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