Computing vibrational spectra beyond the harmonic approximation has become an
active area of research owing to the improved efficiency of computer
techniques.
^{
757
}
J. Chem. Phys.
(2000),
112,
pp. 248.
Link
^{,}
^{
138
}
Spectrochim. Acta A
(2003),
59,
pp. 1881.
Link
^{,}
^{
1215
}
J. Chem. Phys.
(2004),
121,
pp. 1383.
Link
^{,}
^{
62
}
J. Chem. Phys.
(2005),
122,
pp. 014108.
Link
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.
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:
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.
Investigate the effect on the number of configurations employed in a variational calculation.
Avoid using variational theory (due to its expensive computational cost) by using other approximations, for example, perturbation theory.
Obtain the PES indirectly by applying a self-consistent field procedure.
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.
^{
28
}
Phys. Rev.
(1933),
43,
pp. 716.
Link
^{,}
^{
1188
}
J. Chem. Phys.
(1936),
4,
pp. 260.
Link
^{,}
^{
796
}
Phys. Rev.
(1941),
60,
pp. 794.
Link
^{,}
^{
791
}
J. Chem. Phys.
(2003),
118,
pp. 7215.
Link
^{,}
^{
62
}
J. Chem. Phys.
(2005),
122,
pp. 014108.
Link
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.
^{
1179
}
J. Mol. Spect.
(1975),
55,
pp. 356.
Link
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,
^{
671
}
Theor. Chem. Acc.
(2008),
120,
pp. 23.
Link
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.