Srinivasan S. Iyengar Department of Chemistry and Department of Physics, Indiana University

Srinivasan S. IyengarDepartment of Chemistry and Department of Physics,

Indiana University

Atom-centered Density Matrix Atom-centered Density Matrix Propagation (ADMP): Theory and Propagation (ADMP): Theory and

ApplicationsApplications

Iyengar Group, Indiana University

Brief discussion of ab initio molecular dynamics

Atom-centered Density Matrix Propagation (ADMP)• Nut-n-bolts issues

Some Results:• Novel findings for protonated water clusters• QM/MM generalizations: ion channels• Gas phase reaction dynamics

OutlineOutline

http://www.indiana.edu/~ssiweb/


Molecular dynamics on a single potential surfaceMolecular dynamics on a single potential surface

Parameterized force fields (e.g. AMBER, CHARMM)• Energy, forces: parameters obtained from experiment• Molecular motion: 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 (DFT/HF/post-HF) for each nuclear

structure• Nuclei propagated using gradients of energy (forces)• Works for bond-breaking but computationally expensive

Large reactive, polarizable systems: Something like BO, but preferably less expensive.



Circumvent Computational Bottleneck of BO• Avoid repeated SCF: electronic structure, not converged, but propagated• “Simultaneous” propagation of electronic structure and nuclei:

adjustment of time-scales Car-Parrinello (CP) method

• Orbitals expanded in plane waves• Occupied orbital coefficients propagated

– O(N3) computational scaling (traditionally)– O(N) with more recent Wannier representations (?)

Atom-centered Density Matrix Propagation (ADMP) • Atom-centered Gaussian basis functions• Electronic Density Matrix propagated

– Asymptotic linear-scaling with system size

• Allows the use of accurate hybrid density functionals • suitable for clusters

CP: R. Car, M. Parrinello, Phys. Rev. Lett. 55 (22), 2471 (1985). ADMP: Schlegel, et al. JCP, 114, 9758 (2001). Iyengar, et al. JCP, 115,10291 (2001). Iyengar et al. Israel J. Chem. 7, 191, (2002).

Schlegel et al. JCP 114, 8694 (2002). Iyengar and Frisch JCP 121, 5061 (2004).

References…

Extended Lagrangian dynamicsExtended Lagrangian dynamics



Atom-centered Density Matrix Propagation (ADMP)Atom-centered Density Matrix Propagation (ADMP) Construct a classical phase-space {{R,V,M},{P,W,}} The Lagrangian (= Kinetic minus Potential energy)

Nuclear KE

MVVTr21 TL

“Fictitious” KE of P

21/41/4WμμTr21

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

iioccN

i

1

P:matrixdensity particle single of Definition



Euler-Lagrange equations of motion for ADMPEuler-Lagrange equations of motion for ADMP

Equations of motion for density matrix and nuclei

P2

2

RERM

dtd

Classical dynamics in {{R,V,M},{P,W,}} phase space Next few slide: Forces, propagation equations, formal error

analysis

acceleration of density matrix, P

Force on P

“Fictitious” mass of P

PPPEP

R2

2

dtd 2/1μ 2/1μ



Nuclear Forces: What Really makes it workNuclear Forces: What Really makes it work

P

ii

R)P,E(R

P~

dRdSP~FTr

Pulay’s moving basis terms

RV

REP~

dRGd

21P~

dRhdTr xc

NN

Hellman-Feynman contributions

Contributions due to [F,P] 0. Part of non-Hellman-Feynman

dRdUUP~-U

dRdUQ~F,P~Tr

TT1

S=UTU, Cholesky or

Löwdin



Density Matrix Forces:Density Matrix Forces:

Use McWeeny Purified DM (3P2-2P3) in energy expression to obtain

F2P2PFP2FP3PF3FPP

)P,E(R 22

R

ii



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


But Careful - too large : non-physicalAppropriate : approximate BO dynamics But Careful - too large : non-physical


Direction of Increasing Frequency



““Physical” interpretation ofPhysical” interpretation of Bounds Bounds

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”

Iyengar et al. Israel J. Chem. 7, 191, (2002). Reference…



Bounds on the magnitude of Bounds on the magnitude of

fictreal HHHdt

dμdt

dWWμTrdt

d fict1/21/2real HH

PPΛTrP)E(R,WμμTr21MVVTr

21 221/41/4T H

The Conjugate Hamiltonian (Legendre Transform)

PPΛTrP)E(R,WμμTr21MVVTr

21 221/41/4T L

The Lagrangian

Controlling Deviations from BO surface and adiabaticityIyengar et al. JCP. 115,10291 (2001). Reference…



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.

References…Iyengar et al. Israel J. Chem. 7, 191, (2002). Schlegel et al. JCP 114, 8694 (2002). Iyengar and Frisch JCP 121, 5061

(2004).



Propagation of Propagation of P: time-reversible propagation Velocity Verlet propagation of P

2/1iiiii

Ri

ii2/12

ii1i μ PPP

)P,E(Rμ2t-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 (next two slides)

2/1iiiii

Ri

ii2/1i1/2i μ PP

P)P,E(Rμ

2t- W W

2/11i1i1i1i1i

R1i

1i1i2/11/2i1i μ PP

P)P,E(Rμ

2t- W W

Propagation of W



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



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(Rμ2t-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

2t- W W



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…



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, Mike Duncan, M. Okumura• Sum Frequency Generation (SFG) : Y. R. Shen, M. J. Schultz and

coworkers Lots of theory too: Jordan, McCoy, Bowman, Klein, Singer (not

exhaustive by any means..) Variety of medium-sized protonated clusters using ADMP

ADMP treatment of protonated water clusters: Iyengar, et al. JCP, 123, 084309 (2005). Iyengar et al. Int. J. Mass Spec. 241, 197 (2005). Iyengar JCP 123, 084310, (2005).

References…


Iyengar Group, Indiana UniversityProtonated Water Clusters: Protonated Water Clusters: Hopping via the Grotthuss mechanismHopping via 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**



(H(H22O)O)2020HH33OO++: : A recent spectroscopic quandryA recent spectroscopic quandry

J.-W. Shin, N. I. Hammer, E. G. Diken et al., Science 304, 1137 2004.

Theory

Experiment



Spectroscopy: Spectroscopy: A recent quandryA recent quandryWater 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



ADMP Spectrum!! Iyengar et al. JCP, 123 , 084309 (2005)

Spectroscopy: Spectroscopy: A recent quandryA recent quandry



(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**


Iyengar Group, Indiana UniversityLarger Clusters and Larger Clusters and water/vacuum interfaces: Similar resultswater/vacuum interfaces: 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


Iyengar Group, Indiana UniversityProtonated 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)



Experimental results suggest this as wellExperimental results suggest this as 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…



QM/MM treatment: ONIOM ADMPQM/MM treatment: ONIOM ADMP

MMI

QMI

MMfull EEEE

Unified treatment of the full system within ADMP

(This talk will not overview the ONIOM scheme, but the interested reader should look at the reference

below)

N. Rega, S. S. Iyengar, G. A. Voth, H. B. Schlegel, T. Vreven and M. J. Frisch, J. Phys. Chem. B 108 4210 (2004).



Side-chain contribute to hop

“Eigen” like configuration possible using protein backbone

B3LYP and BLYP: qualitatively different results



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


Iyengar Group, Indiana UniversityGlyoxal Glyoxal 3-body Synchronous photo-fragmentation3-body Synchronous photo-fragmentation



ConclusionsConclusions

ADMP: powerful 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



AcknowledgmentAcknowledgment

The work has enormously benefited from my former advisors and collaborators:

– Greg Voth– Berny Schlegel– Gus Scuseria– Mike Frisch

At IU, people contributing to this work are:– Jacek Jakowski (post-doc)– Isaiah Sumner (grad student)– Xiaohu Li (grad student)– Virginia E. Teige (Freshman)


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Srinivasan S. Iyengar Department of Chemistry and Department of Physics, Indiana University