was recognized that the calculated harmonic frequencies significantly overestimate
the observed fundamentals [7]. In the case of the HF procedure, the typical error in
prediction of the vibrational frequencies was found to be up to 15%. Analytic FC
calculations at the HF level were presented in late 1970s [8]. Further development
of efficient algorithms for the correlated QC methods in conjunction with the
outburst of computational facilities made it possible to predict much more accurate
FCs. The calculated frequencies were still too high as compared with fundamentals.
The reasons are analogous to those described in Sect. 2.2.2 for a simple example of
a diatomic molecule:
• the harmonic approximation used in typical applications of the WDC method,
and
• approximations introduced when solving the Schrödinger equation to obtain the
FC matrix (consisting in the incomplete incorporation of the correlation effects
and incompleteness of the basis set used in the calculations). They have the
direct effect on both the quality of the obtained equilibrium geometry and the
curvature of the (approximate) PES with respect to the atomic displacements.
In most cases, harmonic approximation would lead to overestimation of the
observed fundamentals even if exact second energy derivatives at exact equilibrium
geometry were available (which is, of course, impossible). Second item can be
developed as follows. Consider, for example, simple Hartree–Fock approximation.
The approximate wavefunction, i.e., the Slater determinant, is regarded as uncorrelated wavefunction. The lack of bonding–antibonding orbital excitations (which
“remove” a part of bonding charge density from between the nuclei, like in the case
of post-HF methods) leads to too short and, consequently, to strong bonds. The
resultant overestimated FCs lead to overestimated frequencies even in the case of
perfectly harmonic vibrations.
Thus, the calculated frequencies have to be corrected to be applicable in the
interpretation process. The empirical scaling procedures came into being in the
1970s as a consequence of the above-mentioned observations. They constitute an
alternative to the purely QC treatment within variational or perturbation formalisms
analogous to the simple treatment described for diatomic molecules in Sect. 2.2.2.
The formal procedure of the determination of the vibrational spectrum beyond
harmonic approximation is based on solving the Schrödinger equation for the
nuclear motion in an approximate way with at least cubic potential. It is therefore
time-consuming; nowadays, it can be applied to fairly small systems at a reasonably
high computational level in spite of significant progress in computational chemistry.
The review of the relevant literature is beyond the scope of the present chapter; the
reader is referred to the original papers (see, e.g., [9, 10], and references therein). In
such cases, the scaling procedures, in particular multi-parameter ones, will be the
methods of choice for a long time, just as the methods of molecular mechanics are
the methods of choice in the determination of, e.g., the protein structure. The
additional computational time needed to determine the scaled frequencies is negligible as compared with that needed for the determination of the molecular
2 Scaling Procedures in Vibrational Spectroscopy
63
the observed fundamentals [7]. In the case of the HF procedure, the typical error in
prediction of the vibrational frequencies was found to be up to 15%. Analytic FC
calculations at the HF level were presented in late 1970s [8]. Further development
of efficient algorithms for the correlated QC methods in conjunction with the
outburst of computational facilities made it possible to predict much more accurate
FCs. The calculated frequencies were still too high as compared with fundamentals.
The reasons are analogous to those described in Sect. 2.2.2 for a simple example of
a diatomic molecule:
• the harmonic approximation used in typical applications of the WDC method,
and
• approximations introduced when solving the Schrödinger equation to obtain the
FC matrix (consisting in the incomplete incorporation of the correlation effects
and incompleteness of the basis set used in the calculations). They have the
direct effect on both the quality of the obtained equilibrium geometry and the
curvature of the (approximate) PES with respect to the atomic displacements.
In most cases, harmonic approximation would lead to overestimation of the
observed fundamentals even if exact second energy derivatives at exact equilibrium
geometry were available (which is, of course, impossible). Second item can be
developed as follows. Consider, for example, simple Hartree–Fock approximation.
The approximate wavefunction, i.e., the Slater determinant, is regarded as uncorrelated wavefunction. The lack of bonding–antibonding orbital excitations (which
“remove” a part of bonding charge density from between the nuclei, like in the case
of post-HF methods) leads to too short and, consequently, to strong bonds. The
resultant overestimated FCs lead to overestimated frequencies even in the case of
perfectly harmonic vibrations.
Thus, the calculated frequencies have to be corrected to be applicable in the
interpretation process. The empirical scaling procedures came into being in the
1970s as a consequence of the above-mentioned observations. They constitute an
alternative to the purely QC treatment within variational or perturbation formalisms
analogous to the simple treatment described for diatomic molecules in Sect. 2.2.2.
The formal procedure of the determination of the vibrational spectrum beyond
harmonic approximation is based on solving the Schrödinger equation for the
nuclear motion in an approximate way with at least cubic potential. It is therefore
time-consuming; nowadays, it can be applied to fairly small systems at a reasonably
high computational level in spite of significant progress in computational chemistry.
The review of the relevant literature is beyond the scope of the present chapter; the
reader is referred to the original papers (see, e.g., [9, 10], and references therein). In
such cases, the scaling procedures, in particular multi-parameter ones, will be the
methods of choice for a long time, just as the methods of molecular mechanics are
the methods of choice in the determination of, e.g., the protein structure. The
additional computational time needed to determine the scaled frequencies is negligible as compared with that needed for the determination of the molecular
2 Scaling Procedures in Vibrational Spectroscopy
63