184
7 Molecular Structures from Gas-Phase Electron Diffraction
s M(s) =
N
i = j
g i j (s)
r i j
c
exp
−
s
2
2
r
2
i j
c
× sin
s
r i j
c
−
s
3
6
r
3
i j
c
.
(7.38)
The relations between the thermally averaged moments in normal coordinates
and the parameters of potential energy function expanded up to cubic (or quartic)
terms can be obtained, for instance, by means of conventional perturbation theory
(Ischenko et al. 1988).
The topics of this section are considered in detail in the review papers by
Spiridonov (1988) and Spiridonov et al. (2001).
7.8 The Use of Curvilinear Internal Coordinates
For a long time, the Wilson GF matrix formalism (Wilson et al. 1955) was predominantly used for the solution of the vibrational problem in electron diffraction analysis.
Here, F is the matrix of the force constants in the harmonic approximation, and G
is the matrix of kinetic energy in the momentum representation depending on the
nuclear masses and equilibrium geometrical parameters. This method of harmonic
force field calculations is based on the definition of 3N − 6 (3N − 5 for linear
molecules) linearized internal displacement coordinates. However, it became obvious
that the use of natural curvilinear internal coordinates noticeably improves the molecular dynamic description, particularly for a wide range of displacements. For the first
time, it was considered by Morino et al. (1962), Morino and Iijima (1963), and Bartell
(1963). Later, the nonlinear coordinate transformation (from linearized to curvilinear
ones) was used in several electron diffraction models. In these cases, vibrational
corrections to the thermal-average internuclear distances can be presented as:
r g = r e + r 1 + r 2 + r 3 + · · · ,
(7.39)
where r 1 is the perpendicular correction (harmonic term or harmonic vibrational
correction) and r 3 is the anharmonic term (or anharmonic vibrational correction).
The additional term r 2 appears because harmonic (quadratic) potential term in
internal coordinates propagates cubic term in a normal coordinate representation;
i.e., due to use of curvilinear internal coordinates, the part of anharmonicity is taken
into account via the kinetic part of the Hamiltonian. This effect is the so-called
kinematic anharmonicity. The equilibrium molecular structure r e corresponds to
the minimum of the anharmonic potential energy function (surface) in curvilinear
internal coordinates.
Tables 7.1 and 7.2 show that the r 1 and r 2 components can essentially
compensate each other. In such cases, the magnitudes of the total vibrational
corrections are mainly defined by the anharmonicity of a potential energy function
(“dynamic” anharmonicity).
7 Molecular Structures from Gas-Phase Electron Diffraction
s M(s) =
N
i = j
g i j (s)
r i j
c
exp
−
s
2
2
r
2
i j
c
× sin
s
r i j
c
−
s
3
6
r
3
i j
c
.
(7.38)
The relations between the thermally averaged moments in normal coordinates
and the parameters of potential energy function expanded up to cubic (or quartic)
terms can be obtained, for instance, by means of conventional perturbation theory
(Ischenko et al. 1988).
The topics of this section are considered in detail in the review papers by
Spiridonov (1988) and Spiridonov et al. (2001).
7.8 The Use of Curvilinear Internal Coordinates
For a long time, the Wilson GF matrix formalism (Wilson et al. 1955) was predominantly used for the solution of the vibrational problem in electron diffraction analysis.
Here, F is the matrix of the force constants in the harmonic approximation, and G
is the matrix of kinetic energy in the momentum representation depending on the
nuclear masses and equilibrium geometrical parameters. This method of harmonic
force field calculations is based on the definition of 3N − 6 (3N − 5 for linear
molecules) linearized internal displacement coordinates. However, it became obvious
that the use of natural curvilinear internal coordinates noticeably improves the molecular dynamic description, particularly for a wide range of displacements. For the first
time, it was considered by Morino et al. (1962), Morino and Iijima (1963), and Bartell
(1963). Later, the nonlinear coordinate transformation (from linearized to curvilinear
ones) was used in several electron diffraction models. In these cases, vibrational
corrections to the thermal-average internuclear distances can be presented as:
r g = r e + r 1 + r 2 + r 3 + · · · ,
(7.39)
where r 1 is the perpendicular correction (harmonic term or harmonic vibrational
correction) and r 3 is the anharmonic term (or anharmonic vibrational correction).
The additional term r 2 appears because harmonic (quadratic) potential term in
internal coordinates propagates cubic term in a normal coordinate representation;
i.e., due to use of curvilinear internal coordinates, the part of anharmonicity is taken
into account via the kinetic part of the Hamiltonian. This effect is the so-called
kinematic anharmonicity. The equilibrium molecular structure r e corresponds to
the minimum of the anharmonic potential energy function (surface) in curvilinear
internal coordinates.
Tables 7.1 and 7.2 show that the r 1 and r 2 components can essentially
compensate each other. In such cases, the magnitudes of the total vibrational
corrections are mainly defined by the anharmonicity of a potential energy function
(“dynamic” anharmonicity).