180
7 Molecular Structures from Gas-Phase Electron Diffraction
The r
h
e internuclear distances are the parameters of the equilibrium molecular
structure corresponding to the minimum of potential energy function in the harmonic
approximation.
In these equations,
z
2
,
x
2
,
y
2
, xz, and yz are generalized
vibrational amplitudes with z as an axis being along a bond (Morino and Hirota
1955):
z
2
=
n
p=1
A
2
p
Q
2
p
,
y
2
=
n
p=1
C
2
p
Q
2
p
,
x
2
=
n
p=1
B
2
p
Q
2
p
,
yz =
n
p=1
A p C p
Q
2
p
,
and
xz =
n
p=1
A p B p
Q
2
p
(7.27)
with the frequency parameter
Q
2
p
=
h
8π 2 cω p
coth(hcω p /2kT ),
where Q p is the normal coordinate, ω p is the harmonic vibrational frequency, and n
is the number of vibrational degrees of freedom (3N − 6 for nonlinear molecules and
3N − 5 for linear ones). The generalized amplitudes, the coefficients A p , B p , and C p ,
and the frequency parameter can be calculated from the harmonic force constants
and r
h
e values by means of normal coordinate analysis (Cyvin 1968).
7.7.2 Anharmonic Approximation
The anharmonic part of the potential energy function contributes significantly to the
apparent molecular geometry (Bartell 1955).
For a linear XY 2 -type molecule, the anharmonic potential energy function (up to
cubic terms) can be written as:
7 Molecular Structures from Gas-Phase Electron Diffraction
The r
h
e internuclear distances are the parameters of the equilibrium molecular
structure corresponding to the minimum of potential energy function in the harmonic
approximation.
In these equations,
z
2
,
x
2
,
y
2
, xz, and yz are generalized
vibrational amplitudes with z as an axis being along a bond (Morino and Hirota
1955):
z
2
=
n
p=1
A
2
p
Q
2
p
,
y
2
=
n
p=1
C
2
p
Q
2
p
,
x
2
=
n
p=1
B
2
p
Q
2
p
,
yz =
n
p=1
A p C p
Q
2
p
,
and
xz =
n
p=1
A p B p
Q
2
p
(7.27)
with the frequency parameter
Q
2
p
=
h
8π 2 cω p
coth(hcω p /2kT ),
where Q p is the normal coordinate, ω p is the harmonic vibrational frequency, and n
is the number of vibrational degrees of freedom (3N − 6 for nonlinear molecules and
3N − 5 for linear ones). The generalized amplitudes, the coefficients A p , B p , and C p ,
and the frequency parameter can be calculated from the harmonic force constants
and r
h
e values by means of normal coordinate analysis (Cyvin 1968).
7.7.2 Anharmonic Approximation
The anharmonic part of the potential energy function contributes significantly to the
apparent molecular geometry (Bartell 1955).
For a linear XY 2 -type molecule, the anharmonic potential energy function (up to
cubic terms) can be written as: