590 Langevin and generalized Langevin equations
will decay as exp(−ζ′( ̃ω)t/μ). This time scale corresponds toT 1 and is given simply
by
1
T 1
=
ζ′( ̃ω)
μ. (15.4.7)
A comparison of eqns. (15.4.7) and (15.4.1) reveals the prediction ofthe classical GLE
approach namely that the vibrational dephasing and energy relaxation times for a
harmonic oscillator coupled to a bath are related by
1
T 2=
1
2 T 1
. (15.4.8)
Eqn. (15.4.8) is true only for purely harmonic systems. However, real bonds always
involve some degree of anharmonicity, which changes the relation betweenT 1 andT 2.
The more general expression of this relation is
1
T 2=
1
2 T 1
+
1
T 2 ∗
, (15.4.9)
whereT 2 ∗is a pure dephasing time. Now suppose we add to the harmonic potential
μω ̃^2 q^2 /2 a small cubic term of the formgq^3 /6 so that the potential of mean forceW(q)
becomes
W(q) =1
2
μω ̃^2 q^2 +1
6
gq^3. (15.4.10)Theoretical treatments of such a cubic anharmonicity have been presented by Oxtoby
(1979), Levineet al.(1988), Tuckerman and Berne (1993), and Bader and Berne
(1994), all of which lead to explicit expressions for the pure dephasing time. We note,
however, that only direct solution of the full GLE, albeit an approximate one, yields
1 /T 2 in the form of eqn. (15.4.9), as we will now show.
The GLE corresponding to the potential in eqn. (15.4.10) reads
q ̈=−ω ̃^2 q−g
2 μ
q^2 −∫t0dτq ̇(τ)γ(t−τ) +f(t). (15.4.11)As long as the excursions ofqin the cubic potential do not stray too far from the
neighborhood ofq= 0, the motion ofqremains bound between definite turning points.
However, as the energy of the oscillator fluctuations, the time required to move between
the turning points varies. In other words, the period of the motion, and hence the
frequency, varies as a function of the energy. Therefore, we seek a perturbative solution
of eqn. (15.4.11), in which the anharmonicity is treated as an effect that causes the
vibrational frequency to fluctuate in time. Eqn. (15.4.11) is then replaced, to lowest
order in perturbation theory, by an equation of the form
q ̈=−ω^2 (t)q−∫t0dτq ̇(τ)γ(t−τ) +f(t), (15.4.12)whereω(t) = ̃ω+δω(t) andδω(t) =gf(t)/ 2 μω ̃^3. By studying the autocorrelation
functionCqq(t) within perturbation theory, it was shown (Tuckerman and Berne,1993)