Quantum linear response theory 551As a specific example of an infrared spectrum, we show, in Fig. 14.5(a), computed
IR spectra fromab initiomolecular dynamics calculations of pure D 2 O (Lee and Tuck-
erman, 2007), with a comparison to experiment (Bertieet al., 1989; Zelsmann, 1995).
In Fig. 14.5(b), we show computed IR spectra for 1 M and a 13 M aqueous KOD
solutions (Zhu and Tuckerman, 2002). In anab initiomolecular dynamics calculation,
a molecular dynamics trajectory is generated with forces computed from electronic
structure calculations performed “on the fly” as the simulation proceeds. Theab ini-
tiomolecular dynamics technique allows chemical bond-breaking and -forming events
(which occur frequently in KOD solutions as protons are transfered from water to
OD−ions) to be treated implicitly in an unbiased manner. In each of these spectra,
the quantum dipole correlation function is replaced by its classical counterpart using
the harmonic approximation of eqn. (14.4.10). The simulation protocol employed in
Fig. 14.5(a) leads to a small red shift in the OD vibrational band compared to ex-
periment; however, the agreement is generally reasonable. The spectra in Fig. 14.5(b)
show how a strongly red-shifted OD vibrational at 1950 cm−^1 band diminishes with
concentration and disappears in the pure D 2 O case. This band can be assigned to wa-
ter molecules in the first solvation shell of an OD−ion that donate a hydrogen bond
to the OD−oxygen, forming a relatively strong hydrogen bond. The stretch mode of
1000 1500 2000 2500 3000
n (cm-1)a(n)n(n) (arb. units)Theory
Expt.1000 1500 2000 2500 3000
n ( cm-1)a(n)n(n) (arb. units)13 M
1 M(a) (b)Fig. 14.5Computed (solid line) and experimental (dashed line) IR spectra for pure D 2 O
(a) and KOD solutions of 1 M (solid line) and 13 M (dashed line)concentrations.
these OD groups pointing directly to the hydroxyl oxygen are strongly red-shifted. As
the concentration is decreased, it is expected that this band, in particular, will exhibit
diminished intensity in the infrared spectrum. At high concentration, the shoulder
at≈2400 cm−^1 is due to the OD−stretch which is only free for a fraction of the
trajectory and hence is a weak signal in the IR spectrum.
14.5 Quantum linear response theory
In this section, we will show that energy spectrum can be derived directly from the
ensemble density matrix and the quantum Liouville equation without explicit reference