1. They must be uniquely related to both experimentally derived and calculated
normal modes.
2. Each local mode must be independent of the isotope composition of the rest of the
molecule.
3. Each local mode must possess a corresponding local mode force constant,
frequency, mass, and intensity.
4. The local mode force constant must be independent of the internal coordinates
used for the description of the molecular geometry.
3 Local Vibrational Mode Analysis
As will be shown in this section, the Konkoli–Cremer local vibrational modes fulfill
all of these requirements. In 1998, Konkoli and Cremer [78, 224, 225] derived for
the first time local vibrational modes directly from normal vibrational modes by
solving the mass-decoupled Euler–Lagrange equations, i.e., by solving the local
equivalent of the Wilson equation for vibrational spectroscopy [86]. They developed
the leading parameter principle [224], which states that for any internal, symmetry,
curvilinear, etc. coordinate, a local mode can be defined. This mode is independent
of all other internal coordinates used to describe the geometry of a molecule, which
means that it is also independent of using redundant or nonredundant coordinate sets.
The number of local vibrational modes can be larger than N vib (N: number of atoms,
N vib ¼ 3N À 6 for a nonlinear and 3N À 5 for a linear molecule), and therefore, it is
important to determine these local modes, which are essential for the reproduction of
the normal modes. This can be accomplished via an adiabatic connection scheme
(ACS), which relates local vibrational frequencies to normal vibrational frequencies
by increasing a scaling factor λ from zero (local frequencies) to 1(normal frequencies). For a set of redundant internal coordinates and their associated local modes, all
those local mode frequencies converge to zero for λ ! 1, which do not contribute to
the normal modes, so that a set of meaningful N vib local modes remains [85, 226]. In
this way, a 1:1 relationship between local (adiabatically relaxed) vibrational modes
and normal vibrational modes has been established [85].
Cremer and co-workers developed a method for calculating from a complete set
of N vib measured fundamental frequencies, the corresponding local mode frequencies [76]. In this way, one can distinguish between calculated harmonic local mode
frequencies (force constants) and experimentally based local mode frequencies
(force constants), which differ by anharmonicity effects [227, 228]. Zou and
co-workers [85] proved that the reciprocal of the compliance constant of Decius is
identical with the local force constant of Konkoli and Cremer, so that for the first
time the physical meaning of the compliance constants could be established. They
could also show that the local vibrational modes of Konkoli and Cremer are the only
modes, which uniquely relate to the normal vibrational modes [85]. A local
stretching force constant associated with the bond length q n is related to the second
derivative of the molecular energy with regard to q n , i.e., to the curvature of the
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E. Kraka and M. Freindorf
normal modes.
2. Each local mode must be independent of the isotope composition of the rest of the
molecule.
3. Each local mode must possess a corresponding local mode force constant,
frequency, mass, and intensity.
4. The local mode force constant must be independent of the internal coordinates
used for the description of the molecular geometry.
3 Local Vibrational Mode Analysis
As will be shown in this section, the Konkoli–Cremer local vibrational modes fulfill
all of these requirements. In 1998, Konkoli and Cremer [78, 224, 225] derived for
the first time local vibrational modes directly from normal vibrational modes by
solving the mass-decoupled Euler–Lagrange equations, i.e., by solving the local
equivalent of the Wilson equation for vibrational spectroscopy [86]. They developed
the leading parameter principle [224], which states that for any internal, symmetry,
curvilinear, etc. coordinate, a local mode can be defined. This mode is independent
of all other internal coordinates used to describe the geometry of a molecule, which
means that it is also independent of using redundant or nonredundant coordinate sets.
The number of local vibrational modes can be larger than N vib (N: number of atoms,
N vib ¼ 3N À 6 for a nonlinear and 3N À 5 for a linear molecule), and therefore, it is
important to determine these local modes, which are essential for the reproduction of
the normal modes. This can be accomplished via an adiabatic connection scheme
(ACS), which relates local vibrational frequencies to normal vibrational frequencies
by increasing a scaling factor λ from zero (local frequencies) to 1(normal frequencies). For a set of redundant internal coordinates and their associated local modes, all
those local mode frequencies converge to zero for λ ! 1, which do not contribute to
the normal modes, so that a set of meaningful N vib local modes remains [85, 226]. In
this way, a 1:1 relationship between local (adiabatically relaxed) vibrational modes
and normal vibrational modes has been established [85].
Cremer and co-workers developed a method for calculating from a complete set
of N vib measured fundamental frequencies, the corresponding local mode frequencies [76]. In this way, one can distinguish between calculated harmonic local mode
frequencies (force constants) and experimentally based local mode frequencies
(force constants), which differ by anharmonicity effects [227, 228]. Zou and
co-workers [85] proved that the reciprocal of the compliance constant of Decius is
identical with the local force constant of Konkoli and Cremer, so that for the first
time the physical meaning of the compliance constants could be established. They
could also show that the local vibrational modes of Konkoli and Cremer are the only
modes, which uniquely relate to the normal vibrational modes [85]. A local
stretching force constant associated with the bond length q n is related to the second
derivative of the molecular energy with regard to q n , i.e., to the curvature of the
236
E. Kraka and M. Freindorf