Short Time Calculations of Rate Constants for Reactions
With Long-lived Intermediates
Maytal Caspary, Lihu Berman, Uri Peskin
Department of Chemistry and The Lise Meitner Center for Computational Quantum Chemistry,
Technion – Israel Institute of Technology, Haifa 32000, Israel
M. Caspary, L. Berman, U. P eskin, Chem. P hys. Lett. 369 (2003) 232.
M. Caspary, L. Berman, U. P eskin, Isr. J . Chem. 42 (2002) 237.
Defining the problem
Consider a case where there is an intermediate state in a
reaction that is associated with a long-lived resonance state.
Example:
V ( s) = V0 (
− h2 ∂2
ˆ
H=
+ V ( s)
2m ∂s 2
1
1
)
−
2
2
cosh ( s) cosh (5s)
W.H. Miller et. al introduced a powerful expression for the rate
constant calculation:
∞
K (T ) = QR (T ) −1 ∫ Ci ,i (t )dt
0
ˆ
ˆ
ˆ / h ˆ −itHˆ / h
−
β
−
β
/
2
/
2
H
H
itH
ˆ
Ci,i (t ) = tr[e
Fi e
e
Fi e
]
1
ˆ
[ pˆ s ⋅ δ (s − s0 ) + δ (s − s0 ) ⋅ pˆ s ]
F≡
2m
The practical form is:
∞
K (T ) = QR (T ) −1 ∫ ∑ λn ,i < Ψ n ,i (t ) |Fˆi | Ψ n ,i (t ) > dt
e
− β Hˆ / 2
Fˆi e
0 n
− β Hˆ / 2
| Ψ n ,i (0) >= λn ,i | Ψ n ,i (0) >
The Problem: Long computational time
The Solution
The rate can be calculated at any one of the barriers:
∞
K (T ) = QR (T ) −1 ∫ CR ,R (t )dt
0
CR , R (t ) = ∑ λn ,R < Ψ n ,R (t ) |FˆR | Ψ n ,R (t ) >
n
∞
K (T ) = QR (T ) −1 ∫ CR ,P (t )dt
0
CR ,P (t ) = ∑ λn , R < Ψ n ,R (t ) |FˆP | Ψ n ,R (t ) >
n
∞
K (T ) = QR (T ) −1 ∫ CR ,R (t )dt
0
∞
K (T ) = QR (T ) −1 ∫ CR ,P (t )dt
0
The expressions for the rate constant can be represented as
infinite time limits:
t
K (T ) = QR−1 (T ) ⋅ limt →∞ ∫ CR ,R (t ')dt '
0
t
K (T ) = QR−1 (T ) ⋅ limt →∞ ∫ CR ,P (t ')dt '
0
The NEW method
Step 1 – Defining C(t)
Defining a time-dependent weighted average of the two integrals:
t
t
0
0
C (t ) = ω R (t ) ∫ CR ,R (t ')dt ' + ω P (t ) ∫ CR ,P (t ')dt '
ω P (t ) + ω R (t ) = 1
The rate can be written exactly as:
K (T ) = QR (T ) −1 lim t → ∞ C (t )
t
t
0
0
C (t ) = ω R (t ) ∫ CR ,R (t ')dt ' + ω P (t ) ∫ CR ,P (t ')dt '
Rewriting the time integrals
∞
∫C
R ,R
(t )dt = K (T )QR (T )
0
∞
∫C
R ,P
(t )dt = K (T )QR (T )
0
∞
t
∫C
R ,R
0
t
∫C
t
∞
R ,P
0
After substitution:
(t ')dt ' = QR (T ) K (T ) − ∫ CR ,R (t ')dt '
(t ')dt ' = QR (T ) K (T ) − ∫ CR ,P (t ')dt '
t
∞
∞
t
t
C (t ) = K (T )QR (T ) − ω R (t ) ∫ CR ,R (t ')dt ' − ω P (t ) ∫ CR ,P (t ')dt '
Step 2 – determining the weights
∞
∞
t
t
C (t ) = K (T )QR (T ) − ω R (t ) ∫ CR ,R (t ')dt ' − ω P (t ) ∫ CR ,P (t ')dt '
∞
If
∫C
ω P (t )
t
=−∞
ω R (t )
∫C
R ,R
(t ')dt '
R ,P
(t ')dt '
t
then the asymptotic limit is obtained at a finite time
K (T ) = QR (T ) −1 C (t )
But how to calculate the weights at a finite time
?
Lets assume that at t > t0 the dynamics is dominated by the decay of
an isolated resonance. In this case the dynamics is well approximated
by the following expression:
| Ψn, R (t ) >|t > t 0 ≅ e
− i ( E − iΓ / 2 )( t − t 0 ) / h
| Ψn, R (t0 ) >
The flux correlation functions decay asymptotically in time and the
convergence of their time integrals can be accordingly slow:
CR ,R (t ) = ∑ λn , R < Ψ n ,R (t0 ) |FˆR | Ψ n ,R (t0 ) > e − Γ ( t −t0 ) / h = CR ,R (t0 )e − Γ ( t −t0 ) / h
n
CR ,P (t ) = ∑ λn , R < Ψ n ,R (t0 ) |FˆP | Ψ n ,R (t0 ) > e − Γ ( t −t0 ) / h = CR ,P (t0 )e − Γ ( t −t0 ) / h
n
Before
t0
After
t0
Substituting the result,
CR ,R (t ) = CR ,R (t0 )e − Γ ( t −t0 ) / h
CR ,P (t ) = CR ,P (t0 )e − Γ ( t −t0 ) / h
∞
into the chosen relation
∫C
ω P (t )
t
=−∞
ω R (t )
∫C
t
R ,R
(t ')dt '
R ,P
(t ')dt '
gives in the case of a resonance dominating the dynamics
at any t > t ,
0
(1)
(
(2)
CR ,R (t0 )
ω P (t )
=−
ω R (t )
CR ,P (t0 )
ω P (t ) + ω R (t ) = 1
)
Result:The Flux Averaging Method
A “working equation” for the rate constant which is
formally exact:
t
t


 −CR, R (t ) ∫ CR, P (t ')dt ' CR, P (t ) ∫ CR, R (t ')dt ' 

1 
0
0
K (T ) =
+


Q (T )  CR, P (t ) − CR, R (t )
C R, P ( t ) − C R, R ( t ) 



 t >t
0
Numerical Examples:
One-dimensional symmetric potential barriers
The rate constant for the double barrier potential shown
above was calculated in three different ways:
The new expression
converges to the
asymptotic value
much faster than each
one of the time
integrals whose
convergence is limited
by the resonance
decay time.
One-dimensional asymmetric potential barriers
The method is applicable for the more common asymmetrical case.
V ( s) = ν ( s, a 2 , µ 2 ) − ν ( s, a1 , µ1 )
;
x − µa
) + tanh(µ )]2 )
a
µ1 = 0.03
µ 2 = 0.05
v( s, a, µ ) = V0 (e −2 µ − cosh 2 ( µ )[tanh(
V0 = 0.017
The contribution of
each correlation
function to the
weighted average is
non symmetric.
a1 = 0.2
a 2 = 0.8
Multiple resonance states
The method can be generalized for situations in which a number of
resonance states contribute to the reaction rate, and the decay
process is accompanied by an internal dynamics within the quasibound system.


l −1
l −1
−C R , R (l )
C R , P (l )

k (T ) =
lim
∑ C R, P (l ') +
∑ C R, R (l ') 
l
→∞
C

Q(T )
(l ) − C R , R (l ) l '=0
C R, P (l ) − C R, R (l ) l '=0
 R, P

τ
asymmetric potential barriers
More than one dimension
λ = 0.05
λ =0
The 2D Hamiltonian:
Hˆ ( x, q ) = Hˆ ( x) + Hˆ (q) + Hˆ ( x, q)
2
ˆ
p
x
Hˆ ( x) =
+ Vˆ ( x)
2m
pˆ q2
H (q) =
+ Vˆ (q )
2m
mw
H ( x, q ) = λ ⋅ q ⋅ µ ( x ) ⋅
h
λ = 0.005, T = 500 K


1
1
V ( x) = V0 
−

2
2
 cosh ( x) cosh (5 x) 
V0 = 0.0114a.u.
x2
− 2
µ ( x) = e 2α
α = 0.4a.u.
1
V ( q ) = ⋅ m ⋅ w2 ⋅ q 2
2
m = 1834a.u.
w = 10 ⋅ Vmax ( x) = 0.0895a.u.
N =640 M =3
Basis set: {χ m ( q)ϕ n ( x )}n =1 m =1
Conclusions:
• In this work we propose a new expression for the calculation
of the thermal rate constant, which circumvents the problem
of long time dynamics due to resonance states.
• By averaging (“on the fly”) different time-integrals over
flux-flux correlation functions, a formally exact expression is
obtained, which is shown to converge within the time scale of
the direct dynamics, even when a long-lived resonance state is
populated.
• In addition, a generalized flux averaging method is proposed
for cases where the dynamics involve more than a single
resonance state.
• Numerical examples were given in order to demonstrate the
computational efficiency.