10.9.1 Introduction

Computing vibrational spectra beyond the harmonic approximation has become an active area of research owing to the improved efficiency of computer techniques.^{779, 140, 1248, 65} To calculate the exact vibrational spectrum within Born-Oppenheimer approximation, one has to solve the nuclear Schrödinger equation completely using numerical integration techniques, and consider the full configuration interaction of quanta in the vibrational states. This has only been carried out on di- or triatomic system.^{894, 159} The difficulty of this numerical integration arises because solving exact the nuclear Schrödinger equation requires a complete electronic basis set, consideration of all the nuclear vibrational configuration states, and a complete potential energy surface (PES). Simplification of the Nuclear Vibration Theory (NVT) and PES are the doorways to accelerating the anharmonic correction calculations. There are five aspects to simplifying the problem:

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  Expand the potential energy surface using a Taylor series and examine the contribution from higher derivatives. Small contributions can be eliminated, which allows for the efficient calculation of the Hamiltonian.

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  Investigate the effect on the number of configurations employed in a variational calculation.

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  Avoid using variational theory (due to its expensive computational cost) by using other approximations, for example, perturbation theory.

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  Obtain the PES indirectly by applying a self-consistent field procedure.

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  Apply an anharmonic wave function which is more appropriate for describing the distribution of nuclear probability on an anharmonic potential energy surface.

To incorporate these simplifications, new formulae combining information from the Hessian, gradient and energy are used as a default procedure to calculate the cubic and quartic force field of a given potential energy surface.

Here, we also briefly describe various NVT methods. In the early stage of solving the nuclear Schrödinger equation (in the 1930s), second-order Vibrational Perturbation Theory (VPT2) was developed.^{27, 1222, 819, 814, 65} However, problems occur when resonances exist in the spectrum. This becomes more problematic for larger molecules due to the greater chance of accidental degeneracies occurring. To avoid this problem, one can do a direct integration of the secular matrix using Vibrational Configuration Interaction (VCI) theory.¹²¹³ It is the most accurate method and also the least favored due to its computational expense. In Q-Chem 3.0, we introduce a new approach to treating the wave function, transition-optimized shifted Hermite (TOSH) theory,⁶⁹¹ which uses first-order perturbation theory, which avoids the degeneracy problems of VPT2, but which incorporates anharmonic effects into the wave function, thus increasing the accuracy of the predicted anharmonic energies.