Vibrational dephasing 589to a bath with memory, it was shown that, when the frequency of the oscillator is
high compared to the spectral density of the bath, the vibrational relaxation timeT 2
satisfies
1
T 2
=
ζ′( ̃ω)
2 μ. (15.4.1)
T 2 is a measure of the decay time of the velocity and position autocorrelation functions.
In addition toT 2 , there is another relevant time scale, denotedT 1 , which measures
the rate of energy relaxation of the system. In this section, we willshow how the GLE
can be used to develop classical relations betweenT 1 andT 2 for both harmonic and
anharmonic oscillators coupled to a bath. The timesT 1 andT 2 are generally measured
experimentally using nuclear-magnetic resonance techniques and,therefore, relate to
quantum processes. However, we will see that the GLE can nevertheless provide useful
insights into the physical nature of these two time scales.
Using the solutions of the GLE, it is also possible to show that the cross-correlation
functions
Cvq(t) =〈v(0)q(t)〉
Cqv(t) =〈q(0)v(t)〉 (15.4.2)have the same decay time. For this discussion, we will find it convenient to introduce
a change of nomenclature and work with normalized correlation functions:
Cab(t) =〈a(0)b(t)〉
〈a^2 〉. (15.4.3)
In terms of the four normalized correlations functions,Cqq(t),Cvv(t),Cqv(t), and
Cvq(t), the solutions of eqn. (15.3.21) can be expressed as
q(t) =q(0)Cqq(t) + ̇q(0)Cvq(t) +∫t0dτ f(t−τ)Cvq(τ)q ̇(t) = ̇q(0)Cvv(t) +q(0)Cqv(t) +∫t0dτ f(t−τ)Cvv(τ) (15.4.4)(see Problem 15.4). Moreover, if the internal energy of the oscillator
ε(t) =1
2
μq ̇^2 (t) +1
2
μω ̃^2 q^2 (t) (15.4.5)is calculated using the solutions in eqn. (15.4.4), then, as was shown by Tuckerman
and Berne (1993), the autocorrelation function ofε(t) is
Cεε(t) =1
2
Cvv^2 (t) +1
2
Cqq^2 (t) +1
ω ̃^2Cqv^2 (t) (15.4.6)(see Problem 15.4). Since each of the correlation functions appearing in eqn. (15.4.6)
has an exponential decay envelope of the form exp(−ζ′( ̃ω)t/ 2 μ), it follows thatCεε(t)