srinivasan s. iyengar department of chemistry, indiana university atom-centered density matrix...
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Srinivasan S. IyengarDepartment of Chemistry,
Indiana University
Atom-centered Density Matrix Atom-centered Density Matrix Propagation (ADMP): Theory and Propagation (ADMP): Theory and
Application to protonated water clusters Application to protonated water clusters and water/vacuum interfacesand water/vacuum interfaces
Brief outline of ab initio molecular dynamicsAtom-centered Density Matrix Propagation
(ADMP)Results:
• Novel findings for protonated water clusters• Preliminary results for ion-transport through
biological channelsNut-n-bolts issues
This presentation is meant to be a quick outline of ADMP. You should read the related papers to get more complete understanding
Molecular dynamics in ChemistryMolecular dynamics in Chemistry
Molecular motion and structure determine properties:• Spectroscopic properties• Predicting Molecular Reactivity
Computationally molecular dynamics simulates molecular motions: • determine properties from correlation functions• To Simulate molecular motions:
– Need Energy of conformation– Forces to move nuclei: Simulate nuclear motion
Methods for molecular dynamics Methods for molecular dynamics on a single potential surfaceon a single potential surface
Parameterized force fields (e.g. AMBER, CHARMM)• Energy, forces: parameters obtained from experiment• Molecules moved: Newton’s laws • Works for large systems
– But hard to parameterize bond-breaking/formation (chemical reactions)
– Issues with polarization/charge transfer/dynamical effects Born-Oppenheimer (BO) Dynamics
• Solve electronic Schrödinger eqn within some approximation for each nuclear structure
• Nuclei are propagated using gradients (forces)• Works for bond-breaking but computationally expensive
Large reactive, polarizable systems: We need something like BO, but less expensive.
Atom-centered Density Matrix Propagation Atom-centered Density Matrix Propagation (ADMP) : An Extended Lagrangian approach(ADMP) : An Extended Lagrangian approach
Circumvent Computational Bottleneck of BOAvoid repeated SCF for electronic SE electronic structure, not converged, but
propagated “Simultaneous” propagation of electronic
structure with nuclei: an adjustment of time-scales
Atom-centered Density Matrix Atom-centered Density Matrix Propagation (ADMP)Propagation (ADMP)
Construct a classical phase-space {{R,V,M},{P,W,}}
The Lagrangian (= Kinetic minus Potential energy)
Nuclear KE
MVVTr2
1 TL
“Fictitious” KE of P
21/41/4WμμTr2
1
Energy functional
P)E(R,
Lagrangian Constraint for N-representability of P: Idempotency and Particle number
PPΛTr 2
P : represented using atom-centered gaussian basis sets
Euler-Lagrange equations of motionEuler-Lagrange equations of motion
Equations of motion for density matrix and nuclei
P2
2
R
ERM
dt
d
Classical dynamics in {{R,V,M},{P,W,}} phase space Solutions obtained using velocity Verlet integrator
acceleration of density matrix, P
Force on P
“Fictitious” mass of P
PPP
EP
R2
2
dt
d2/1μ 2/1μ
effects an adjustment of time-scales:effects an adjustment of time-scales:
Bounds for : From a Hamiltonian formalism : alsoalso related to deviations from the BO surface related to deviations from the BO surface
Consequence of : P changes slower with time: characteristic frequency adjusted
Consequence of : P changes slower with time: characteristic frequency adjusted
But Careful - too large : non-physicalAppropriate : approximate BO dynamics
But Careful - too large : non-physical
Consequence of : P changes slower with time: characteristic frequency adjusted
Direction of Increasing Frequency
““Physical” interpretation ofPhysical” interpretation of
21/41/4
FF
WμμTrWP,
1PF,
Commutator of the electronic Hamiltonian and density matrix: bounded by magnitude of
Magnitude of : represents deviation from BO surface
acts as an “adiabatic control parameter”
Bounds on the magnitude of Bounds on the magnitude of
fictreal HHHdt
dμ
dt
dWWμTr
dt
d fict1/21/2real HH
PPΛTrP)E(R,WμμTr2
1MVVTr
2
1 221/41/4T H
The Conjugate Hamiltonian (Legendre Transform)
PPΛTrP)E(R,WμμTr2
1MVVTr
2
1 221/41/4T L
The Lagrangian
By controlling control deviations from BO surface and adiabaticity
Nuclear Forces: What Really makes it workNuclear Forces: What Really makes it work
P
ii
R
)P,E(R
P
~
dR
dSP~
FTr
Pulay’s moving basis terms
R
V
R
EP~
dR
Gd
2
1P~
dR
hdTr xc
NN
Hellman-Feynman contributions
Contributions due to [F,P] 0. Part of non-Hellman-Feynman
dR
dUUP
~-U
dR
dUQ~
F,P~
TrT
T1
S=UTU, Cholesky or
Löwdin
Some Advantages of ADMPSome Advantages of ADMP
ADMP:– Currently 3-4 times faster
than BO dynamics– Improvements will allow ADMP ~ 10 times faster– Computational scaling O(N)
– Hybrid functionals (more
accurate) : routine
– Smaller Greater adiabatic control
– QM/MM: localized bases: natural
Comparison with BO dynamicsComparison with BO dynamics
Born-Oppenheimer dynamics:• Converged electronic
states.
• Approx. 8-12 SCF cycles / nuclear config.
• dE/dR not same in both methods
ADMP:
• Electronic state propagated classically : no convergence reqd.
• 1 SCF cycle : for Fock matrix -> dE/dP
• Current: 3-4 times faster. 10 times
Reference…
H. B. Schlegel, S. S. Iyengar, X. Li, J. M. Millam, G. A. Voth, G. E. Scuseria, M. J. Frisch, JCP, In Press.
Atom-centered Density Matrix Propagation (ADMP) approach using Gaussian basis sets• Atom-centered Gaussian basis functions
– Fewer basis functions for molecular systems
• Electronic Density Matrix propagated– Asymptotic linear-scaling with system size
Car-Parrinello (CP) method• Orbitals expanded in plane waves• Occupied orbital coefficients propagated
– O(N3) computational scaling
CP: R. Car, M. Parrinello, Phys. Rev. Lett. 55 (22), 2471 (1985). ADMP:H. B. Schlegel, J. Millam, S. S. Iyengar, G. A. Voth, A. D. Daniels, G. E. Scuseria, M. J. Frisch, JCP, 114, 9758 (2001). S. S. Iyengar, H. B. Schlegel, J. Millam, G. A. Voth, G. E. Scuseria, M. J. Frisch, JCP, 115,10291 (2001).
References…
Comparison with Car-Parrinello : Slide 0Comparison with Car-Parrinello : Slide 0
Comparison with Car-Parrinello : Slide 1Comparison with Car-Parrinello : Slide 1
Plane-wave CP:• Computational scaling O(N3)
• Pure functionals (e.g. BLYP)
Hybrid (B3LYP): expensive
• Adiabatic control limited : larger : D2O for H2O
• Properties depend on §
ADMP:– Computational scaling O(N) – Hybrid functionals (more
accurate) : routine– Smaller Greater adiabatic control: can use H2O
– Properties independent of #
References…
§ Scandolo and Tangney, JCP. 116, 14 (2002).# Schlegel, Iyengar, Li, Millam, Voth, Scuseria, Frisch, JCP, 117, 8694 (2002).
Comparison with Car-Parrinello : Slide 2Comparison with Car-Parrinello : Slide 2
Plane-wave CP:• Larger no. of basis fns.
• QM/MM: Plane-waves enter MM region
• Pseudopotentials required for core
ADMP:• Fewer basis fns.
• QM/MM: localized bases: natural
• Pseudopotentials not required for core
– Important for metals e.g., redox species and enzyme active sites
Propagation of Propagation of P: a time-reversible propagation scheme Velocity Verlet propagation of P
2/1iiiii
Ri
ii2/12
ii1i μ PPP
)P,E(Rμ
2
t-t W P P
Classical dynamics in {{R,V},{P,W}} phase spacei and i+1 obtained iteratively:
– Conditions: Pi+1 2 = Pi+1 and WiPi + PiWi = Wi
2/1iiiii
Ri
ii2/1i1/2i μ PP
P
)P,E(Rμ
2
t- W W
2/11i1i1i1i1i
R1i
1i1i2/11/2i1i μ PP
P
)P,E(Rμ
2
t- W W
Propagation of W
Idempotency: To obtain Idempotency: To obtain PPi+1i+1
Given Pi2 = Pi, need to find indempotent Pi+1
Guess:
Or guess: Iterate Pi+1 to satisfy Pi+1
2 = Pi+1
Rational for choice PiTPi + QiTQi above:
2/1
Ri
ii2/12
ii*
1i μ P
)P,E(Rμ
2
t-t W P P
2/1iiii
2/1*1i1i μ TQQTPPμ P P
2/1*1i1i
2/1 μ PP~
μ T
iiiiiiiiiii QQPP PP
t W-t 2W P P 1/2-iii*
1i
Idempotency: To obtain Idempotency: To obtain WWi+1i+1
Given WiPi + PiWi = Wi, find appropriate Wi+1
Guess:
Iterate Wi+1 to satisfy Wi+1Pi+1 + Pi+1Wi+1 = Wi+1
2/11i1i1i1i
2/1*1i1i μ QT
~QPT
~Pμ W W
2/1*1i1i
2/1 μ WW~
μ T~
2/1
R1i
1i1i2/11/2i
*1i μ
P
)P,E(Rμ
2
t- W W
Density Matrix Forces:Density Matrix Forces:
Use McWeeny Purified DM (3P2-2P3) in energy expression to obtain
F2P2PFP2FP3PF3FPP
)P,E(R 22
R
ii
Nuclear Forces: What Really makes it workNuclear Forces: What Really makes it work
P
ii
R
)P,E(R
P
~
dR
dSP~
FTr
Pulay’s moving basis terms
R
V
R
EP~
dR
Gd
2
1P~
dR
hdTr xc
NN
Hellman-Feynman contributions
Contributions due to [F,P] 0. Part of non-Hellman-Feynman
dR
dUUP
~-U
dR
dUQ~
F,P~
TrT
T1
S=UTU, Cholesky or
Löwdin
Idempotency (N-Representibility of DM):Idempotency (N-Representibility of DM):
Given Pi2 = Pi, need i to find idempotent
Pi+1
Solve iteratively: Pi+12 = Pi+1
Given Pi, Pi+1, Wi, Wi+1/2, need i+1 to find Wi+1
Solve iteratively: Wi+1 Pi+1 + Pi+1 Wi+1 = Wi+1
How it all works …How it all works …
Initial config.: R(0). Converged SCF: P(0) Initial velocities V(0) and W(0) : flexible P(t), W(t) : from analytical gradients and
idempotency Similarly for R(t)And the loop continues…
ResultsResults
For Comparison with Born-Oppenheimer dynamics• Formaldehyde photo-dissociation
• Glyoxal photo-dissociation
New Results for Protonated Water clusters Protonated water wire Ion transport through gramicidin ion channels
Protonated Water ClustersProtonated Water Clusters
Important systems for:• Ion transport in biological and condensed systems• Enzyme kinetics• Acidic water clusters: Atmospheric interest• Electrochemistry
Experimental work: • Mass Spec.: Castleman• IR: M. A. Johnson, M. Okumura• Sum Frequency Generation (SFG) : Y. R. Shen, M. J. Schultz
and coworkers Variety of medium-sized protonated clusters using
ADMP
Protonated Water Clusters: Hopping Protonated Water Clusters: Hopping via the Grotthuss mechanismvia the Grotthuss mechanism
True for 20, 30, 40, 50 and larger clusters…
(H(H22O)O)2020HH33OO++: : Magic numberMagic number cluster cluster
Castleman’s experimental results:• 10 “dangling” hydrogens
in cluster– Found by absorption of
trimethylamine (TMA)
• 10 “dangling” hydrogens: consistent with our ADMP simulations
But: hydronium on the surface
Hydronium goes to surface: 150K, 200K and 300K: B3LYP/6-31+G** and BPBE/6-31+G**
Larger Clusters and water/vacuum Larger Clusters and water/vacuum interfaces: Similar resultsinterfaces: Similar results
Predicting New Chemistry: TheoreticallyPredicting New Chemistry: Theoretically
A Quanlitative explanation to the remarkable Sum Frequency Generation (SFG) of Y. R. Shen, M. J. Schultz and coworkers
Protonated Water Cluster: Conceptual Protonated Water Cluster: Conceptual Reasons for “hopping” to surfaceReasons for “hopping” to surface
H3O+ has reduced density aroundReduction of entropy of surrounding waters
H2O coordination 4 H3O+ coordination =3
Is Hydronium hydrophobic ?
Hydrophobic and hydrophillic regions: Directional hydrophobicity (it is amphiphilic)
Spectroscopy: Spectroscopy: A recent quandryA recent quandry
Water Clusters: Important in Atmospheric Chemistry
Bottom-right spectrumFrom ADMP agrees well with expt: dynamical effects in IR spectroscopy
Explains the experiments of M. A. Johnson
Experimental results seem to suggest this Experimental results seem to suggest this as wellas well
Y. R. Shen: Sum Frequency Generation (SFG) • IR for water/vapor interface shows dangling O-H bonds
• intensity substantially diminishes as acid conc. is increased
• Consistent with our results– Hydronium on surface: lone pair outwards, instead of dangling O-H
• acid concentration is higher on the surface
Schultz and coworkers: acidic moieties alter the structure of water/vapor interfaces
P. B. Miranda and Y. R. Shen, J. Phys. Chem. B, 103, 3292-3307 (1999). M. J. Schultz, C. Schnitzer, D. Simonelli and S. Baldelli, Int. Rev. Phys. Chem. 19, 123-153 (2000)
References…
Protonated Water Cluster: Conceptual Protonated Water Cluster: Conceptual Reasons for “hopping” to surfaceReasons for “hopping” to surface
H3O+ has reduced density aroundReduction of entropy of surrounding waters
H2O coordination 4 H3O+ coordination =3
Is Hydronium hydrophobic ?
Hydrophobic and hydrophillic regions: Directional hydrophobicity
Protonated Water Clusters: progress Protonated Water Clusters: progress of the protonof the proton
Most protonated water closer to the surface as simulation progresses
3 ang
Protonated Water Cluster: Radial Protonated Water Cluster: Radial Distribution FunctionsDistribution Functions
Zundel [H5O2+]: ~2.45
Eigen [H9O4+]: ~2.55
BLYP : Zundel and Eigen
B3LYP: ZundelBLYP : proton more
delocalized
O*-O Radial Distribution function peaks: • BLYP : ~2.45 Angstrom and ~2.55 Angstrom
• B3LYP : ~2.45
Protonated Water WireProtonated Water Wire Proton hopping across “water wire”
• Model for proton transfer in: – ion channels– Enzymes– liquids
DFT - B3LYP / 6-31+G** / 300K / ~1 ps Basis set / functional: good water-dimer properties
Protonated Water WireProtonated Water Wire
Protonated Oxygen peak ~ 2.4 Angstrom
Non-protonated Oxygen peaks : spread (about 2.8 Ang.)
Results consistent with Brewer, Schmidt and Voth using EVB model
Water wire to Ion Channels: QM/MM Water wire to Ion Channels: QM/MM ADMPADMP
Proton transport through ion-channel
QM/MM approach to AIMD
QM/MM treatment of bio-systemsQM/MM treatment of bio-systems
MMI
QMI
MMfull EEEE
Unified treatment of the full system within ADMP
ONIOM: Energy partitioningONIOM: Energy partitioning
MMI
QMI
MMII EEEE
MM
j
QM
i ji
jiMMself I,
MMI
RR
ZZ EE
Link atom coordinates are expressed in terms of their neighbors: Link atoms factor out
MM
j
QM
i ji
jiMM
j j
jQMself I,
QMI
RR
Z Z
Rr
ZH H QM
IQMI FE
Preliminary results:
Side-chain contributions to hop:
B3LYP and BLYP: qualitatively different results
Protonated Water Cluster v/s Protonated Protonated Water Cluster v/s Protonated Water WireWater Wire
Cluster: Proton goes to surfaceWire: Proton tends to centerWhy?Cluster:
• H3O+ coordination number 3
• Lone pair has reduced water density around
Wire:• 2 H-bonds at center: 1 H-bond at end
• H3O+ lone pair has reduced density: center and edge
• Reduced density not a factor: Number of H-bonds is
Photolysis at 29500 cm-1 : To S1 state• Returns to ground state vibrationally hot• Product: rotationally cold, vibrationally excited H2
• And CO broad rotational distr: <J> = 42. Very little vib. Excitation H2CO H2 + CO: BO and ADMP at HF/3-21G, HF/6-31G**
HCHO photodissociationHCHO photodissociation
Glyoxal 3-body Synchronous photo-Glyoxal 3-body Synchronous photo-fragmentationfragmentation
What about BSSE? What about BSSE?
Due to:• difference in instantaneous incompleteness in basis set. • Atom centered nature of basis set (not present in plane-
waves). Worst when neighbouring atoms leave completely (ie,
total dissociation). Present case: proton hopping, no complete dissociation
(replaced by new proton). Expected to be less. Dominant sources of errors:
• Off the BO surface• DFT functional
What about BSSE? What about BSSE?
Difference in completeness of basis set. Worst when neighbouring atoms leave completely (ie,
total dissociation). Dynamics without total dissociation:
• Effect expected to be less. Dominant sources of errors:
• DFT functional
Chloride-Water ClusterChloride-Water Cluster
Conservation Properties :
Fictitious KE =
Change in Fict. KE ~ 0.0002% of total Energy 21/41/4Wμμ
2
1Tr
Chloride-Water Clusters: Chloride-Water Clusters: Red-shiftsRed-shifts
Bend: ~ 1600 cm-1, Stretch ~3400 & ~3600 cm-1
Exptal. O-H Red Shift for ClCl-- (H (H22O)O)11 :– 3130 cm -1 Ar matrix : M. A.
Johnson, Yale University
– 3285 cm -1 CCl4 matrix : M. Okumura, CalTech
Critical to use hydrogens in these simulations
DFT – B3LYP / 6-31G*
Chloride-Water Cluster: ClChloride-Water Cluster: Cl-- (H (H22O)O)2525
ADMP dynamics oscillates about the BO result.
Protonated Water Cluster: IR SpectrumProtonated Water Cluster: IR Spectrum
Bending ~ 1600-1700 cm-1. Stretch: broad: 3000 – 3700 cm-1. Libration modes at less than 800 cm-1
Broad Stretching band: due to proton affecting the H-bond network
ConclusionsConclusions
ADMP is powerful new approach to ab initio molecular dynamics• Linear scaling with system size• Hybrid (more accurate) density functionals• Smaller values for fictitious mass allow
– treatment of systems with hydrogens is easy (no deuteriums required)
– greater adiabatic control (closer to BO surface)
Examples bear out the accuracy of the method