Post on 13-Feb-2016
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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
Iyengar Group, Indiana University
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.
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
Density Matrix Forces:Density Matrix Forces:
Use McWeeny Purified DM (3P2-2P3) in energy expression to obtain
F2P2PFP2FP3PF3FPP
)P,E(R 22
R
ii
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
““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…
Iyengar Group, Indiana University
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…
Iyengar Group, Indiana University
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).
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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…
Iyengar Group, Indiana University
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…
Iyengar Group, Indiana University
(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 University
(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
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
ADMP Spectrum!! Iyengar et al. JCP, 123 , 084309 (2005)
Spectroscopy: Spectroscopy: A recent quandryA recent quandry
Iyengar Group, Indiana University
(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
Iyengar Group, Indiana University
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)
Iyengar Group, Indiana University
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…
Iyengar Group, Indiana University
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).
Iyengar Group, Indiana University
Side-chain contribute to hop
“Eigen” like configuration possible using protein backbone
B3LYP and BLYP: qualitatively different results
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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)