that, if integrated over the van der Waals surface of ligand L, can be related to the
TEP and Tolman’s cone angle, as was demonstrated for phosphines and phosphites.
However, this approach turned out to be only reliable for ligands (L) with high
polarizability [214].
Although the TEP is still today one of the most popular measures used by the
experimental and computational chemistry community for quantifying the catalytic
activity of transition metal complexes based on vibrational spectroscopy, criticism
on the TEP has been raised by several authors in the recent literature [139, 177, 192,
205, 206, 215–221], in particular with regard to the validity of Tolman’s original
assumption of uncoupled CO stretching modes. The normal vibrational modes in a
molecule always couple [86]. There are only a few examples for uncoupled,
non-delocalized, i.e., local, vibrational modes. The bending vibration of the water
molecule is such an example of a local vibration, where the frequency is not
contaminated by coupling contributions. In general, mode–mode coupling depends
on the orientation of the mode vectors: vibrational modes with orthogonal mode
vectors do not couple. Also, difference in the atomic masses can suppress coupling.
For example, for the light–heavy–light arrangement of an acyclic three-atom molecule, the central atom can function as a “wall,” thus largely suppressing mode–mode
coupling [222].
The TEP is based on measured or calculated normal mode CO stretching frequencies, which may be effected by mode–mode coupling between the CO
stretchings or even between the CO and the MC stretching modes. There are two
different coupling mechanisms between vibrational modes as a consequence of the
fact that there is a kinetic and a potential contribution to the energy of a vibrational
mode [86]. The electronic coupling between modes is reflected by the off-diagonal
elements of the force constant matrix. By diagonalizing the force constant matrix F
q
expressed in terms of internal coordinates q n , i.e., a transformation to normal
coordinates and related normal modes, the electronic mode–mode coupling is
eliminated. However, the resulting normal mode force constants are still contaminated by kinematic mode–mode coupling, and as described above, they depend on
the internal coordinates chosen to describe the molecular geometry. Already in the
1960s, Decius [117] attempted to solve the force constant problem by using the
inverse force constant matrix Γ ¼ F
q
ð Þ
À1 and introducing the so-called compliance
constants Γ nn as bond strength descriptors. However, the relationship of the compliance constants to normal or other vibrational modes was unclear. Hence, the
compliance constants remained force constants without a mode and a frequency.
Also a given Γ nn is related to off-diagonal elements Γ mn (m 6 ¼ n), the physical
meaning of which is unclear. This led to criticism and questions about the usefulness
of compliance constants [223]. For example, why should one only use the diagonal
Γ nn terms without considering the role of the off-diagonal Γ mn terms when chemical
bonds were described. There were also questions about the physical meaning of
compliance constants related to redundant internal coordinates. Therefore, needed
for an advanced TEP model are local vibrational modes, which fulfill the following
requirements:
Characterizing the Metal–Ligand Bond Strength via Vibrational Spectroscopy:. . .
235
TEP and Tolman’s cone angle, as was demonstrated for phosphines and phosphites.
However, this approach turned out to be only reliable for ligands (L) with high
polarizability [214].
Although the TEP is still today one of the most popular measures used by the
experimental and computational chemistry community for quantifying the catalytic
activity of transition metal complexes based on vibrational spectroscopy, criticism
on the TEP has been raised by several authors in the recent literature [139, 177, 192,
205, 206, 215–221], in particular with regard to the validity of Tolman’s original
assumption of uncoupled CO stretching modes. The normal vibrational modes in a
molecule always couple [86]. There are only a few examples for uncoupled,
non-delocalized, i.e., local, vibrational modes. The bending vibration of the water
molecule is such an example of a local vibration, where the frequency is not
contaminated by coupling contributions. In general, mode–mode coupling depends
on the orientation of the mode vectors: vibrational modes with orthogonal mode
vectors do not couple. Also, difference in the atomic masses can suppress coupling.
For example, for the light–heavy–light arrangement of an acyclic three-atom molecule, the central atom can function as a “wall,” thus largely suppressing mode–mode
coupling [222].
The TEP is based on measured or calculated normal mode CO stretching frequencies, which may be effected by mode–mode coupling between the CO
stretchings or even between the CO and the MC stretching modes. There are two
different coupling mechanisms between vibrational modes as a consequence of the
fact that there is a kinetic and a potential contribution to the energy of a vibrational
mode [86]. The electronic coupling between modes is reflected by the off-diagonal
elements of the force constant matrix. By diagonalizing the force constant matrix F
q
expressed in terms of internal coordinates q n , i.e., a transformation to normal
coordinates and related normal modes, the electronic mode–mode coupling is
eliminated. However, the resulting normal mode force constants are still contaminated by kinematic mode–mode coupling, and as described above, they depend on
the internal coordinates chosen to describe the molecular geometry. Already in the
1960s, Decius [117] attempted to solve the force constant problem by using the
inverse force constant matrix Γ ¼ F
q
ð Þ
À1 and introducing the so-called compliance
constants Γ nn as bond strength descriptors. However, the relationship of the compliance constants to normal or other vibrational modes was unclear. Hence, the
compliance constants remained force constants without a mode and a frequency.
Also a given Γ nn is related to off-diagonal elements Γ mn (m 6 ¼ n), the physical
meaning of which is unclear. This led to criticism and questions about the usefulness
of compliance constants [223]. For example, why should one only use the diagonal
Γ nn terms without considering the role of the off-diagonal Γ mn terms when chemical
bonds were described. There were also questions about the physical meaning of
compliance constants related to redundant internal coordinates. Therefore, needed
for an advanced TEP model are local vibrational modes, which fulfill the following
requirements:
Characterizing the Metal–Ligand Bond Strength via Vibrational Spectroscopy:. . .
235