188
7 Molecular Structures from Gas-Phase Electron Diffraction
rectilinear corrections), the derived structure is denoted as r h0 , r
h
e or r α (equilibrium structure in harmonic approximation). The rectilinear vibrational corrections
alone as being overestimated for the bonded distances and underestimated for the
non-bonded distances lead to strong deformation of the refined structure.
Besides the vibrational corrections, the centrifugal distortion effects δ cent are
usually taken into account in the transformation of thermal-average internuclear
distances to equilibrium ones. Although these effects turned out to be small compared
to vibrational effects (see data of Tables 7.2 and 7.3 as an example), they can be nevertheless significant. Centrifugal distortions of the molecular geometry are caused
by centrifugal forces due to the rotation of the molecule as whole (Iwasaki and
Hedberg 1962), δ cent,rot , as well as due to vibrations (Bartell 1963), so-called local
centrifugal distortion δ cent,vib . Therefore, δ cent = δ cent,rot + δ cent,vib . The centrifugal
effects of molecular rotations vanish at T = 0 K (molecular rotation is frozen),
whereas the δ cent,vib values at 0 K are larger than δ cent,rot at high temperature (for
bonded internuclear distances) (Sipachev 2001).
It is worth noting that the structures, derived from experimental data by taking into
account vibrational corrections calculated from quantum-chemical force constants,
are not completely experimental. In spectroscopy, such structures are specified as
semiexperimental (r
se
e ). In electron diffraction literature, this point is usually not
emphasized, although all structures derived from experimental data using quantumchemical force fields indeed belong to this category.
7.9 Large-Amplitude Motions
The methods of structure determination considered above (see Sect. 7.7) were
successfully used for the study of relatively rigid molecules (without low-frequency
internal motions). However, they can fail for molecules with large-amplitude internal
motions, such as inversion, ring puckering, pseudo-rotation, and hindered internal
rotation. The terms semirigid, non-rigid, flexible, etc., are used to specify such
molecules.
Adiabatic approximation, i.e., separate consideration of high-frequency rigid
frame vibrations and large-amplitude motions, which have noticeably different
periods and energies, provides the simplest basis for treating the large-amplitude
vibrational effects. Following Bastiansen et al. (1979), many important types of lowfrequency motions were considered using method, which is known as the method of
pseudo-conformers (see, for instance, the review paper by Novikov and Vilkov 2000).
In this method, the molecular model is described by a set of pseudo-conformers, i.e.,
by a set of quasi-static rigid molecular configurations (rigid frame) different by the
magnitude of the fixed large-amplitude coordinate φ. In this case, the probability
distribution function is expressed as:
P(r ) =
P r (r, φ) · P(φ)dφ,
(7.41)
7 Molecular Structures from Gas-Phase Electron Diffraction
rectilinear corrections), the derived structure is denoted as r h0 , r
h
e or r α (equilibrium structure in harmonic approximation). The rectilinear vibrational corrections
alone as being overestimated for the bonded distances and underestimated for the
non-bonded distances lead to strong deformation of the refined structure.
Besides the vibrational corrections, the centrifugal distortion effects δ cent are
usually taken into account in the transformation of thermal-average internuclear
distances to equilibrium ones. Although these effects turned out to be small compared
to vibrational effects (see data of Tables 7.2 and 7.3 as an example), they can be nevertheless significant. Centrifugal distortions of the molecular geometry are caused
by centrifugal forces due to the rotation of the molecule as whole (Iwasaki and
Hedberg 1962), δ cent,rot , as well as due to vibrations (Bartell 1963), so-called local
centrifugal distortion δ cent,vib . Therefore, δ cent = δ cent,rot + δ cent,vib . The centrifugal
effects of molecular rotations vanish at T = 0 K (molecular rotation is frozen),
whereas the δ cent,vib values at 0 K are larger than δ cent,rot at high temperature (for
bonded internuclear distances) (Sipachev 2001).
It is worth noting that the structures, derived from experimental data by taking into
account vibrational corrections calculated from quantum-chemical force constants,
are not completely experimental. In spectroscopy, such structures are specified as
semiexperimental (r
se
e ). In electron diffraction literature, this point is usually not
emphasized, although all structures derived from experimental data using quantumchemical force fields indeed belong to this category.
7.9 Large-Amplitude Motions
The methods of structure determination considered above (see Sect. 7.7) were
successfully used for the study of relatively rigid molecules (without low-frequency
internal motions). However, they can fail for molecules with large-amplitude internal
motions, such as inversion, ring puckering, pseudo-rotation, and hindered internal
rotation. The terms semirigid, non-rigid, flexible, etc., are used to specify such
molecules.
Adiabatic approximation, i.e., separate consideration of high-frequency rigid
frame vibrations and large-amplitude motions, which have noticeably different
periods and energies, provides the simplest basis for treating the large-amplitude
vibrational effects. Following Bastiansen et al. (1979), many important types of lowfrequency motions were considered using method, which is known as the method of
pseudo-conformers (see, for instance, the review paper by Novikov and Vilkov 2000).
In this method, the molecular model is described by a set of pseudo-conformers, i.e.,
by a set of quasi-static rigid molecular configurations (rigid frame) different by the
magnitude of the fixed large-amplitude coordinate φ. In this case, the probability
distribution function is expressed as:
P(r ) =
P r (r, φ) · P(φ)dφ,
(7.41)