Vibrational analysis is an extremely important tool for the quantum chemist,
supplying a molecular fingerprint which is invaluable for aiding identification
of molecular species in many experimental studies. Q-Chem includes a
vibrational analysis package that can calculate vibrational frequencies and
their Raman^{Johnson:1995} and infrared activities. Vibrational
frequencies are calculated by either using an analytic Hessian (if available;
see Table 9.1) or, numerical finite difference of the
gradient. The default setting in Q-Chem is to use the highest analytical
derivative order available for the requested theoretical method.

When calculating analytic frequencies at the HF and DFT levels of theory, the
coupled-perturbed SCF equations must be solved. This is the most time-consuming
step in the calculation, and also consumes the most memory. The amount of
memory required is $\mathcal{O}({N}^{2}M)$ where $N$ is the number of basis functions,
and $M$ the number of atoms. This is an order more memory than is required for
the SCF calculation, and is often the limiting consideration when treating
larger systems analytically. Q-Chem incorporates a new approach to this
problem that avoids this memory bottleneck by solving the CPSCF equations in
segments.^{Korambath:2002} Instead of solving for all the perturbations at
once, they are divided into several segments, and the CPSCF is applied for one
segment at a time, resulting in a memory scaling of $\mathcal{O}({N}^{2}M/{N}_{\mathrm{seg}})$,
where ${N}_{\mathrm{seg}}$ is the number of segments. This option is invoked
automatically by the program.

Following a vibrational analysis, Q-Chem computes useful statistical thermodynamic properties at standard temperature and pressure, including: zero-point vibration energy (ZPVE) and, translational, rotational and vibrational, entropies and enthalpies.

The performance of various *ab initio* theories in determining vibrational
frequencies has been well documented; see
Refs. Murray:1992, Scott:1996, Johnson:1993d.