(3.2.11). V N (Q) is the potential energy of the nuclei (3.2.12), E
v
k is the energy of
the oscillations of the k-th state described by the wave function v(Q) (3.2.12).
If one take rotation into account, the total energy of a molecule in the adiabatic
approximation is:
E k ¼ E
el
k þ E
vr
k ;
ð3:2:13aÞ
here, E
vr
k is the energy of the rotational-vibrational excitation.
In a very rough approximation
E k ¼ E
el
k þ E
v
k þ E
r
k ;
ð3:2:13bÞ
and
W evr ¼ U Á v Q Á v r
ð3:2:14Þ
One should note, that (3.2.13b, 3.2.14) are rough approximations; in particular,
they do not suit for a description of degenerate electronic states. However, symmetry types (species, irreducible representations), C, of the vibronic, W ev , and
rovibronic, W evr , wave functions are equal to the direct products of the U,Áv Q , and
v r .wave functions:
C W ev
ð Þ ¼ C U
ð Þ Â C v Q
À Á ;
ð3:2:15Þ
C W evr
ð
Þ ¼C U
ð Þ Â C v Q
À Á Â C v r
ð Þ:
ð3:2:16Þ
If we digress from the PEC of a diatomic molecule and return to the consideration of the general case, we see the following:
The adiabatic approximation allows us to solve the problem of the motion of
nuclei, considering the motion of the slow subsystem, only but its own for each fast,
i.e., motion on PECs, corresponding to fast subsystems. If the slow subsystem has
s degrees of freedom, then, as we already understand, the function U k (Q) can be
represented by an s-dimensional hypersurface in s + 1-dimensional space. For one
degree of freedom (interatomic distance) using PECs in two-dimensional space.
Under what conditions is the adiabatic approximation applicable? The exact
wave function U(r, t), corresponding to the motion of the slow subsystem given by
the parameters Q(t), is a solution of the nonstationary Schrödinger equation:
i h
dUðr; tÞ
dt
¼ b
T e ðrÞ þ b
V e ðr; QÞ
h
i
U k ðr; QÞ
ð 3:2:17Þ
This solution can be represented as an expansion in adiabatic wave functions,
but the coefficients a k of the expansion can depend on time.
3.2 Adiabatic Approximations. Potential Energy Curves and Surfaces
47
v
k is the energy of
the oscillations of the k-th state described by the wave function v(Q) (3.2.12).
If one take rotation into account, the total energy of a molecule in the adiabatic
approximation is:
E k ¼ E
el
k þ E
vr
k ;
ð3:2:13aÞ
here, E
vr
k is the energy of the rotational-vibrational excitation.
In a very rough approximation
E k ¼ E
el
k þ E
v
k þ E
r
k ;
ð3:2:13bÞ
and
W evr ¼ U Á v Q Á v r
ð3:2:14Þ
One should note, that (3.2.13b, 3.2.14) are rough approximations; in particular,
they do not suit for a description of degenerate electronic states. However, symmetry types (species, irreducible representations), C, of the vibronic, W ev , and
rovibronic, W evr , wave functions are equal to the direct products of the U,Áv Q , and
v r .wave functions:
C W ev
ð Þ ¼ C U
ð Þ Â C v Q
À Á ;
ð3:2:15Þ
C W evr
ð
Þ ¼C U
ð Þ Â C v Q
À Á Â C v r
ð Þ:
ð3:2:16Þ
If we digress from the PEC of a diatomic molecule and return to the consideration of the general case, we see the following:
The adiabatic approximation allows us to solve the problem of the motion of
nuclei, considering the motion of the slow subsystem, only but its own for each fast,
i.e., motion on PECs, corresponding to fast subsystems. If the slow subsystem has
s degrees of freedom, then, as we already understand, the function U k (Q) can be
represented by an s-dimensional hypersurface in s + 1-dimensional space. For one
degree of freedom (interatomic distance) using PECs in two-dimensional space.
Under what conditions is the adiabatic approximation applicable? The exact
wave function U(r, t), corresponding to the motion of the slow subsystem given by
the parameters Q(t), is a solution of the nonstationary Schrödinger equation:
i h
dUðr; tÞ
dt
¼ b
T e ðrÞ þ b
V e ðr; QÞ
h
i
U k ðr; QÞ
ð 3:2:17Þ
This solution can be represented as an expansion in adiabatic wave functions,
but the coefficients a k of the expansion can depend on time.
3.2 Adiabatic Approximations. Potential Energy Curves and Surfaces
47