# 11.10 Harmonic Vibrational Analysis

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 Raman428 and infrared activities. Vibrational frequencies are calculated by either using an analytic Hessian (if available; see Table 10.1) 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 ${\cal{O}}({N^{2}M})$ 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.487 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 ${\cal{O}}({N^{2}M/N_{\mathrm{seg}}})$, 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. 651, 819, 431.