Ab initio path integrals and applications of AIMD to problems of aqueous ion

solvation and transport

Mark E. Tuckerman

Dept. of Chemistry

and Courant Institute of Mathematical Sciences

New York University, 100 Washington Sq. East

New York, NY 10003

Illustration of hydride transfer in dihydrofolate reductaseFrom Agarwal, Billeter, Hammes-Schiffer, J. Phys. Chem. B (2002).

Nuclear quantum effects critical for describing this reaction!

ˆ /( ) (0)iHtt e

ˆ /( ) (0)iHtx t x e

ˆ /( ) ' ' ' (0)iHtx t dx x e x x

ˆ /( , ) ' ' ( ',0)iHtx t dx x e x x

ˆ /( , '; ) 'iHtU x x t x e x

Time-dependent quantum mechanics

Real-time quantum propagator:

ˆ ˆ( ) ( )O t O t

Expectation value:

ˆ ( ) ( )H t i tt

Electron

source

x

1 2( ) ( ) ( )A x A x A x

1

2

2 2 2

1 2 1 2 1 2 1 2( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) cos ( ) ( )P x A x A x A x A x A x A x x x 1 2 3 4( ) ( ) ( ) ( ) ( )A x A x A x A x A x

Heuristic Derivation of the Path Integral

Electron

source

x

Heuristic Derivation of the Path Integral

( ) ( )ii

A x A x

Electron

source

x

Heuristic Derivation of the Path Integral

( ) ( )ii

A x A x

x0

( )x t

0( ; ) [ ( )]A x x A x t0 pathpaths

( ; ) [ ( )]A x x A x t

2ˆ ˆ ˆ ˆ ˆˆ( ) , 02

pH V x T V T V

m

ˆ ˆ ˆ( )Tr H T VZ e dx x e x

ˆ ˆ ˆ ˆ( ) / /limP

T V V P T P

Pe e e

ˆ ˆ/ /lim P

V P T P

PZ dx x e e x

Derivation of the path integral

Hamiltonian:

Trotter Theorem:

ˆ ˆ/ /

1 1 1lim P

V P T P

PZ dx x e e x

1 1 1ˆlim P

PZ dx x x

1 1 1

factors

ˆ ˆ ˆ ˆlim P

P

Z dx x x

1 2 3 1 2 2 3 3 1ˆ ˆ ˆ ˆlim P P P

PZ dx dx dx dx x x x x x x x x

1 11 2 3 1

1

ˆlim P

P

P i i x xPi

Z dx dx dx dx x x

I dx x x

Derivation of the path integral (cont’d)

Coordinate-space completeness relation:

ˆ ˆ/ /

1 1 T P T P

i i i ix e x dp x p p e x

2 /2

1

p mP

i idp x p p x e

2

1( )/ /21

2i iip x x p mPdp e e

2 2

1

1/2

/2

22i imP x xmP

e

ˆ ˆ/ /

1 1ˆ V P T P

i i i ix x x e e x

ˆ ( )//

1iV x PT P

i ix e x e

Derivation of the path integral (cont’d)

Matrix elements of Ω

1 1

/2

2

1 12 21

lim exp ( )2 2

P

PP

P i i iP

ix x

mP mPZ dx dx x x V x

P

2 2

1

1/2

/2 ( )/

1 2ˆ

2i i imP x x V x P

i i

mPx x e e

2

12

1 1

1/2

( )/21 2 3 2

1

lim 2

i ii

P

mPP x xV x P

PP

ix x

mPZ dx dx dx dx e e

Derivation of the path integral (cont’d)

Reassemble:

Discrete path integral for the canonical partition function:

. . .

...

...

.

.

...

Classical Isomorphism

1 2

3P

P-1( )V x

Chandler and Wolyner, J. Chem. Phys. 74, 4078 (1981)

Interaction between two

cyclic polymer chains

“Classical” cyclic polymer

chain in an potential V(x)

Ab initio path integrals

Partition function for N particles on ground-state surface (Path-integral BO approximation):

. ... . . .......

1

2

3

P-1

P

MET, et al. JCP 99, 2796 (1993); Marx and Parrinello, JCP 104, 4077 (1996); MET, et al. JCP 104, 5579 (1996)

3 /2

2

, 1 , 0 1, ,2 21 11

lim exp ,...,2 2

PN P N

NPI II k I k k N k

Pk II

M P M PZ d E

P

R R R R R

Trace condition:, 1 ,1I P I R R

MET and D. Marx, Phys. Rev. Lett. 86, 4946 (2001)

c

Proton transfer in malonaldehydeMET and D. Marx, Phys. Rev. Lett. 86, 4946 (2001)

Classical

Proton transfer in malonaldehydeMET and D. Marx, Phys. Rev. Lett. 86, 4946 (2001)

Quantum

1d 2d

1 2d d Reaction coordinate:

Path integral molecular dynamics

• Path integrals can be evaluated via Monte Carlo or molecular

dynamics.

• Molecular dynamics offers certain advantages in terms of

parallelization since in each step, the entire system is moved.

• Ab initio path integrals (path integrals with potential energies and

forces derived “on the fly” from electronic structure calculations are

considerably more efficient with molecular dynamics.

• Naïve molecular dynamics suffers from severe sampling problems,

so how to we create a molecular dynamics approach that is as

efficient as Monte Carlo?

Time scales in path integral molecular dynamics

Write the partition function as follows:

P

P

1 1

222

11

1 1( )

2 2P

Pi

P i i ii i x x

pm x x V x

m P

H

Naïve choice of Hamiltonian for molecular dynamics

1 1

222

11

1 1exp ( )

2 2P

PP P i

P i i ii i

x x

pZ d pd x m x x V x

m P

Problems with naïve approach

• As the system becomes more quantum, P → ∞ and ωP → ∞.

However, potential is attenuated by a factor of 1/P , and harmonic

term dominates. System will stay close to closed orbits and not

sample configuration space.

• Harmonic term has a spectrum of frequencies. Highest frequency

determines the time step, which means slow, large-scale chain

motions and breathing modes will not be sampled efficiently.

• Need to sample the canonical distribution, so, at the very least, the

system needs to be coupled to a thermostat.

Path integral molecular dynamics

Martyna, Tuckerman, Berne JCP 99, 2796 (1993)

The path integral is just a bunch of integrals, so we can just change variables.

Introduce a linear transformation:

k kl ll

u T x

whose effect is to diagonalize the harmonic nearest-neighbor coupling:

22 2 2

11 1

P P

P k k k M kk k

m x x m u

• Ensure all modes move on same time scale by choosing: 1 k km m m m

• Couple each degree of freedom to a heat bath (Langevin, Nosé-Hoover chains,….)

Transformations

1 11 1

( 1) k

k k

k x xu x u x

k

Staging:

1 0 1

k

km m m

k

Normal Modes:

1

2 ( 1)( 1)exp

P

k ll

i k lx a

P

1 11

( 2)/2 2 2 2 1

1 2 1 2 2

1

Re Im

2 ( 1)0 2 1 cos

P

kk

P P k k k k

k k k k

u a xP

u a u a u a

km m m P

P

Results for harmonic oscillator2

2 21

2 2

15.8 0.03 400

pH m x

m

mP

No transformations

PIMD (staging)

PIMC (staging)

1806:

PEM vs. AAEM fuel cells

(AAEM=Alkali-anion exchange membrane)

From Varcoe and Slade,

Fuel Cells 5, 198 (2005)

Structures of the excess proton in water

Structures of the excess proton in water

H9O4+

H5O2+

H3O+

+ + +

Grotthuss Mechanism (1806)

Vehicle Mechanism

DFT (BLYP) proton diffusion constants

D(H3O+) = 7.2 x 10-9 m2/s complete DVR basis

[Berkelbach, Lee, Tuckerman (in preparation)]

D(H3O+) = 6.7 x 10-9 m2/s

(Exp: Halle and Karlström, JCSFT II 70, 1031

(1983))

Complete DVR basis set:

System specifics: 31 H2O + 1 H3O+ in a 10 Å periodic box

60 ps simulation

DVR grid size = 753

Troullier-Martins pseudopotentials

The Grotthuss mechanism in waterMET, et al,JPC, 99, 5749 (1995); JCP 103, 150 (1995)

D. Marx, MET, J. Hutter, M. Parrinello, Nature 397, 601 (1999).

N. Agmon, Chem. Phys. Lett. 244, 456 (1995)

T. J. F. Day, et al. J. Am. Chem. Soc. 122, 12027 (2000)

Solvent coordinate view:

P. M. Kiefer, J. T. Hynes

J. Phys. Chem. A 108, 11793 (2004)

The Grotthuss mechanism in water

Second solvation shell H-bond breaking followed

by formation of intermediate Zundel complex:

P

Presolvation Concept:

Proton-receiving species must be

“pre-solvated” like the species into

which it will be transformed in the

proton-transfer reaction.

MET, et al ,Nature 417, 925 (2002)

The Grotthuss mechanism in water

Computed transfer timeτ = 1.5 ps

NMR: 1.3 ps

Transfer of proton resulting in “diffusion’’ of

solvation structure:

A. Chandra, MET, D. Marx

Phys. Rev. Lett. 99, 145901 (2007)

Probability distribution functions

Quantum Classical(P=8 Trotter points)

Quantum delocalization of structural defect

D. Marx, MET, J. Hutter and M. Parrinello Nature 397, 601 (1999)

“Proton hole” mechanism of hydroxide mobility

N. Agmon, Chem. Phys. Lett. 319, 247 (2000)

OH-

H+

Spectra of 14 M KOH

IR

Raman

Librovich and Maiorov, Russian J. Phys. Chem. 56, 624 (1982)

Identified in neutron scattering of concentrated NaOH and KOH solutions:

A. K. Soper and coworkers, JCP 120, 10154 (2004); JCP122, 194509 (2005).

Also in other CPMD studies: B. Chen, et al. JPCB 106, 8009 (2002); JACS 124, 8534 (2002).

And in X-ray absorption spectroscopy: C. D. Cappa, et al. J. Phys. Chem. A 111, 4776 (2007)

System specifics:

31 H2O + 1 OH- in 10 Å periodic box

Plane-wave basis, 70 Ry cutoff

Simulation time: 60 ps

BLYP functional, Troullier-Martins PPs

Weak H-bond donated by hydroxide also identified in neutron scattering of concentrated NaOH and KOH solutions:

A. K. Soper and coworkers, JCP 120, 10154 (2004); JCP122, 194509 (2005).

M. Smiechowski and J. Stangret, JPCA 111, 2889 (2007).

T. Megyes, et al. JCP 128, 044501 (2008).

B. Winter, et al. Nature (2008)

Hydronium:

Water:

Hydroxide:

d1 d2 = d1 - d2

> 0.5 Å < 0.1 Å

O*

H’

O*H/O*O

H’O

MET, D. Marx, M. Parrinello

Nature 417, 925 (2002)

(P=8 Trotter points)

90o

105o

Geometry of relevant solvation complexes

H 9O5-

H7O4-

θ

MET, et al. Science 275, 817 (1997)

Classical Quantum

Selected References

1. M. E. Tuckerman, et al. J. Chem. Phys. 103, 150 (1995); J. Phys. Chem. 99, 5749 (1995)

2. N. Agmon, Chem. Phys. Lett. 244, 456 (1995).

3. M. E. Tuckerman, et al. J. Chem. Phys. 99, 2796 (1993)

4. D. Marx and M. Parrinello, Z. Phys. B 95, 143 (1994)

5. D. Marx and M. Parrinello, J. Chem. Phys. 104, 4077 (1996)

6. M. E. Tuckerman, et al. J. Chem. Phys. 104, 5579 (1996)

7. D. Marx, et al. Nature 367, 601 (1999)

8. N. Agmon, Chem. Phys. Lett. 319, 247 (2000).

9. M. E. Tuckerman, et al. Nature 417, 925 (2002)

10. M. E. Tuckerman, et al. Acc. Chem. Res. 39, 151 (2006)

11. A. Chandra, et al. Phys. Rev. Lett. 99, 145901 (2007)

12. A. K. Soper, et al., J. Chem. Phys. 120, 10154 (2004); J. Chem. Phys. 122, 194509 (2005)

13. M. Smiechowski and J. Stangret, J. Phys. Chem. A 111, 2889 (2007).

14. C. D. Cappa, et al. J. Phys. Chem. A 111, 4776 (2007)

15. T. Megyes, et al. J. Chem. Phys. 128, 044501 (2008).

16. E. F. Aziz, et al. Nature, 455 89 (2008).