Outline
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
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Molecular 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.
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Extended Lagrangian dynamics
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
References
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).
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Atom-centered Density Matrix Propagation (ADMP)
Construct a classical phase-space {{R,V,M},{P,W,μ}}
The Lagrangian (= Kinetic minus Potential energy)
P : represented using atom-centered gaussian basis sets
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Euler-Lagrange equations of motion for ADMP
Equations of motion for density matrix and nuclei
Classical dynamics in {{R,V,M},{P,W,μ}} phase space
Next few slide: Forces, propagation equations, formal error analysis
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Nuclear Forces: What Really makes it work
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Density Matrix Forces:
Use McWeeny Purified DM (3P2-2P3) in energy expression to obtain
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μ effects an adjustment of time-scales:
Consequence of μ : P changes slower with time: characteristic frequency adjusted
But Careful - too large μ: non-physical
Appropriate μ: approximate BO dynamics
Bounds for μ: From a Hamiltonian formalism
μ: also related to deviations from the BO surface
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“Physical” interpretation of μ : Bounds
Commutator of the electronic Hamiltonian and density matrix: bounded by magnitude of μ
Magnitude of μ : represents deviation from BO surface
μ acts as an “adiabatic control parameter”
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Bounds on the magnitude of μ :
The Lagrangian
The Conjugate Hamiltonian (Legendre Transform)
Controlling μ: Deviations from BO surface and adiabaticity
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Comparison with BO dynamics
Born-Oppenheimer dynamics:
Converged electronic states.
dE/dR not same in both methods
ADMP:
1 SCF cycle : for Fock matrix -> dE/dP
Current: 3-4 times faster.
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Propagation of P: time-reversible propagation
Velocity Verlet propagation of P
Propagation of W
Classical dynamics in {{R,V},{P,W}} phase space
Λi and Λi+1 obtained iteratively:
Conditions: Pi+1 2 = Pi+1 and WiPi + PiWi = Wi (next two slides)
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Idempotency (N-Representibility of DM):
Given Pi2 = Pi, need Λi to find idempotent Pi+1
Solve iteratively: Pi+12 = Pi+1
Given Pi, Pi+1, Wi, Wi+1/2, need Λi+1 to find Wi+1
Solve iteratively: Wi+1 Pi+1 + Pi+1 Wi+1 = Wi+1
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Idempotency: To obtain Pi+1
Given Pi2 = Pi, need to find indempotent Pi+1
Guess:
Or guess:
Iterate Pi+1 to satisfy Pi+12 = Pi+1
Rational for choice PiTPi + QiTQi above:
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How it all works …
Initial config.: R(0). Converged SCF:P(0)
Initial velocities V(0) and W(0) : flexible
P(Δt), W(Δt) : from analytical gradients and idempotency
Similarly for R(Δt)
And the loop continues…
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Protonated 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
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Protonated Water Clusters:
Hopping via the Grotthuss mechanism
True for 20, 30, 40, 50 and larger clusters…
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(H2O)20H3O+: Magic number cluster
Hydronium goes to surface: 150K, 200K and 300K: B3LYP/6-31+G** and BPBE/6-31+G**
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
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(H2O)20H3O+: A recent spectroscopic quandry
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Spectroscopy: A recent quandry
Water Clusters: Important in Atmospheric Chemistry
Explains the experiments of M. A. Johnson
Bottom-right spectrum
From ADMP agrees well with expt: dynamical effects in IR spectroscopy
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Spectroscopy: A recent quandry
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(H2O)20H3O+: Magic number cluster
Hydronium goes to surface: 150K, 200K and 300K: B3LYP/6-31+G** and BPBE/6-31+G**
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
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Larger Clusters and water/vacuum interfaces: Similar results
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Predicting 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 Reasons for “hopping” to surface
Hydrophobic and hydrophillic regions: Directional hydrophobicity (it is amphiphilic)
H3O+ has reduced density around Reduction of entropy of surrounding waters
Is Hydronium hydrophobic ?
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Experimental 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
References
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)
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QM/MM treatment: ONIOM ADMP
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).
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Side-chain contribute to hop
“Eigen” like configuration possible using protein backbone
B3LYP and BLYP: qualitatively different results
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HCHO photodissociation
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**
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Glyoxal 3-body Synchronous photo-fragmentation
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Conclusions
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