1549380323-Statistical Mechanics Theory and Molecular Simulation

Approximations 557

=

i

̄h

[

Tr

(

ρˆ 0 AˆBˆ(t)

)

−Tr

(

ρˆ 0 Bˆ(t)Aˆ

)]

=

i

̄h

〈

AˆBˆ(t)−Bˆ(t)Aˆ

〉

=−

i

̄h

〈

[Bˆ(t),Aˆ]

〉

=−ΦBA(t). (14.5.27)

Thus, the effect of time reversal on a general after-effect function is to reverse the order
of the operators and change the overall sign. IfAˆ=Bˆ, then the after-effect function
only picks up an overall change of sign upon time reversal. When eqn.(14.5.27) is
introduced into eqn. (14.5.24), the energy spectrum becomes

Q(ω) =iω|F(ω)|^2

[

−

∫∞

0

dteiωtφVV(t) +

∫∞

0

dte−iωtΦVV(t)

]

= 2ω|F(ω)|^2

∫∞

0

dtsin(ωt)φVV(t)

= 2ω|F(ω)|^2 Im [χVV(ω)]. (14.5.28)

Thus, the net absorption spectrum is related to the imaginary partof the frequency-
dependent susceptibility. Note that eqns. (14.5.28) and (14.3.22) are equivalent, demon-
strating that the spectrum is expressible in terms either of Hermitian or anti-Hermitian
quantum time correlation functions. This derivation establishes theequivalence be-
tween the wave-function approach, leading to the Fermi Golden rule treatment of
spectra, and the statistical-mechanical approach, which startswith the ensemble and
its density matrix ˆρ(t) and makes no explicit reference to the eigenstates ofHˆ 0. This
is significant, as the former approach is manifestlyeigenstate resolved, meaning that
it explicitly considers the transitions between eigenstates ofHˆ 0 , which is closer to the
experimental view. The latter, which builds directly from the densitymatrix, is closer
in spirit to the path-integral perspective.

14.6 Approximations to quantum time correlation functions

In this section, we will discuss the general problem of calculating quantum time corre-
lation functions for condensed-phase systems. We will first show how to formulate the
correlation function in terms of the eigenstates ofHˆ 0. While the eigenstate formulation
is useful for analyzing the properties of time correlation functions, we have already
alluded, in Section 10.1, to the computational intractability of solvingthe eigenvalue
problem ofHˆ 0 for systems containing more than just a few degrees of freedom.Thus,
we will also express the quantum time correlation function using the path-integral for-
mulation of quantum mechanics from Chapter 12. Although, as we willshow, even the
path-integral representation suffers from severe numerical difficulties, it serves as a use-
ful starting point for the development of computationally manageable approximation
schemes.