550 Quantum time-dependent statistical mechanics
for approximating the quantum position autocorrelation function of an anharmonic
system using the corresponding classical autocorrelation function. The latter can
be obtained directly from a molecular dynamics calculation. This approximation is
known as theharmonic approximation, within which the quantum-mechanical pref-
actor (β ̄hω 0 /2)tanh(β ̄hω 0 /2) serves to capture at least some of the true quantum
character of the system (Bader and Berne, 1994; Skinner and Park, 2001). The utility
of this approximation depends on how well a system can be represented as a collection
of harmonic oscillators.
14.4.2 The infrared spectrum
One of the most commonly used approaches to probe the vibrational energy levels of a
system is infrared spectroscopy, in which electromagnetic radiation of frequency in the
near infrared part of the spectrum (10^12 to 10^14 Hz) is used to induce transitions be-
tween the vibrational levels. By sweeping through this frequency range, the technique
records the frequencies at which the transitions occur and the intensities associated
with each transition.
Infrared spectroscopy makes use of the fact that the total electric dipole moment
operator of a systemμˆcouples to the electric field component of an electromagnetic
wave via
Hˆ 1 (t) =−μˆ·E(t). (14.4.11)
If we orient the coordinate system such thatE(t) = (0, 0 ,E(t)) and recall that the
wavelength of infrared radiation is long compared to a typical samplesize, then
E(t) =E(ω)e−iωt, and the perturbation HamiltonianHˆ 1 (t) is of the form given in
eqn. (14.2.37). For this perturbation, the energy spectrum is given by
Q(ω) =ω
̄h|E(ω)|^2 tanh(β ̄hω/2)∫∞
−∞dte−iωt〈
[ˆμz(0),μˆz(t)]+〉
. (14.4.12)
Since we could have chosen any direction for the electric fieldE(t), we may compute
the spectrum by averaging over the three spatial directions and obtain
Q(ω) =ω
3 ̄h|E(ω)|^2 tanh(β ̄hω/2)∫∞
−∞dte−iωt〈μˆ(t)·μˆ(0) +μˆ(0)·μˆ(t)〉. (14.4.13)What is actually measured in an infrared experiment is the absorptivityα(ω) from
the Beer–Lambert law. The product ofα(ω) with the frequency-dependent index of
refractionn(ω) is directly proportional toQ(ω) in eqn. (14.4.13),Q(ω)∝α(ω)n(ω). If
the quantum dipole-moment autocorrelation function is replaced bya classical auto-
correlation function, withμ(t) =
∑
iqiri(t) the classical dipole moment for a system
ofNchargesq 1 ,...,qN, then the approximation in eqn. (14.4.10) can be employed.
Through the use of the Kramers–Kr ̈onig relations (see eqn. (14.5.20) in Section 14.5),
a straightforward computational procedure can be employed to computen(ω) (Iftimie
and Tuckerman, 2005). The examples of water and ice considered by Iftimie and Tuck-
erman show thatn(ω) has only a weak dependence on frequency so thatα(ω)n(ω) is
a reasonable representation of the experimental observable.