Short Time Calculations of Rate Constants for Reactions With Long-lived

IntermediatesMaytal 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. Peskin, Chem. Phys. Lett. 369 (2003) 232.

M. Caspary, L. Berman, U. Peskin, Isr. J. Chem. 42 (2002) 237.

Defining the problemConsider a case where there is an intermediate state in a reaction that is associated with a long-lived resonance state.

Example:1 1

( ) ( )0 2 2cosh ( ) cosh (5 )V s V

s s

)(2

ˆ2

22

sVsm

H

W.H. Miller et. al introduced a powerful expression for the rate constant calculation:

1,

0

( ) ( ) ( )R i iK T Q T C t dt

ˆ ˆ ˆ ˆ/ 2 / 2 / /ˆ ˆ( ) [ ],

H H itH itHC t tr e F e e F ei i i i

The practical form is: 1

, , ,

0

ˆ( ) ( ) ( ) | | ( )R n i n i i n in

K T Q T t F t dt

ˆ ˆ/ 2 / 2

, , ,ˆ | (0) | (0)H Hi n i n i n ie Fe

The Problem: Long computational time

0

,1 )()()( dttCTQTK RRR

0

,1 )()()( dttCTQTK PRR

)(|ˆ|)()( ,,,, tFttC RnRn

RnRnRR

)(|ˆ|)()( ,,,, tFttC RnPn

RnRnPR

The rate can be calculated at any one of the barriers:

The Solution

t

PRt

t

RRtR dttCdttCTQTK0

,

0

, ')'(lim')'(lim)()(

The expressions for the rate constant can be represented as infinite time limits:

The NEW method

t

RRR

t

PRP dttCtdttCttC0

,

0

, ')'()(')'()()(

1)()( tt RP

Defining a time-dependent weighted average of the two integrals:

The rate can be written exactly as:

)(lim)()( 1 tCTQTK tR

t

RRR

t

RR dttCTKTQdttC ')'()()(')'( ,

0

,

t

PRR

t

PR dttCTKTQdttC ')'()()(')'( ,

0

,

t

PRP

t

RRRR dttCtdttCtTQTKtC ')'()(')'()()()()( ,,

After substitution:

Rewriting the time integrals

t

PR

t

RR

R

P

dttC

dttC

t

t

')'(

')'(

)(

)(

,

,

If the asymptotic limit is

obtained at a finite time )()()( 1 tCTQTK R

Lets assume that at the dynamics is dominated by the decay of the resonance:

)(||)(| 0,

/))(2/(,

0

0tet Rn

ttiEittRn

/)(0,

/)(0,0,,,

00 )()(|ˆ|)()( ttRR

ttRnR

nRnRnRR etCetFttC

/)(0,

/)(0,0,,,

00 )()(|ˆ|)()( ttPR

ttRnP

nRnRnPR etCetFttC

0tt

The flux correlation functions decay asymptotically in time and the convergence of their time integrals can be accordingly slow:

In the case of a resonance dominating the dynamics at any

0tt ,

)(

)(

)(

)(

0,

0,

tC

tC

t

t

PR

RR

R

P

0

( ) ( ') ' ( ) ( ') ', , , ,1 0 0( )( ) ( ) ( ) ( ) ( ), , , ,

t tC t C t dt C t C t dtR R R P R P R R

K TQ T C t C t C t C tR P R R R P R R

t t

Result:The Flux Averaging Method

A “working equation” for the rate constant which is formally exact:

Numerical Examples:One-dimensional symmetric potential barriersThe 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

))]tanh())[tanh((cosh(),,(

;),,(),,()(

2220

1122

a

axeVasv

asassV

017.00 V 2.01 a 8.02 a 03.01 05.02

The contribution of each correlation function to the weighted average is non symmetric.

The method is applicable for the more common asymmetrical case.

Multiple resonance statesThe 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 quasi-bound system.

, ,, ,

, , , ,

1 1( ) ( )( ) lim ( ') ( ')

( ) ( ) ( ) ( ) ( )' 0 ' 0

R R R PR P R Rl

R P R R R P R R

l lC l C lk T C l C l

Q T C l C l C l C ll l

asymmetric potential barriers

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.