10.7.1 Introduction

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 infrared and Raman activities. ⁵⁷⁸ Johnson B. G., Florián J.
Chem. Phys. Lett.
(1995), 247, pp. 120. Link Vibrational frequencies are calculated by either using an analytic Hessian (if available; see Table 9.2) 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. The performance of various ab initio theories in determining vibrational frequencies has been well documented; see Refs.  867 Murray C. W. et al.
Chem. Phys. Lett.
(1992), 199, pp. 551. Link , 1102 Scott A. P., Radom L.
J. Phys. Chem.
(1996), 100, pp. 16502. Link , 581 Johnson B. G., Gill P. M. W., Pople J. A.
J. Chem. Phys.
(1993), 98, pp. 5612. Link .

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 $𝒪(N^{2}M)$ 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. ⁶⁴¹ Korambath P. P. et al.
Mol. Phys.
(2002), 100, pp. 1755. Link 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 $𝒪(N^{2}M/N_{\mathrm{seg}})$, where $N_{\mathrm{seg}}$ is the number of segments. This option is invoked automatically by the program.

After a vibrational analysis, Q-Chem computes useful statistical thermodynamic properties at standard temperature and pressure following the rigid-rotor-harmonic-oscillator (RRHO) approach. These include the zero-point vibration energy (ZPVE) and, translational, rotational and vibrational, entropies and enthalpies. Note: in the Q-Chem output the “total enthalpy” actually means the total enthalpy correction to the internal energy. One must add this “total enthalpy” to the internal energy to obtain the total enthalpy in common sense. In addition to these thermal corrections, Q-Chem also prints the interpolated vibrational entropy and enthalpy according to the quasi-RRHO (qRRHO) approach by Head-Gordon and co-workers ⁷⁴³ Li Y.-P. et al.
J. Phys. Chem. C
(2015), 119, pp. 1840. Link which extends Grimme’s previous scheme ⁴³⁹ Grimme S.
Chem. Eur. J
(2012), 18, pp. 9955. Link to address low-frequency vibrations (see Section 10.7.2).

In order to carry out a frequency analysis users must at a minimum provide a molecule within the $molecule keyword and define an appropriate level of theory within the $rem keyword using the $rem variables EXCHANGE, CORRELATION (if required) (Chapter 4) and BASIS (Chapter 8). Since the default type of job (JOBTYPE) is a single point energy (SP) calculation, the JOBTYPE $rem variable must be set to FREQ.

It is very important to note that a vibrational frequency analysis must be performed at a stationary point on the potential surface that has been optimized at the same level of theory. Therefore a vibrational frequency analysis most naturally follows a geometry optimization in the same input deck, where the molecular geometry is obtained (see examples).

Users should also be aware that the quality of the quadrature grid used in DFT calculations is more important when calculating second derivatives. The default grid for some atoms has changed in Q-Chem 3.0 (see Section 5.5) and for this reason vibrational frequencies may vary slightly form previous versions. It is recommended that a grid larger than the default grid is used when performing frequency calculations.

The standard output from a frequency analysis includes the following.

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  Vibrational frequencies.

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  Raman and IR activities and intensities (requires $rem DORAMAN).

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  Atomic masses.

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  Zero-point vibrational energy.

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  Translational, rotational, and vibrational, entropies and enthalpies.

Several other $rem variables are available that control the vibrational frequency analysis. In detail, they are: