Atom-centered Density Matrix
Propagation (ADMP): Theory and
Application to protonated water clusters
and water/vacuum interfaces
Srinivasan S. Iyengar
Department of Chemistry,
Indiana University
This presentation is meant to be a quick outline of ADMP. You
should read the related papers to get more complete
understanding
 Brief
outline of ab initio molecular dynamics
 Atom-centered Density Matrix Propagation
(ADMP)
 Results:
• Novel findings for protonated water clusters
• Preliminary results for ion-transport through
biological channels
 Nut-n-bolts
issues
Molecular 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
on 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
(ADMP) : An Extended Lagrangian approach
 Circumvent
Computational Bottleneck of BO
repeated SCF for electronic SE
 electronic structure, not converged, but
propagated
 “Simultaneous” propagation of electronic
structure with nuclei: an adjustment of timescales
 Avoid
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
P : represented using atom-centered gaussian basis sets
Euler-Lagrange equations of motion

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
 Solutions obtained using velocity Verlet integrator
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

“Physical” interpretation of m :
 Commutator of
the electronic Hamiltonian
and density matrix: bounded by magnitude of
m
F, P F

1
1/4
1/4 2

Tr μ Wμ 
P, W F
 Magnitude of

m : represents deviation from
BO surface
 m acts as an “adiabatic control parameter”
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


By controlling m: control deviations from BO
surface and adiabaticity
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
Some 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 m : Greater
adiabatic control
– QM/MM: localized bases:
natural
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.
 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.
Comparison with Car-Parrinello : Slide 0

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
References…
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).
Comparison with Car-Parrinello : Slide 1
Plane-wave CP:
ADMP:
• Computational scaling O(N3)
• Pure functionals (e.g. BLYP)
– Computational scaling O(N)
Hybrid (B3LYP): expensive
Adiabatic control limited :
larger m : D2O for H2O
Properties depend on m §
accurate) : routine
– Smaller m : Greater adiabatic
control: can use H2O
– Properties independent of m #
•
•
– Hybrid functionals (more
References…
§ Scandolo and Tangney, JCP. 116, 14 (2002).
# Schlegel, Iyengar, Li, Millam, Voth, Scuseria, Frisch, JCP, 117, 8694 (2002).
Comparison 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 P: a time-reversible
propagation scheme

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
R
 Pi

 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
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
Idempotency: To obtain Wi+1

Given WiPi + PiWi = Wi, find appropriate Wi+1
 Guess:
*
i 1
W
t 1/ 2  E(R i 1 , Pi 1 )  1/ 2
 Wi 1/2 - μ 
μ
2
Pi 1

R

Wi 1  W
*
i 1
μ
1/ 2



~
~
Pi 1TPi 1  Qi 1TQi 1 μ 1/ 2

~
~
*
T  μ1/ 2 Wi 1  Wi 1 μ1/ 2

Iterate Wi+1 to satisfy Wi+1Pi+1 + Pi+1Wi+1 = Wi+1
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
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
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 …
 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…
Results
 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 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
via the Grotthuss mechanism
True for 20,
30, 40, 50
and larger
clusters…
(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
Larger Clusters and water/vacuum
interfaces: Similar results
Predicting 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
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
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
Experimental results seem to 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)
Protonated Water Cluster: Conceptual
Reasons for “hopping” to surface
Hydrophobic and hydrophillic regions: Directional hydrophobicity
H3O+ has reduced density around
Reduction of entropy of surrounding waters
Is Hydronium hydrophobic ?
H2O coordination 4
H3O+ coordination =3
Protonated Water Clusters: progress
of the proton
Most protonated water closer
to the surface as simulation
progresses
3 ang
Protonated Water Cluster: Radial
Distribution Functions

O*-O Radial Distribution function peaks:
•
•





BLYP : ~2.45 Angstrom and ~2.55 Angstrom
B3LYP : ~2.45
Zundel [H5O2+]: ~2.45
Eigen [H9O4+]: ~2.55
BLYP : Zundel and
Eigen
B3LYP: Zundel
BLYP : proton more
delocalized
Protonated 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 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
ADMP
 Proton transport
through ion-channel
 QM/MM
AIMD
approach to
QM/MM treatment of bio-systems
EE
MM
full
E
QM
I
E
MM
I
I
Unified treatment of the full
system within ADMP
ONIOM: Energy partitioning
EE
MM
II
E
QM
QM
QM
E QM

F

H

H
I
I
I
I, self
MM
E MM

E
I
I, self
QM
I
E
 MM Z
j

  
 j r R
j

MM
I
  MM QM Z Z 
i
j 

  
  
j
i Ri  R j
 

 MM QM Z Z 
i
j
     
 j i R R 
i
j 

Link atom coordinates are expressed in terms of
their neighbors: Link atoms factor out
Preliminary
results:
Side-chain
contributions
to hop:
B3LYP and
BLYP:
qualitatively
different
results
Protonated Water Cluster v/s Protonated
Water Wire
 Cluster:
Proton goes to surface
 Wire: Proton tends to center
 Why?
 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
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**
Glyoxal 3-body Synchronous photofragmentation
What about BSSE?

Due to:
•
•
difference in instantaneous incompleteness in basis set.
Atom centered nature of basis set (not present in planewaves).

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?

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 Cluster

Conservation Properties :
1
1/4
1/4 2
Fictitious KE = Tr μ Wμ 
2
Change in Fict. KE ~ 0.0002% of total Energy


Chloride-Water Clusters:
Red-shifts


Bend: ~ 1600 cm-1, Stretch ~3400 & ~3600 cm-1
Exptal. O-H Red Shift for
Cl- (H2O)1 :
– 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: Cl- (H2O)25
ADMP dynamics oscillates about the BO result.
Protonated 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
Conclusions
 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
method
bear out the accuracy of the