Atom-centered Density Matrix
Propagation (ADMP): Theory and
Applications
Srinivasan S. Iyengar
Department of Chemistry and Department of Physics,
Indiana University
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
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.
Iyengar Group, Indiana University
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).
Iyengar Group, Indiana University
Atom-centered Density Matrix Propagation (ADMP)

Construct a classical phase-space {{R,V,M},{P,W,m}}

The Lagrangian (= Kinetic minus Potential energy)



1
1
T
L  Tr V MV  Tr μ1/4Wμ1/4
2
2

2
Nuclear KE
“Fictitious”
KE of P
Energy
functional

 E(R, P)  Tr Λ P 2  P

Lagrangian Constraint
for N-representability
of P: Idempotency and
Particle number
N occ
Definition of single particle density matrix : P   i i
i 1

P : represented using atom-centered gaussian basis sets
Iyengar Group, Indiana University
Euler-Lagrange equations of motion for ADMP

Equations of motion for density matrix and nuclei
“Fictitious”
mass of P
acceleration of
density matrix, P
d 2P
dt 2

 μ
d 2R
M 2
dt


1/ 2
 E
 1/ 2
 P  P    μ

 P R


E

R
Force on P
P
Classical dynamics in {{R,V,M},{P,W,m}} phase space
Next few slide: Forces, propagation equations, formal error
analysis
Iyengar Group, Indiana University
Nuclear Forces: What Really makes it work
Hellman-Feynman
contributions
E(R i , Pi )

R
P
 dh ~ 1 dG ~  E xc VNN
Tr 
P 
P 

2 dR  R
R
 dR
 
~
 Tr  P, F

Pulay’s moving
basis terms
 ~ dS ~ 
 Tr FP
P 
dR 

 ~ dU 1 ~ T dU T 
 Q

U - PU
dR 
 dR
Contributions due to [F,P]  0. Part of
non-Hellman-Feynman
S=UTU,
Cholesky or
Löwdin
Iyengar Group, Indiana University
Density Matrix Forces:
McWeeny Purified DM (3P2-2P3) in
energy expression to obtain
 Use
E(R i , Pi )
 3FP  3PF  2FP2  2PFP  2P 2 F
P
R
Iyengar Group, Indiana University
m effects an adjustment of time-scales:
Direction of Increasing Frequency
of m : P changes slower with time:
characteristic frequency adjusted
 But Careful - too large m: non-physical
 Appropriate m: approximate BO dynamics
 Consequence
Bounds for m: From a Hamiltonian formalism
 m: also related to deviations from the BO surface

Iyengar Group, Indiana University
“Physical” interpretation of m :
Bounds
 Commutator of
the electronic Hamiltonian
and density matrix: bounded by magnitude of
m
F, P F
 Magnitude

1
1/4
1/4 2

Tr μ Wμ 
P, W F

of m : represents deviation from
BO surface
 m acts as an “adiabatic control parameter”
Reference… Iyengar et al. Israel J. Chem. 7, 191, (2002).
Iyengar Group, Indiana University
Bounds on the magnitude of m
 The
Lagrangian



1
1
T
L  Tr V MV  Tr μ1/4Wμ1/4
2
2

:
  E(R, P)  Tr ΛP
2
2
P

The Conjugate Hamiltonian (Legendre Transform)



1
1
T
H  Tr V MV  Tr μ1/4Wμ1/4
2
2
H  H real  H fict

  E(R, P)  Tr ΛP
2
2
P

dH real
dH fict

1/2 dW 1/2 
 Tr  Wμ
μ 
dt
dt
dt


Controlling m: Deviations from BO surface and adiabaticity
Reference… Iyengar et al. JCP. 115,10291 (2001).
Iyengar Group, Indiana University
Comparison with BO dynamics
 Born-Oppenheimer
dynamics:
• Converged electronic
 ADMP:
• Electronic state
states.
• Approx. 8-12 SCF
•
cycles / nuclear config.
dE/dR not same in
both methods
•
•
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 P: time-reversible propagation

Velocity Verlet propagation of P
 1/ 2
t 2 1/ 2  E(R i , Pi )
Pi 1  Pi  Wi t μ 
  i Pi  Pi  i   i  μ
2
 Pi

R

Propagation of W
 1/ 2
t 1/ 2  E(R i , Pi )
Wi 1/2  Wi - μ 
  i Pi  Pi  i   i  μ
2
 Pi

R
 1/ 2
t 1/ 2  E(R i 1 , Pi 1 )
Wi 1  Wi 1/2 - μ 
  i 1Pi 1  Pi 1 i 1   i 1  μ
2
Pi 1
R


 Classical dynamics in {{R,V},{P,W}} phase space

i and i+1 obtained iteratively:
– Conditions: Pi+1 2 = Pi+1 and WiPi + PiWi = Wi (next two slides)
Iyengar Group, Indiana University
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 Pi+1
Given Pi2 = Pi, need to find indempotent Pi+1
 Guess:
t 2 1/ 2  E(R i , Pi )  1/ 2
*

Pi 1  Pi  Wi t -
2
μ


Pi
μ
R


Or guess: Pi1*  Pi  2Wi t - Wi-1/2 t
 Iterate Pi+1 to satisfy Pi+12 = Pi+1
Pi 1  Pi 1  μ 1/ 2 Pi TPi  Qi TQ i  μ 1/ 2
~
*
T  μ1/ 2 Pi 1  Pi 1 μ1/ 2
*



Rational for choice PiTPi + QiTQi above:
i Pi  Pi i  i  Pi i Pi  Qi i Qi
Iyengar Group, Indiana University
Idempotency: To obtain Wi+1

Given WiPi + PiWi = Wi, find appropriate Wi+1
 Guess:
 E(R , P ) 
t
Wi 1  Wi 1/2 *
μ 1/ 2 
2

i 1
i 1
Pi 1

1/ 2
μ

R


~
~
*
Wi 1  Wi 1  μ 1/ 2 Pi 1TPi 1  Qi 1TQi 1 μ 1/ 2


~ 1/ 2 ~
*
T  μ Wi 1  Wi 1 μ1/ 2

Iterate Wi+1 to satisfy Wi+1Pi+1 + Pi+1Wi+1 = Wi+1
Iyengar Group, Indiana University
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 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
References…
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).
Protonated Water Clusters: Iyengar Group, Indiana University
Hopping via the Grotthuss mechanism
True for 20,
30, 40, 50
and larger
clusters…
Iyengar Group, Indiana University
(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
Iyengar Group, Indiana University
(H2O)20H3O+: A recent spectroscopic quandry
Theory
Experiment
J.-W. Shin, N. I. Hammer, E. G. Diken et al.,
Science 304, 1137 2004.
Iyengar Group, Indiana University
Spectroscopy: A recent quandry
Water Clusters: Important in Atmospheric Chemistry
Bottom-right spectrum
From ADMP agrees
well with expt:
dynamical effects in IR
spectroscopy
Explains the experiments of M. A. Johnson
Iyengar Group, Indiana University
Spectroscopy: A recent quandry
ADMP Spectrum!!
Iyengar et al. JCP, 123 ,
084309 (2005)
Iyengar Group, Indiana University
(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
Iyengar Group, Indiana University
Larger Clusters and
water/vacuum interfaces: Similar results
Iyengar Group, Indiana University
Predicting 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 University
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 ?
H2O coordination 4
H3O+ coordination =3
Iyengar Group, Indiana University
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)
Iyengar Group, Indiana University
QM/MM treatment: ONIOM ADMP
EE
MM
full
E
QM
I
E
MM
I
Unified treatment of the full
system within ADMP
I
(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
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**
Iyengar Group, Indiana University
Glyoxal
3-body Synchronous photo-fragmentation
Iyengar Group, Indiana University
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
method
bear out the accuracy of the
Iyengar Group, Indiana University
Acknowledgment

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)