Ab initio path integrals and applications of AIMD to ... · New York University, 100 Washington Sq....
Transcript of Ab initio path integrals and applications of AIMD to ... · New York University, 100 Washington Sq....
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).