Molecular dynamics 591thatCqq(t) is an oscillatory function with an exponential decay envelope. Thus, the
general approximate form ofCqq(t) is
Cqq(±)(t) =C(qq±,0)(t)〈
ei∫t
0 dτδω(τ)〉
≈Cqq(±,0)(t)e−∫t
0 dτ(t−τ)〈δω(0)δω(τ)〉, (15.4.13)whereC(qq±,0)(t) are purely harmonic autocorrelation functions similar to those in eqn.
(15.3.34). The exponential decay term in eqn. (15.4.13) is the resultof a cumulant
expansion applied to〈exp(i
∫t
0 dτδω(τ)〉(see eqn. (4.7.21)). Combining the decay of
Cqq(±,0)(t) with the long-time behavior of the integral in eqn. (15.4.13) leads tothe
vibrational dephasing time
1
T 2=
ζ′( ̃ω)
2 μ+
g^2 kT
4 μ^3 ̃ω^6̃γ(0), (15.4.14)where ̃γ(0) is the Laplace transform ofγ(t) ats= 0, which is also the static friction
coefficient. The second term in eqn. (15.4.14) is a consequence of the anharmonicity.
Since 1/T 2 ∗is a pure dephasing time, 1/T 1 is stillζ′( ̃ω)/μto the same order in pertur-
bation theory. Hence, eqn. (15.4.14) implies that 1/T 2 ≥ 1 / 2 T 1 , where equality holds
forg= 0. This inequality betweenT 1 andT 2 is usually true for anharmonic systems.
An analysis by Skinner and coworkers using a higher order in perturbation theory
suggested possible violations of this inequality under special circumstances (Budimir
and Skinner, 1987; Laird and Skinner, 1991). Further analysis of such violations and
potential difficulties with their detection were subsequently discussed by Reichman
and Silbey (1996).
15.5 Molecular dynamics with the Langevin equation
Because the Langevin and generalized Langevin equations replace alarge number of
bath degrees of freedom with the much simpler memory integral andrandom force
terms, simulations based on these equations are convenient and often very useful.
They have a much lower computational overhead than a full bath calculation and
can, therefore, access much longer time scales. The Langevin equation can also be em-
ployed as a simple and efficient thermostatting method for generating the canonical
distribution, which is one of the most common applications. The generalized Langevin
equation can also be used as a thermostatting method; however, the need to input a
dynamic friction kernelζ(t) renders the GLE less convenient for this purpose. Under
certain conditions, a friction kernel can be generated from a molecular dynamics sim-
ulation (Straubet al., 1988; Berneet al., 1990), but some subtleties arise, which we
will discuss in Section 15.7. When such a friction kernel is available, theGLE can,
to a good approximation, yield the same dynamical properties as thefull molecular
dynamics calculation. Because of this important property, the GLEhas been employed
in the development of low-dimensional or “coarse-grained” models derived from fully
atomistic potential functions. Employing the GLE helps to ensure that the coarse-
grained model can more faithfully reproduce the dynamics of the more detailed model
from which it is obtained (Izvekov and Voth, 2006).