Download Chemical Dynamics

Chemical Dynamics
Telluride
July 2009
J. Tully
Telluride
July 2009
J. Tully
Chemical Dynamics
Time-dependent processes:

i   t   H  t 
t
2 classes:
1. H depends on time
E. Schrödinger
2. H independent of time
a. Initial state: let
i
 (t0 )   j
where

 t   E j  t 
t
 (t )   (t0 )
2
H  j  Ej j
 (t )  exp(iE j t / )  (t0 )
2
time-independent
Telluride
July 2009
J. Tully
Chemical Dynamics
Time-dependent processes:

i   t   H  t 
t
2 classes:
1. H depends on time
E. Schrödinger
2. H independent of time
b. Initial state: let
 (t0 )   c j j
j
where
H j  Ej j
 (t )   c j exp(iE j t / )  j
j
 (t )   | c j | |  j | 
2
2
j
2
*
*


c
c
exp
i
(
E
E
)
t
/




 k j  j k  k j
jk
All Done!
time-dependent
Chemical Dynamics
Objective:
i

  r, R, t   H  r, R    r, R, t 
t
r = electrons
H  

Telluride
July 2009
J. Tully
R = nuclei
2
2 2
2
2
 R  
 r  V  r, R    
 2R  H el  r; R 
2M 
 2M 
i 2 me
H el  r; R   j  r; R   E j  R   j  r; R 
Adiabatic (Born-Oppenheimer)
Potential Energy Surface
Chemical Dynamics
Born-Oppenheimer Approximation:
Max Born
J. Robert Oppenheimer
Telluride
July 2009
J. Tully
Chemical Dynamics
Telluride
July 2009
J. Tully
Born-Oppenheimer Approximation:
  r , R , t    j  r; R   j  R , t 
Substitute into TDSE, multiply from left by  j  r; R  , integrate over r:
*


i  j  R , t    
t


2 2
 R  E j  R    j  R , t 

 2M 

student problem: why is this not quite right?
Max Born
J. Robert Oppenheimer
Chemical Dynamics
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
Quantum Dynamics
Semiclassical Dynamics
aside: tutorial on classical mechanics
The Classical Limit via the Bohm Equations
Classical Molecular Dynamics
Adiabatic “on-the-fly” Dynamics
Car-Parrinello Dynamics
Infrequent Events
aside: transition state theory and re-crossing
Beyond Born Oppenheimer
Ehrenfest Dynamics
Surface Hopping
Dynamics at Metal Surfaces
Mixed Quantum-Classical Nuclear Dynamics
Telluride
July 2009
J. Tully
I. Quantum Dynamics


i  j  R , t    
t

wavefunction for nuclear motion
Telluride
July 2009
J. Tully

2 2
 R  E j  R    j  R , t 

 2M 

Born-Oppenheimer P.E.S
(or diabatic P. E. S.)
1. Zero-point energy
2. Quantized energy levels
3. Tunneling
4. Phase interference
(coherence)
I. Quantum Dynamics


i  j  R , t    
t

ln(Rate)
wavefunction for nuclear motion
Quantum
Arrhenius
(classical)
1/T
Telluride
July 2009
J. Tully

2 2
 R  E j  R    j  R , t 

 2M 

Born-Oppenheimer P.E.S
I. Quantum Dynamics


i  j  R , t    
t

Wave packet methods:
Telluride
July 2009
J. Tully

2 2


E
R
  j  R, t 

R
j
 2M 

Initial wave function: Gaussian wave packet
1/4
 2 
( x, 0)   2  exp(ik0 x) exp[( x  x0 ) 2 / a 2 ]
a 
k0  p0  initial avg momentum
student problem: what is momentum distribution?
I. Quantum Dynamics


i  j  R , t    
t

Wave packet methods:
Telluride
July 2009
J. Tully

2 2


E
R
  j  R, t 

R
j
 2M 

Initial wave function: Gaussian wave packet
1/4
 2 
( x, 0)   2  exp(ik0 x) exp[( x  x0 ) 2 / a 2 ]
a 
k0  p0  initial avg momentum
1/2
| ( x, 0) |2
 2 
  2  exp[2( x  x0 ) 2 / a 2 ]
a 
I. Quantum Dynamics
Telluride
July 2009
J. Tully
 2 d 2



E
  x, t 
i    x, t    
x



2
t
 2 M dx

Wave packet methods: Propagate wave function
i[  x, t       x, t ]

  x, t   
(x,t)
 2 d 2

 

E
x


   x, t 
2
2
M
dx


i
 ( x, t ) 

 2 d 2

  2 M dx 2  E  x     x, t 


crude!
d 2  ( x, t )
( x  x, t )  ( x  x, t )  2( x, t )

dx 2
(x) 2
finite difference approximation ?
Telluride
July 2009
J. Tully
I. Quantum Dynamics
 2 d 2



E
  x, t 
i    x, t    
x



2
t
 2 M dx

Wave packet methods: Propagate wave function: Fourier Transform method
1. Compute F.T.
1
 ( p, t ) 
2
ipx / 

x
t
dx
(
,
)
e

R. Kosloff
2. Inverse F.T.
 2 d 2  ( x, t )
1
p2
 ipx / 

(
,
)
e

p
t
dp
2

2M
dx
2  2 M
student problem: derive Eq. 2
(x,t)
I. Quantum Dynamics


i  j  R , t    
t

Telluride
July 2009
J. Tully

2 2
E
R


 j  R, t 



R
j

 2M 

Angular and velocity distributions
of the reaction F + H2  FH + H
Experiment
Theory
M. Qiu et al., Science 311, 1440 (2006)
I. Quantum Dynamics
Problem: Scaling with Size is Prohibitive: 103N-6
(x,t)
1D
2D
Telluride
July 2009
J. Tully
Quantum Dynamics: Path Integrals
I. The Propagator

i
t
Telluride
July 2009
J. Tully
(H indep of t)
 it 
 (t )  exp   H   (t  0)
  
 H
Rewrite using position representation:
define:
 (r0 )   r0 |  (r ) 
| r0    (r  r0 )
and
 ( x, t )   x | e
it
 H

 dr | r
0
|  (t  0)    dy  x | e
0
it
 H

  dy  x | e
The contribution to the wave function at
x and t from the wavefunction at y and t=0.
 r0 |  1
it
 H

| y  y |  (t  0) 
| y   ( y, t  0)
the propagator
Quantum Dynamics: Path Integrals
II. Evaluating the Propagator
e  [e
A
A/ N N
]
 x|e
it
 H

Telluride
July 2009
J. Tully
|y
N
 it 
 iH 
t  , with t  t / N
exp   H    exp  
k

1
  
 

Aside: functions of operators? functions of matrices?
1 2 1 3
e  1  A  A  A 
2!
3!
1
1
e A B  1  A  B  ( A2  AB  BA  B 2 )  ( A3  A2 B  ABA 
2!
3!
A
Student Problem: evaluate
  / 4  / 4 
sin 



/
4

/
4


Telluride
July 2009
J. Tully
Quantum Dynamics: Path Integrals
II. Evaluating the Propagator
e  [e
A
A/ N N
]
 x|e
it
 H

|y
N
 it 
 iH 
t  , with t  t / N
exp   H    exp  
k

1
  
 

 iH

 iH

exp  
(tk 1  tk )  exp  
(tk  tk 1 ) 
 

 

 iH

 iH

(tk 1  tk )  | xk  xk | exp  
(tk  tk 1 ) 
  dxk exp  
 

 

 iH
 xN | exp  
 

t  | x0  

already looks like a path integral:
 iH 
dx
dx
x
...
|
exp


 1  N 1 k 1 k
   t  | xk 1 
N
x
x0
xN
t
Quantum Dynamics: Path Integrals
Split Operators:
Trotter’s formula
p2
H ( x, p ) 
 V ( x)  T  V
2m
(T and V do not commute)
 i H t 
 iV t 
 iT t 
 iV t 
exp  

exp

exp

exp







 
 

 2 

 2 
 xk | e

iH t

| xk 1    xk | e

iT t

 it

| xk 1  exp  
(Vk  Vk 1 ) 
 2

Telluride
July 2009
J. Tully
Quantum Dynamics: Path Integrals
Telluride
July 2009
J. Tully
Evaluate T-integral analytically:
 xk | 
1
i
dp

p

pxk ) and
|
exp(


2 
 xk | e

iT t

| xk 1  
| xk 1  
1
i
dp

p ' xk 1 ) | p ' 
'exp(


2 
m
i m

exp 
( xk  xk 1 ) 2 
2 it
  2t

Telluride
July 2009
J. Tully
Quantum Dynamics: Path Integrals
Put pieces back together:
 xN | e

iH t

 i N  mN
t

| x0   C  dx1... dxN 1 exp   
( xk  xk 1 ) 2 
(Vk  Vk 1 ) 
2N

  k 1  2t
The Classical Action S is the integral of the Lagrangian:
S

 L dt

t 
 m
 N  mN
2
2





(
)
(
)
S   t 
x
x
V
x
x
Vk 

k
k 1
k
k
k 1
2

N 
 2 t
 k 1  2t
k 1
N
remember Rig’s discussion of the classical action
 (T  V ) dt
Telluride
July 2009
J. Tully
Quantum Dynamics: Path Integrals
Put pieces back together:
 xN | e

iH t

 i N  mN
t

| x0   C  dx1... dxN 1 exp   
( xk  xk 1 ) 2 
(Vk  Vk 1 ) 
2N

  k 1  2t
The Classical Action S is the integral of the Lagrangian:
S

 L dt

 (T  V ) dt
t 
 m
 N  mN
2
2





(
)
(
)
S   t 
x
x
V
x
x
Vk 

k
k 1
k
k
k 1
2

N 
 2 t
 k 1  2t
k 1
N
Feynman Path Integral:
 xN | e

iH t

i

| x0   C  dx1... dxN 1 exp  S ( x0 , x1 ,...xN ) 


Richard Feynman
Telluride
July 2009
J. Tully
Quantum Dynamics: Path Integrals
Feynman Path Integral:
S(x0,x1,…)
 xN | e

iH t

i

| x0   C  dx1... dxN 1 exp  S ( x0 , x1 ,...xN ) 


Far from classical path:
rapidly oscillating phases cancel
Near classical path: phases do not cancel
path coordinates x0, x1, …
least action = stationary phase = classical path
Nancy Makri
Aside: Quantum Statistical Mechanics:
Canonical Partition Function: Q 

allstates n
Eigenstate representation:
Telluride
July 2009
J. Tully
e  En / kT   e   En
n
 n | e   En | n    n | e   H | n 
Q    n | e   H | n   Tr e   H 
Valid for any representation
n
In particular, in position representation:
Q   dx  x | e   H | x 
Compare to path integral evaluation of propagator:
Set
t   i 
and y = x (closed loop)
 x | e  iHt /  | y 
 x | e  iHt /  | x    x | e   H | x 
Evaluate Q via path integral in imaginary time
Aside: Quantum Statistical Mechanics:
 xN | e

iH t

i



x
C
dx
dx
S
x
x
x
| 0
 1... N 1 exp   ( 0 , 1 ,... N )
S 
where
Telluride
July 2009
J. Tully
t 
 mN
2


(
)
x
x
Vk 

k 1
 2t k
N 
k 1
N
now want:
i 
 x | e   H | x   C  dx1... dxN 1 exp  S 
 
where path is specified by x, x1, …xN-1, x
 mN
 
( xk  xk 1 ) 2 
S (t  i  )  i  
Vk   i S R
N 
k 1  2  
N
where SR is real
Necklace of Classical Beads
Telluride
July 2009
J. Tully
Aside: Quantum Statistical Mechanics:
Necklace of Classical Beads
i 
 x | e   H | x   C  dx1... dxN 1 exp  S 
 
where path is specified by x, x1, …xN-1, x
 mN
 
( xk  xk 1 ) 2 
S (t  i  )  i  
Vk   i S R
N 
k 1  2  
N
Q  C  dx
where SR is real
 S R (path) 
exp


 


all loop
paths
potential energy function for a
classical mechanical “ring polymer”
decaying: no phase
cancellation problems
x6
x5
x
x4
x3
x1
x2
Aside: Quantum Statistical Mechanics:
How does a purely classical mechanical formalism introduce
quantum effects like zero-point energy and tunneling?
zero-point energy
tunneling
Telluride
July 2009
J. Tully
Telluride
July 2009
J. Tully
Quantum Dynamics: Path Integrals
CMD: Centroid Variable
Molecular Dynamics
x6
RPMD: Ring Polymer
Molecular Dynamics
x6
x5
x
x4
x5
x
x4
x3
x1
x3
x1
x2
x2
solves scaling problem
but: short times only
Greg Voth
David Manolopoulos
Chemical Dynamics
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
Quantum Dynamics
(Semiclassical Dynamics)
aside: tutorial on classical mechanics
The Classical Limit via the Bohm Equations
Classical Molecular Dynamics
Adiabatic “on-the-fly” Dynamics
Car-Parrinello Dynamics
Infrequent Events
aside: transition state theory and re-crossing
Beyond Born Oppenheimer
Ehrenfest Dynamics
Surface Hopping
Dynamics at Metal Surfaces
Mixed Quantum-Classical Nuclear Dynamics
Telluride
July 2009
J. Tully
II. Semiclassical Dynamics
Telluride
July 2009
J. Tully
Semiclassical Path Integral Approach:
S(x0,x1,…)
 xN | e

iH t

i

| x0   C  dx1... dxN 1 exp  S ( x0 , x1 ,...xN ) 


Far from classical path:
rapidly oscillating phases cancel
Near classical path: phases do not cancel
path coordinates x0, x1, …
least action = stationary phase = classical path
Bill Miller
Telluride
July 2009
J. Tully
II. Semiclassical Dynamics
Frozen Gaussian:
1/4
 2 
( x, 0)   2  exp(ik0 x) exp[( x  x0 ) 2 / a 2 ]
 a 
fixed
Center of Gaussian, x0, follows classical trajectory
Advantages:
1. Simple
2. Includes delocalization of wavepacket
3. Includes quantum interference effects
(add amplitudes, not probabilities)
Rick Heller
II. Semiclassical Dynamics
Frozen Gaussian (E. J. Heller)
1/ 4
 2 
( x, 0)   2  exp(ik0 x) exp[( x  x0 ) 2 / a 2 ]
 a 
Center of Gaussian, x0, follows classical trajectory
Disadvantages:
Telluride
July 2009
J. Tully
Chemical Dynamics
Classical Molecular Dynamics:
Aside: brief tutorial on classical mechanics
Telluride
July 2009
J. Tully
Chemical Dynamics
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
Quantum Dynamics
Semiclassical Dynamics
aside: tutorial on classical mechanics
The Classical Limit via the Bohm Equations
Classical Molecular Dynamics
Adiabatic “on-the-fly” Dynamics
Car-Parrinello Dynamics
Infrequent Events
aside: transition state theory and re-crossing
Beyond Born Oppenheimer
Ehrenfest Dynamics
Surface Hopping
Dynamics at Metal Surfaces
Mixed Quantum-Classical Nuclear Dynamics
Telluride
July 2009
J. Tully
Tutorial on Classical Mechanics
Newton’s equations:
F
 ma  mq 
p
Force is the gradient of the potential:
F
  V (q) q
mq 
p  V (q) q
Newton’s Equations
Telluride
July 2009
J. Tully
Tutorial on Classical Mechanics
Telluride
July 2009
J. Tully
Hamilton’s equations:
Define total energy = Hamiltonian, H:
q  H ( p, q) p
p   H (q, p) q
if
Hamilton’s Equations
H ( p, q )  T ( p )  V ( q ) 
q 
p 2 / 2m  V ( q )
p/m
p   V (q) q
William
Rowan
Hamilton.
Tutorial on Classical Mechanics
Telluride
July 2009
J. Tully
Lagrangian Mechanics:
Define Lagrangian L:
L (q, q )  T (q )  V (q)
d  L 
L

 
dt  q 
q
if
Euler-Lagrange Equations
L (q, q )  T (q )  V (q) 
mq 
1
2
mq 2  V (q)
p  V (q) q
Joseph-Louis Lagrange
Tutorial on Classical Mechanics
Telluride
July 2009
J. Tully
Hamilton-Jacobi Equation:
Define Hamilton’s Principle Function S:
S

 p dq
S
= -H
t
 S 
 q, 
 q 
Hamilton-Jacobi Equations
 S  
  S
 H  q,   

q  t
 q  
p 
V (q)
q
 0
Karl Gustav Jacob Jacobi
Chemical Dynamics
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
Quantum Dynamics
Semiclassical Dynamics
aside: tutorial on classical mechanics
The Classical Limit via the Bohm Equations
Classical Molecular Dynamics
Adiabatic “on-the-fly” Dynamics
Car-Parrinello Dynamics
Infrequent Events
aside: transition state theory and re-crossing
Beyond Born Oppenheimer
Ehrenfest Dynamics
Surface Hopping
Dynamics at Metal Surfaces
Mixed Quantum-Classical Nuclear Dynamics
Telluride
July 2009
J. Tully
III. Classical Limit via Bohm Eqs.


i  j  R , t    
t

 j  R, t  

2 2
 R  E j  R    j  R , t 

 2M 

i
Aj  R , t  exp[ S j (R , t )]

(1)
(2)
Substitute (2) into (1) and separate real and imaginary parts:

Sj

Aj
2
2

1

R A j
2
[ R S j ]  E j  R   
 
Aj
 2M 
 2M 

1
2[ R Aj ]  [ R S j ]  Aj  2R S j 


 2M 
(3)
(4)
Telluride
July 2009
J. Tully
III. Classical Limit via Bohm Eqs.
Telluride
July 2009
J. Tully
Compare Eq. (3) with Hamilton-Jacobi Equation:
S
= -H
t

Sj
 S 
 q, 
 q 
where
S
=
q
p
2
2

1

R A j
2
 
[ R S j ]  E j  R   
Aj
 2M 
 2M 
(3)
the “quantum potential”
  0:

Sj
 

1
[ R S j ]2  E j  R 
2M 
David Bohm
Chemical Dynamics
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
Quantum Dynamics
Semiclassical Dynamics
aside: tutorial on classical mechanics
The Classical Limit via the Bohm Equations
Classical Molecular Dynamics
Adiabatic “on-the-fly” Dynamics
Car-Parrinello Dynamics
Infrequent Events
aside: transition state theory and re-crossing
Beyond Born Oppenheimer
Ehrenfest Dynamics
Surface Hopping
Dynamics at Metal Surfaces
Mixed Quantum-Classical Nuclear Dynamics
Telluride
July 2009
J. Tully
IV. Classical Molecular Dynamics
Basic Assumptions:
1. Born-Oppenheimer Approximation
2. Classical mechanical nuclear motion
Unavoidable Additional Approximations:
1. Approximate potential energy surface
2. Incomplete sampling
3. (often) Extrapolate to longer timescales
4. Too few atoms
5. Continuum solvent and other shortcuts
Telluride
July 2009
J. Tully
IV. Classical Molecular Dynamics
Numerical propagation of equations of motion:
Assume we know x(t) and v(t) at some time t.
How do we find x(t+) and v(t+) at later time t+ 
Taylor’s series:
dx
d 2x 2
d 3x 3
x(t   )  x(t )   
 
  ...
2
3
dt
2dt
3!dt
1
d 3x 3
2
 x(t )  v(t )  a(t ) 
  ...
3
2
3!dt
1
d 3x 3
2
  ...
x(t   )  x(t )  v(t )  a (t ) 
3
2
3!dt
Add 
where
x(t   )  2 x(t )  x(t   )  a(t ) 2  O( 4 )
Ma (t )   E ( x) x
Verlet algorithm
Telluride
July 2009
J. Tully
IV. Classical Molecular Dynamics
Telluride
July 2009
J. Tully
Choice of time step :
The highest frequency (fastest)
Motions are stiff vibrational modes
t
 << 1/
t+
e.g, O-H stretch: 3700 cm-1   = 1x1014 s-1
Test: Energy Conservation
remember Rig: “weak test”
Telluride
July 2009
J. Tully
H + ClF
HF + Cl
Telluride
July 2009
J. Tully
IV. Classical Molecular Dynamics
Inelastic scattering of NO molecules from a gold surface
0.0
2.0
time (ps)
4.0
IV. Classical Molecular Dynamics
Protein in water
do we need all of these
water molecules?
dielectric continuum
approximation
Telluride
July 2009
J. Tully