k

ful¬lling the rate equation

dcR

= ’kcR + k cP (5.57)

dt

on a coarse-grained time scale. In terms of simulation results, the rate

constant k can be found by observing the ensemble-averaged change in pop-

ulation ”ρ11 of the R-state, starting at ρ11 (0) = 1, over a time ”t that

is large with respect to detailed ¬‚uctuations but small with respect to the

inverse of k:

”ρ11

k=’ . (5.58)

”t

In terms of the variable z, starting with z = 1, the rate constant is expressed

as

”z

k=’ . (5.59)

2”t

For the two-level system there are analytical solutions to the response to

stochastic perturbations in certain simpli¬ed cases. Such analytical solutions

can give insight into the ongoing processes, but in simulations there is no

need to approximate the description of the processes in order to allow for

analytical solutions. The full wave function evolution or “ preferably “ the

density matrix evolution (5.43) can be followed on the ¬‚y during a dynamical

simulation. This applies also to the multilevel system (next section), for

which analytical solutions do not exist. We now give an example of an

analytical solution.

The perturbation ω(t) is a stochastic vector, i.e., its components are ¬‚uc-

tuating functions of time. Analytical solutions can only be obtained when

the ¬‚uctuations of the perturbations decay fast with respect to the change

of r. This is the limit considered by Borgis et al. (1989) and by Borgis

and Hynes (1991) to arrive at an expression for the proton transfer rate

in a double-well potential; it is also the Red¬eld limit in the treatment of

126 Dynamics of mixed quantum/classical systems

relaxation in spin systems (Red¬eld, 1965). Now consider the case that the

o¬-diagonal perturbation is real, with

2C(t)

ωx = (5.60)

ωy = 0. (5.61)

We also de¬ne the diagonal perturbation in terms of its integral over time,

which is a ¬‚uctuating phase:

t

def

φ(t) = ωz (t ) dt . (5.62)

0

We start at t = 0 with x, y = 0 and z = 1. Since we consider a time interval

in which z is nearly constant, we obtain from (5.55) the following equations

to ¬rst order in t by approximating z = 1:

x = yωz ,

™ (5.63)

y = ωx ’ xωz ,

™ (5.64)

z = ’yωx .

™ (5.65)

¿From the ¬rst two equations a solution for y is obtained by substituting

g(t) = (x + iy)eiφ , g(0) = 0, (5.66)

which yields

g = iωx eiφ ,

™ (5.67)

with solution

t

ωx (t )eiφ(t ) dt .

g(t) = i (5.68)

0

¿From (5.66) and (5.68) y(t) is recovered as

t

ωx (t ) cos[φ(t ) ’ φ(t)] dt .

y(t) = (5.69)

0

Finally, the rate constant is given by

1 1

k=’ z = yωx ,

™ (5.70)

2 2

which, with „ = t ’ t and extending the integration limit to ∞ because t is

much longer than the decay time of the correlation functions, leads to the

following expression:

∞

1

ωx (t)ωx (t ’ „ ) cos[φ(t) ’ φ(t ’ „ )] d„.

k= (5.71)

2 0

5.2 Quantum dynamics in a non-stationary potential 127

After inserting (5.60), this expression is equivalent to the one used by Borgis

et al. (1989):

∞ t

2

d„ C(t)C(t ’ „ ) cos

k= ωz (t ) dt . (5.72)

2

0 t’„

This equation teaches us a few basic principles of perturbation theory. First

consider what happens when ωz is constant or nearly constant, as is the

case when the level splitting is large. Then the cosine term in (5.72) equals

cos ωz „ and (5.72) represents the Fourier transform or spectral density (see

Chapter 12, Eq. (12.72) on page 326) of the correlation function of the

¬‚uctuating coupling term C(t) at the angular frequency ωz , which is the fre-

quency corresponding to the energy di¬erence of the two levels. An example

is the transition rate between ground and excited state resulting from an os-

cillating external electric ¬eld when the system has a nonzero o¬-diagonal

transition dipole moment μ12 (see (2.92) on page 35). This applies to op-

tical absorption and emission, but also to proton transfer in a double-well

potential resulting from electric ¬eld ¬‚uctuations due to solvent dynamics.

Next consider what happens in the case of level crossing. In that case, at

the crossing point, ωz = 0 and the transfer rate is determined by the integral

of the correlation function of C(t), i.e., the zero-frequency component of its

spectral density. However, during the crossing event the diagonal elements

are not identically zero and the transfer rate is determined by the time-

dependence of both the diagonal and o¬-diagonal elements of the Hamilton-

ian, according to (5.72). A simplifying assumption is that the ¬‚uctuation of

the o¬-diagonal coupling term is not correlated with the ¬‚uctuation of the

diagonal splitting term. The transfer rate is then determined by the integral

of the product of two correlation functions fx („ ) and fz („ ):

fx („ ) = C(t)C(t ’ „ ) , (5.73)

t

fz („ ) = cos φ(„ ) , with φ(„ ) = ωz (t ) dt . (5.74)

t’„

For stationary stochastic processes φ is a function of „ only. When ωz (t)

is a memoryless random process, φ(„ ) is a Wiener process (see page 253),

representing a di¬usion along the φ-axis, starting at φ = 0, with di¬usion

constant D:

∞

D= ωz (0)ωz (t) dt. (5.75)

0

128 Dynamics of mixed quantum/classical systems

This di¬usion process leads to a distribution function after a time „ of

φ2

1

p(φ, „ ) = √ exp ’ , (5.76)

4D„

4πD„

which implies an exponentially decaying average cosine function:

∞

p(φ, „ ) cos φ dφ = e’D„ .

cos φ(„ ) = (5.77)

’∞