewise quadratic double-wellpotential defining the incoming and outgoing boundary con-ditions in the parabolic wells These asymptotically divergingsolutions were determined from the quantum Hamilton-Jac-obi equation [25] resulting in impractical implementationbesides being limited by the functional form of the potential
Dynamics the probability of finding a particle initiallylocalized in the reactant well on the product side oscillatesin time without converging to a well-defined value unlessenergy dissipation and interaction of the reactive systemswith the environment are fully included
We will consider the rectangular double-well potentialwhich is easy to analyze in terms of the plane-wave solutionsFor such a potential 119881 shown in Figure 1
119881 (119909) =
infin 119909 le 1199090 119909 ge 119909
3walls
0 1199090lt 119909 lt 119909
1reactant region
119881119887 1199091le 119909 le 119909
2barrier region
1198810 1199092lt 119909 lt 119909
3product region
(8)
the energy eigenstates and the corresponding transmissionprobabilities for the open scattering system defined by (4)and for the full double-well potential of (8) are the sameat the energy levels 119864
119899of the full potential in the region
of finite 119881 Another feature of this bound potential is thatthe reactant and product regions are uniquely defined Thusthe expression for the thermal reaction rates of (2) can bemodified (i) using the transmission probability in the opensystem having the same barrier region as the double-wellpotential and (ii) using the energy eigenstates of the double-well potential
119896 (120573) =
1
2120587
sum119899120588119899119879 (119864119899) 119890
minus120573119864119899
sum119899120588119899119890
minus120573119864119899
(9)
minus4 minus2 0 2 4
Coordinate x [a0]
Ener
gyE
[har
tree]
4
2
0
x0 x1 x2 x3
n = 0dwn = 1dwn = 2dw
n = 0 rwn = 1 rw
Ψ1
Ψ0
Ψ2
Figure 1 The asymmetric rectangular double-well potential ofSection 31 Its three lowest eigenfunctions are shown as dashesThehorizontal lines mark the corresponding three energy levels (labeled119899 = 0 1 2 dw) and also two lowest energy levels of the reactant wellwith infinitely high walls (labeled 1198991015840 = 0 1 rw)
The summation goes over the energy eigenstates 120601119899of the
bound system
119867120601119899= 119864119899120601119899 (10)
and 120588119899is the probability of finding a particle on the reactant
side for each state 119899
120588119899= ⟨120601119899
10038161003816100381610038161003816
119875119903
10038161003816100381610038161003816
120601119899⟩ = int
119909119903
minusinfin
1003816100381610038161003816
120601119899(119909)
1003816100381610038161003816
2119889119909 (11)
Operator 119875119903projects a wavefunction onto the reactant region
defined as 119909 lt 119909119903
In case of the rectangular double well of (8) the projec-tions of the eigenstates on the reactant well are unambiguous(119909119903= 1199091)
120588119899= int
1199091
1199090
1003816100381610038161003816
120601119899 (119909)
1003816100381610038161003816
2119889119909 (12)
Generally a definition of 119875119903requires specification of the
reactant region Some possibilities are as follows (i) 119909119903is a
position of the barrier top for any 119899 and (ii) 119909119903is a classical
turning point that is 119909(119899)119903
is a root of 119881(119909(119899)119903) = 119864
119899in the
reactant wellThe first option is reasonable for simple barrierswith a singlemaximum while the second one is more generaland consistent with the WKB tunneling expression
To summarize the formalism for general potentials withreactant and product well regions the transmission probabil-ity is defined by the properties of the open scattering systemthat is by the barrier with the 119881(119909) set to constants beyondthe reactantproduct well minima 119909
0and 119909
3 respectively
4 Journal of Theoretical Chemistry
minus2 minus1 0 1 2
0
05
x1 xL xR x2
Vb
V0
Pote
ntia
l [Eh
]
Coordinate [a0]
Figure 2 A double-well potential (solid line) and the correspondingto it scattering potential (dash) Coordinates 119909
1and 119909
2are the
positions of the minima in the reactant and product regionsrespectively and also define the barrier region 119909
119871and 119909
119877are
inflection points of the potential given by (20)
This is shown in Figure 2 for a piecewise quadratic continuouspotential The double-well potential simply specifies whichenergies out of the continuum of the scattering energy statesto include into the Boltzmann averaging in (9) For anasymmetric potential the reactant well may have eigenstatesbelow the asymptote119881
0of the product wellThose eigenstates
should be included into the Boltzmann averaging in (9)with the zero transmission probability The same argumentsand expressions are applicable to more complicated boundpotentials as long as the reactant and product regions arespecified
3 Examples and Discussion
In this section we examine reaction probabilities and rateconstants computed from (9) for several model systems withthe emphasis on comparison of the QM andWKB tunnelingprobabilities We also consider better-than-WKB approxi-mations relevant to each model a ldquominimalisticrdquo two-statedescription for a symmetric rectangular well (Section 31)a mixed QCQM calculation for an asymmetric well (Sec-tion 32) and a parabolic barrier approximation for a piece-wise quadratic potential (Section 33) Accuracy of transmis-sion probabilities and the energy level splittings followingfrom the quasiclassical WKB expression is examined Exam-ple of the KIE calculation for a model proton transfer systemis described in Section 34
31 Comparison of the QMWKB and Two-State Model Prob-abilities for a Symmetric Double Well Let us apply (9) toa symmetric rectangular double well and compare reactionprobabilities computed using the exact QM and approximateenergy levels and transmission probabilities For a general
rectangular barrier given by (8) without the infinitely highwalls the transmission probability is
119879 (119864)
=
4119896119903119896
2
119887119896119901
(119896119903+ 119896119901)
2
119896
2
119887+[119896
2
119903119896
2
119901+119896
2
119887(119896
2
119887minus 119896
2
119903minus 119896
2
119901)] sin2 (119896
119887119871)
for 119864 gt 119881119887
4119896119903119896
2
119887119896119901
(119896119903+ 119896119901)
2
119896
2
119887+[119896
2
119903119896
2
119901+119896
2
119887(119896
2
119887+ 119896
2
119903+ 119896
2
119901)] sinh2 (119896
119887119871)
for 1198810lt 119864 lt 119881
119887
(13)
In the expression above 119896119887= radic2119898|119881
119887minus 119864| and 119871 = 119909
2minus 1199091
is the barrier width 119896119903and 119896
119901are given in (5) Analytical
expression for QM transmission probability 119879(119864) allowshigh precision calculations performed with Maple [26] Theparticle mass is119898 = 1 a u
We are considering a symmetric barrier of fixed height119881119887
and variablewidthThewidth of thewells119889119889 = 1199091minus1199090= 1199093minus
1199092 is fixed at 119889 = 2119886
0 The barrier height 119881
119887is set to 4119864
ℎ The
barrier width 119871 is in the range of 119871 = [025 80]1198860 In the limit
of infinitely wide barrier the reactant and product wells havetwo bound states The transmission probabilities 119879(119864) andthe energy level splitting Δ associated with both asymptoticenergy levels are computed by considering two lowest pairsof energy eigenstates of the finite-width potentials ExactQM transmission probabilities are compared with thoseobtained using the quasiclassical WKB approach (labeledQC) and the two-state description (labeled 2119904) of this systemThe WKB tunneling probabilities are used in comparisoneven though the WKB method is inaccurate for potentialswith discontinuous derivatives because of its routine use intunneling calculations for chemical systems
In a symmetric potential the ground and first excitedstates have evenodd symmetry so the half-sumhalf-difference of the two states represents a particle located on thereactantproduct side Therefore for consistent comparisonof exact and approximate expressions the QM transmission119879
QM is defined as
119879
QM=
1205880119879 (1198640) + 1205881119879 (1198641)
1205880+ 1205881
(14)
to account for localization of the ldquoinitialrdquo wavefunction inthe reactant well The difference between (9) and (14) isthat the latter neglects the effect of temperature on theoccupation of 119899 = 0 and 119899 = 1 states This omission isaccurate in the limit of small energy level splittings Equation(14) is also consistent with the two-state representation of awavefunction in the basis formed by the ground eigenstatesof reactant and product wells 120601
119903and 120601
119901 respectively These
functions are the lowest energy eigenfunctions of the reactantand product wells in the limit of an infinitely wide barrier(The eigenfunctions of the isolated wells with infinitely highwalls cannot be used for a rectangular potential becausesuch eigenfunctions will have zero overlap) In the two-state
Journal of Theoretical Chemistry 5
6 80
0
05
1
15
2
25
42Barrier width L [a0]
TQCTQM
T2sTQM
Figure 3 Tunneling in the symmetric rectangular double-wellpotential of Section 31 Ratios of approximate transmission prob-abilities that is the quasiclassical WKB (squaresdash) and the two-state representation (circlessolid lines) to the exact QM probabilityare given as functions of the barrier width
representation the energies of the ground and excited statesare the generalized eigenvalues of the 2-by-2 HamiltonianmatrixHwith the overlapmatrix S For a symmetric potentialthe matrix elements are
ℎ11= ℎ22= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119903⟩ ℎ12= ℎ21= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119901⟩ (15)
11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)
H1205952119904119894= 119864
2119904
119894S1205952119904119894119894 = 0 1 (17)
The energy level splitting is Δ2119904 = 11986421199041minus 119864
2119904
0
Following [24] we use the quasiclassical relation betweenthe energy level splitting and the tunneling probability
1198641minus 1198640= Δ =
21198640
120587
radic119879 (1198640) (18)
with the ground state energy used instead of the frequency ofthe harmonic reactant well In the two-state representationthe energy level splitting Δ2119904 following from (17) yields anestimate for the tunneling 1198792119904 The same (18) gives estimatesof the quasiclassical energy level splitting ΔQC once thequasiclassical WKB tunneling probability 119879QC is computedfrom (3) Two pairs of states of the full potential 119899 = 0 119899 = 1and 119899 = 2 119899 = 3 are analyzed the higher energy states 119899 = 2and 119899 = 3 are described in the basis of the first excited statesof the reactant and product wells
Table 1 lists the ground and first excited energy levels ob-tained exactly using quasiclassical WKB approximation andwithin the two-state approximation labeled QM QC and2119904 respectively are listed in the upper half of Table 1The samequantities for the next highest pair of levels (Δ = 119864
3minus1198642) are
given in the lower half of Table 1 In the quasiclassical WKBtreatment the energy levels are defined by the width of the
1 2 3 4
Energy [Eh]
00001
1e minus 08
Tunn
elin
gT
(E)
L = 2 QML = 2 QC
E0
E1
Figure 4 Transmission through the symmetric rectangular barrierof Section 31 of the width 119871 = 2119886
0 Exact QM and QC WKB
probabilities are shown as a solid line and a dash respectively Theenergies of the ground and first excited states for the well width119889 = 2119886
0are indicated with dot-dashes
reactant well and therefore do not depend on the barrierwidth The ground state energy in the two-state approx-imation is remarkably accurate even for narrow barrierswhile the accuracy of 119899 = 2 level deteriorates for smallerbarrier widths The ratios of the quasiclassical and two-stateestimates of tunneling probability to the exact QM values asa function of the barrier width are plotted in Figure 3 for thelowest pair of states The agreement between 1198792119904 and 119879QM isexcellent for the barrier width 119871 gt 10 119886
0 The quasiclassical
WKB results underestimate the tunneling probability by afactor of 2This discrepancy can be understood by comparingquasiclassical WKB and QM tunneling at the energies ofthe asymptotic eigenstates shown in Figure 4 for the barrierwidth 119871 = 2119886
0 The transmission probabilities at these ener-
gies differ by a factor of 2 For a wider well of the width119889 = 4119886
0(not shown here) the ground state energy happens to
be near the intersection of the QM and quasiclassical WKBtunneling curves at 119864 = 02599119864
ℎ which coincidentally
yields much better agreement than the results shown inFigure 3 Therefore the accuracy of the quasiclassical resultsdepends on the reactant well width even if the accuracy ofthe quasiclassical tunneling probability does not At very lowenergies the discrepancy between the QM and QC tunnelingis in orders of magnitude but fortunately the energy regimebelow the zero-point energy of reactants does not contributeto the tunneling in a double-well system
To summarize the two-state tunneling probabilities forthe lowest pair of eigenstates are remarkably accurate exceptfor very narrow barriers and may provide useful estimatesif the rest of the states are of much higher energy Thebarrier can be considered sufficiently wide when the overlapof the reactant and product well eigenstates ⟨120601
119901|120601119903⟩ is less
than a few percent (5 percent in the current example)The QC expression (18) relating the energy level splitting
6 Journal of Theoretical Chemistry
Table1Energylevelssplittin
gsand
tunn
elingp
robabilityfor
the119899=01and119899=23pairofstatesofthesym
metric
rectangu
lard
ouble-wellpotentia
lofSectio
n31Th
ereactantp
rodu
ctstates
overlap
ofthetwo-stated
escriptio
nisgivenin
thelastcolum
nTh
eenergylevelsandlevelsplittings
areg
iven
in119864ℎN
umbersin
parenthesesa
rethep
owerso
f10
119871[1198860]
119864
QM0
Δ
QM
119879
QM
119864
QC0
Δ
QC
119879
QC
119864
2s 0Δ
2119904
119879
2119904
⟨120601119901|120601119903⟩
Forthe119899=01pairof
energy
states
808800
1919
(minus9)
1205
(minus17)
08800
1174
(minus9)
439
(minus18)
08800
1913
(minus9)
117(minus17)
3554(minus9)
408800
4192(minus5)
5752(minus9)
08800
2564(minus5)
2095(minus9)
08800
4210(minus5)
564
6(minus9)
440
8(minus5)
208770
6196(minus3)
1257
(minus4)
08800
3790(minus3)
4577(minus5)
08769
6224(minus3)
1243
(minus4)
4035(minus3)
108422
00753
00185
08800
4608(minus3)
6765(minus3)
08425
7564
(minus2)
1989
(minus2)
3398(minus2)
05
07433
02618
02105
08800
01607
8225(minus2)
07483
02637
03064
00922
025
060
9804881
05900
08800
03000
02868
06289
04906
1502
01476
Forthe119899=23pairof
energy
states
833053
1099
(minus4)
1481
(minus8)
33053
1690
(minus4)
6450(minus9)
33053
0974(minus4)
2141(minus9)
4712(minus4)
432993
1227
(minus2)
1848
(minus4)
33053
1886
(minus2)
8031(minus5)
32999
1087
(minus2)
2676(minus5)
2790(minus2)
232477
01334
2144(minus2)
33053
01992
8962(minus3)
32472
0114
63071(minus3)
01644
131337
04856
02267
33053
064
7400947
30860
03487
00315
03225
05
300
6810
078
06166
33053
11672
03077
28262
05329
00877
03905
025
29086
14864
08529
33053
15672
05547
25604
06101
01401
03961
Journal of Theoretical Chemistry 7
Table 2 Thermal rate constants 119896(119879) for the asymmetric rectangular double well of Section 32 The first column lists the temperature thatis 119896119861119879 in units of the barrier height119881
119887The thermal rate constants obtained using (9) using exact energy levels and QM and QC transmission
probabilities are given in the second and third columns respectively The last column contains the thermal rate constants obtained from QCtunneling probability and the energy levels of the reactant potential with infinitely high walls
119896119861119879119881119887
119879
QM119864
QM119899
119879
QC119864
QM119899
119879
QC119864
QC119899
001 691 (minus13) 265 (minus13) 738 (minus15)002 101 (minus8) 380 (minus9) 778 (minus10)004 123 (minus6) 464 (minus7) 252 (minus7)01 399 (minus5) 267 (minus5) 196 (minus5)02 131 (minus3) 197 (minus3) 241 (minus3)04 195 (minus2) 266 (minus2) 364 (minus2)10 0134 0159 019820 0286 0320 037140 0450 0483 0533
and tunneling is inaccurate for the higher energy pairs ofldquosplitrdquo levelsThe accuracy of the QC approximation dependson both the accuracy of QC tunneling probability anddiscrepancy between the energy levels of the full potentialand of the asymptotic reactant well For one-dimensionalsystems the use of exact energy eigenstates of the full potentialrather than those of an isolated reactant well in (9) ismore appropriate even with approximate calculation of thetunneling probability
32 Asymmetric Rectangular Barrier Now let us consider anasymmetric rectangular double-well potential given by (8)and sketched in Figure 1 There are 7 states with energiesbelow the barrier top The ground and first excited statesare localized on the reactant and product sides respectivelyThe eigenstates of the full potential with even quantum num-bers correlate with the eigenstates of the reactant well thedifference in energies of the full and asymptotic potentialsincreases with the quantum number The parameter valuesfor the potential are 119881
119887= 40119864
ℎ 1198810= 04119864
ℎ and 119871 = 10119886
0
The well width is 119889 = 401198860 The particle mass is119898 = 1 a u
Exact QM energy levels and eigenfunctions needed tocompute projections on the reactant well are obtained aseigenvalues and eigenvectors of the Hamiltonian matrix inthe Colbert-Miller discrete variable representation [27]
Exact QM transmission probability is given by (13) Qua-siclassical transmission probability is defined by (3) and set to(1) for energies above the barrier top The exact and approxi-mate rate constants computed from (9) are listed inTable 2 fortemperatures measured in units of the barrier top 119881
119887 Results
of the fully QM fully QC and mixed QCQM (QC trans-mission evaluated at exact QM energy levels) descriptions areshown in Figure 5 and in Table 2 At the lowest temperatureequivalent to 1 of the barrier height the QC rate constantis 93 times lower than the QM result while the mixedQCQM estimate is only 2 times lower At high temperatureswhen the high energy eigenstates significantly contribute tothe reaction rate constant the discrepancy between exactand approximate rate constants becomes smaller with themixed QCQM estimate being closer to the exact result at allenergies One concludes that generally the accuracy of 119896(119879)
0 005 01 015 02
Rate
cons
tant
Exact QM QM levelsQC T(E)Reactant levelsQC T(E)
k(T
)
0
02
04
06
08
Temperature [Vb]
0 10 20 30
00001
1e minus 08
1e minus 12
1e minus 16
Figure 5 Transmission through the asymmetric rectangular barrierof Section 32 The thermal reaction rate constant is given as afunction of temperature measured in units of the barrier height[119896119861119879119881119887] where119881
119887= 4119864ℎThe rate constants are obtained using the
exact QM transmission probability and energy levels (circlessolidline) the QC transmission probability evaluated at the exact QMenergy levels (trianglesdot-dash) and at the QC energy levels of theisolated reactant well (squaresdash) The insert shows 119896(119879) on thelinear scale for wide range of temperatures
depends both on the accuracy ofQC transmission probability119879
QC and on the difference between exact and asymptotic-wellenergy levels Using exact QM energy levels and functions inconjunction with quasiclassical transmission probabilities ismore accurate than the fully QC description
Note that for the rectangular potential considered here119879
QC(119864) = 1 for 119864 gt 119881
119887is inaccurate because QC approxi-
mation does not describe above-the-barrier reflection leadingto oscillations of 119879QM For general potentials the reactionprobability may be computed using for example the time-dependent QM wave packet approach [28] used in the next
8 Journal of Theoretical Chemistry
example yielding reaction probabilities for a range of ener-gies from a single calculation or with a time-independentmethod for a few energy values
33 Piecewise Quadratic Potential Next we examine a piece-wise quadratic potential sketched in Figure 2 The reactantwell and barrier are quadratic functions in 119909 The reactionrate constants are obtained using the QM and quasiclassicalWKB expressions and using the analytical QM transmission
119879
QMpar (119864) = (1 + exp(minus
2120587119864
119908119887
))
minus1
119908119887=radic
119896119887
119898
(19)
through a parabolic barrier Equation (19) is more accuratethan the QC expression near the barrier top [16] A continu-ous piecewise quadratic potential is defined as follows
119881 (119909)
=
119896(119909 minus 1199091)
2
2
119909 lt 119909119871reactant region
minus
119896119887119909
2
2
+ 119881119887
119909119871le 119909 le 119909
119877barrier top region
119896(119909 minus 1199092)
2
2
+ 1198810119909 gt 119909
119877product region
(20)
with the proper choice of parametersmdash119881119887 1198810 119909119871 and 119909
119877mdash
the potential and its first derivative is continuous function at119909 = 119909
119871and 119909 = 119909
119877 Here a symmetric double well 119881
0= 0
is considered the remaining parameter values are 1199092= minus1199091
= 11198860 119896 = 119896
119887= 80119864
ℎ119886
minus2
0 119881119887= 20119864
ℎ and 119909
119877= minus119909119871= 05119886
0
The particle mass is 119898 = 1 a u (If the Hamiltonian werescaled by the mass of the proton 119881
119887would be equivalent to
03 eV or 7 kcalmol)The barrier parameters are chosen suchthat there are only few energy levels under the barrier topwhich is typical for a chemical reaction at low temperature
The time-dependent wave packet correlation approach[28] has been used to calculate QM transmission probability119879(119864) = |119878
119903119901(119864)|
2 entering (9) since we examine dependenceof the reaction rate on the accuracy of the energy levels usedin (9) The scattering matrix element 119878
119903119901(119864) describes trans-
mission from reactants to products
119878119903119901(119864) =
(2120587)
minus1
120578
lowast
119901(119864) 120578119903 (
119864)
int
infin
minusinfin
⟨120601
minus
119901
100381610038161003816100381610038161003816
119890
minus120580119905100381610038161003816100381610038161003816
120601
+
119903⟩ 119890
120580119864119905119889119905 (21)
The subscripts 119903119901 refer to reactantproduct reaction chan-nels and plusmn refers to the asymptotically incomingoutgoingwave relative to the barrier An incoming wave packet |120601+
119903⟩
taken as a Gaussian function in the left asymptotic regionof 119881 is evolved in time in the ldquounfoldedrdquo potential shownin Figure 2 with a dash Another Gaussian function |120601minus
119901⟩
is placed in the product region of the potential The time-dependent overlap of evolving |120601+
119903⟩ with the stationary ⟨120601minus
119901|
or correlation function 119862(119905) = ⟨120601
minus
119901(119909 0) | 120601
+
119903(119909 119905)⟩ is
computed and Fourier-transformed into the energy domain
The denominator in (21) accounts for the distributions ofenergy eigenstates in the reactant and product wave packetsat time 119905 = 0 these distributions depend on the wave packetlocalization and kinetic energy
120578119903119901(119864) = radic
119898
2120587119901
int
infin
minusinfin
119890
minus120580119901119909120601
plusmn
119903119901(119909) 119889119909 119901 =
radic2119898119864
(22)
A single wave packet propagation accomplished on a gridusing the split-operator method [29 30] gives transmissionprobability for a range of energies represented in the reac-tantproduct wave packets which is convenient if multipleeigenstates contribute to the rate constant At low energiesit is hard to converge transmission probability using time-dependent dynamics methods but the energies below thezero-point energy of the bound potential do not contributeto 119896(119879)The exact QM energy levels and projections of eigen-functions on the reactant well are performed as in Section 32There are five energy levels below the barrier top but manymore states contribute to 119896(119879) at higher temperatures
The energy-resolved exactQM parabolic barrier approxi-mation and QC transmission probabilities are shown in Fig-ure 6(a) Tunneling probabilities are underestimated in theQC approximation while the parabolic barrier expressiongives accurate results for energies above 119881
1198875 = 4119864
ℎ(the
barrier is in fact parabolic in this energy range) The discrep-ancy grows at energy below 4119864
ℎ This energy range however
is below the zero-point energy and does not contributeto the thermal rate constants shown in Figure 6(b) Theapproximate rate constants differ from exact QM results byat most 25ndash30
34 Proton Transfer As a chemically relevant model we con-sider the proton transfer in the HOndashHndashCH
3system for a
constrained collinear OndashHndashC geometry In this model sys-tem the proton is transferred from a donor carbon to ac-ceptor oxygenThe potential energy surface is obtained fromdensity function theory (DFT) electronic structure calcula-tions in particular at the B3LYP6-31G(dp) level of theory[31] The energies were compared to those obtained at theCCSD(T)aug-cc-pVDZ theory level for the same geome-tries and were found in excellent agreement In this sys-tem collinear donor-proton-acceptor arrangement is a goodapproximation to the fully optimized reaction path as deter-mined by a set of constrained geometry optimizations Inthese optimizations119877OH was fixed to a value in the range 08ndash20 A the average deviation of angCHO from linearity was lessthan 2∘ and the maximum deviation was 10∘
One-dimensional potential energy surfaces of 40 pointswere generated as functions of 119877OH for a fixed 119877CO distanceProton transfer on three surfaces for119877CO = 27 28 and 29 Ais analyzed below The surfaces were parametrized as sixthdegree polynomials in 119877OH The potential energy curvesshown in Figure 7(a) have characteristic asymmetric double-well shape
We examine the effect of donor-acceptor distance 119877COon the tunneling rate constants using QM and QC transmis-sion probabilities and exact and approximate energy levels
Journal of Theoretical Chemistry 9
00001
001
1
QMQCParabolic
0 10 20 30
1
1e minus 08
1e minus 06
Energy [Eh]
T(E
)
(a)
0 05 1 15 20
05
Rate
cons
tant
0 01 02
00001
001
Temperature [Vb]
(b)
Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896
119861119879119881119887] where 119881
119887= 20119864
ℎ
0
10
20
30
40
50
08 1 12 14 16 18 2
272829
Ener
gy (m
E h)
ROH (A)
(a)
0 500 1000 1500
103
102
101
100
KIE
Temperature (K)
(b)
Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO
distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896
119867119896119863(KIE) using exact QM formulation
in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top
The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896
119867119896119863should be largest at low
temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Quantum Chemistry
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ElectrochemistryInternational Journal of
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CatalystsJournal of
Journal of Theoretical Chemistry 3
reactants to products have the following asymptotic form for119864 gt 119881
0(119898 is the particle mass)
120595119864(119909)
=
exp (120580119896119903119909)
+119903 (119864) exp (minus120580119896119903119909) 119909 lt 1199091 119896119903=radic2119898119864
119905 (119864) exp (120580119896119901119909) 119909 gt 119909
2 119896119901= radic2119898 (119864 minus 119881
0)
(5)
The wavefunction in the barrier region
120595119864(119909) = 119891
119864(119909) 119909
1le 119909 le 119909
2 (6)
is an unspecified solution to the one-dimensional Schro-dinger equation which smoothly connects solutions in thetwo asymptotic regions and does not explicitly enter the ex-pression for the transmission probability 119879(119864)
119879 (119864) equiv 119873 (119864) = |119905 (119864)|
2119896119901
119896119903
(7)
For the double-well potential the energy eigenstateswith scat-tering boundary conditions [11] are undefined making (5)and (7) inapplicable Weiner [17 24] extended the conceptof scattering states to a piecewise quadratic double-wellpotential defining the incoming and outgoing boundary con-ditions in the parabolic wells These asymptotically divergingsolutions were determined from the quantum Hamilton-Jac-obi equation [25] resulting in impractical implementationbesides being limited by the functional form of the potential
Dynamics the probability of finding a particle initiallylocalized in the reactant well on the product side oscillatesin time without converging to a well-defined value unlessenergy dissipation and interaction of the reactive systemswith the environment are fully included
We will consider the rectangular double-well potentialwhich is easy to analyze in terms of the plane-wave solutionsFor such a potential 119881 shown in Figure 1
119881 (119909) =
infin 119909 le 1199090 119909 ge 119909
3walls
0 1199090lt 119909 lt 119909
1reactant region
119881119887 1199091le 119909 le 119909
2barrier region
1198810 1199092lt 119909 lt 119909
3product region
(8)
the energy eigenstates and the corresponding transmissionprobabilities for the open scattering system defined by (4)and for the full double-well potential of (8) are the sameat the energy levels 119864
119899of the full potential in the region
of finite 119881 Another feature of this bound potential is thatthe reactant and product regions are uniquely defined Thusthe expression for the thermal reaction rates of (2) can bemodified (i) using the transmission probability in the opensystem having the same barrier region as the double-wellpotential and (ii) using the energy eigenstates of the double-well potential
119896 (120573) =
1
2120587
sum119899120588119899119879 (119864119899) 119890
minus120573119864119899
sum119899120588119899119890
minus120573119864119899
(9)
minus4 minus2 0 2 4
Coordinate x [a0]
Ener
gyE
[har
tree]
4
2
0
x0 x1 x2 x3
n = 0dwn = 1dwn = 2dw
n = 0 rwn = 1 rw
Ψ1
Ψ0
Ψ2
Figure 1 The asymmetric rectangular double-well potential ofSection 31 Its three lowest eigenfunctions are shown as dashesThehorizontal lines mark the corresponding three energy levels (labeled119899 = 0 1 2 dw) and also two lowest energy levels of the reactant wellwith infinitely high walls (labeled 1198991015840 = 0 1 rw)
The summation goes over the energy eigenstates 120601119899of the
bound system
119867120601119899= 119864119899120601119899 (10)
and 120588119899is the probability of finding a particle on the reactant
side for each state 119899
120588119899= ⟨120601119899
10038161003816100381610038161003816
119875119903
10038161003816100381610038161003816
120601119899⟩ = int
119909119903
minusinfin
1003816100381610038161003816
120601119899(119909)
1003816100381610038161003816
2119889119909 (11)
Operator 119875119903projects a wavefunction onto the reactant region
defined as 119909 lt 119909119903
In case of the rectangular double well of (8) the projec-tions of the eigenstates on the reactant well are unambiguous(119909119903= 1199091)
120588119899= int
1199091
1199090
1003816100381610038161003816
120601119899 (119909)
1003816100381610038161003816
2119889119909 (12)
Generally a definition of 119875119903requires specification of the
reactant region Some possibilities are as follows (i) 119909119903is a
position of the barrier top for any 119899 and (ii) 119909119903is a classical
turning point that is 119909(119899)119903
is a root of 119881(119909(119899)119903) = 119864
119899in the
reactant wellThe first option is reasonable for simple barrierswith a singlemaximum while the second one is more generaland consistent with the WKB tunneling expression
To summarize the formalism for general potentials withreactant and product well regions the transmission probabil-ity is defined by the properties of the open scattering systemthat is by the barrier with the 119881(119909) set to constants beyondthe reactantproduct well minima 119909
0and 119909
3 respectively
4 Journal of Theoretical Chemistry
minus2 minus1 0 1 2
0
05
x1 xL xR x2
Vb
V0
Pote
ntia
l [Eh
]
Coordinate [a0]
Figure 2 A double-well potential (solid line) and the correspondingto it scattering potential (dash) Coordinates 119909
1and 119909
2are the
positions of the minima in the reactant and product regionsrespectively and also define the barrier region 119909
119871and 119909
119877are
inflection points of the potential given by (20)
This is shown in Figure 2 for a piecewise quadratic continuouspotential The double-well potential simply specifies whichenergies out of the continuum of the scattering energy statesto include into the Boltzmann averaging in (9) For anasymmetric potential the reactant well may have eigenstatesbelow the asymptote119881
0of the product wellThose eigenstates
should be included into the Boltzmann averaging in (9)with the zero transmission probability The same argumentsand expressions are applicable to more complicated boundpotentials as long as the reactant and product regions arespecified
3 Examples and Discussion
In this section we examine reaction probabilities and rateconstants computed from (9) for several model systems withthe emphasis on comparison of the QM andWKB tunnelingprobabilities We also consider better-than-WKB approxi-mations relevant to each model a ldquominimalisticrdquo two-statedescription for a symmetric rectangular well (Section 31)a mixed QCQM calculation for an asymmetric well (Sec-tion 32) and a parabolic barrier approximation for a piece-wise quadratic potential (Section 33) Accuracy of transmis-sion probabilities and the energy level splittings followingfrom the quasiclassical WKB expression is examined Exam-ple of the KIE calculation for a model proton transfer systemis described in Section 34
31 Comparison of the QMWKB and Two-State Model Prob-abilities for a Symmetric Double Well Let us apply (9) toa symmetric rectangular double well and compare reactionprobabilities computed using the exact QM and approximateenergy levels and transmission probabilities For a general
rectangular barrier given by (8) without the infinitely highwalls the transmission probability is
119879 (119864)
=
4119896119903119896
2
119887119896119901
(119896119903+ 119896119901)
2
119896
2
119887+[119896
2
119903119896
2
119901+119896
2
119887(119896
2
119887minus 119896
2
119903minus 119896
2
119901)] sin2 (119896
119887119871)
for 119864 gt 119881119887
4119896119903119896
2
119887119896119901
(119896119903+ 119896119901)
2
119896
2
119887+[119896
2
119903119896
2
119901+119896
2
119887(119896
2
119887+ 119896
2
119903+ 119896
2
119901)] sinh2 (119896
119887119871)
for 1198810lt 119864 lt 119881
119887
(13)
In the expression above 119896119887= radic2119898|119881
119887minus 119864| and 119871 = 119909
2minus 1199091
is the barrier width 119896119903and 119896
119901are given in (5) Analytical
expression for QM transmission probability 119879(119864) allowshigh precision calculations performed with Maple [26] Theparticle mass is119898 = 1 a u
We are considering a symmetric barrier of fixed height119881119887
and variablewidthThewidth of thewells119889119889 = 1199091minus1199090= 1199093minus
1199092 is fixed at 119889 = 2119886
0 The barrier height 119881
119887is set to 4119864
ℎ The
barrier width 119871 is in the range of 119871 = [025 80]1198860 In the limit
of infinitely wide barrier the reactant and product wells havetwo bound states The transmission probabilities 119879(119864) andthe energy level splitting Δ associated with both asymptoticenergy levels are computed by considering two lowest pairsof energy eigenstates of the finite-width potentials ExactQM transmission probabilities are compared with thoseobtained using the quasiclassical WKB approach (labeledQC) and the two-state description (labeled 2119904) of this systemThe WKB tunneling probabilities are used in comparisoneven though the WKB method is inaccurate for potentialswith discontinuous derivatives because of its routine use intunneling calculations for chemical systems
In a symmetric potential the ground and first excitedstates have evenodd symmetry so the half-sumhalf-difference of the two states represents a particle located on thereactantproduct side Therefore for consistent comparisonof exact and approximate expressions the QM transmission119879
QM is defined as
119879
QM=
1205880119879 (1198640) + 1205881119879 (1198641)
1205880+ 1205881
(14)
to account for localization of the ldquoinitialrdquo wavefunction inthe reactant well The difference between (9) and (14) isthat the latter neglects the effect of temperature on theoccupation of 119899 = 0 and 119899 = 1 states This omission isaccurate in the limit of small energy level splittings Equation(14) is also consistent with the two-state representation of awavefunction in the basis formed by the ground eigenstatesof reactant and product wells 120601
119903and 120601
119901 respectively These
functions are the lowest energy eigenfunctions of the reactantand product wells in the limit of an infinitely wide barrier(The eigenfunctions of the isolated wells with infinitely highwalls cannot be used for a rectangular potential becausesuch eigenfunctions will have zero overlap) In the two-state
Journal of Theoretical Chemistry 5
6 80
0
05
1
15
2
25
42Barrier width L [a0]
TQCTQM
T2sTQM
Figure 3 Tunneling in the symmetric rectangular double-wellpotential of Section 31 Ratios of approximate transmission prob-abilities that is the quasiclassical WKB (squaresdash) and the two-state representation (circlessolid lines) to the exact QM probabilityare given as functions of the barrier width
representation the energies of the ground and excited statesare the generalized eigenvalues of the 2-by-2 HamiltonianmatrixHwith the overlapmatrix S For a symmetric potentialthe matrix elements are
ℎ11= ℎ22= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119903⟩ ℎ12= ℎ21= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119901⟩ (15)
11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)
H1205952119904119894= 119864
2119904
119894S1205952119904119894119894 = 0 1 (17)
The energy level splitting is Δ2119904 = 11986421199041minus 119864
2119904
0
Following [24] we use the quasiclassical relation betweenthe energy level splitting and the tunneling probability
1198641minus 1198640= Δ =
21198640
120587
radic119879 (1198640) (18)
with the ground state energy used instead of the frequency ofthe harmonic reactant well In the two-state representationthe energy level splitting Δ2119904 following from (17) yields anestimate for the tunneling 1198792119904 The same (18) gives estimatesof the quasiclassical energy level splitting ΔQC once thequasiclassical WKB tunneling probability 119879QC is computedfrom (3) Two pairs of states of the full potential 119899 = 0 119899 = 1and 119899 = 2 119899 = 3 are analyzed the higher energy states 119899 = 2and 119899 = 3 are described in the basis of the first excited statesof the reactant and product wells
Table 1 lists the ground and first excited energy levels ob-tained exactly using quasiclassical WKB approximation andwithin the two-state approximation labeled QM QC and2119904 respectively are listed in the upper half of Table 1The samequantities for the next highest pair of levels (Δ = 119864
3minus1198642) are
given in the lower half of Table 1 In the quasiclassical WKBtreatment the energy levels are defined by the width of the
1 2 3 4
Energy [Eh]
00001
1e minus 08
Tunn
elin
gT
(E)
L = 2 QML = 2 QC
E0
E1
Figure 4 Transmission through the symmetric rectangular barrierof Section 31 of the width 119871 = 2119886
0 Exact QM and QC WKB
probabilities are shown as a solid line and a dash respectively Theenergies of the ground and first excited states for the well width119889 = 2119886
0are indicated with dot-dashes
reactant well and therefore do not depend on the barrierwidth The ground state energy in the two-state approx-imation is remarkably accurate even for narrow barrierswhile the accuracy of 119899 = 2 level deteriorates for smallerbarrier widths The ratios of the quasiclassical and two-stateestimates of tunneling probability to the exact QM values asa function of the barrier width are plotted in Figure 3 for thelowest pair of states The agreement between 1198792119904 and 119879QM isexcellent for the barrier width 119871 gt 10 119886
0 The quasiclassical
WKB results underestimate the tunneling probability by afactor of 2This discrepancy can be understood by comparingquasiclassical WKB and QM tunneling at the energies ofthe asymptotic eigenstates shown in Figure 4 for the barrierwidth 119871 = 2119886
0 The transmission probabilities at these ener-
gies differ by a factor of 2 For a wider well of the width119889 = 4119886
0(not shown here) the ground state energy happens to
be near the intersection of the QM and quasiclassical WKBtunneling curves at 119864 = 02599119864
ℎ which coincidentally
yields much better agreement than the results shown inFigure 3 Therefore the accuracy of the quasiclassical resultsdepends on the reactant well width even if the accuracy ofthe quasiclassical tunneling probability does not At very lowenergies the discrepancy between the QM and QC tunnelingis in orders of magnitude but fortunately the energy regimebelow the zero-point energy of reactants does not contributeto the tunneling in a double-well system
To summarize the two-state tunneling probabilities forthe lowest pair of eigenstates are remarkably accurate exceptfor very narrow barriers and may provide useful estimatesif the rest of the states are of much higher energy Thebarrier can be considered sufficiently wide when the overlapof the reactant and product well eigenstates ⟨120601
119901|120601119903⟩ is less
than a few percent (5 percent in the current example)The QC expression (18) relating the energy level splitting
6 Journal of Theoretical Chemistry
Table1Energylevelssplittin
gsand
tunn
elingp
robabilityfor
the119899=01and119899=23pairofstatesofthesym
metric
rectangu
lard
ouble-wellpotentia
lofSectio
n31Th
ereactantp
rodu
ctstates
overlap
ofthetwo-stated
escriptio
nisgivenin
thelastcolum
nTh
eenergylevelsandlevelsplittings
areg
iven
in119864ℎN
umbersin
parenthesesa
rethep
owerso
f10
119871[1198860]
119864
QM0
Δ
QM
119879
QM
119864
QC0
Δ
QC
119879
QC
119864
2s 0Δ
2119904
119879
2119904
⟨120601119901|120601119903⟩
Forthe119899=01pairof
energy
states
808800
1919
(minus9)
1205
(minus17)
08800
1174
(minus9)
439
(minus18)
08800
1913
(minus9)
117(minus17)
3554(minus9)
408800
4192(minus5)
5752(minus9)
08800
2564(minus5)
2095(minus9)
08800
4210(minus5)
564
6(minus9)
440
8(minus5)
208770
6196(minus3)
1257
(minus4)
08800
3790(minus3)
4577(minus5)
08769
6224(minus3)
1243
(minus4)
4035(minus3)
108422
00753
00185
08800
4608(minus3)
6765(minus3)
08425
7564
(minus2)
1989
(minus2)
3398(minus2)
05
07433
02618
02105
08800
01607
8225(minus2)
07483
02637
03064
00922
025
060
9804881
05900
08800
03000
02868
06289
04906
1502
01476
Forthe119899=23pairof
energy
states
833053
1099
(minus4)
1481
(minus8)
33053
1690
(minus4)
6450(minus9)
33053
0974(minus4)
2141(minus9)
4712(minus4)
432993
1227
(minus2)
1848
(minus4)
33053
1886
(minus2)
8031(minus5)
32999
1087
(minus2)
2676(minus5)
2790(minus2)
232477
01334
2144(minus2)
33053
01992
8962(minus3)
32472
0114
63071(minus3)
01644
131337
04856
02267
33053
064
7400947
30860
03487
00315
03225
05
300
6810
078
06166
33053
11672
03077
28262
05329
00877
03905
025
29086
14864
08529
33053
15672
05547
25604
06101
01401
03961
Journal of Theoretical Chemistry 7
Table 2 Thermal rate constants 119896(119879) for the asymmetric rectangular double well of Section 32 The first column lists the temperature thatis 119896119861119879 in units of the barrier height119881
119887The thermal rate constants obtained using (9) using exact energy levels and QM and QC transmission
probabilities are given in the second and third columns respectively The last column contains the thermal rate constants obtained from QCtunneling probability and the energy levels of the reactant potential with infinitely high walls
119896119861119879119881119887
119879
QM119864
QM119899
119879
QC119864
QM119899
119879
QC119864
QC119899
001 691 (minus13) 265 (minus13) 738 (minus15)002 101 (minus8) 380 (minus9) 778 (minus10)004 123 (minus6) 464 (minus7) 252 (minus7)01 399 (minus5) 267 (minus5) 196 (minus5)02 131 (minus3) 197 (minus3) 241 (minus3)04 195 (minus2) 266 (minus2) 364 (minus2)10 0134 0159 019820 0286 0320 037140 0450 0483 0533
and tunneling is inaccurate for the higher energy pairs ofldquosplitrdquo levelsThe accuracy of the QC approximation dependson both the accuracy of QC tunneling probability anddiscrepancy between the energy levels of the full potentialand of the asymptotic reactant well For one-dimensionalsystems the use of exact energy eigenstates of the full potentialrather than those of an isolated reactant well in (9) ismore appropriate even with approximate calculation of thetunneling probability
32 Asymmetric Rectangular Barrier Now let us consider anasymmetric rectangular double-well potential given by (8)and sketched in Figure 1 There are 7 states with energiesbelow the barrier top The ground and first excited statesare localized on the reactant and product sides respectivelyThe eigenstates of the full potential with even quantum num-bers correlate with the eigenstates of the reactant well thedifference in energies of the full and asymptotic potentialsincreases with the quantum number The parameter valuesfor the potential are 119881
119887= 40119864
ℎ 1198810= 04119864
ℎ and 119871 = 10119886
0
The well width is 119889 = 401198860 The particle mass is119898 = 1 a u
Exact QM energy levels and eigenfunctions needed tocompute projections on the reactant well are obtained aseigenvalues and eigenvectors of the Hamiltonian matrix inthe Colbert-Miller discrete variable representation [27]
Exact QM transmission probability is given by (13) Qua-siclassical transmission probability is defined by (3) and set to(1) for energies above the barrier top The exact and approxi-mate rate constants computed from (9) are listed inTable 2 fortemperatures measured in units of the barrier top 119881
119887 Results
of the fully QM fully QC and mixed QCQM (QC trans-mission evaluated at exact QM energy levels) descriptions areshown in Figure 5 and in Table 2 At the lowest temperatureequivalent to 1 of the barrier height the QC rate constantis 93 times lower than the QM result while the mixedQCQM estimate is only 2 times lower At high temperatureswhen the high energy eigenstates significantly contribute tothe reaction rate constant the discrepancy between exactand approximate rate constants becomes smaller with themixed QCQM estimate being closer to the exact result at allenergies One concludes that generally the accuracy of 119896(119879)
0 005 01 015 02
Rate
cons
tant
Exact QM QM levelsQC T(E)Reactant levelsQC T(E)
k(T
)
0
02
04
06
08
Temperature [Vb]
0 10 20 30
00001
1e minus 08
1e minus 12
1e minus 16
Figure 5 Transmission through the asymmetric rectangular barrierof Section 32 The thermal reaction rate constant is given as afunction of temperature measured in units of the barrier height[119896119861119879119881119887] where119881
119887= 4119864ℎThe rate constants are obtained using the
exact QM transmission probability and energy levels (circlessolidline) the QC transmission probability evaluated at the exact QMenergy levels (trianglesdot-dash) and at the QC energy levels of theisolated reactant well (squaresdash) The insert shows 119896(119879) on thelinear scale for wide range of temperatures
depends both on the accuracy ofQC transmission probability119879
QC and on the difference between exact and asymptotic-wellenergy levels Using exact QM energy levels and functions inconjunction with quasiclassical transmission probabilities ismore accurate than the fully QC description
Note that for the rectangular potential considered here119879
QC(119864) = 1 for 119864 gt 119881
119887is inaccurate because QC approxi-
mation does not describe above-the-barrier reflection leadingto oscillations of 119879QM For general potentials the reactionprobability may be computed using for example the time-dependent QM wave packet approach [28] used in the next
8 Journal of Theoretical Chemistry
example yielding reaction probabilities for a range of ener-gies from a single calculation or with a time-independentmethod for a few energy values
33 Piecewise Quadratic Potential Next we examine a piece-wise quadratic potential sketched in Figure 2 The reactantwell and barrier are quadratic functions in 119909 The reactionrate constants are obtained using the QM and quasiclassicalWKB expressions and using the analytical QM transmission
119879
QMpar (119864) = (1 + exp(minus
2120587119864
119908119887
))
minus1
119908119887=radic
119896119887
119898
(19)
through a parabolic barrier Equation (19) is more accuratethan the QC expression near the barrier top [16] A continu-ous piecewise quadratic potential is defined as follows
119881 (119909)
=
119896(119909 minus 1199091)
2
2
119909 lt 119909119871reactant region
minus
119896119887119909
2
2
+ 119881119887
119909119871le 119909 le 119909
119877barrier top region
119896(119909 minus 1199092)
2
2
+ 1198810119909 gt 119909
119877product region
(20)
with the proper choice of parametersmdash119881119887 1198810 119909119871 and 119909
119877mdash
the potential and its first derivative is continuous function at119909 = 119909
119871and 119909 = 119909
119877 Here a symmetric double well 119881
0= 0
is considered the remaining parameter values are 1199092= minus1199091
= 11198860 119896 = 119896
119887= 80119864
ℎ119886
minus2
0 119881119887= 20119864
ℎ and 119909
119877= minus119909119871= 05119886
0
The particle mass is 119898 = 1 a u (If the Hamiltonian werescaled by the mass of the proton 119881
119887would be equivalent to
03 eV or 7 kcalmol)The barrier parameters are chosen suchthat there are only few energy levels under the barrier topwhich is typical for a chemical reaction at low temperature
The time-dependent wave packet correlation approach[28] has been used to calculate QM transmission probability119879(119864) = |119878
119903119901(119864)|
2 entering (9) since we examine dependenceof the reaction rate on the accuracy of the energy levels usedin (9) The scattering matrix element 119878
119903119901(119864) describes trans-
mission from reactants to products
119878119903119901(119864) =
(2120587)
minus1
120578
lowast
119901(119864) 120578119903 (
119864)
int
infin
minusinfin
⟨120601
minus
119901
100381610038161003816100381610038161003816
119890
minus120580119905100381610038161003816100381610038161003816
120601
+
119903⟩ 119890
120580119864119905119889119905 (21)
The subscripts 119903119901 refer to reactantproduct reaction chan-nels and plusmn refers to the asymptotically incomingoutgoingwave relative to the barrier An incoming wave packet |120601+
119903⟩
taken as a Gaussian function in the left asymptotic regionof 119881 is evolved in time in the ldquounfoldedrdquo potential shownin Figure 2 with a dash Another Gaussian function |120601minus
119901⟩
is placed in the product region of the potential The time-dependent overlap of evolving |120601+
119903⟩ with the stationary ⟨120601minus
119901|
or correlation function 119862(119905) = ⟨120601
minus
119901(119909 0) | 120601
+
119903(119909 119905)⟩ is
computed and Fourier-transformed into the energy domain
The denominator in (21) accounts for the distributions ofenergy eigenstates in the reactant and product wave packetsat time 119905 = 0 these distributions depend on the wave packetlocalization and kinetic energy
120578119903119901(119864) = radic
119898
2120587119901
int
infin
minusinfin
119890
minus120580119901119909120601
plusmn
119903119901(119909) 119889119909 119901 =
radic2119898119864
(22)
A single wave packet propagation accomplished on a gridusing the split-operator method [29 30] gives transmissionprobability for a range of energies represented in the reac-tantproduct wave packets which is convenient if multipleeigenstates contribute to the rate constant At low energiesit is hard to converge transmission probability using time-dependent dynamics methods but the energies below thezero-point energy of the bound potential do not contributeto 119896(119879)The exact QM energy levels and projections of eigen-functions on the reactant well are performed as in Section 32There are five energy levels below the barrier top but manymore states contribute to 119896(119879) at higher temperatures
The energy-resolved exactQM parabolic barrier approxi-mation and QC transmission probabilities are shown in Fig-ure 6(a) Tunneling probabilities are underestimated in theQC approximation while the parabolic barrier expressiongives accurate results for energies above 119881
1198875 = 4119864
ℎ(the
barrier is in fact parabolic in this energy range) The discrep-ancy grows at energy below 4119864
ℎ This energy range however
is below the zero-point energy and does not contributeto the thermal rate constants shown in Figure 6(b) Theapproximate rate constants differ from exact QM results byat most 25ndash30
34 Proton Transfer As a chemically relevant model we con-sider the proton transfer in the HOndashHndashCH
3system for a
constrained collinear OndashHndashC geometry In this model sys-tem the proton is transferred from a donor carbon to ac-ceptor oxygenThe potential energy surface is obtained fromdensity function theory (DFT) electronic structure calcula-tions in particular at the B3LYP6-31G(dp) level of theory[31] The energies were compared to those obtained at theCCSD(T)aug-cc-pVDZ theory level for the same geome-tries and were found in excellent agreement In this sys-tem collinear donor-proton-acceptor arrangement is a goodapproximation to the fully optimized reaction path as deter-mined by a set of constrained geometry optimizations Inthese optimizations119877OH was fixed to a value in the range 08ndash20 A the average deviation of angCHO from linearity was lessthan 2∘ and the maximum deviation was 10∘
One-dimensional potential energy surfaces of 40 pointswere generated as functions of 119877OH for a fixed 119877CO distanceProton transfer on three surfaces for119877CO = 27 28 and 29 Ais analyzed below The surfaces were parametrized as sixthdegree polynomials in 119877OH The potential energy curvesshown in Figure 7(a) have characteristic asymmetric double-well shape
We examine the effect of donor-acceptor distance 119877COon the tunneling rate constants using QM and QC transmis-sion probabilities and exact and approximate energy levels
Journal of Theoretical Chemistry 9
00001
001
1
QMQCParabolic
0 10 20 30
1
1e minus 08
1e minus 06
Energy [Eh]
T(E
)
(a)
0 05 1 15 20
05
Rate
cons
tant
0 01 02
00001
001
Temperature [Vb]
(b)
Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896
119861119879119881119887] where 119881
119887= 20119864
ℎ
0
10
20
30
40
50
08 1 12 14 16 18 2
272829
Ener
gy (m
E h)
ROH (A)
(a)
0 500 1000 1500
103
102
101
100
KIE
Temperature (K)
(b)
Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO
distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896
119867119896119863(KIE) using exact QM formulation
in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top
The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896
119867119896119863should be largest at low
temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Analytical ChemistryInternational Journal of
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Quantum Chemistry
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CatalystsJournal of
4 Journal of Theoretical Chemistry
minus2 minus1 0 1 2
0
05
x1 xL xR x2
Vb
V0
Pote
ntia
l [Eh
]
Coordinate [a0]
Figure 2 A double-well potential (solid line) and the correspondingto it scattering potential (dash) Coordinates 119909
1and 119909
2are the
positions of the minima in the reactant and product regionsrespectively and also define the barrier region 119909
119871and 119909
119877are
inflection points of the potential given by (20)
This is shown in Figure 2 for a piecewise quadratic continuouspotential The double-well potential simply specifies whichenergies out of the continuum of the scattering energy statesto include into the Boltzmann averaging in (9) For anasymmetric potential the reactant well may have eigenstatesbelow the asymptote119881
0of the product wellThose eigenstates
should be included into the Boltzmann averaging in (9)with the zero transmission probability The same argumentsand expressions are applicable to more complicated boundpotentials as long as the reactant and product regions arespecified
3 Examples and Discussion
In this section we examine reaction probabilities and rateconstants computed from (9) for several model systems withthe emphasis on comparison of the QM andWKB tunnelingprobabilities We also consider better-than-WKB approxi-mations relevant to each model a ldquominimalisticrdquo two-statedescription for a symmetric rectangular well (Section 31)a mixed QCQM calculation for an asymmetric well (Sec-tion 32) and a parabolic barrier approximation for a piece-wise quadratic potential (Section 33) Accuracy of transmis-sion probabilities and the energy level splittings followingfrom the quasiclassical WKB expression is examined Exam-ple of the KIE calculation for a model proton transfer systemis described in Section 34
31 Comparison of the QMWKB and Two-State Model Prob-abilities for a Symmetric Double Well Let us apply (9) toa symmetric rectangular double well and compare reactionprobabilities computed using the exact QM and approximateenergy levels and transmission probabilities For a general
rectangular barrier given by (8) without the infinitely highwalls the transmission probability is
119879 (119864)
=
4119896119903119896
2
119887119896119901
(119896119903+ 119896119901)
2
119896
2
119887+[119896
2
119903119896
2
119901+119896
2
119887(119896
2
119887minus 119896
2
119903minus 119896
2
119901)] sin2 (119896
119887119871)
for 119864 gt 119881119887
4119896119903119896
2
119887119896119901
(119896119903+ 119896119901)
2
119896
2
119887+[119896
2
119903119896
2
119901+119896
2
119887(119896
2
119887+ 119896
2
119903+ 119896
2
119901)] sinh2 (119896
119887119871)
for 1198810lt 119864 lt 119881
119887
(13)
In the expression above 119896119887= radic2119898|119881
119887minus 119864| and 119871 = 119909
2minus 1199091
is the barrier width 119896119903and 119896
119901are given in (5) Analytical
expression for QM transmission probability 119879(119864) allowshigh precision calculations performed with Maple [26] Theparticle mass is119898 = 1 a u
We are considering a symmetric barrier of fixed height119881119887
and variablewidthThewidth of thewells119889119889 = 1199091minus1199090= 1199093minus
1199092 is fixed at 119889 = 2119886
0 The barrier height 119881
119887is set to 4119864
ℎ The
barrier width 119871 is in the range of 119871 = [025 80]1198860 In the limit
of infinitely wide barrier the reactant and product wells havetwo bound states The transmission probabilities 119879(119864) andthe energy level splitting Δ associated with both asymptoticenergy levels are computed by considering two lowest pairsof energy eigenstates of the finite-width potentials ExactQM transmission probabilities are compared with thoseobtained using the quasiclassical WKB approach (labeledQC) and the two-state description (labeled 2119904) of this systemThe WKB tunneling probabilities are used in comparisoneven though the WKB method is inaccurate for potentialswith discontinuous derivatives because of its routine use intunneling calculations for chemical systems
In a symmetric potential the ground and first excitedstates have evenodd symmetry so the half-sumhalf-difference of the two states represents a particle located on thereactantproduct side Therefore for consistent comparisonof exact and approximate expressions the QM transmission119879
QM is defined as
119879
QM=
1205880119879 (1198640) + 1205881119879 (1198641)
1205880+ 1205881
(14)
to account for localization of the ldquoinitialrdquo wavefunction inthe reactant well The difference between (9) and (14) isthat the latter neglects the effect of temperature on theoccupation of 119899 = 0 and 119899 = 1 states This omission isaccurate in the limit of small energy level splittings Equation(14) is also consistent with the two-state representation of awavefunction in the basis formed by the ground eigenstatesof reactant and product wells 120601
119903and 120601
119901 respectively These
functions are the lowest energy eigenfunctions of the reactantand product wells in the limit of an infinitely wide barrier(The eigenfunctions of the isolated wells with infinitely highwalls cannot be used for a rectangular potential becausesuch eigenfunctions will have zero overlap) In the two-state
Journal of Theoretical Chemistry 5
6 80
0
05
1
15
2
25
42Barrier width L [a0]
TQCTQM
T2sTQM
Figure 3 Tunneling in the symmetric rectangular double-wellpotential of Section 31 Ratios of approximate transmission prob-abilities that is the quasiclassical WKB (squaresdash) and the two-state representation (circlessolid lines) to the exact QM probabilityare given as functions of the barrier width
representation the energies of the ground and excited statesare the generalized eigenvalues of the 2-by-2 HamiltonianmatrixHwith the overlapmatrix S For a symmetric potentialthe matrix elements are
ℎ11= ℎ22= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119903⟩ ℎ12= ℎ21= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119901⟩ (15)
11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)
H1205952119904119894= 119864
2119904
119894S1205952119904119894119894 = 0 1 (17)
The energy level splitting is Δ2119904 = 11986421199041minus 119864
2119904
0
Following [24] we use the quasiclassical relation betweenthe energy level splitting and the tunneling probability
1198641minus 1198640= Δ =
21198640
120587
radic119879 (1198640) (18)
with the ground state energy used instead of the frequency ofthe harmonic reactant well In the two-state representationthe energy level splitting Δ2119904 following from (17) yields anestimate for the tunneling 1198792119904 The same (18) gives estimatesof the quasiclassical energy level splitting ΔQC once thequasiclassical WKB tunneling probability 119879QC is computedfrom (3) Two pairs of states of the full potential 119899 = 0 119899 = 1and 119899 = 2 119899 = 3 are analyzed the higher energy states 119899 = 2and 119899 = 3 are described in the basis of the first excited statesof the reactant and product wells
Table 1 lists the ground and first excited energy levels ob-tained exactly using quasiclassical WKB approximation andwithin the two-state approximation labeled QM QC and2119904 respectively are listed in the upper half of Table 1The samequantities for the next highest pair of levels (Δ = 119864
3minus1198642) are
given in the lower half of Table 1 In the quasiclassical WKBtreatment the energy levels are defined by the width of the
1 2 3 4
Energy [Eh]
00001
1e minus 08
Tunn
elin
gT
(E)
L = 2 QML = 2 QC
E0
E1
Figure 4 Transmission through the symmetric rectangular barrierof Section 31 of the width 119871 = 2119886
0 Exact QM and QC WKB
probabilities are shown as a solid line and a dash respectively Theenergies of the ground and first excited states for the well width119889 = 2119886
0are indicated with dot-dashes
reactant well and therefore do not depend on the barrierwidth The ground state energy in the two-state approx-imation is remarkably accurate even for narrow barrierswhile the accuracy of 119899 = 2 level deteriorates for smallerbarrier widths The ratios of the quasiclassical and two-stateestimates of tunneling probability to the exact QM values asa function of the barrier width are plotted in Figure 3 for thelowest pair of states The agreement between 1198792119904 and 119879QM isexcellent for the barrier width 119871 gt 10 119886
0 The quasiclassical
WKB results underestimate the tunneling probability by afactor of 2This discrepancy can be understood by comparingquasiclassical WKB and QM tunneling at the energies ofthe asymptotic eigenstates shown in Figure 4 for the barrierwidth 119871 = 2119886
0 The transmission probabilities at these ener-
gies differ by a factor of 2 For a wider well of the width119889 = 4119886
0(not shown here) the ground state energy happens to
be near the intersection of the QM and quasiclassical WKBtunneling curves at 119864 = 02599119864
ℎ which coincidentally
yields much better agreement than the results shown inFigure 3 Therefore the accuracy of the quasiclassical resultsdepends on the reactant well width even if the accuracy ofthe quasiclassical tunneling probability does not At very lowenergies the discrepancy between the QM and QC tunnelingis in orders of magnitude but fortunately the energy regimebelow the zero-point energy of reactants does not contributeto the tunneling in a double-well system
To summarize the two-state tunneling probabilities forthe lowest pair of eigenstates are remarkably accurate exceptfor very narrow barriers and may provide useful estimatesif the rest of the states are of much higher energy Thebarrier can be considered sufficiently wide when the overlapof the reactant and product well eigenstates ⟨120601
119901|120601119903⟩ is less
than a few percent (5 percent in the current example)The QC expression (18) relating the energy level splitting
6 Journal of Theoretical Chemistry
Table1Energylevelssplittin
gsand
tunn
elingp
robabilityfor
the119899=01and119899=23pairofstatesofthesym
metric
rectangu
lard
ouble-wellpotentia
lofSectio
n31Th
ereactantp
rodu
ctstates
overlap
ofthetwo-stated
escriptio
nisgivenin
thelastcolum
nTh
eenergylevelsandlevelsplittings
areg
iven
in119864ℎN
umbersin
parenthesesa
rethep
owerso
f10
119871[1198860]
119864
QM0
Δ
QM
119879
QM
119864
QC0
Δ
QC
119879
QC
119864
2s 0Δ
2119904
119879
2119904
⟨120601119901|120601119903⟩
Forthe119899=01pairof
energy
states
808800
1919
(minus9)
1205
(minus17)
08800
1174
(minus9)
439
(minus18)
08800
1913
(minus9)
117(minus17)
3554(minus9)
408800
4192(minus5)
5752(minus9)
08800
2564(minus5)
2095(minus9)
08800
4210(minus5)
564
6(minus9)
440
8(minus5)
208770
6196(minus3)
1257
(minus4)
08800
3790(minus3)
4577(minus5)
08769
6224(minus3)
1243
(minus4)
4035(minus3)
108422
00753
00185
08800
4608(minus3)
6765(minus3)
08425
7564
(minus2)
1989
(minus2)
3398(minus2)
05
07433
02618
02105
08800
01607
8225(minus2)
07483
02637
03064
00922
025
060
9804881
05900
08800
03000
02868
06289
04906
1502
01476
Forthe119899=23pairof
energy
states
833053
1099
(minus4)
1481
(minus8)
33053
1690
(minus4)
6450(minus9)
33053
0974(minus4)
2141(minus9)
4712(minus4)
432993
1227
(minus2)
1848
(minus4)
33053
1886
(minus2)
8031(minus5)
32999
1087
(minus2)
2676(minus5)
2790(minus2)
232477
01334
2144(minus2)
33053
01992
8962(minus3)
32472
0114
63071(minus3)
01644
131337
04856
02267
33053
064
7400947
30860
03487
00315
03225
05
300
6810
078
06166
33053
11672
03077
28262
05329
00877
03905
025
29086
14864
08529
33053
15672
05547
25604
06101
01401
03961
Journal of Theoretical Chemistry 7
Table 2 Thermal rate constants 119896(119879) for the asymmetric rectangular double well of Section 32 The first column lists the temperature thatis 119896119861119879 in units of the barrier height119881
119887The thermal rate constants obtained using (9) using exact energy levels and QM and QC transmission
probabilities are given in the second and third columns respectively The last column contains the thermal rate constants obtained from QCtunneling probability and the energy levels of the reactant potential with infinitely high walls
119896119861119879119881119887
119879
QM119864
QM119899
119879
QC119864
QM119899
119879
QC119864
QC119899
001 691 (minus13) 265 (minus13) 738 (minus15)002 101 (minus8) 380 (minus9) 778 (minus10)004 123 (minus6) 464 (minus7) 252 (minus7)01 399 (minus5) 267 (minus5) 196 (minus5)02 131 (minus3) 197 (minus3) 241 (minus3)04 195 (minus2) 266 (minus2) 364 (minus2)10 0134 0159 019820 0286 0320 037140 0450 0483 0533
and tunneling is inaccurate for the higher energy pairs ofldquosplitrdquo levelsThe accuracy of the QC approximation dependson both the accuracy of QC tunneling probability anddiscrepancy between the energy levels of the full potentialand of the asymptotic reactant well For one-dimensionalsystems the use of exact energy eigenstates of the full potentialrather than those of an isolated reactant well in (9) ismore appropriate even with approximate calculation of thetunneling probability
32 Asymmetric Rectangular Barrier Now let us consider anasymmetric rectangular double-well potential given by (8)and sketched in Figure 1 There are 7 states with energiesbelow the barrier top The ground and first excited statesare localized on the reactant and product sides respectivelyThe eigenstates of the full potential with even quantum num-bers correlate with the eigenstates of the reactant well thedifference in energies of the full and asymptotic potentialsincreases with the quantum number The parameter valuesfor the potential are 119881
119887= 40119864
ℎ 1198810= 04119864
ℎ and 119871 = 10119886
0
The well width is 119889 = 401198860 The particle mass is119898 = 1 a u
Exact QM energy levels and eigenfunctions needed tocompute projections on the reactant well are obtained aseigenvalues and eigenvectors of the Hamiltonian matrix inthe Colbert-Miller discrete variable representation [27]
Exact QM transmission probability is given by (13) Qua-siclassical transmission probability is defined by (3) and set to(1) for energies above the barrier top The exact and approxi-mate rate constants computed from (9) are listed inTable 2 fortemperatures measured in units of the barrier top 119881
119887 Results
of the fully QM fully QC and mixed QCQM (QC trans-mission evaluated at exact QM energy levels) descriptions areshown in Figure 5 and in Table 2 At the lowest temperatureequivalent to 1 of the barrier height the QC rate constantis 93 times lower than the QM result while the mixedQCQM estimate is only 2 times lower At high temperatureswhen the high energy eigenstates significantly contribute tothe reaction rate constant the discrepancy between exactand approximate rate constants becomes smaller with themixed QCQM estimate being closer to the exact result at allenergies One concludes that generally the accuracy of 119896(119879)
0 005 01 015 02
Rate
cons
tant
Exact QM QM levelsQC T(E)Reactant levelsQC T(E)
k(T
)
0
02
04
06
08
Temperature [Vb]
0 10 20 30
00001
1e minus 08
1e minus 12
1e minus 16
Figure 5 Transmission through the asymmetric rectangular barrierof Section 32 The thermal reaction rate constant is given as afunction of temperature measured in units of the barrier height[119896119861119879119881119887] where119881
119887= 4119864ℎThe rate constants are obtained using the
exact QM transmission probability and energy levels (circlessolidline) the QC transmission probability evaluated at the exact QMenergy levels (trianglesdot-dash) and at the QC energy levels of theisolated reactant well (squaresdash) The insert shows 119896(119879) on thelinear scale for wide range of temperatures
depends both on the accuracy ofQC transmission probability119879
QC and on the difference between exact and asymptotic-wellenergy levels Using exact QM energy levels and functions inconjunction with quasiclassical transmission probabilities ismore accurate than the fully QC description
Note that for the rectangular potential considered here119879
QC(119864) = 1 for 119864 gt 119881
119887is inaccurate because QC approxi-
mation does not describe above-the-barrier reflection leadingto oscillations of 119879QM For general potentials the reactionprobability may be computed using for example the time-dependent QM wave packet approach [28] used in the next
8 Journal of Theoretical Chemistry
example yielding reaction probabilities for a range of ener-gies from a single calculation or with a time-independentmethod for a few energy values
33 Piecewise Quadratic Potential Next we examine a piece-wise quadratic potential sketched in Figure 2 The reactantwell and barrier are quadratic functions in 119909 The reactionrate constants are obtained using the QM and quasiclassicalWKB expressions and using the analytical QM transmission
119879
QMpar (119864) = (1 + exp(minus
2120587119864
119908119887
))
minus1
119908119887=radic
119896119887
119898
(19)
through a parabolic barrier Equation (19) is more accuratethan the QC expression near the barrier top [16] A continu-ous piecewise quadratic potential is defined as follows
119881 (119909)
=
119896(119909 minus 1199091)
2
2
119909 lt 119909119871reactant region
minus
119896119887119909
2
2
+ 119881119887
119909119871le 119909 le 119909
119877barrier top region
119896(119909 minus 1199092)
2
2
+ 1198810119909 gt 119909
119877product region
(20)
with the proper choice of parametersmdash119881119887 1198810 119909119871 and 119909
119877mdash
the potential and its first derivative is continuous function at119909 = 119909
119871and 119909 = 119909
119877 Here a symmetric double well 119881
0= 0
is considered the remaining parameter values are 1199092= minus1199091
= 11198860 119896 = 119896
119887= 80119864
ℎ119886
minus2
0 119881119887= 20119864
ℎ and 119909
119877= minus119909119871= 05119886
0
The particle mass is 119898 = 1 a u (If the Hamiltonian werescaled by the mass of the proton 119881
119887would be equivalent to
03 eV or 7 kcalmol)The barrier parameters are chosen suchthat there are only few energy levels under the barrier topwhich is typical for a chemical reaction at low temperature
The time-dependent wave packet correlation approach[28] has been used to calculate QM transmission probability119879(119864) = |119878
119903119901(119864)|
2 entering (9) since we examine dependenceof the reaction rate on the accuracy of the energy levels usedin (9) The scattering matrix element 119878
119903119901(119864) describes trans-
mission from reactants to products
119878119903119901(119864) =
(2120587)
minus1
120578
lowast
119901(119864) 120578119903 (
119864)
int
infin
minusinfin
⟨120601
minus
119901
100381610038161003816100381610038161003816
119890
minus120580119905100381610038161003816100381610038161003816
120601
+
119903⟩ 119890
120580119864119905119889119905 (21)
The subscripts 119903119901 refer to reactantproduct reaction chan-nels and plusmn refers to the asymptotically incomingoutgoingwave relative to the barrier An incoming wave packet |120601+
119903⟩
taken as a Gaussian function in the left asymptotic regionof 119881 is evolved in time in the ldquounfoldedrdquo potential shownin Figure 2 with a dash Another Gaussian function |120601minus
119901⟩
is placed in the product region of the potential The time-dependent overlap of evolving |120601+
119903⟩ with the stationary ⟨120601minus
119901|
or correlation function 119862(119905) = ⟨120601
minus
119901(119909 0) | 120601
+
119903(119909 119905)⟩ is
computed and Fourier-transformed into the energy domain
The denominator in (21) accounts for the distributions ofenergy eigenstates in the reactant and product wave packetsat time 119905 = 0 these distributions depend on the wave packetlocalization and kinetic energy
120578119903119901(119864) = radic
119898
2120587119901
int
infin
minusinfin
119890
minus120580119901119909120601
plusmn
119903119901(119909) 119889119909 119901 =
radic2119898119864
(22)
A single wave packet propagation accomplished on a gridusing the split-operator method [29 30] gives transmissionprobability for a range of energies represented in the reac-tantproduct wave packets which is convenient if multipleeigenstates contribute to the rate constant At low energiesit is hard to converge transmission probability using time-dependent dynamics methods but the energies below thezero-point energy of the bound potential do not contributeto 119896(119879)The exact QM energy levels and projections of eigen-functions on the reactant well are performed as in Section 32There are five energy levels below the barrier top but manymore states contribute to 119896(119879) at higher temperatures
The energy-resolved exactQM parabolic barrier approxi-mation and QC transmission probabilities are shown in Fig-ure 6(a) Tunneling probabilities are underestimated in theQC approximation while the parabolic barrier expressiongives accurate results for energies above 119881
1198875 = 4119864
ℎ(the
barrier is in fact parabolic in this energy range) The discrep-ancy grows at energy below 4119864
ℎ This energy range however
is below the zero-point energy and does not contributeto the thermal rate constants shown in Figure 6(b) Theapproximate rate constants differ from exact QM results byat most 25ndash30
34 Proton Transfer As a chemically relevant model we con-sider the proton transfer in the HOndashHndashCH
3system for a
constrained collinear OndashHndashC geometry In this model sys-tem the proton is transferred from a donor carbon to ac-ceptor oxygenThe potential energy surface is obtained fromdensity function theory (DFT) electronic structure calcula-tions in particular at the B3LYP6-31G(dp) level of theory[31] The energies were compared to those obtained at theCCSD(T)aug-cc-pVDZ theory level for the same geome-tries and were found in excellent agreement In this sys-tem collinear donor-proton-acceptor arrangement is a goodapproximation to the fully optimized reaction path as deter-mined by a set of constrained geometry optimizations Inthese optimizations119877OH was fixed to a value in the range 08ndash20 A the average deviation of angCHO from linearity was lessthan 2∘ and the maximum deviation was 10∘
One-dimensional potential energy surfaces of 40 pointswere generated as functions of 119877OH for a fixed 119877CO distanceProton transfer on three surfaces for119877CO = 27 28 and 29 Ais analyzed below The surfaces were parametrized as sixthdegree polynomials in 119877OH The potential energy curvesshown in Figure 7(a) have characteristic asymmetric double-well shape
We examine the effect of donor-acceptor distance 119877COon the tunneling rate constants using QM and QC transmis-sion probabilities and exact and approximate energy levels
Journal of Theoretical Chemistry 9
00001
001
1
QMQCParabolic
0 10 20 30
1
1e minus 08
1e minus 06
Energy [Eh]
T(E
)
(a)
0 05 1 15 20
05
Rate
cons
tant
0 01 02
00001
001
Temperature [Vb]
(b)
Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896
119861119879119881119887] where 119881
119887= 20119864
ℎ
0
10
20
30
40
50
08 1 12 14 16 18 2
272829
Ener
gy (m
E h)
ROH (A)
(a)
0 500 1000 1500
103
102
101
100
KIE
Temperature (K)
(b)
Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO
distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896
119867119896119863(KIE) using exact QM formulation
in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top
The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896
119867119896119863should be largest at low
temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CatalystsJournal of
Journal of Theoretical Chemistry 5
6 80
0
05
1
15
2
25
42Barrier width L [a0]
TQCTQM
T2sTQM
Figure 3 Tunneling in the symmetric rectangular double-wellpotential of Section 31 Ratios of approximate transmission prob-abilities that is the quasiclassical WKB (squaresdash) and the two-state representation (circlessolid lines) to the exact QM probabilityare given as functions of the barrier width
representation the energies of the ground and excited statesare the generalized eigenvalues of the 2-by-2 HamiltonianmatrixHwith the overlapmatrix S For a symmetric potentialthe matrix elements are
ℎ11= ℎ22= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119903⟩ ℎ12= ℎ21= ⟨120601119903
10038161003816100381610038161003816
119867
10038161003816100381610038161003816
120601119901⟩ (15)
11990411= 11990422= ⟨120601119903| 120601119903⟩ 11990412= 11990421= ⟨120601119903| 120601119903⟩ (16)
H1205952119904119894= 119864
2119904
119894S1205952119904119894119894 = 0 1 (17)
The energy level splitting is Δ2119904 = 11986421199041minus 119864
2119904
0
Following [24] we use the quasiclassical relation betweenthe energy level splitting and the tunneling probability
1198641minus 1198640= Δ =
21198640
120587
radic119879 (1198640) (18)
with the ground state energy used instead of the frequency ofthe harmonic reactant well In the two-state representationthe energy level splitting Δ2119904 following from (17) yields anestimate for the tunneling 1198792119904 The same (18) gives estimatesof the quasiclassical energy level splitting ΔQC once thequasiclassical WKB tunneling probability 119879QC is computedfrom (3) Two pairs of states of the full potential 119899 = 0 119899 = 1and 119899 = 2 119899 = 3 are analyzed the higher energy states 119899 = 2and 119899 = 3 are described in the basis of the first excited statesof the reactant and product wells
Table 1 lists the ground and first excited energy levels ob-tained exactly using quasiclassical WKB approximation andwithin the two-state approximation labeled QM QC and2119904 respectively are listed in the upper half of Table 1The samequantities for the next highest pair of levels (Δ = 119864
3minus1198642) are
given in the lower half of Table 1 In the quasiclassical WKBtreatment the energy levels are defined by the width of the
1 2 3 4
Energy [Eh]
00001
1e minus 08
Tunn
elin
gT
(E)
L = 2 QML = 2 QC
E0
E1
Figure 4 Transmission through the symmetric rectangular barrierof Section 31 of the width 119871 = 2119886
0 Exact QM and QC WKB
probabilities are shown as a solid line and a dash respectively Theenergies of the ground and first excited states for the well width119889 = 2119886
0are indicated with dot-dashes
reactant well and therefore do not depend on the barrierwidth The ground state energy in the two-state approx-imation is remarkably accurate even for narrow barrierswhile the accuracy of 119899 = 2 level deteriorates for smallerbarrier widths The ratios of the quasiclassical and two-stateestimates of tunneling probability to the exact QM values asa function of the barrier width are plotted in Figure 3 for thelowest pair of states The agreement between 1198792119904 and 119879QM isexcellent for the barrier width 119871 gt 10 119886
0 The quasiclassical
WKB results underestimate the tunneling probability by afactor of 2This discrepancy can be understood by comparingquasiclassical WKB and QM tunneling at the energies ofthe asymptotic eigenstates shown in Figure 4 for the barrierwidth 119871 = 2119886
0 The transmission probabilities at these ener-
gies differ by a factor of 2 For a wider well of the width119889 = 4119886
0(not shown here) the ground state energy happens to
be near the intersection of the QM and quasiclassical WKBtunneling curves at 119864 = 02599119864
ℎ which coincidentally
yields much better agreement than the results shown inFigure 3 Therefore the accuracy of the quasiclassical resultsdepends on the reactant well width even if the accuracy ofthe quasiclassical tunneling probability does not At very lowenergies the discrepancy between the QM and QC tunnelingis in orders of magnitude but fortunately the energy regimebelow the zero-point energy of reactants does not contributeto the tunneling in a double-well system
To summarize the two-state tunneling probabilities forthe lowest pair of eigenstates are remarkably accurate exceptfor very narrow barriers and may provide useful estimatesif the rest of the states are of much higher energy Thebarrier can be considered sufficiently wide when the overlapof the reactant and product well eigenstates ⟨120601
119901|120601119903⟩ is less
than a few percent (5 percent in the current example)The QC expression (18) relating the energy level splitting
6 Journal of Theoretical Chemistry
Table1Energylevelssplittin
gsand
tunn
elingp
robabilityfor
the119899=01and119899=23pairofstatesofthesym
metric
rectangu
lard
ouble-wellpotentia
lofSectio
n31Th
ereactantp
rodu
ctstates
overlap
ofthetwo-stated
escriptio
nisgivenin
thelastcolum
nTh
eenergylevelsandlevelsplittings
areg
iven
in119864ℎN
umbersin
parenthesesa
rethep
owerso
f10
119871[1198860]
119864
QM0
Δ
QM
119879
QM
119864
QC0
Δ
QC
119879
QC
119864
2s 0Δ
2119904
119879
2119904
⟨120601119901|120601119903⟩
Forthe119899=01pairof
energy
states
808800
1919
(minus9)
1205
(minus17)
08800
1174
(minus9)
439
(minus18)
08800
1913
(minus9)
117(minus17)
3554(minus9)
408800
4192(minus5)
5752(minus9)
08800
2564(minus5)
2095(minus9)
08800
4210(minus5)
564
6(minus9)
440
8(minus5)
208770
6196(minus3)
1257
(minus4)
08800
3790(minus3)
4577(minus5)
08769
6224(minus3)
1243
(minus4)
4035(minus3)
108422
00753
00185
08800
4608(minus3)
6765(minus3)
08425
7564
(minus2)
1989
(minus2)
3398(minus2)
05
07433
02618
02105
08800
01607
8225(minus2)
07483
02637
03064
00922
025
060
9804881
05900
08800
03000
02868
06289
04906
1502
01476
Forthe119899=23pairof
energy
states
833053
1099
(minus4)
1481
(minus8)
33053
1690
(minus4)
6450(minus9)
33053
0974(minus4)
2141(minus9)
4712(minus4)
432993
1227
(minus2)
1848
(minus4)
33053
1886
(minus2)
8031(minus5)
32999
1087
(minus2)
2676(minus5)
2790(minus2)
232477
01334
2144(minus2)
33053
01992
8962(minus3)
32472
0114
63071(minus3)
01644
131337
04856
02267
33053
064
7400947
30860
03487
00315
03225
05
300
6810
078
06166
33053
11672
03077
28262
05329
00877
03905
025
29086
14864
08529
33053
15672
05547
25604
06101
01401
03961
Journal of Theoretical Chemistry 7
Table 2 Thermal rate constants 119896(119879) for the asymmetric rectangular double well of Section 32 The first column lists the temperature thatis 119896119861119879 in units of the barrier height119881
119887The thermal rate constants obtained using (9) using exact energy levels and QM and QC transmission
probabilities are given in the second and third columns respectively The last column contains the thermal rate constants obtained from QCtunneling probability and the energy levels of the reactant potential with infinitely high walls
119896119861119879119881119887
119879
QM119864
QM119899
119879
QC119864
QM119899
119879
QC119864
QC119899
001 691 (minus13) 265 (minus13) 738 (minus15)002 101 (minus8) 380 (minus9) 778 (minus10)004 123 (minus6) 464 (minus7) 252 (minus7)01 399 (minus5) 267 (minus5) 196 (minus5)02 131 (minus3) 197 (minus3) 241 (minus3)04 195 (minus2) 266 (minus2) 364 (minus2)10 0134 0159 019820 0286 0320 037140 0450 0483 0533
and tunneling is inaccurate for the higher energy pairs ofldquosplitrdquo levelsThe accuracy of the QC approximation dependson both the accuracy of QC tunneling probability anddiscrepancy between the energy levels of the full potentialand of the asymptotic reactant well For one-dimensionalsystems the use of exact energy eigenstates of the full potentialrather than those of an isolated reactant well in (9) ismore appropriate even with approximate calculation of thetunneling probability
32 Asymmetric Rectangular Barrier Now let us consider anasymmetric rectangular double-well potential given by (8)and sketched in Figure 1 There are 7 states with energiesbelow the barrier top The ground and first excited statesare localized on the reactant and product sides respectivelyThe eigenstates of the full potential with even quantum num-bers correlate with the eigenstates of the reactant well thedifference in energies of the full and asymptotic potentialsincreases with the quantum number The parameter valuesfor the potential are 119881
119887= 40119864
ℎ 1198810= 04119864
ℎ and 119871 = 10119886
0
The well width is 119889 = 401198860 The particle mass is119898 = 1 a u
Exact QM energy levels and eigenfunctions needed tocompute projections on the reactant well are obtained aseigenvalues and eigenvectors of the Hamiltonian matrix inthe Colbert-Miller discrete variable representation [27]
Exact QM transmission probability is given by (13) Qua-siclassical transmission probability is defined by (3) and set to(1) for energies above the barrier top The exact and approxi-mate rate constants computed from (9) are listed inTable 2 fortemperatures measured in units of the barrier top 119881
119887 Results
of the fully QM fully QC and mixed QCQM (QC trans-mission evaluated at exact QM energy levels) descriptions areshown in Figure 5 and in Table 2 At the lowest temperatureequivalent to 1 of the barrier height the QC rate constantis 93 times lower than the QM result while the mixedQCQM estimate is only 2 times lower At high temperatureswhen the high energy eigenstates significantly contribute tothe reaction rate constant the discrepancy between exactand approximate rate constants becomes smaller with themixed QCQM estimate being closer to the exact result at allenergies One concludes that generally the accuracy of 119896(119879)
0 005 01 015 02
Rate
cons
tant
Exact QM QM levelsQC T(E)Reactant levelsQC T(E)
k(T
)
0
02
04
06
08
Temperature [Vb]
0 10 20 30
00001
1e minus 08
1e minus 12
1e minus 16
Figure 5 Transmission through the asymmetric rectangular barrierof Section 32 The thermal reaction rate constant is given as afunction of temperature measured in units of the barrier height[119896119861119879119881119887] where119881
119887= 4119864ℎThe rate constants are obtained using the
exact QM transmission probability and energy levels (circlessolidline) the QC transmission probability evaluated at the exact QMenergy levels (trianglesdot-dash) and at the QC energy levels of theisolated reactant well (squaresdash) The insert shows 119896(119879) on thelinear scale for wide range of temperatures
depends both on the accuracy ofQC transmission probability119879
QC and on the difference between exact and asymptotic-wellenergy levels Using exact QM energy levels and functions inconjunction with quasiclassical transmission probabilities ismore accurate than the fully QC description
Note that for the rectangular potential considered here119879
QC(119864) = 1 for 119864 gt 119881
119887is inaccurate because QC approxi-
mation does not describe above-the-barrier reflection leadingto oscillations of 119879QM For general potentials the reactionprobability may be computed using for example the time-dependent QM wave packet approach [28] used in the next
8 Journal of Theoretical Chemistry
example yielding reaction probabilities for a range of ener-gies from a single calculation or with a time-independentmethod for a few energy values
33 Piecewise Quadratic Potential Next we examine a piece-wise quadratic potential sketched in Figure 2 The reactantwell and barrier are quadratic functions in 119909 The reactionrate constants are obtained using the QM and quasiclassicalWKB expressions and using the analytical QM transmission
119879
QMpar (119864) = (1 + exp(minus
2120587119864
119908119887
))
minus1
119908119887=radic
119896119887
119898
(19)
through a parabolic barrier Equation (19) is more accuratethan the QC expression near the barrier top [16] A continu-ous piecewise quadratic potential is defined as follows
119881 (119909)
=
119896(119909 minus 1199091)
2
2
119909 lt 119909119871reactant region
minus
119896119887119909
2
2
+ 119881119887
119909119871le 119909 le 119909
119877barrier top region
119896(119909 minus 1199092)
2
2
+ 1198810119909 gt 119909
119877product region
(20)
with the proper choice of parametersmdash119881119887 1198810 119909119871 and 119909
119877mdash
the potential and its first derivative is continuous function at119909 = 119909
119871and 119909 = 119909
119877 Here a symmetric double well 119881
0= 0
is considered the remaining parameter values are 1199092= minus1199091
= 11198860 119896 = 119896
119887= 80119864
ℎ119886
minus2
0 119881119887= 20119864
ℎ and 119909
119877= minus119909119871= 05119886
0
The particle mass is 119898 = 1 a u (If the Hamiltonian werescaled by the mass of the proton 119881
119887would be equivalent to
03 eV or 7 kcalmol)The barrier parameters are chosen suchthat there are only few energy levels under the barrier topwhich is typical for a chemical reaction at low temperature
The time-dependent wave packet correlation approach[28] has been used to calculate QM transmission probability119879(119864) = |119878
119903119901(119864)|
2 entering (9) since we examine dependenceof the reaction rate on the accuracy of the energy levels usedin (9) The scattering matrix element 119878
119903119901(119864) describes trans-
mission from reactants to products
119878119903119901(119864) =
(2120587)
minus1
120578
lowast
119901(119864) 120578119903 (
119864)
int
infin
minusinfin
⟨120601
minus
119901
100381610038161003816100381610038161003816
119890
minus120580119905100381610038161003816100381610038161003816
120601
+
119903⟩ 119890
120580119864119905119889119905 (21)
The subscripts 119903119901 refer to reactantproduct reaction chan-nels and plusmn refers to the asymptotically incomingoutgoingwave relative to the barrier An incoming wave packet |120601+
119903⟩
taken as a Gaussian function in the left asymptotic regionof 119881 is evolved in time in the ldquounfoldedrdquo potential shownin Figure 2 with a dash Another Gaussian function |120601minus
119901⟩
is placed in the product region of the potential The time-dependent overlap of evolving |120601+
119903⟩ with the stationary ⟨120601minus
119901|
or correlation function 119862(119905) = ⟨120601
minus
119901(119909 0) | 120601
+
119903(119909 119905)⟩ is
computed and Fourier-transformed into the energy domain
The denominator in (21) accounts for the distributions ofenergy eigenstates in the reactant and product wave packetsat time 119905 = 0 these distributions depend on the wave packetlocalization and kinetic energy
120578119903119901(119864) = radic
119898
2120587119901
int
infin
minusinfin
119890
minus120580119901119909120601
plusmn
119903119901(119909) 119889119909 119901 =
radic2119898119864
(22)
A single wave packet propagation accomplished on a gridusing the split-operator method [29 30] gives transmissionprobability for a range of energies represented in the reac-tantproduct wave packets which is convenient if multipleeigenstates contribute to the rate constant At low energiesit is hard to converge transmission probability using time-dependent dynamics methods but the energies below thezero-point energy of the bound potential do not contributeto 119896(119879)The exact QM energy levels and projections of eigen-functions on the reactant well are performed as in Section 32There are five energy levels below the barrier top but manymore states contribute to 119896(119879) at higher temperatures
The energy-resolved exactQM parabolic barrier approxi-mation and QC transmission probabilities are shown in Fig-ure 6(a) Tunneling probabilities are underestimated in theQC approximation while the parabolic barrier expressiongives accurate results for energies above 119881
1198875 = 4119864
ℎ(the
barrier is in fact parabolic in this energy range) The discrep-ancy grows at energy below 4119864
ℎ This energy range however
is below the zero-point energy and does not contributeto the thermal rate constants shown in Figure 6(b) Theapproximate rate constants differ from exact QM results byat most 25ndash30
34 Proton Transfer As a chemically relevant model we con-sider the proton transfer in the HOndashHndashCH
3system for a
constrained collinear OndashHndashC geometry In this model sys-tem the proton is transferred from a donor carbon to ac-ceptor oxygenThe potential energy surface is obtained fromdensity function theory (DFT) electronic structure calcula-tions in particular at the B3LYP6-31G(dp) level of theory[31] The energies were compared to those obtained at theCCSD(T)aug-cc-pVDZ theory level for the same geome-tries and were found in excellent agreement In this sys-tem collinear donor-proton-acceptor arrangement is a goodapproximation to the fully optimized reaction path as deter-mined by a set of constrained geometry optimizations Inthese optimizations119877OH was fixed to a value in the range 08ndash20 A the average deviation of angCHO from linearity was lessthan 2∘ and the maximum deviation was 10∘
One-dimensional potential energy surfaces of 40 pointswere generated as functions of 119877OH for a fixed 119877CO distanceProton transfer on three surfaces for119877CO = 27 28 and 29 Ais analyzed below The surfaces were parametrized as sixthdegree polynomials in 119877OH The potential energy curvesshown in Figure 7(a) have characteristic asymmetric double-well shape
We examine the effect of donor-acceptor distance 119877COon the tunneling rate constants using QM and QC transmis-sion probabilities and exact and approximate energy levels
Journal of Theoretical Chemistry 9
00001
001
1
QMQCParabolic
0 10 20 30
1
1e minus 08
1e minus 06
Energy [Eh]
T(E
)
(a)
0 05 1 15 20
05
Rate
cons
tant
0 01 02
00001
001
Temperature [Vb]
(b)
Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896
119861119879119881119887] where 119881
119887= 20119864
ℎ
0
10
20
30
40
50
08 1 12 14 16 18 2
272829
Ener
gy (m
E h)
ROH (A)
(a)
0 500 1000 1500
103
102
101
100
KIE
Temperature (K)
(b)
Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO
distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896
119867119896119863(KIE) using exact QM formulation
in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top
The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896
119867119896119863should be largest at low
temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
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Carbohydrate Chemistry
International Journal of
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Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
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Analytical Methods in Chemistry
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Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
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Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
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Quantum Chemistry
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Organic Chemistry International
ElectrochemistryInternational Journal of
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CatalystsJournal of
6 Journal of Theoretical Chemistry
Table1Energylevelssplittin
gsand
tunn
elingp
robabilityfor
the119899=01and119899=23pairofstatesofthesym
metric
rectangu
lard
ouble-wellpotentia
lofSectio
n31Th
ereactantp
rodu
ctstates
overlap
ofthetwo-stated
escriptio
nisgivenin
thelastcolum
nTh
eenergylevelsandlevelsplittings
areg
iven
in119864ℎN
umbersin
parenthesesa
rethep
owerso
f10
119871[1198860]
119864
QM0
Δ
QM
119879
QM
119864
QC0
Δ
QC
119879
QC
119864
2s 0Δ
2119904
119879
2119904
⟨120601119901|120601119903⟩
Forthe119899=01pairof
energy
states
808800
1919
(minus9)
1205
(minus17)
08800
1174
(minus9)
439
(minus18)
08800
1913
(minus9)
117(minus17)
3554(minus9)
408800
4192(minus5)
5752(minus9)
08800
2564(minus5)
2095(minus9)
08800
4210(minus5)
564
6(minus9)
440
8(minus5)
208770
6196(minus3)
1257
(minus4)
08800
3790(minus3)
4577(minus5)
08769
6224(minus3)
1243
(minus4)
4035(minus3)
108422
00753
00185
08800
4608(minus3)
6765(minus3)
08425
7564
(minus2)
1989
(minus2)
3398(minus2)
05
07433
02618
02105
08800
01607
8225(minus2)
07483
02637
03064
00922
025
060
9804881
05900
08800
03000
02868
06289
04906
1502
01476
Forthe119899=23pairof
energy
states
833053
1099
(minus4)
1481
(minus8)
33053
1690
(minus4)
6450(minus9)
33053
0974(minus4)
2141(minus9)
4712(minus4)
432993
1227
(minus2)
1848
(minus4)
33053
1886
(minus2)
8031(minus5)
32999
1087
(minus2)
2676(minus5)
2790(minus2)
232477
01334
2144(minus2)
33053
01992
8962(minus3)
32472
0114
63071(minus3)
01644
131337
04856
02267
33053
064
7400947
30860
03487
00315
03225
05
300
6810
078
06166
33053
11672
03077
28262
05329
00877
03905
025
29086
14864
08529
33053
15672
05547
25604
06101
01401
03961
Journal of Theoretical Chemistry 7
Table 2 Thermal rate constants 119896(119879) for the asymmetric rectangular double well of Section 32 The first column lists the temperature thatis 119896119861119879 in units of the barrier height119881
119887The thermal rate constants obtained using (9) using exact energy levels and QM and QC transmission
probabilities are given in the second and third columns respectively The last column contains the thermal rate constants obtained from QCtunneling probability and the energy levels of the reactant potential with infinitely high walls
119896119861119879119881119887
119879
QM119864
QM119899
119879
QC119864
QM119899
119879
QC119864
QC119899
001 691 (minus13) 265 (minus13) 738 (minus15)002 101 (minus8) 380 (minus9) 778 (minus10)004 123 (minus6) 464 (minus7) 252 (minus7)01 399 (minus5) 267 (minus5) 196 (minus5)02 131 (minus3) 197 (minus3) 241 (minus3)04 195 (minus2) 266 (minus2) 364 (minus2)10 0134 0159 019820 0286 0320 037140 0450 0483 0533
and tunneling is inaccurate for the higher energy pairs ofldquosplitrdquo levelsThe accuracy of the QC approximation dependson both the accuracy of QC tunneling probability anddiscrepancy between the energy levels of the full potentialand of the asymptotic reactant well For one-dimensionalsystems the use of exact energy eigenstates of the full potentialrather than those of an isolated reactant well in (9) ismore appropriate even with approximate calculation of thetunneling probability
32 Asymmetric Rectangular Barrier Now let us consider anasymmetric rectangular double-well potential given by (8)and sketched in Figure 1 There are 7 states with energiesbelow the barrier top The ground and first excited statesare localized on the reactant and product sides respectivelyThe eigenstates of the full potential with even quantum num-bers correlate with the eigenstates of the reactant well thedifference in energies of the full and asymptotic potentialsincreases with the quantum number The parameter valuesfor the potential are 119881
119887= 40119864
ℎ 1198810= 04119864
ℎ and 119871 = 10119886
0
The well width is 119889 = 401198860 The particle mass is119898 = 1 a u
Exact QM energy levels and eigenfunctions needed tocompute projections on the reactant well are obtained aseigenvalues and eigenvectors of the Hamiltonian matrix inthe Colbert-Miller discrete variable representation [27]
Exact QM transmission probability is given by (13) Qua-siclassical transmission probability is defined by (3) and set to(1) for energies above the barrier top The exact and approxi-mate rate constants computed from (9) are listed inTable 2 fortemperatures measured in units of the barrier top 119881
119887 Results
of the fully QM fully QC and mixed QCQM (QC trans-mission evaluated at exact QM energy levels) descriptions areshown in Figure 5 and in Table 2 At the lowest temperatureequivalent to 1 of the barrier height the QC rate constantis 93 times lower than the QM result while the mixedQCQM estimate is only 2 times lower At high temperatureswhen the high energy eigenstates significantly contribute tothe reaction rate constant the discrepancy between exactand approximate rate constants becomes smaller with themixed QCQM estimate being closer to the exact result at allenergies One concludes that generally the accuracy of 119896(119879)
0 005 01 015 02
Rate
cons
tant
Exact QM QM levelsQC T(E)Reactant levelsQC T(E)
k(T
)
0
02
04
06
08
Temperature [Vb]
0 10 20 30
00001
1e minus 08
1e minus 12
1e minus 16
Figure 5 Transmission through the asymmetric rectangular barrierof Section 32 The thermal reaction rate constant is given as afunction of temperature measured in units of the barrier height[119896119861119879119881119887] where119881
119887= 4119864ℎThe rate constants are obtained using the
exact QM transmission probability and energy levels (circlessolidline) the QC transmission probability evaluated at the exact QMenergy levels (trianglesdot-dash) and at the QC energy levels of theisolated reactant well (squaresdash) The insert shows 119896(119879) on thelinear scale for wide range of temperatures
depends both on the accuracy ofQC transmission probability119879
QC and on the difference between exact and asymptotic-wellenergy levels Using exact QM energy levels and functions inconjunction with quasiclassical transmission probabilities ismore accurate than the fully QC description
Note that for the rectangular potential considered here119879
QC(119864) = 1 for 119864 gt 119881
119887is inaccurate because QC approxi-
mation does not describe above-the-barrier reflection leadingto oscillations of 119879QM For general potentials the reactionprobability may be computed using for example the time-dependent QM wave packet approach [28] used in the next
8 Journal of Theoretical Chemistry
example yielding reaction probabilities for a range of ener-gies from a single calculation or with a time-independentmethod for a few energy values
33 Piecewise Quadratic Potential Next we examine a piece-wise quadratic potential sketched in Figure 2 The reactantwell and barrier are quadratic functions in 119909 The reactionrate constants are obtained using the QM and quasiclassicalWKB expressions and using the analytical QM transmission
119879
QMpar (119864) = (1 + exp(minus
2120587119864
119908119887
))
minus1
119908119887=radic
119896119887
119898
(19)
through a parabolic barrier Equation (19) is more accuratethan the QC expression near the barrier top [16] A continu-ous piecewise quadratic potential is defined as follows
119881 (119909)
=
119896(119909 minus 1199091)
2
2
119909 lt 119909119871reactant region
minus
119896119887119909
2
2
+ 119881119887
119909119871le 119909 le 119909
119877barrier top region
119896(119909 minus 1199092)
2
2
+ 1198810119909 gt 119909
119877product region
(20)
with the proper choice of parametersmdash119881119887 1198810 119909119871 and 119909
119877mdash
the potential and its first derivative is continuous function at119909 = 119909
119871and 119909 = 119909
119877 Here a symmetric double well 119881
0= 0
is considered the remaining parameter values are 1199092= minus1199091
= 11198860 119896 = 119896
119887= 80119864
ℎ119886
minus2
0 119881119887= 20119864
ℎ and 119909
119877= minus119909119871= 05119886
0
The particle mass is 119898 = 1 a u (If the Hamiltonian werescaled by the mass of the proton 119881
119887would be equivalent to
03 eV or 7 kcalmol)The barrier parameters are chosen suchthat there are only few energy levels under the barrier topwhich is typical for a chemical reaction at low temperature
The time-dependent wave packet correlation approach[28] has been used to calculate QM transmission probability119879(119864) = |119878
119903119901(119864)|
2 entering (9) since we examine dependenceof the reaction rate on the accuracy of the energy levels usedin (9) The scattering matrix element 119878
119903119901(119864) describes trans-
mission from reactants to products
119878119903119901(119864) =
(2120587)
minus1
120578
lowast
119901(119864) 120578119903 (
119864)
int
infin
minusinfin
⟨120601
minus
119901
100381610038161003816100381610038161003816
119890
minus120580119905100381610038161003816100381610038161003816
120601
+
119903⟩ 119890
120580119864119905119889119905 (21)
The subscripts 119903119901 refer to reactantproduct reaction chan-nels and plusmn refers to the asymptotically incomingoutgoingwave relative to the barrier An incoming wave packet |120601+
119903⟩
taken as a Gaussian function in the left asymptotic regionof 119881 is evolved in time in the ldquounfoldedrdquo potential shownin Figure 2 with a dash Another Gaussian function |120601minus
119901⟩
is placed in the product region of the potential The time-dependent overlap of evolving |120601+
119903⟩ with the stationary ⟨120601minus
119901|
or correlation function 119862(119905) = ⟨120601
minus
119901(119909 0) | 120601
+
119903(119909 119905)⟩ is
computed and Fourier-transformed into the energy domain
The denominator in (21) accounts for the distributions ofenergy eigenstates in the reactant and product wave packetsat time 119905 = 0 these distributions depend on the wave packetlocalization and kinetic energy
120578119903119901(119864) = radic
119898
2120587119901
int
infin
minusinfin
119890
minus120580119901119909120601
plusmn
119903119901(119909) 119889119909 119901 =
radic2119898119864
(22)
A single wave packet propagation accomplished on a gridusing the split-operator method [29 30] gives transmissionprobability for a range of energies represented in the reac-tantproduct wave packets which is convenient if multipleeigenstates contribute to the rate constant At low energiesit is hard to converge transmission probability using time-dependent dynamics methods but the energies below thezero-point energy of the bound potential do not contributeto 119896(119879)The exact QM energy levels and projections of eigen-functions on the reactant well are performed as in Section 32There are five energy levels below the barrier top but manymore states contribute to 119896(119879) at higher temperatures
The energy-resolved exactQM parabolic barrier approxi-mation and QC transmission probabilities are shown in Fig-ure 6(a) Tunneling probabilities are underestimated in theQC approximation while the parabolic barrier expressiongives accurate results for energies above 119881
1198875 = 4119864
ℎ(the
barrier is in fact parabolic in this energy range) The discrep-ancy grows at energy below 4119864
ℎ This energy range however
is below the zero-point energy and does not contributeto the thermal rate constants shown in Figure 6(b) Theapproximate rate constants differ from exact QM results byat most 25ndash30
34 Proton Transfer As a chemically relevant model we con-sider the proton transfer in the HOndashHndashCH
3system for a
constrained collinear OndashHndashC geometry In this model sys-tem the proton is transferred from a donor carbon to ac-ceptor oxygenThe potential energy surface is obtained fromdensity function theory (DFT) electronic structure calcula-tions in particular at the B3LYP6-31G(dp) level of theory[31] The energies were compared to those obtained at theCCSD(T)aug-cc-pVDZ theory level for the same geome-tries and were found in excellent agreement In this sys-tem collinear donor-proton-acceptor arrangement is a goodapproximation to the fully optimized reaction path as deter-mined by a set of constrained geometry optimizations Inthese optimizations119877OH was fixed to a value in the range 08ndash20 A the average deviation of angCHO from linearity was lessthan 2∘ and the maximum deviation was 10∘
One-dimensional potential energy surfaces of 40 pointswere generated as functions of 119877OH for a fixed 119877CO distanceProton transfer on three surfaces for119877CO = 27 28 and 29 Ais analyzed below The surfaces were parametrized as sixthdegree polynomials in 119877OH The potential energy curvesshown in Figure 7(a) have characteristic asymmetric double-well shape
We examine the effect of donor-acceptor distance 119877COon the tunneling rate constants using QM and QC transmis-sion probabilities and exact and approximate energy levels
Journal of Theoretical Chemistry 9
00001
001
1
QMQCParabolic
0 10 20 30
1
1e minus 08
1e minus 06
Energy [Eh]
T(E
)
(a)
0 05 1 15 20
05
Rate
cons
tant
0 01 02
00001
001
Temperature [Vb]
(b)
Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896
119861119879119881119887] where 119881
119887= 20119864
ℎ
0
10
20
30
40
50
08 1 12 14 16 18 2
272829
Ener
gy (m
E h)
ROH (A)
(a)
0 500 1000 1500
103
102
101
100
KIE
Temperature (K)
(b)
Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO
distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896
119867119896119863(KIE) using exact QM formulation
in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top
The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896
119867119896119863should be largest at low
temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal ofPhotoenergy
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Carbohydrate Chemistry
International Journal of
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Journal of
Chemistry
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Advances in
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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Medicinal ChemistryInternational Journal of
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Chromatography Research International
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Analytical ChemistryInternational Journal of
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Journal of
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Quantum Chemistry
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Organic Chemistry International
ElectrochemistryInternational Journal of
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CatalystsJournal of
Journal of Theoretical Chemistry 7
Table 2 Thermal rate constants 119896(119879) for the asymmetric rectangular double well of Section 32 The first column lists the temperature thatis 119896119861119879 in units of the barrier height119881
119887The thermal rate constants obtained using (9) using exact energy levels and QM and QC transmission
probabilities are given in the second and third columns respectively The last column contains the thermal rate constants obtained from QCtunneling probability and the energy levels of the reactant potential with infinitely high walls
119896119861119879119881119887
119879
QM119864
QM119899
119879
QC119864
QM119899
119879
QC119864
QC119899
001 691 (minus13) 265 (minus13) 738 (minus15)002 101 (minus8) 380 (minus9) 778 (minus10)004 123 (minus6) 464 (minus7) 252 (minus7)01 399 (minus5) 267 (minus5) 196 (minus5)02 131 (minus3) 197 (minus3) 241 (minus3)04 195 (minus2) 266 (minus2) 364 (minus2)10 0134 0159 019820 0286 0320 037140 0450 0483 0533
and tunneling is inaccurate for the higher energy pairs ofldquosplitrdquo levelsThe accuracy of the QC approximation dependson both the accuracy of QC tunneling probability anddiscrepancy between the energy levels of the full potentialand of the asymptotic reactant well For one-dimensionalsystems the use of exact energy eigenstates of the full potentialrather than those of an isolated reactant well in (9) ismore appropriate even with approximate calculation of thetunneling probability
32 Asymmetric Rectangular Barrier Now let us consider anasymmetric rectangular double-well potential given by (8)and sketched in Figure 1 There are 7 states with energiesbelow the barrier top The ground and first excited statesare localized on the reactant and product sides respectivelyThe eigenstates of the full potential with even quantum num-bers correlate with the eigenstates of the reactant well thedifference in energies of the full and asymptotic potentialsincreases with the quantum number The parameter valuesfor the potential are 119881
119887= 40119864
ℎ 1198810= 04119864
ℎ and 119871 = 10119886
0
The well width is 119889 = 401198860 The particle mass is119898 = 1 a u
Exact QM energy levels and eigenfunctions needed tocompute projections on the reactant well are obtained aseigenvalues and eigenvectors of the Hamiltonian matrix inthe Colbert-Miller discrete variable representation [27]
Exact QM transmission probability is given by (13) Qua-siclassical transmission probability is defined by (3) and set to(1) for energies above the barrier top The exact and approxi-mate rate constants computed from (9) are listed inTable 2 fortemperatures measured in units of the barrier top 119881
119887 Results
of the fully QM fully QC and mixed QCQM (QC trans-mission evaluated at exact QM energy levels) descriptions areshown in Figure 5 and in Table 2 At the lowest temperatureequivalent to 1 of the barrier height the QC rate constantis 93 times lower than the QM result while the mixedQCQM estimate is only 2 times lower At high temperatureswhen the high energy eigenstates significantly contribute tothe reaction rate constant the discrepancy between exactand approximate rate constants becomes smaller with themixed QCQM estimate being closer to the exact result at allenergies One concludes that generally the accuracy of 119896(119879)
0 005 01 015 02
Rate
cons
tant
Exact QM QM levelsQC T(E)Reactant levelsQC T(E)
k(T
)
0
02
04
06
08
Temperature [Vb]
0 10 20 30
00001
1e minus 08
1e minus 12
1e minus 16
Figure 5 Transmission through the asymmetric rectangular barrierof Section 32 The thermal reaction rate constant is given as afunction of temperature measured in units of the barrier height[119896119861119879119881119887] where119881
119887= 4119864ℎThe rate constants are obtained using the
exact QM transmission probability and energy levels (circlessolidline) the QC transmission probability evaluated at the exact QMenergy levels (trianglesdot-dash) and at the QC energy levels of theisolated reactant well (squaresdash) The insert shows 119896(119879) on thelinear scale for wide range of temperatures
depends both on the accuracy ofQC transmission probability119879
QC and on the difference between exact and asymptotic-wellenergy levels Using exact QM energy levels and functions inconjunction with quasiclassical transmission probabilities ismore accurate than the fully QC description
Note that for the rectangular potential considered here119879
QC(119864) = 1 for 119864 gt 119881
119887is inaccurate because QC approxi-
mation does not describe above-the-barrier reflection leadingto oscillations of 119879QM For general potentials the reactionprobability may be computed using for example the time-dependent QM wave packet approach [28] used in the next
8 Journal of Theoretical Chemistry
example yielding reaction probabilities for a range of ener-gies from a single calculation or with a time-independentmethod for a few energy values
33 Piecewise Quadratic Potential Next we examine a piece-wise quadratic potential sketched in Figure 2 The reactantwell and barrier are quadratic functions in 119909 The reactionrate constants are obtained using the QM and quasiclassicalWKB expressions and using the analytical QM transmission
119879
QMpar (119864) = (1 + exp(minus
2120587119864
119908119887
))
minus1
119908119887=radic
119896119887
119898
(19)
through a parabolic barrier Equation (19) is more accuratethan the QC expression near the barrier top [16] A continu-ous piecewise quadratic potential is defined as follows
119881 (119909)
=
119896(119909 minus 1199091)
2
2
119909 lt 119909119871reactant region
minus
119896119887119909
2
2
+ 119881119887
119909119871le 119909 le 119909
119877barrier top region
119896(119909 minus 1199092)
2
2
+ 1198810119909 gt 119909
119877product region
(20)
with the proper choice of parametersmdash119881119887 1198810 119909119871 and 119909
119877mdash
the potential and its first derivative is continuous function at119909 = 119909
119871and 119909 = 119909
119877 Here a symmetric double well 119881
0= 0
is considered the remaining parameter values are 1199092= minus1199091
= 11198860 119896 = 119896
119887= 80119864
ℎ119886
minus2
0 119881119887= 20119864
ℎ and 119909
119877= minus119909119871= 05119886
0
The particle mass is 119898 = 1 a u (If the Hamiltonian werescaled by the mass of the proton 119881
119887would be equivalent to
03 eV or 7 kcalmol)The barrier parameters are chosen suchthat there are only few energy levels under the barrier topwhich is typical for a chemical reaction at low temperature
The time-dependent wave packet correlation approach[28] has been used to calculate QM transmission probability119879(119864) = |119878
119903119901(119864)|
2 entering (9) since we examine dependenceof the reaction rate on the accuracy of the energy levels usedin (9) The scattering matrix element 119878
119903119901(119864) describes trans-
mission from reactants to products
119878119903119901(119864) =
(2120587)
minus1
120578
lowast
119901(119864) 120578119903 (
119864)
int
infin
minusinfin
⟨120601
minus
119901
100381610038161003816100381610038161003816
119890
minus120580119905100381610038161003816100381610038161003816
120601
+
119903⟩ 119890
120580119864119905119889119905 (21)
The subscripts 119903119901 refer to reactantproduct reaction chan-nels and plusmn refers to the asymptotically incomingoutgoingwave relative to the barrier An incoming wave packet |120601+
119903⟩
taken as a Gaussian function in the left asymptotic regionof 119881 is evolved in time in the ldquounfoldedrdquo potential shownin Figure 2 with a dash Another Gaussian function |120601minus
119901⟩
is placed in the product region of the potential The time-dependent overlap of evolving |120601+
119903⟩ with the stationary ⟨120601minus
119901|
or correlation function 119862(119905) = ⟨120601
minus
119901(119909 0) | 120601
+
119903(119909 119905)⟩ is
computed and Fourier-transformed into the energy domain
The denominator in (21) accounts for the distributions ofenergy eigenstates in the reactant and product wave packetsat time 119905 = 0 these distributions depend on the wave packetlocalization and kinetic energy
120578119903119901(119864) = radic
119898
2120587119901
int
infin
minusinfin
119890
minus120580119901119909120601
plusmn
119903119901(119909) 119889119909 119901 =
radic2119898119864
(22)
A single wave packet propagation accomplished on a gridusing the split-operator method [29 30] gives transmissionprobability for a range of energies represented in the reac-tantproduct wave packets which is convenient if multipleeigenstates contribute to the rate constant At low energiesit is hard to converge transmission probability using time-dependent dynamics methods but the energies below thezero-point energy of the bound potential do not contributeto 119896(119879)The exact QM energy levels and projections of eigen-functions on the reactant well are performed as in Section 32There are five energy levels below the barrier top but manymore states contribute to 119896(119879) at higher temperatures
The energy-resolved exactQM parabolic barrier approxi-mation and QC transmission probabilities are shown in Fig-ure 6(a) Tunneling probabilities are underestimated in theQC approximation while the parabolic barrier expressiongives accurate results for energies above 119881
1198875 = 4119864
ℎ(the
barrier is in fact parabolic in this energy range) The discrep-ancy grows at energy below 4119864
ℎ This energy range however
is below the zero-point energy and does not contributeto the thermal rate constants shown in Figure 6(b) Theapproximate rate constants differ from exact QM results byat most 25ndash30
34 Proton Transfer As a chemically relevant model we con-sider the proton transfer in the HOndashHndashCH
3system for a
constrained collinear OndashHndashC geometry In this model sys-tem the proton is transferred from a donor carbon to ac-ceptor oxygenThe potential energy surface is obtained fromdensity function theory (DFT) electronic structure calcula-tions in particular at the B3LYP6-31G(dp) level of theory[31] The energies were compared to those obtained at theCCSD(T)aug-cc-pVDZ theory level for the same geome-tries and were found in excellent agreement In this sys-tem collinear donor-proton-acceptor arrangement is a goodapproximation to the fully optimized reaction path as deter-mined by a set of constrained geometry optimizations Inthese optimizations119877OH was fixed to a value in the range 08ndash20 A the average deviation of angCHO from linearity was lessthan 2∘ and the maximum deviation was 10∘
One-dimensional potential energy surfaces of 40 pointswere generated as functions of 119877OH for a fixed 119877CO distanceProton transfer on three surfaces for119877CO = 27 28 and 29 Ais analyzed below The surfaces were parametrized as sixthdegree polynomials in 119877OH The potential energy curvesshown in Figure 7(a) have characteristic asymmetric double-well shape
We examine the effect of donor-acceptor distance 119877COon the tunneling rate constants using QM and QC transmis-sion probabilities and exact and approximate energy levels
Journal of Theoretical Chemistry 9
00001
001
1
QMQCParabolic
0 10 20 30
1
1e minus 08
1e minus 06
Energy [Eh]
T(E
)
(a)
0 05 1 15 20
05
Rate
cons
tant
0 01 02
00001
001
Temperature [Vb]
(b)
Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896
119861119879119881119887] where 119881
119887= 20119864
ℎ
0
10
20
30
40
50
08 1 12 14 16 18 2
272829
Ener
gy (m
E h)
ROH (A)
(a)
0 500 1000 1500
103
102
101
100
KIE
Temperature (K)
(b)
Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO
distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896
119867119896119863(KIE) using exact QM formulation
in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top
The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896
119867119896119863should be largest at low
temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
8 Journal of Theoretical Chemistry
example yielding reaction probabilities for a range of ener-gies from a single calculation or with a time-independentmethod for a few energy values
33 Piecewise Quadratic Potential Next we examine a piece-wise quadratic potential sketched in Figure 2 The reactantwell and barrier are quadratic functions in 119909 The reactionrate constants are obtained using the QM and quasiclassicalWKB expressions and using the analytical QM transmission
119879
QMpar (119864) = (1 + exp(minus
2120587119864
119908119887
))
minus1
119908119887=radic
119896119887
119898
(19)
through a parabolic barrier Equation (19) is more accuratethan the QC expression near the barrier top [16] A continu-ous piecewise quadratic potential is defined as follows
119881 (119909)
=
119896(119909 minus 1199091)
2
2
119909 lt 119909119871reactant region
minus
119896119887119909
2
2
+ 119881119887
119909119871le 119909 le 119909
119877barrier top region
119896(119909 minus 1199092)
2
2
+ 1198810119909 gt 119909
119877product region
(20)
with the proper choice of parametersmdash119881119887 1198810 119909119871 and 119909
119877mdash
the potential and its first derivative is continuous function at119909 = 119909
119871and 119909 = 119909
119877 Here a symmetric double well 119881
0= 0
is considered the remaining parameter values are 1199092= minus1199091
= 11198860 119896 = 119896
119887= 80119864
ℎ119886
minus2
0 119881119887= 20119864
ℎ and 119909
119877= minus119909119871= 05119886
0
The particle mass is 119898 = 1 a u (If the Hamiltonian werescaled by the mass of the proton 119881
119887would be equivalent to
03 eV or 7 kcalmol)The barrier parameters are chosen suchthat there are only few energy levels under the barrier topwhich is typical for a chemical reaction at low temperature
The time-dependent wave packet correlation approach[28] has been used to calculate QM transmission probability119879(119864) = |119878
119903119901(119864)|
2 entering (9) since we examine dependenceof the reaction rate on the accuracy of the energy levels usedin (9) The scattering matrix element 119878
119903119901(119864) describes trans-
mission from reactants to products
119878119903119901(119864) =
(2120587)
minus1
120578
lowast
119901(119864) 120578119903 (
119864)
int
infin
minusinfin
⟨120601
minus
119901
100381610038161003816100381610038161003816
119890
minus120580119905100381610038161003816100381610038161003816
120601
+
119903⟩ 119890
120580119864119905119889119905 (21)
The subscripts 119903119901 refer to reactantproduct reaction chan-nels and plusmn refers to the asymptotically incomingoutgoingwave relative to the barrier An incoming wave packet |120601+
119903⟩
taken as a Gaussian function in the left asymptotic regionof 119881 is evolved in time in the ldquounfoldedrdquo potential shownin Figure 2 with a dash Another Gaussian function |120601minus
119901⟩
is placed in the product region of the potential The time-dependent overlap of evolving |120601+
119903⟩ with the stationary ⟨120601minus
119901|
or correlation function 119862(119905) = ⟨120601
minus
119901(119909 0) | 120601
+
119903(119909 119905)⟩ is
computed and Fourier-transformed into the energy domain
The denominator in (21) accounts for the distributions ofenergy eigenstates in the reactant and product wave packetsat time 119905 = 0 these distributions depend on the wave packetlocalization and kinetic energy
120578119903119901(119864) = radic
119898
2120587119901
int
infin
minusinfin
119890
minus120580119901119909120601
plusmn
119903119901(119909) 119889119909 119901 =
radic2119898119864
(22)
A single wave packet propagation accomplished on a gridusing the split-operator method [29 30] gives transmissionprobability for a range of energies represented in the reac-tantproduct wave packets which is convenient if multipleeigenstates contribute to the rate constant At low energiesit is hard to converge transmission probability using time-dependent dynamics methods but the energies below thezero-point energy of the bound potential do not contributeto 119896(119879)The exact QM energy levels and projections of eigen-functions on the reactant well are performed as in Section 32There are five energy levels below the barrier top but manymore states contribute to 119896(119879) at higher temperatures
The energy-resolved exactQM parabolic barrier approxi-mation and QC transmission probabilities are shown in Fig-ure 6(a) Tunneling probabilities are underestimated in theQC approximation while the parabolic barrier expressiongives accurate results for energies above 119881
1198875 = 4119864
ℎ(the
barrier is in fact parabolic in this energy range) The discrep-ancy grows at energy below 4119864
ℎ This energy range however
is below the zero-point energy and does not contributeto the thermal rate constants shown in Figure 6(b) Theapproximate rate constants differ from exact QM results byat most 25ndash30
34 Proton Transfer As a chemically relevant model we con-sider the proton transfer in the HOndashHndashCH
3system for a
constrained collinear OndashHndashC geometry In this model sys-tem the proton is transferred from a donor carbon to ac-ceptor oxygenThe potential energy surface is obtained fromdensity function theory (DFT) electronic structure calcula-tions in particular at the B3LYP6-31G(dp) level of theory[31] The energies were compared to those obtained at theCCSD(T)aug-cc-pVDZ theory level for the same geome-tries and were found in excellent agreement In this sys-tem collinear donor-proton-acceptor arrangement is a goodapproximation to the fully optimized reaction path as deter-mined by a set of constrained geometry optimizations Inthese optimizations119877OH was fixed to a value in the range 08ndash20 A the average deviation of angCHO from linearity was lessthan 2∘ and the maximum deviation was 10∘
One-dimensional potential energy surfaces of 40 pointswere generated as functions of 119877OH for a fixed 119877CO distanceProton transfer on three surfaces for119877CO = 27 28 and 29 Ais analyzed below The surfaces were parametrized as sixthdegree polynomials in 119877OH The potential energy curvesshown in Figure 7(a) have characteristic asymmetric double-well shape
We examine the effect of donor-acceptor distance 119877COon the tunneling rate constants using QM and QC transmis-sion probabilities and exact and approximate energy levels
Journal of Theoretical Chemistry 9
00001
001
1
QMQCParabolic
0 10 20 30
1
1e minus 08
1e minus 06
Energy [Eh]
T(E
)
(a)
0 05 1 15 20
05
Rate
cons
tant
0 01 02
00001
001
Temperature [Vb]
(b)
Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896
119861119879119881119887] where 119881
119887= 20119864
ℎ
0
10
20
30
40
50
08 1 12 14 16 18 2
272829
Ener
gy (m
E h)
ROH (A)
(a)
0 500 1000 1500
103
102
101
100
KIE
Temperature (K)
(b)
Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO
distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896
119867119896119863(KIE) using exact QM formulation
in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top
The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896
119867119896119863should be largest at low
temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Theoretical Chemistry 9
00001
001
1
QMQCParabolic
0 10 20 30
1
1e minus 08
1e minus 06
Energy [Eh]
T(E
)
(a)
0 05 1 15 20
05
Rate
cons
tant
0 01 02
00001
001
Temperature [Vb]
(b)
Figure 6 Exact and approximate transmission probabilities (a) and thermal rate constants (b) for the piecewise quadratic double well ofSection 33 The results obtained in the parabolic approximation to the barrier (trianglesdot-dash) are in close agreement with the QMresults (circlessolid line) compared to the quasiclassical WKB results (squaresdash) The vertical dashes in (a) mark positions of 6 lowestenergy levels (119899 = 0 and 119899 = 1 are indistinguishable on the plot) The insert in (b) shows the thermal rate constant on the logarithmic scaleThe temperature is given in the units of the barrier height [119896
119861119879119881119887] where 119881
119887= 20119864
ℎ
0
10
20
30
40
50
08 1 12 14 16 18 2
272829
Ener
gy (m
E h)
ROH (A)
(a)
0 500 1000 1500
103
102
101
100
KIE
Temperature (K)
(b)
Figure 7 (a) Three potential energy surfaces for the collinear proton transfer in HOndashHndashCH3 One curve is generated for each fixed 119877CO
distance (indicated by the line type in A) The curves have been aligned by setting the acceptor (OndashH) minimum to 0mEh (b) Ratio of thethermal rate constants 119896
119867119896119863(KIE) using exact QM formulation
in (9) QM transmission probability is calculated using thewave packet correlation approach outlined in Section 33 QCtransmission is defined by (3) The calculation of reactionrate constants using either of these approaches requires ei-genstate projections on the reactant region The eigenstatesare computed exactly as outlined in Section 32 defining thereactant region to the right of the barrier top
The results are shown in Figure 7(b) as a function of tem-perature At low temperatures proton rate constants are sev-eral orders of magnitude higher than those for the deuteronand this gap decreases as temperature rises As a result the ki-netic isotope effect KIE = 119896
119867119896119863should be largest at low
temperatures and this trend is shown in Figure 7(b) TheKIE calculated for each surface remains nearly constant in
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
10 Journal of Theoretical Chemistry
Table 3 Contribution of the ground state to the rate constants for the HOndashHndashCH3proton transfer model obtained using fully quantum and
fully quasiclassical approaches H and D label quantities relevant to reactions with hydrogen (proton) and deuterium respectively KIE =119896H119896D The ground state energies of the QM and QC descriptions are listed in the last two columns
119877CO 119896
QM119896
QC119896
QC119896
QM119864
QM0
[119898119864ℎ] 119864
QC0
[119898119864ℎ]
27 AH 09909 (minus1) 04463 (minus1) 04505 160194 150527D 01125 (minus1) 03232 (minus2) 02874 147793 138555KIE 88114 138086 15671
lowast mdash mdash
28 AH 02037 (minus2) 05481 (minus3) 02691 177501 162825D 02383 (minus4) 05669 (minus5) 02379 161081 149856KIE 854646 966882 11313
lowast mdash mdash
29 AH 01262 (minus4) 02804 (minus5) 02222 184041 167282D 01894 (minus7) 03310 (minus8) 01748 166112 153731KIE 6660938 8470630 12716
lowast mdash mdashAsterisk marks KIEQCKIEQM
the low-temperature (0ndash300K) region and then begins toapproach 1 as the temperatures rises These very large KIEvalues point to a reaction dominated by quantum tunneling atlow temperaturesThe largest discrepancy in theKIE betweendifferent surfaces is seen at very low temperatures as well andby increasing 119877CO by only 01 A the KIE is enhanced by anorder of magnitude
When calculating QC rates the energy levels of the iso-lated donor well are traditionally used as 119864
119899rather than the
energy corresponding to 120588119899as we have defined it in (11) In
the three double-well potentials considered here the QCground state energy is lower than that of theQM calculationsThus the QC calculations yield lower rate constants by upto a factor of 5 as shown in Table 3 The table compares theground state contributions to 119896QM and 119896QC Despite thediscrepancy in rate constants the KIE predicted by the QCmethod is within a factor of 2 of the QM results due to can-cellation of error
4 Conclusions
Reliable estimates of the QM tunneling probabilities througha barrier along the reaction path are often used in studies ofreactions proceeding in condensed phase The formal defini-tion of the QM reaction probability and rate constant basedon asymptotic scattering states [11] cannot be used for boundpotentials representing such processes In dynamics of a wavepacket representing reactants this aspect manifests itself aspersisting-in-time oscillations of the reaction probabilitiesA proposed modification of the QM expression for boundpotentials (9) which addresses this problem is based onthe analysis of a rectangular double-well potential For thispotential the rate constant expression separates into (i) thereactantproduct transmission probability through a barrierof a scattering system with the same barrier region as thefull potential and (ii) the eigenstate energies and eigenstateprojections on the reactant region of the full bound potentialExact QM or quasiclassical (or other approximate) methodsmay be used to estimate the tunneling probability and energy
levels For example for the rectangular double-well potentialthe two-state representation gave fairly accurate estimatesof tunneling probabilities derived from the quasiclassicalrelationship between the energy level splitting and tunneling(18)The relationshipworked for the lowest pair of eigenstateseven for narrow barriers but did not hold for higher energypairs of eigenstates
The quasiclassical WKB estimates of rate constants areshown to depend on both the accuracy of the transmissionprobabilities and positions of the energy levels use of exactQM energy eigenstates is preferred The QC rate constantsare lower at low energy and higher at energies comparable tothe barrier top in comparison to QM results Performing QMscattering calculations to obtain transmission probabilities ismore expensive thanQC estimates but it makes the approachgeneralizable to more than one dimension For a smoothpotential (piecewise quadratic potential) the approximatetransmission probabilities were quite accurate The accuracyof the reaction rate constant in the parabolic approximationto the barrier was better than 4 and better than 40 forthe quasiclassical WKB approximation In all cases we findthat the accuracy of rate constants is improved when exacteigenstates are usedwith approximate probabilities in (9) Forthe proton transfer model for the HOndashHndashCH
3system with
constrained donor-acceptor distance the QC approximationgave reasonable estimates of the tunneling the QC reactionrate constants were approximately 4 times smaller than theexact QM counterparts and their ratio the KIE was within50 of the exact QM value due to cancellation of errorsWhilemultidimensional dynamics is preferable for a rigoroustheoretical study of a reaction in condensed phase this simpleapproach of computing reaction probabilities and thermalrate constants in bound potentials may be used to analyzebarriers and to assess importance ofQM tunneling for a givensystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Theoretical Chemistry 11
Acknowledgments
This material is based upon work supported by the NationalScience Foundation under Grant no CHE-1056188 Theauthors thank Vitaly Rassolov for active discussions
References
[1] X-Z Li BWalker andAMichaelides ldquoQuantumnature of thehydrogen bondrdquoProceedings of theNational Academy of Sciencesof the United States of America vol 108 no 16 pp 6369ndash63732011
[2] M J Sutclie and N S Scrutton ldquoA new conceptual frameworkfor enzyme catalysisrdquo European Journal of Biochemistry vol269 no 13 pp 3096ndash3102 2002
[3] A P Jardine E Y M Lee D J Ward et al ldquoDetermination ofthe quantum contribution to the activated motion of hydrogenon a metal surface HPt(111)rdquo Physical Review Letters vol 105no 13 Article ID 136101 2010
[4] S S Iyengar I Sumner and J Jakowski ldquoHydrogen tunnelingin an enzyme active site a quantum wavepacket dynamicalperspectiverdquo The Journal of Physical Chemistry B vol 112 no25 pp 7601ndash7613 2008
[5] M J Knapp K Rickert and J P Klinman ldquoTemperature-dependent isotope effects in soybean Lipoxygenase-1 correlat-ing hydrogen tunneling with protein dynamicsrdquo Journal of theAmericanChemical Society vol 124 no 15 pp 3865ndash3874 2002
[6] P R Schreiner H P Reisenauer D Ley D Gerbig C Wuand W D Allen ldquoMethylhydroxycarbene tunneling control ofa chemical reactionrdquo Science vol 332 no 6035 pp 1300ndash13032011
[7] R J Shannon M A Blitz A Goddard and D E HeardldquoAccelerated chemistry in the reaction between the hydroxylradical and methanol at interstellar temperatures facilitated bytunnellingrdquo Nature Chemistry vol 5 no 9 pp 745ndash749 2013
[8] S Hay C R Pudney T A McGrory J Pang M J Sut-cliffe and N S Scrutton ldquoBarrier compression enhances anenzymatic hydrogen-transfer reactionrdquo Angewandte ChemiemdashInternational Edition vol 48 no 8 pp 1452ndash1454 2009
[9] S Hay L O Johannissen M J Sutclie and N S ScruttonldquoBarrier compression and its contribution to both classical andquantum mechanical aspects of enzyme catalysisrdquo BiophysicalJournal vol 98 no 1 pp 121ndash128 2010
[10] S Hay and N S Scrutton ldquoGood vibrations in enzyme-catalysed reactionsrdquoNature Chemistry vol 4 no 3 pp 161ndash1682012
[11] R G Newton Scattering Theory of Waves and ParticlesSpringer New York NY USA 1982
[12] W H Miller S D Schwartz and J W Tromp ldquoQuantummechanical rate constants for bimolecular reactionsrdquoThe Jour-nal of Chemical Physics vol 79 no 10 pp 4889ndash4898 1983
[13] D E Makarov and H Metiu ldquoThe reaction rate constant ina system with localized trajectories in the transition regionclassical and quantum dynamicsrdquo The Journal of ChemicalPhysics vol 107 no 19 pp 7787ndash7799 1997
[14] J Y Ge and J Zhang ldquoQuantummechanical tunneling througha time-dependent barrierrdquoThe Journal of Chemical Physics vol105 no 19 pp 8628ndash8632 1996
[15] W H Miller ldquoQuantummechanical transition state theory anda new semiclassical model for reaction rate constantsrdquo TheJournal of Chemical Physics vol 61 no 5 pp 1823ndash1834 1974
[16] L D Landau and E M Lifshitz Quantum MechanicsButterworth-Heinemann Oxford UK 1999
[17] J HWeiner ldquoQuantum rate theory for a symmetric double-wellpotentialrdquo The Journal of Chemical Physics vol 68 no 5 pp2492ndash2506 1978
[18] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927
[19] B Poirier ldquoReconciling semiclassical and Bohmian mechanicsI Stationary statesrdquoThe Journal of Chemical Physics vol 121 no10 pp 4501ndash4515 2004
[20] N Rom E Engdahl and N Moiseyev ldquoTunneling rates inbound systems using smooth exterior complex scaling withinthe framework of the finite basis set approximationrdquoThe Journalof Chemical Physics vol 93 no 5 pp 3413ndash3419 1990
[21] J Vanicek W H Miller J F Castillo and F J Aoiz ldquoQuantum-instanton evaluation of the kinetic isotope effectsrdquo The Journalof Chemical Physics vol 123 no 5 Article ID 054108 2005
[22] I R Craig and D E Manolopoulos ldquoChemical reaction ratesfrom ring polymer molecular dynamicsrdquoThe Journal of Chem-ical Physics vol 122 no 8 Article ID 084106 2005
[23] A V Turbiner ldquoDouble well potential perturbation theorytunneling WKB (beyond instantons)rdquo International Journal ofModern Physics A vol 25 no 2-3 pp 647ndash658 2010
[24] J H Weiner ldquoTransmission function vs energy splitting intunneling calculationsrdquoThe Journal of Chemical Physics vol 69no 11 pp 4743ndash4849 1978
[25] D Bohm ldquoA suggested interpretation of the quantum theory interms of ldquohiddenrdquo variables Irdquo Physical Review vol 85 no 2pp 166ndash193 1952
[26] Maple 14 and 16 Maplesoft a division of Waterloo Maple IncWaterloo Canada httpwwwmaplesoftcom
[27] D T Colbert and W H Miller ldquoA novel discrete variablerepresentation for quantum mechanical reactive scattering viathe S-matrix Kohn methodrdquo The Journal of Chemical Physicsvol 96 no 3 pp 1982ndash1991 1992
[28] D J Tannor and D E Weeks ldquoWave packet correlationfunction formulation of scattering theory the quantum analogof classical S-matrix theoryrdquo The Journal of Chemical Physicsvol 98 no 5 pp 3884ndash3893 1993
[29] R Koslo ldquoTime-dependent quantum-mechanical methods formolecular dynamicsrdquoThe Journal of Physical Chemistry vol 92no 8 pp 2087ndash2100 1988
[30] M D Feit J A Fleck Jr and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982
[31] Y Shao L F Molnar Y Jung et al ldquoAdvances in methods andalgorithms in a modern quantum chemistry program packagerdquoPhysical ChemistryChemical Physics vol 8 no 27 pp 3172ndash31912006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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