research article quantum-dynamical theory of electron...

Hindawi Publishing CorporationAdvances in Physical ChemistryVolume 2013 Article ID 497267 8 pageshttpdxdoiorg1011552013497267

Research ArticleQuantum-Dynamical Theory of Electron Exchange Correlation

Burke Ritchie12 and Charles A Weatherford3

1 Lawrence Livermore National Laboratory Livermore CA 94550 USA2 Livermore Software Technology Corporation Livermore CA 94550 USA3Department of Physics Florida AampM University Tallahassee FL 32307 USA

Correspondence should be addressed to Burke Ritchie ritchielstccom

Received 4 November 2012 Revised 13 January 2013 Accepted 14 January 2013

Academic Editor Benjaram M Reddy

Copyright copy 2013 B Ritchie and C A Weatherford This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The relationship between the spin of an individual electron and Fermi-Dirac statistics (FDS) which is obeyed by electrons in theaggregate is elucidatedThe relationship depends on the use of spin-dependent quantum trajectories (SDQT) to evaluate Coulombrsquoslaw between any two electrons as an instantaneous interaction in space and time rather than as a quantum-mean interaction inthe form of screening and exchange potentials Hence FDS depends in an ab initio sense on the inference of SDQT from Diracrsquosequation which provides for relativistic Lorentz invariance and a permanent magnetic moment (or spin) in the electronrsquos equationof motion Schroedingerrsquos time-dependent equation can be used to evaluate the SDQT in the nonrelativistic regime of electronvelocity Remarkably FDS is a relativistic property of an ensemble of electron even though it is of order 1198880 in the nonrelativistic limitin agreement with experimental observation Finally it is shown that covalent versus separated-atoms limits can be characterizedby the SDQT As an example of the use of SDQT in a canonical structure problem the energies of the 1Σ

119892and 3Σ

119906states of H

2are

calculated and compared with the accurate variational energies of Kolos and Wolniewitz

1 Introduction

One may consider that quantum chemistry is dominated bytheoretical and computational efforts to achieve an accuratedescription of electron exchange correlation evolving suchworkhorse methodologies as Hartree-Fock-ConfigurationInteraction Density Functional Theory and numerous vari-ations on the theme of nonrelativistic quantum mechanicsapplied to problems of chemical interest But yet owingto historical happenstance more heat than light has beengenerated concerning the fundamental physical understand-ing of exchange-correlation Even in early calculations inwhich correlation was built into the wave function it wasrecognized that the concept of exchange tended to losemeaning in a calculation in which correlation was treated tohigh accuracy [1] As another example it was shown that ahigh-order perturbation calculation in which the electron-electron interaction is treated as the perturbation is able toachieve order by order the correct permutation symmetryof the wave function starting in zeroth order with a simpleunsymmetrized product of orbitals [2] This conundrum

can be easily understood for the two-electron one-nucleusproblem by a simple change from nucleus centered to Jacobicoordinates in which one vector connects the two electronsand the other connects the center of mass of the two electronsto the nucleus (where of course it is understood that theBorn-Oppenheimer picture is being used) Then the wavefunction separates naturally into two linearly-independentwave functions which are either even or odd in the exchangeof electrons In other words the new choice of coordinates inwhich correlation is achieved through a physically judiciouschoice of coordinates also achieves the correct exchangesymmetry which is guaranteed by the symmetry of theHamiltonian with respect to the electron exchange

Nonrelativistic quantum theory fails us however evenfor two-electron problems According to experimental obser-vation the electrons have intrinsic angular momentumcomprising two-spin-12 states such that the total wavefunctionmust be antisymmetric on electron exchange whichis satisfied either by a product of the even spatial wavefunction adduced above with a spin state which is odd onelectron exchange (singlet state) or by a product of the

2 Advances in Physical Chemistry

adduced odd spatial wave function with a spin state whichis even on electron exchange (triplet state) The general-ization is of course the Slater determinantal wave functionof spin orbitals which guarantees the correct antisymmetricpermutation symmetry for N spin-12 particles Hence thestandard methodologies [3] for all their success simulatemany-electron quantum states in an ad hoc manner basedon experimental observation but tell us nothing fundamentalabout the physical basis of how the spin state of an individualelectron is related to the Pauli exclusion principle and theobserved Fermi-Dirac statistical behavior of an aggregate ofelectrons

2 Spin-Dependent Quantum Trajectories andElectron Exchange Correlation

Readers should recognize that an electronrsquos spin state and itsspatial correlation with another electron are strongly relatedas suggested by our observation that a successful simulationof correlation is also accompanied by the correct exchangesymmetry Notice however that such a relationship betweenelectron spin and the electron-electron Coulomb interactionappears to be grossly at oddswith our intuitive understandingof quantum chemistry likely due to the absence of a particle-trajectory picture in the standard methodologies Bohm for-mulated a quantum dynamical approach in which the phase-amplitude solution of Schroedingerrsquos equation has a formalrelationship to classical hydrodynamics [4 5] Bohmiandynamics has recently undergone a renewed interest [6]It is a method of solving the time-dependent Schrodingerequation by assuming a form for the wavefunction Ψ(119909 119905) =

119877(119909 119905)119890119894119878(119909119905) where R (such that R is greater than or equal

to zero for all x) is the amplitude and S is the actionfunction Then upon substitution into the time-dependentSchroedinger equation two coupled equations (continuityequation and the quantum Hamilton-Jacobi equation) areobtained The net result is to produce quantum trajectoriesThese equations have proven to be difficult to solve accuratelyparticularly for multi electrons [6] In fact it is difficult tosolve in three spatial dimensions for one electron Note thatin the context of the Bohmian equations the discretizationis done in terms of what are called pseudo particles Thuswhen it is said that say fifty particles are propagated ittypically means fifty pseudo particles are used to describethe dynamics of one real particle As far as we are aware noone has computed the dynamics of two real particles usingBohmian dynamics and in fact it does not seem possible todo so with current algorithms and computer hardware

We therefore seek a quantum trajectory approach forreal electrons in which the spin of an individual electron iscorrectly accounted for which means that we must look toDiracrsquos rather than to Schroedingerrsquos equation One of thedesiderata of Diracrsquos program [7] for incorporating Einsteinrsquosspecial theory of relativity into quantum mechanics was thatthe quantum analog of the classical equation of continuity

120597120588

120597119905

+ nabla sdot = 0 (1)

should be inferred from the equation of motion for therelativistic electron

(119894ℎ

120597

120597119905

minus 119881 minus 1198981198882)120595 ( 119903 119905) + 119894ℎ119888 sdot nabla120594 ( 119903 119905) = 0 (2a)

(119894ℎ

120597

120597119905

minus 119881 + 1198981198882)120594 ( 119903 119905) + 119894ℎ119888 sdot nabla120595 ( 119903 119905) = 0 (2b)

whereupon [7]

119895 ( 119903 119905) = 119888 [120595+( 119903 119905) 120594 ( 119903 119905) + 120594

+( 119903 119905) 120595 ( 119903 119905)] (3)

where is Paulirsquos vector and the quantum density is

120588 ( 119903 119905) = 120595+( 119903 119905) 120595 ( 119903 119905) + 120594

+( 119903 119905) 120594 ( 119903 119905) (4)

where 120595( 119903 119905) and 120594( 119903 119905) are the large and small Diracspinors respectively and the superscripts denote Hermi-tian conjugates On eliminating (2b) in favor of (2a) anddropping all terms of order 119888minus2 which is the nonrelativisticlimit Schroedingerrsquos equation is recovered for 120595 which wasanother desideratum in Diracrsquos program for a relativistic-electron theory Now notice that a velocity field can beinferred from the current by writing

119895 ( 119903 119905) = 119888 [120595+( 119903 119905) 120594 ( 119903 119905) + 120594

+( 119903 119905) 120595 ( 119903 119905)]

= ( 119903 119905) 120588 ( 119903 119905)

(5)

and solving for ( 119903 119905) from which a trajectory 119904(119905) can becalculated from the time integration of the velocity field tofind a position field

( 119903 119905) = int

119905

0

1198891199051015840 ( 119903 119905

1015840) (6)

and finally by finding the quantum expectation value of theposition field

119904 (119905) = int119889 119903 [120595+( 119903 119905) ( 119903 119905) 120595 ( 119903 119905)

+120594+( 119903 119905) ( 119903 119905) 120594 ( 119903 119905)]

(7)

One more step is needed to show the relationship ofelectron spin and electron exchange correlation namelythe evaluation of the electron-electron Coulomb interactionusing quantum trajectories Considering the case of twoelectrons one of which is described by (1)ndash(7) The Coulombinteraction for an electron with position vector 119903 with theother electron with position vector 119903

1015840 can then be written asfollows

1198811198901198901015840 ( 119903 119905) =

1198902

10038161003816100381610038161003816119903 minus 1199041015840(119905)

10038161003816100381610038161003816

(8)

Similarly the electron at 1199031015840 is described by (1)ndash(7) in which 119903

is replaced by 1199031015840 and the interaction is given by

1198811198901015840119890( 1199031015840 119905) =

1198902

100381610038161003816100381610038161199031015840minus 119904 (119905)

10038161003816100381610038161003816

(9)

Advances in Physical Chemistry 3

minus20

minus10

0

10

20

Sy (a

u)

0 50 100 150 200119905 (au)

Figure 1 Quantum trajectories in the y direction at 119877 = 14 aushowing the correlation of the two electrons in the 1Σ

119892state of H

2

Solid spin-up electron Dotted spin-down electronThe trajectoriesare calculated from (7) in the nonrelativistic limit using spectralwave functions

Thus the electron-electron potential is evaluated as aninstantaneous interaction in space and time rather thanas a quantum-mean interaction in the form of Coulomband exchange integrals This is a critical step Normally theinteraction is written as 120592

1198901198901015840 = 119890

2| 119903 minus 119903

1015840| which is a func-

tion in 6-space well known to be incompatible with Lorentz-invariance [8] which is a requirement for a correct rela-tivistic theory The authorsrsquo comment [8] following theirequation (384) ldquothe Coulomb term [e-e1015840 interaction] in(383) however is not even approximately Lorentz invariantand relativistic corrections to the interaction between the twoelectrons are furnished by quantum electrodynamicsrdquo refersto the Lorentz-invariant form of the e-e1015840 interaction given byits representation in QED in terms of the exchange of virtualphotons Notice that in the present paper we are claiming thatthese relativistic corrections at least for nonradiative interac-tions are given by representing the e-e1015840 or e1015840-e interactions inthe forms given by (8)-(9) and by calculating the trajectoriesusing (2a)ndash(7) This is so because the 4-space form of (8)-(9) is fully compatible with the Lorentz invariance of Diracrsquosone-electron equation as given by (2a) and (2b) in which Vis explicitly time dependent and can be written as 119881( 119903 119905) Inother words the time-independent 6-space potential 120592

1198901198901015840 =

1198902| 119903 minus 119903

1015840| is represented in 3-space and the time by using

the position vector of one electron and the trajectory of theother Readers will note that Diracrsquos one-electron equationas given for example in the literature for the hydrogen atomis exact and therefore Lorentz invariant for the Coulombattraction to the nucleus 119881(119903) = minus119885119890

2119903 It is still exact and

Lorentz invariant for the general scalar potential 119881( 119903 119905) Thisis a true statement because in the electronrsquos 4-momentum(120574119898119888 120574119898) = (120574119898119888 ) = (120574119898119888 minus119894ℎnabla) the scalar part can

be expressed as terms of (1119888)[119894ℎ(120597120597119905) minus 119881( 119903 119905)] on usingthe operator form of the classical relativistic Hamiltonian119894ℎ(120597120597119905) = 120574119898119888

2+ 119881( 119903 119905) where 120574 is the Lorentz factor

120574 = radic1 + (119901211989821198882) We can only guess that workers have

not used the present methodology to find a fully relativisticLorentz-invariant theory for many fermions due historicallyto the dominance of time-independent quantum theory inabsence of an explicitly time-dependent interaction suchas the vector potential ( 119903 119905) in matter-radiation physicsin which case the 4-momentum is written (120574119898119888 minus119894ℎnabla minus

(119890119888)( 119903 119905)) Equations (8)-(9) are functions of 3-space andthe time and are fully compatible with the Lorentz invarianceof Diracrsquos one-electron Hamiltonian and with the Lorentzcovariance of his one-electron wave equation [7] Notice thatwe do not venture onto the ice of two-body Dirac theory [9]Proceeding heuristically we believe that it is possible to derivea correct many-electron relativistic quantum theory simplyby replacingNewtonrsquos equation ofmotion for each electron inan aggregate of electrons by Diracrsquos equation of each electronin the aggregate thusly for an electron whose position vectoris 119903 interacting with another electron whose position vectoris 1199031015840

119889 (119905)

119889119905

= minusnabla

1198902

10038161003816100381610038161003816119903 (119905) minus 119903

1015840(119905)

10038161003816100381610038161003816

997888rarr

119894ℎ

120597120595119863 (

119903 119905)

120597119905

= [ minus 119894ℎ119888 sdot nabla + 1205731198981198882

+

1198902

10038161003816100381610038161003816119903 minus 1199041015840(119905)

10038161003816100381610038161003816

] 120595119863( 119903 119905)

(10)

where = 120574119898(119889 119903(119905)119889119905) 120574 = radic1 + (119901211989821198882) 120595119863

= (120595

120594 ) = (

0

0) 120573 = (

119868 0

0 minus119868) and I is the 2 times 2 identity matrix

Equations (2a) and (2b) are identical to the right-hand side ofthe arrow in (10) which is just another way of writing Diracrsquosequation

Paulirsquos exclusion principle is obeyed if (2a) and (2b) andtheir counterparts for a second electron are evolved in thetime for identical spatial functions and opposite spin states atinitial time (Figure 1) which corresponds to a singlet stateor for different spatial functions and the same spin states atinitial time (Figure 2) which corresponds to a triplet stateNotice that Figure 1 shows quantum trajectories with slightlydifferent frequencies which can happen because spatial wavefunctions which are identical at initial time can developdifferences over time in analogy to the use of unrestrictedHartree-Fock theory in standard time-independent theoryIt should be noted that although one must choose a set ofinitial conditions for the time integration of the equationsthe problem is not an initial-value problem in the same senseas in classical mechanics in which due to the deterministicnature of the trajectories one must find the physical resultas an average over a set of different initial conditions sincewe cannot know a unique set of initial conditions forthe particles The reason can be found in the Heisenberguncertainty principle Δ119864Δ119905 ge ℎ which governs the time


integration of the quantum wave equations near initial timethe energy is very broad and narrows with increasing timeto give a spectrum of energy levels (Figure 3) Quantumdynamics however does share with classical dynamics theproperty that stationary (ie constant energy) solutions areobtained

The envelop of trajectory amplitudes shown in Figure 1is observed to increase over time because the quantum tra-jectories (hereafter called spectral trajectories) are calculatedusing the time-dependent solutions (hereafter called thespectral solutions [10]) which are a superposition of quantumstates including the continuum such that the envelop growthover time reflects the excited and continuum content of thewave function On the other hand the envelop of trajectoryamplitudes shown in Figure 4 is not observed to increaseover time because the trajectories (hereafter called eigen-trajectories) are calculated using eigenfunctions as filteredfrom the spectral solutions at a specific stationary energy asdetermined fromFigure 3 as described in [10] In some of thecalculations Hamiltonian expectation values calculated usingeigenfunctions and eigentrajectories (Figure 5) still exhibit anoscillation in the time In these cases the physical energiescan be found from the time average of the Hamiltonianexpectation values (Figures 5 and 6) in other words the timeaverage is sensibly stationary

3 Quantum Trajectories in the NonrelativisticLimit

The time-independent solution of (2b) can be written as

120594 ( 119903 119905) =

minus119894ℎ119888 sdot nabla120595 ( 119903 119905)

119864 minus 119881 + 1198981198882

(11)

Evaluating 120594( 119903 119905) in the nonrelativistic limit for which (119864 minus

119881) ≪ 1198981198882 we immediately see that the current given by

(3) is of order 1198880 in this limit Remarkably the 119888

0 order ofthe quantum trajectories is in agreement with experimentalobservation that Fermi-Dirac statistics is obeyed for allelectron velocities including those whose Dirac current iscalculated in the regime of nonrelativistic velocity

The trajectories are quantum mechanical since they canbe calculated either using Diracrsquos equation in the regime ofrelativistic velocities or Schrodingerrsquos equation in the regimeof nonrelativistic velocitiesThe quantum trajectories are spindependent since they depend explicitly on Paulirsquos vector Ourresult for the first time gives a relativistic correction which isof order 1198880 all other relativistic corrections being of order 119888minus2This relativistic correction is due exclusively to the spin of theelectron and accounts for Fermi-Dirac statistics Traditionallyrelativistic corrections have been considered to start at order119888minus2 for atomic fine structure Readers will appreciate theapparent paradox that we had to appeal to Diracrsquos equationto explain the omnipresence of electron spin in all fermionicphenomena notwithstanding the relativistic or nonrelativisticnature of the motion

We believe that this unforeseen Dirac correction of order1198880 which is of course the same order as all Schroedingerrsquoscontributions to the Hamiltonian is due to the Lorentz

0

1

2

3

4

5

6

minus1

0 50 100 150 200119905 (au)

Sze (

au)

Figure 2 Quantum eigentrajectories in the z direction showinghow the two electrons of H

2correlate with increasing time in the

formation of an antibonding state Upper spin-up electron Lowerspin-up electronThe eigentrajectories are calculated from (7) in thenonrelativistic limit using eigenfunctions for two parallel spin statesof the 3Σ

119906state of H

2

9119890 + 03

8119890 + 03

7119890 + 03

6119890 + 03

5119890 + 03

4119890 + 03

3119890 + 03

2119890 + 03

1119890 + 03

0119890 + 00

minus2 minus1 0 1

119864 (au)

Spec

trum

(au)

Figure 3 Spectrum at119877 = 25 au for the 1Σ119892state of H

2 Solid spin-

up electron Dotted spin-down electron

covariance [7] of Diracrsquos equation appropriate for a spin-12 particle as discussed in the last section This surmiseis supported by the fact that the Klein-Gordon relativisticequation appropriate for a spin-0 particle although it isLorentz invariant it fails to satisfy the equation of continuitygiven by (1) for a physically acceptable current


0 50 100 150 200119905 (au)

minus3

minus2

minus1

0

1

2

3

Sze (

au)

Figure 4 Quantum eigentrajectories in the z direction Innercurves 119877 = 14 au Outer curves 119877 = 30 au The inner curves showhow the two electrons correlate with increasing time and eventuallyfind the region of covalent bonding located between the two protonsfixed at 119911 = plusmn07 au while the outer curves show that the electronsremain in the vicinity of the separated atoms for all times Theeigentrajectories are calculated from (7) in the nonrelativistic limitusing eigenfunctions for the up and down spin states for the 1Σ

119892

state of H2

150140 160 170 180 190 200119905 (au)

119864(a

u)

minus15

minus1

minus05

Figure 5 Expectation values of the Hamiltonian and their timeaverages for 119877 = 14 au toward the end of the time integrationLower 1Σ

119892state Upper 3Σ

119906state These energies especially for

the 1Σ119892state have some numerical noise (high-frequency hair

superimposed on the wave oscillation) but nevertheless the timeaverages at maximum time agree well with the time-independentresults which are minus1174 and minus07842 for the 1Σ

119892and 3Σ

119906states

respectively

minus12

minus115

minus11

minus105

minus1

119864(a

u)

150 160 170 180 190 200119905 (au)

Figure 6 Time average of the Hamiltonian expectation values forthe 1Σ

119892state From bottom to top for 119877 = 14 au 119877 = 25 au

119877 = 30 au and 119877 = 40 au respectively The KW energiesare respectively minus1174 au minus1094 au minus1057 au and minus1016 au Theagreement with KW at the larger internuclear distances is not asgood owing to smaller binding energies whose magnitude falls intothe last digit of accuracy of the calculation

In the nonrelativistic regime of electron velocity (2a)ndash(9) may be evaluated using Schroedingerrsquos equation whichfollows on exactly eliminating (2b) in favor of (2a) anddropping contributions in the resulting equation for the largecomponent 120595( 119903 119905) of order 119888minus2 The current is evaluated inthe nonrelativistic limit using 120595( 119903 119905) = 120595

119878( 119903 119905)120594

119898119904

and 120594( 119903 119905)

given by (11) in the regime (119864minus119881) ≪ 1198981198882 where120595

119878( 119903 119905)obeys

the time-dependent Schroedingerrsquos equation as follows

119894ℎ

120597120595119878( 119903 119905)

120597119905

= [minus

ℎ2

2119898

nabla2+ 119881 ( 119903 119905)] 120595

119878 (119903 119905) (12)

120594119898119904

has up (plus sign) or down (minus sign) magneticsubstates denoted by 119898

119904= plusmn12 (ie 120572 or 120573 spin states)

respectively Written out explicitly in terms of the largecomponent the current given by (3) becomes

( 119903 119905) cong

ℎ

2119898

[minus119894 (120595lowast

119878nabla120595119878minus 120595119878nabla120595lowast

119878)

+ 120595lowast

119878120594+

119898119904

(nabla times ) 120595119878120594119898119904

minus120595119878120594+

119898119904

( times nabla) 120595lowast

119878120594119898119904

]

(13)


where we have used += and the identity ( sdot )( sdot ) =

sdot+119894sdot(times) fromwhich the identities useful in evaluatingthe current can be inferred as follows

( sdot nabla) = nabla + 119894 (nabla times ) (14a)

( sdot nabla) = nabla + 119894 ( times nabla) (14b)

Written out explicitly for up (upper sign) or down (lowersign) spin states the nonrelativistic current is

119895119899119903( 119903 119905) =

ℎ

119898

[ Im120595lowast

119878( 119903 119905) nabla120595119878 (

119903 119905)

plusmn Re120595lowast119878( 119903 119905)

120597

120597119910

120595119878 (

119903 119905)

∓Re120595lowast119878( 119903 119905)

120597

120597119909

120595119878 (

119903 119905)]

(15)

The first term on the right side of (15) which is independentof spin is the familiar term contributed by Schroedingerrsquosequation while the second and third terms which aretransverse to the axis of quantization along z are contributeduniquely by Dirac theory Notice that all contributions to thecurrent are of order 1198880 as discussed earlier in this section

4 Calculations and Results

The time-dependent Schroedingerrsquos equation [(12)] is solvedin the time and three Cartesian coordinates for each electronof H2using an algorithm described previously [11] (Prelim-

inary He-atom calculations were presented previously [12])The computational grid box is square with a uniform meshof 323 The accuracy of the calculation could be improved bymesh refinement of the uniformmesh or by use of an adaptivemesh (which we do not have at our disposal) however wefind that for 119885 = 1 or 119885 = 2 the present mesh is adequateespecially for the proof of principle calculations presentedhere The two Schroedingerrsquos equations were integrated inthe time for a length of 200 au using 4000 mesh points (forinternuclear distance 119877 = 14 au calculations) or 8000 meshpoints (for all other internuclear distance calculations) Atinitial time a Slater-type orbital with form 119903119890

minus120577119903 120577 = 12

centered on each proton is used for each electron for the 1Σ119892

state and with forms 119903119890minus120577119903

120577 = 1 and 1199032119890minus120577119903 120577 = 12 for

the electrons of the 3Σ119906state The Pauli principle is enforced

at initial time for electrons in the same spin state and theelectrons obey the Pauli principle for all subsequent times dueto their mutual correlation If the Pauli principle is violated atinitial time by assigning the same spatial orbital to electronsin the same spin state then the electrons fail to correlate forsubsequent times

The time-dependent wave function for each electron isevolved and its spectrum of eigenvalues and eigenfunctionsis obtained using the methods described in the pioneeringpaper of Feit et al [10] on the spectral solution of the time-dependent Schroedinger equationThe results are sensitive to

initial conditions only in the sense that a poor choice (use of1199030 and larger values of 120577 eg) was found to lead to trajectoryrun away and an unphysical solution As in classical dynamicscare should be taken to use energy-conserving numericalmethods to integrate the trajectory fields using (6) Trajectoryrun away is easily recognized from a plot of trajectory versustime in which the trajectory is not periodic and grows withtime representing an unbound electron (self ionization) orin which it is periodic at large distances from the nucleirepresenting a spurious excited state (self excitation) Theenergies of runaway trajectories are observed to lie below thetrue energies due to insufficient strength of the interelectronicpotential When sufficient care is taken to obtain energy-conserving trajectories calculations tend to converge frombelow rather than from above experimental energies

Readers should be reminded that the evolution of thewave function by solving (12) is not an initial-value problemalthough it may appear to be from the methodology justoutlined This is a point of confusion for workers whoseexperience is in time-independent structure theory In factthe Heisenberg uncertainty principle Δ119864Δ119905 ge ℎ guaranteesthat at initial time the spectral width of a line is infinitelybroad and narrows with the length of the time integrationwhose measure is Δ119905 The positions of the spectral peaks(Figure 3) are accurately predicted even for early times andwide spectral lines as measured by Δ119864 but the integrationshould be continued for long enough times that the spectralpeaks for different eigenvalues are well separated This maybe hard to accomplish for integrating (12) for heavy particles

Figure 1 shows quantum trajectories in the y directionfor up and down spin states and identical spatial orbitalsat initial time All calculations are performed for protonslocated at plusmn1198772 along the 119911 axis These are trajectorieswhich belong to the 1Σ

119892state It is easy to see that the

two opposite-spin trajectories are correlated in the sensethat they keep apart and on opposite sides of the centers ofattraction most of the time Notice the expanding envelopsof the amplitudes with increasing time which indicates theincreasing excited and continuum-orbital contribution to thewave functionswith increasing timeThis behavior is possiblebecause the time-dependent or spectral wave functionswhichare used to calculate the currents whence the trajectoriesare a superposition of the complete set of states of theHamiltonians as emphasized in [10]

Figure 3 shows the spectra at an internuclear distance of119877 = 25 au as calculated from the temporal Fourier transformof the overlaps of the wave functions at time t with the initial-time wave functions as discussed in [10] Having located thespectral energies at the positions of the peaks eigenfunctions(Figures 7 and 8) are calculated from the Fourier transformof the spectral wave function precisely at the spectral peakpositions A window function [10]119908(119905) = (1minuscos(2120587119905119905max)is used in both the eigenvalue and eigenfunction calculationsin order to suppress side-lobe numerical artifacts

Figure 7 shows a z-slice at 119909 = 119910 = 0 of the eigen-functions for 119877 = 14 au and Figure 8 similarly shows theeigenfunctions for 119877 = 25 au The fixed locations of theprotons are clearly depicted in both plots as the twin peaks


0

01

02

03

04

WF

(au)

minus4 minus3 minus2 minus1 0 1 2 3 4119911 (au)

Figure 7 Wave functions along z at 119909 = 119910 = 0 at 119877 = 14 au forthe 1Σ

119892state of H

2 Solid spin-up electron Dotted spin-down

electron Convergence is shown for the solid curves at 075 119905max(lower) and for 119905max (upper) and for the upper curves at 075 119905max(upper) and for 119905max (lower) where 119905max = 200 au

of the eigenfunctions The nascent separation into H disso-ciation products is evident in Figure 8 by the dissymmetryof the wave functions with respect to the nuclear centersThe wave function which is symmetric about the protoncenters has the interelectronic potential set equal to zero thusthe correct separation into He-atom dissociation productsdepends on the correlation of the two electrons Figure 4shows corresponding quantum trajectories in the z directioncalculated using (7) in the nonrelativistic limit as in Figure 1but with the spectral wave function which is a superpositionof states replaced by eigenfunctions obtained as mentionedpreviously and described in [10] from the temporal Fouriertransform of the spectral wave function at a spectral eigen-value These may be considered to be eigentrajectories asopposed to the spectral trajectories appearing in Figure 1Notice that at later times the eigentrajectories for 119877 =

14 au are pulled into the internuclear region reflecting thecovalent nature of the bond In contrast the eigentrajectoriesfor 119877 = 25 au remain in the separated-atoms region for alltimes reflecting the weak covalency of the bond for largeinternuclear separation We regard this behavior as strongjustification for the present methodology in which the corre-lation of the two electrons guarantees the correct dissociationproducts automatically without having to build this behaviorinto the basis set in a conventional variational calculation

Figure 2 shows eigentrajectories for electrons having thesame spin states and different initial orbitals These aretrajectories belonging to the 3Σ

119906state

0 50

01

02

03

WF

(au)

minus5

119911 (au)

Figure 8 Wave functions along z at 119909 = 119910 = 0 at 119877 = 30 aufor the 1Σ

119892state of H


electronThe dotted curve which is symmetric about the two protonpositions is calculated for the interelectronic interaction artificiallyset equal to zero demonstrating that the correct dissociation limitinto two hydrogen atoms as shown by the solid and dottedcurves asymmetric about the two protons depends critically on theinterelectronic interaction

Notice that the trajectories are separated into a ground-state trajectory centered about one nucleus and an excited-state trajectorymoving at large distances from the two centersof attraction Such a state is known in quantum chemistry asantibonding

Figure 5 shows the late-time energies of the two states andthe time averages of the energies The energies are calculatedfrom the expectation value of the two-electron Hamiltonianusing the eigenfunctions inferred from the spectral solutionas discussed previously and in [10]The interelectronic poten-tials are calculated using the eigentrajectories as discussedpreviously and the sum over two electrons is divided by2 in order to avoid double counting Notice that the timedependence of the potential as given by (8)-(9) means thatthe Hamiltonian expectation values tend to oscillate about astationary time average as shown in Figures 5 and 6

Finally Figure 6 shows the energy time averages forfour different internuclear distances The agreement of thetime averages with the variational calculations of Kolos andWolniewicz [13] is good whose energies for the bondingand antibonding states are minus1174 and minus07842 respectivelyReaders are reminded that the binding energy is only 1745of the total electronic energy of minus1174 for the ground state ofmolecular hydrogenThe agreement at the larger internucleardistances is not as good owing to smaller binding energieswhose magnitude falls into the last digit of accuracy of thecalculation


5 Conclusions

One cannot extract more than three significant figures fromthe calculations at the present level of accuracy but we areconfident that the results can be systematically improvedfor accuracy by further refinements in the grid meshes andespecially by using improved energy-conserving numericalmethods to integrate the quantum trajectories in order toavoid run away the underestimation of the interelectronicpotential energy and as a consequence the overestimationof the binding energy These problems notwithstanding webelieve that the quantum-trajectory theory (a) establishesthe relationship between the spin of an individual electronand the Fermi-Dirac statistical behavior of an ensembleof electrons and (b) achieves the pair-wise correlation ofelectrons by evaluating Coulombrsquos law as an instantaneousinteraction in the time rather than as a quantum stationary-state mean or exchange interaction

Acknowledgments

The author is grateful to T Scott Carman for his support ofthis workThis work was performed under the auspices of theLawrence Livermore National Security LLC (LLNS) underContract no DE-AC52-07NA273

References

[1] H M James and A S Coolidge ldquoThe ground state of thehydrogen moleculerdquoThe Journal of Chemical Physics vol 1 no12 pp 825ndash835 1933

[2] M E Riley J M Schulman and J I Musher ldquoNonsymmetricperturbation calculations of excited states of heliumrdquo PhysicalReview A vol 5 no 5 pp 2255ndash2259 1972

[3] R J Bartlett and M Musial ldquoCoupled-cluster theory in quan-tum chemistryrdquo Reviews of Modern Physics vol 79 no 1 pp291ndash352 2007

[4] D Bohm Physical Review A vol 85 p 166 1952[5] P Holland Quantum Theory of Motion Cambridge Press

Cambridge UK 1993[6] R EWyattQuantumDynamics with TrajecTories Introduction

to Quantum Hydrodynamics Springer New York NY USA2005

[7] J D Bjorken and S D Drell Relativistic Quantum MechanicsMcGraw-Hill New York NY USA 1964

[8] H A Bethe and E E Salpeter QuantumMechanics of One- andTwo-Electron Atoms Dover New York NY USA 2008

[9] H A Bethe and E E SalpeterQuantumMechanics of One- andTwo-Electron Atoms Dover New York NY USA 2008

[10] M D Feit J A Fleck and A Steiger ldquoSolution of theSchrodinger equation by a spectral methodrdquo Journal of Com-putational Physics vol 47 no 3 pp 412ndash433 1982

[11] M E Riley and B Ritchie ldquoNumerical time-dependentSchrodinger description of charge-exchange collisionsrdquo Physi-cal Review A vol 59 no 5 pp 3544ndash3547 1999

[12] B Ritchie ldquoQuantum molecular dynamicsrdquo International Jour-nal of Quantum Chemistry vol 111 no 1 pp 1ndash7 2011

[13] W Kolos and L Wolniewicz ldquoPotentialmdashenergy curves for the1198831Σ119892 1198873Σ119906 and 119862

1Π119906states of the hydrogen moleculerdquo The

Journal of Chemical Physics vol 43 no 7 pp 2429ndash2441 1965

Submit your manuscripts athttpwwwhindawicom

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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adduced odd spatial wave function with a spin state whichis even on electron exchange (triplet state) The general-ization is of course the Slater determinantal wave functionof spin orbitals which guarantees the correct antisymmetricpermutation symmetry for N spin-12 particles Hence thestandard methodologies [3] for all their success simulatemany-electron quantum states in an ad hoc manner basedon experimental observation but tell us nothing fundamentalabout the physical basis of how the spin state of an individualelectron is related to the Pauli exclusion principle and theobserved Fermi-Dirac statistical behavior of an aggregate ofelectrons

2 Spin-Dependent Quantum Trajectories andElectron Exchange Correlation

Readers should recognize that an electronrsquos spin state and itsspatial correlation with another electron are strongly relatedas suggested by our observation that a successful simulationof correlation is also accompanied by the correct exchangesymmetry Notice however that such a relationship betweenelectron spin and the electron-electron Coulomb interactionappears to be grossly at oddswith our intuitive understandingof quantum chemistry likely due to the absence of a particle-trajectory picture in the standard methodologies Bohm for-mulated a quantum dynamical approach in which the phase-amplitude solution of Schroedingerrsquos equation has a formalrelationship to classical hydrodynamics [4 5] Bohmiandynamics has recently undergone a renewed interest [6]It is a method of solving the time-dependent Schrodingerequation by assuming a form for the wavefunction Ψ(119909 119905) =

119877(119909 119905)119890119894119878(119909119905) where R (such that R is greater than or equal

to zero for all x) is the amplitude and S is the actionfunction Then upon substitution into the time-dependentSchroedinger equation two coupled equations (continuityequation and the quantum Hamilton-Jacobi equation) areobtained The net result is to produce quantum trajectoriesThese equations have proven to be difficult to solve accuratelyparticularly for multi electrons [6] In fact it is difficult tosolve in three spatial dimensions for one electron Note thatin the context of the Bohmian equations the discretizationis done in terms of what are called pseudo particles Thuswhen it is said that say fifty particles are propagated ittypically means fifty pseudo particles are used to describethe dynamics of one real particle As far as we are aware noone has computed the dynamics of two real particles usingBohmian dynamics and in fact it does not seem possible todo so with current algorithms and computer hardware

We therefore seek a quantum trajectory approach forreal electrons in which the spin of an individual electron iscorrectly accounted for which means that we must look toDiracrsquos rather than to Schroedingerrsquos equation One of thedesiderata of Diracrsquos program [7] for incorporating Einsteinrsquosspecial theory of relativity into quantum mechanics was thatthe quantum analog of the classical equation of continuity

120597120588

120597119905

+ nabla sdot = 0 (1)

should be inferred from the equation of motion for therelativistic electron

(119894ℎ

120597

120597119905

minus 119881 minus 1198981198882)120595 ( 119903 119905) + 119894ℎ119888 sdot nabla120594 ( 119903 119905) = 0 (2a)

(119894ℎ

120597

120597119905

minus 119881 + 1198981198882)120594 ( 119903 119905) + 119894ℎ119888 sdot nabla120595 ( 119903 119905) = 0 (2b)

whereupon [7]

119895 ( 119903 119905) = 119888 [120595+( 119903 119905) 120594 ( 119903 119905) + 120594

+( 119903 119905) 120595 ( 119903 119905)] (3)

where is Paulirsquos vector and the quantum density is

120588 ( 119903 119905) = 120595+( 119903 119905) 120595 ( 119903 119905) + 120594

+( 119903 119905) 120594 ( 119903 119905) (4)

where 120595( 119903 119905) and 120594( 119903 119905) are the large and small Diracspinors respectively and the superscripts denote Hermi-tian conjugates On eliminating (2b) in favor of (2a) anddropping all terms of order 119888minus2 which is the nonrelativisticlimit Schroedingerrsquos equation is recovered for 120595 which wasanother desideratum in Diracrsquos program for a relativistic-electron theory Now notice that a velocity field can beinferred from the current by writing

119895 ( 119903 119905) = 119888 [120595+( 119903 119905) 120594 ( 119903 119905) + 120594

+( 119903 119905) 120595 ( 119903 119905)]

= ( 119903 119905) 120588 ( 119903 119905)

(5)

and solving for ( 119903 119905) from which a trajectory 119904(119905) can becalculated from the time integration of the velocity field tofind a position field

( 119903 119905) = int

119905

0

1198891199051015840 ( 119903 119905

1015840) (6)

and finally by finding the quantum expectation value of theposition field

119904 (119905) = int119889 119903 [120595+( 119903 119905) ( 119903 119905) 120595 ( 119903 119905)

+120594+( 119903 119905) ( 119903 119905) 120594 ( 119903 119905)]

(7)

One more step is needed to show the relationship ofelectron spin and electron exchange correlation namelythe evaluation of the electron-electron Coulomb interactionusing quantum trajectories Considering the case of twoelectrons one of which is described by (1)ndash(7) The Coulombinteraction for an electron with position vector 119903 with theother electron with position vector 119903

1015840 can then be written asfollows

1198811198901198901015840 ( 119903 119905) =

1198902

10038161003816100381610038161003816119903 minus 1199041015840(119905)

10038161003816100381610038161003816

(8)

Similarly the electron at 1199031015840 is described by (1)ndash(7) in which 119903

is replaced by 1199031015840 and the interaction is given by

1198811198901015840119890( 1199031015840 119905) =

1198902

100381610038161003816100381610038161199031015840minus 119904 (119905)

10038161003816100381610038161003816

(9)


minus20

minus10

0

10

20

Sy (a

u)

0 50 100 150 200119905 (au)


119892state of H

2



1198901198901015840 = 119890

2| 119903 minus 119903



1198901198901015840 =

1198902| 119903 minus 119903










119889 (119905)

119889119905

= minusnabla

1198902

10038161003816100381610038161003816119903 (119905) minus 119903

1015840(119905)

10038161003816100381610038161003816

997888rarr

119894ℎ

120597120595119863 (

119903 119905)

120597119905


+

1198902

10038161003816100381610038161003816119903 minus 1199041015840(119905)

10038161003816100381610038161003816

] 120595119863( 119903 119905)

(10)

where = 120574119898(119889 119903(119905)119889119905) 120574 = radic1 + (119901211989821198882) 120595119863

= (120595

120594 ) = (

0

0) 120573 = (

119868 0









120594 ( 119903 119905) =


119864 minus 119881 + 1198981198882

(11)







0

1

2

3

4

5

6

minus1

0 50 100 150 200119905 (au)

Sze (

au)




119906state of H

2

9119890 + 03

8119890 + 03

7119890 + 03

6119890 + 03

5119890 + 03

4119890 + 03

3119890 + 03

2119890 + 03

1119890 + 03

0119890 + 00

minus2 minus1 0 1

119864 (au)

Spec

trum

(au)


2 Solid spin-




0 50 100 150 200119905 (au)

minus3

minus2

minus1

0

1

2

3

Sze (

au)


119892

state of H2

150140 160 170 180 190 200119905 (au)

119864(a

u)

minus15

minus1

minus05






119892and 3Σ

119906states

respectively

minus12

minus115

minus11

minus105

minus1

119864(a

u)

150 160 170 180 190 200119905 (au)





119878( 119903 119905)120594

119898119904

and 120594( 119903 119905)


119878( 119903 119905)obeys


119894ℎ

120597120595119878( 119903 119905)

120597119905

= [minus

ℎ2

2119898

nabla2+ 119881 ( 119903 119905)] 120595

119878 (119903 119905) (12)

120594119898119904




( 119903 119905) cong

ℎ

2119898



119878)

+ 120595lowast

119878120594+

119898119904

(nabla times ) 120595119878120594119898119904

minus120595119878120594+

119898119904


119878120594119898119904

]

(13)







119895119899119903( 119903 119905) =

ℎ

119898

[ Im120595lowast

119878( 119903 119905) nabla120595119878 (

119903 119905)

plusmn Re120595lowast119878( 119903 119905)

120597

120597119910

120595119878 (

119903 119905)

∓Re120595lowast119878( 119903 119905)

120597

120597119909

120595119878 (

119903 119905)]

(15)





minus120577119903 120577 = 12



120577 = 1 and 1199032119890minus120577119903 120577 = 12 for












0

01

02

03

04

WF

(au)



119892state of H






119906state

0 50

01

02

03

WF

(au)

minus5

119911 (au)


119892state of H







5 Conclusions


Acknowledgments


References

























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


minus20

minus10

0

10

20

Sy (a

u)

0 50 100 150 200119905 (au)


119892state of H

2



1198901198901015840 = 119890

2| 119903 minus 119903



1198901198901015840 =

1198902| 119903 minus 119903










119889 (119905)

119889119905

= minusnabla

1198902

10038161003816100381610038161003816119903 (119905) minus 119903

1015840(119905)

10038161003816100381610038161003816

997888rarr

119894ℎ

120597120595119863 (

119903 119905)

120597119905


+

1198902

10038161003816100381610038161003816119903 minus 1199041015840(119905)

10038161003816100381610038161003816

] 120595119863( 119903 119905)

(10)

where = 120574119898(119889 119903(119905)119889119905) 120574 = radic1 + (119901211989821198882) 120595119863

= (120595

120594 ) = (

0

0) 120573 = (

119868 0









120594 ( 119903 119905) =


119864 minus 119881 + 1198981198882

(11)







0

1

2

3

4

5

6

minus1

0 50 100 150 200119905 (au)

Sze (

au)




119906state of H

2

9119890 + 03

8119890 + 03

7119890 + 03

6119890 + 03

5119890 + 03

4119890 + 03

3119890 + 03

2119890 + 03

1119890 + 03

0119890 + 00

minus2 minus1 0 1

119864 (au)

Spec

trum

(au)


2 Solid spin-




0 50 100 150 200119905 (au)

minus3

minus2

minus1

0

1

2

3

Sze (

au)


119892

state of H2

150140 160 170 180 190 200119905 (au)

119864(a

u)

minus15

minus1

minus05






119892and 3Σ

119906states

respectively

minus12

minus115

minus11

minus105

minus1

119864(a

u)

150 160 170 180 190 200119905 (au)





119878( 119903 119905)120594

119898119904

and 120594( 119903 119905)


119878( 119903 119905)obeys


119894ℎ

120597120595119878( 119903 119905)

120597119905

= [minus

ℎ2

2119898

nabla2+ 119881 ( 119903 119905)] 120595

119878 (119903 119905) (12)

120594119898119904




( 119903 119905) cong

ℎ

2119898



119878)

+ 120595lowast

119878120594+

119898119904

(nabla times ) 120595119878120594119898119904

minus120595119878120594+

119898119904


119878120594119898119904

]

(13)







119895119899119903( 119903 119905) =

ℎ

119898

[ Im120595lowast

119878( 119903 119905) nabla120595119878 (

119903 119905)

plusmn Re120595lowast119878( 119903 119905)

120597

120597119910

120595119878 (

119903 119905)

∓Re120595lowast119878( 119903 119905)

120597

120597119909

120595119878 (

119903 119905)]

(15)





minus120577119903 120577 = 12



120577 = 1 and 1199032119890minus120577119903 120577 = 12 for












0

01

02

03

04

WF

(au)



119892state of H






119906state

0 50

01

02

03

WF

(au)

minus5

119911 (au)


119892state of H







5 Conclusions


Acknowledgments


References

























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of






120594 ( 119903 119905) =


119864 minus 119881 + 1198981198882

(11)







0

1

2

3

4

5

6

minus1

0 50 100 150 200119905 (au)

Sze (

au)




119906state of H

2

9119890 + 03

8119890 + 03

7119890 + 03

6119890 + 03

5119890 + 03

4119890 + 03

3119890 + 03

2119890 + 03

1119890 + 03

0119890 + 00

minus2 minus1 0 1

119864 (au)

Spec

trum

(au)


2 Solid spin-




0 50 100 150 200119905 (au)

minus3

minus2

minus1

0

1

2

3

Sze (

au)


119892

state of H2

150140 160 170 180 190 200119905 (au)

119864(a

u)

minus15

minus1

minus05






119892and 3Σ

119906states

respectively

minus12

minus115

minus11

minus105

minus1

119864(a

u)

150 160 170 180 190 200119905 (au)





119878( 119903 119905)120594

119898119904

and 120594( 119903 119905)


119878( 119903 119905)obeys


119894ℎ

120597120595119878( 119903 119905)

120597119905

= [minus

ℎ2

2119898

nabla2+ 119881 ( 119903 119905)] 120595

119878 (119903 119905) (12)

120594119898119904




( 119903 119905) cong

ℎ

2119898



119878)

+ 120595lowast

119878120594+

119898119904

(nabla times ) 120595119878120594119898119904

minus120595119878120594+

119898119904


119878120594119898119904

]

(13)







119895119899119903( 119903 119905) =

ℎ

119898

[ Im120595lowast

119878( 119903 119905) nabla120595119878 (

119903 119905)

plusmn Re120595lowast119878( 119903 119905)

120597

120597119910

120595119878 (

119903 119905)

∓Re120595lowast119878( 119903 119905)

120597

120597119909

120595119878 (

119903 119905)]

(15)





minus120577119903 120577 = 12



120577 = 1 and 1199032119890minus120577119903 120577 = 12 for












0

01

02

03

04

WF

(au)



119892state of H






119906state

0 50

01

02

03

WF

(au)

minus5

119911 (au)


119892state of H







5 Conclusions


Acknowledgments


References

























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 50 100 150 200119905 (au)

minus3

minus2

minus1

0

1

2

3

Sze (

au)


119892

state of H2

150140 160 170 180 190 200119905 (au)

119864(a

u)

minus15

minus1

minus05






119892and 3Σ

119906states

respectively

minus12

minus115

minus11

minus105

minus1

119864(a

u)

150 160 170 180 190 200119905 (au)





119878( 119903 119905)120594

119898119904

and 120594( 119903 119905)


119878( 119903 119905)obeys


119894ℎ

120597120595119878( 119903 119905)

120597119905

= [minus

ℎ2

2119898

nabla2+ 119881 ( 119903 119905)] 120595

119878 (119903 119905) (12)

120594119898119904




( 119903 119905) cong

ℎ

2119898



119878)

+ 120595lowast

119878120594+

119898119904

(nabla times ) 120595119878120594119898119904

minus120595119878120594+

119898119904


119878120594119898119904

]

(13)







119895119899119903( 119903 119905) =

ℎ

119898

[ Im120595lowast

119878( 119903 119905) nabla120595119878 (

119903 119905)

plusmn Re120595lowast119878( 119903 119905)

120597

120597119910

120595119878 (

119903 119905)

∓Re120595lowast119878( 119903 119905)

120597

120597119909

120595119878 (

119903 119905)]

(15)





minus120577119903 120577 = 12



120577 = 1 and 1199032119890minus120577119903 120577 = 12 for












0

01

02

03

04

WF

(au)



119892state of H






119906state

0 50

01

02

03

WF

(au)

minus5

119911 (au)


119892state of H







5 Conclusions


Acknowledgments


References

























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of







119895119899119903( 119903 119905) =

ℎ

119898

[ Im120595lowast

119878( 119903 119905) nabla120595119878 (

119903 119905)

plusmn Re120595lowast119878( 119903 119905)

120597

120597119910

120595119878 (

119903 119905)

∓Re120595lowast119878( 119903 119905)

120597

120597119909

120595119878 (

119903 119905)]

(15)





minus120577119903 120577 = 12



120577 = 1 and 1199032119890minus120577119903 120577 = 12 for












0

01

02

03

04

WF

(au)



119892state of H






119906state

0 50

01

02

03

WF

(au)

minus5

119911 (au)


119892state of H







5 Conclusions


Acknowledgments


References

























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0

01

02

03

04

WF

(au)



119892state of H






119906state

0 50

01

02

03

WF

(au)

minus5

119911 (au)


119892state of H







5 Conclusions


Acknowledgments


References

























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


5 Conclusions


Acknowledgments


References

























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of










Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

research article quantum-dynamical theory of electron...

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