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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
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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
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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
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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.
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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
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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
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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
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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
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““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”
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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
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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
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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
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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.
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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
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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).
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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
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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
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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
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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
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Density Matrix Forces:Density Matrix Forces:
Use McWeeny Purified DM (3P2-2P3) in energy expression to obtain
F2P2PFP2FP3PF3FPP
)P,E(R 22
R
ii
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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
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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
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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…
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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
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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
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Protonated Water Clusters: Hopping Protonated Water Clusters: Hopping via the Grotthuss mechanismvia the Grotthuss mechanism
True for 20, 30, 40, 50 and larger clusters…
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(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**
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Larger Clusters and water/vacuum Larger Clusters and water/vacuum interfaces: Similar resultsinterfaces: Similar results
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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
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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)
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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
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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…
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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
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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
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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
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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
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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
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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
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QM/MM treatment of bio-systemsQM/MM treatment of bio-systems
MMI
QMI
MMfull EEEE
Unified treatment of the full system within ADMP
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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
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Preliminary results:
Side-chain contributions to hop:
B3LYP and BLYP: qualitatively different results
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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
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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
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Glyoxal 3-body Synchronous photo-Glyoxal 3-body Synchronous photo-fragmentationfragmentation
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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
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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
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Chloride-Water ClusterChloride-Water Cluster
Conservation Properties :
Fictitious KE =
Change in Fict. KE ~ 0.0002% of total Energy 21/41/4Wμμ
2
1Tr
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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*
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Chloride-Water Cluster: ClChloride-Water Cluster: Cl-- (H (H22O)O)2525
ADMP dynamics oscillates about the BO result.
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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
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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