modest computational effort. This effectively eliminates the first disadvantage if the
identity of the species can be narrowed down to a small number of candidates.
Thus, vibrational spectroscopy supported by quantum chemistry calculations
appears to be increasingly effective structural tool in modern chemistry.
Scaling procedures are now routinely used to improve agreement between the
calculated harmonic frequencies and the observed fundamentals. In this chapter, a
short review of the available scaling methods will be given. This includes the
simplest uniform scaling, wavenumber linear scaling, scaled quantum mechanical
force field, and quite recently proposed effective scaling frequency factor approaches. This is not the authors’ intention to provide a detailed review of the up-to-date
literature. In the subsequent sections, we will briefly summarize the main “theory”
of each method and then describe methodology development from the historical
point of view, citing the methodological references. No papers with applications
only will be cited. In addition, we will also make no citations to quantum chemistry
methods we are referring to, basis sets, etc., to make the reference list as short as
possible. We will use a number of acronyms to keep the text concise. The first time
(apart from this section) a given term appears, the acronym will be given in
parentheses. Then it will be used throughout the chapter in most cases. A list of
acronyms extended by the explanation of mathematical symbols used is provided in
a separate section in the beginning of this chapter for the reader’s convenience.
2.2 Fundamentals
In this section, fundamentals of vibrational spectroscopy will be given. Basic ideas,
including the concept of the potential energy curve and (hyper)surface, harmonic
approximation, anharmonicity treatment, will be reminded. Finally, an outline of
the Wilson–Decius–Cross method of polyatomic molecules vibrations treatment
will be given.
2.2.1 Potential Energy Surface
Within Born–Oppenheimer approximation the potential energy curve (PEC) U(r) of
a diatomic molecule, where r is the internuclear distance, has a Morse-like shape
shown in Fig. 2.1a. It can be approximated using various quantum chemistry
(QC) methods, like the Hartree–Fock (HF) method, post-Hartree–Fock methods
(MCSCF, MP2, CI, CC…), DFT methods, etc. The accuracy of each approximation
is different with respect to the dissociation energy D e , curvature of PEC at minimum, etc. It can be found, say, in the pointwise way by calculating energies for a
number of internuclear distances, which affords the curve as a set of points
r p ; U r p
À Á
À
Á
; p ¼ 1; 2; . . . From the point of view of vibrational spectroscopy, the
52
O. Bąk and P. Borowski
identity of the species can be narrowed down to a small number of candidates.
Thus, vibrational spectroscopy supported by quantum chemistry calculations
appears to be increasingly effective structural tool in modern chemistry.
Scaling procedures are now routinely used to improve agreement between the
calculated harmonic frequencies and the observed fundamentals. In this chapter, a
short review of the available scaling methods will be given. This includes the
simplest uniform scaling, wavenumber linear scaling, scaled quantum mechanical
force field, and quite recently proposed effective scaling frequency factor approaches. This is not the authors’ intention to provide a detailed review of the up-to-date
literature. In the subsequent sections, we will briefly summarize the main “theory”
of each method and then describe methodology development from the historical
point of view, citing the methodological references. No papers with applications
only will be cited. In addition, we will also make no citations to quantum chemistry
methods we are referring to, basis sets, etc., to make the reference list as short as
possible. We will use a number of acronyms to keep the text concise. The first time
(apart from this section) a given term appears, the acronym will be given in
parentheses. Then it will be used throughout the chapter in most cases. A list of
acronyms extended by the explanation of mathematical symbols used is provided in
a separate section in the beginning of this chapter for the reader’s convenience.
2.2 Fundamentals
In this section, fundamentals of vibrational spectroscopy will be given. Basic ideas,
including the concept of the potential energy curve and (hyper)surface, harmonic
approximation, anharmonicity treatment, will be reminded. Finally, an outline of
the Wilson–Decius–Cross method of polyatomic molecules vibrations treatment
will be given.
2.2.1 Potential Energy Surface
Within Born–Oppenheimer approximation the potential energy curve (PEC) U(r) of
a diatomic molecule, where r is the internuclear distance, has a Morse-like shape
shown in Fig. 2.1a. It can be approximated using various quantum chemistry
(QC) methods, like the Hartree–Fock (HF) method, post-Hartree–Fock methods
(MCSCF, MP2, CI, CC…), DFT methods, etc. The accuracy of each approximation
is different with respect to the dissociation energy D e , curvature of PEC at minimum, etc. It can be found, say, in the pointwise way by calculating energies for a
number of internuclear distances, which affords the curve as a set of points
r p ; U r p
À Á
À
Á
; p ¼ 1; 2; . . . From the point of view of vibrational spectroscopy, the
52
O. Bąk and P. Borowski