Centrifugal distortion, which can be considered from one point of view as a perturbation of rigid-rotor rotational energy levels, may be treated in diatomic molecules as follows. We rewrite eq (1.10), making two changes.

(4.11) |

First, careful attention is paid to units, and dimensionless angular momentum
operators are obtained by considering
rather than . Second, the quantity *B* is no longer taken to be
a constant, but is rather given its true functional form
[3] (p. 13) in terms of the vibrational
coordinate *Q = r - r _{e}*

(4.12) |

Note that *µ* is the reduced mass
*m*_{1}*m*_{2}/(*m*_{1} +
*m*_{2}) of the diatomic molecule and that *B*(*Q*) has
the dimensions of energy, i.e., *ML*^{2}*T*^{-2}.

The next step is to apply a Van Vleck transformation (see
sect. 4.5) to the problem. The complete
Hamiltonian = + is considered to be divided into two
parts = *H*_{0} + *H*_{1},
such that *H*_{0} = and
*H*_{1} = . Energies of the unperturbed states are thus equal to
energies of the nonrotating-molecule states. Wave functions for the unperturbed
states (the basis set functions) have one of the forms (see
sect. 1.4 and
sect. 1.7): |Ω; *υ*;
Ω*J M*⟩, |Λ*S*Σ; *υ*;
Ω*J M*⟩, or
|*L*Λ*S*Σ; *υ*; Ω*J M*⟩.
Since centrifugal distortion involves an interaction between rotation and
vibration, a vibrational quantum number *υ* is included in these basis set functions. They are still taken
to be simple products, however, of the form |Ω⟩ |*υ*⟩ , |*υ*⟩ ,
or
|*υ*⟩ , respectively. We consider, in the
submatrix to be diagonalized exactly (sect. 4.5),
states characterized by fixed *J* and *M*, by fixed vibrational
quantum number *υ*, and by fixed
|Ω| or by fixed |Λ| and *S*, with
Σ = *S*, *S* - 1, ...,
- *S*, or by fixed *L* and *S* with
Λ = *L*, *L* - 1,
... , - *L* and Σ = *S*,
*S* - 1, ... , - *S*, respectively,
for the three types of wave functions given above. For the purpose of
calculating centrifugal distortion corrections to the rotational energy levels,
the distant perturbing states (sect. 4.5) are
taken to be states characterized by the same fixed value of *J* and
*M*, by different values of the vibrational quantum number
*υ*, and by electronic quantum
numbers from the same set Ω, Λ*S*Σ, or
*L*Λ*S*Σ, respectively.

Consider now centrifugal distortion corrections for a situation in which the
(2*L* + 1)(2*S* + 1) states of fixed *L*,
*S*, *J*, *M*, and *υ* having wave functions of the form
|*L*Λ*S*Σ; *υ*; Ω*J M*⟩ lie close together in energy
compared to the separation between adjacent vibrational levels. Under these
circumstances the fourth term of (4.8) can
be written

(4.13) |

where the energy separation between two states
|*L*Λ*S*Σ; *υ*; Ω*J M*⟩ and
|*L*Λ″*S*Σ″; *υ*″; Ω″*J M*⟩ has been
approximated by the purely vibrational energy separation *E*_{υ} - *E*_{υ″} ≡ (*υ*-*υ*″)*h*ν, and where the summation indices take on
the values *υ*″ ≠ *υ*; Λ″ = *L*,
*L* - 1, ... , - *L*; and
Σ″ = *S*,
*S* - 1, ... , - *S*.

Since the basis set functions are simple products of a vibrational part, and a
rotational and electronic part, and since of (4.11) is also a simple product of
a vibrational part and a rotational-electronic part, (4.13) can be written as
the product of a vibrational part and a rotational-electronic part. If we make
a further approximation and consider only the constant and linear terms in the
series expansion (4.12) for *B*(*Q*), we find
that (4.13) can be written as

(4.14) |

The sum over *υ*″ in
(4.14), i.e., the vibrational factor in the product, can be evaluated
explicitly since [4] (pp. 67-82)

(4.15) |

Representing the sum over *υ*″ by the symbol
- *D _{e}*, we find for the vibrational factor in (4.14)

(4.16) |

where *D _{e}*,

(4.17) |

The sum over Λ″ and Σ″ in (4.14),
i.e., the rotational-electronic factor in the product, can also be evaluated
explicitly. It can be shown by considering various commutation relations
[7] (pp. 45-78), [8],
that the operator when acting on a function of the form
|*L*Λ′*S*Σ′;
Ω′*J M*⟩ transforms it into a linear combination of
functions characterized by the same *L*, *S*, *J*, and *M*,
but by various values of Λ′, Σ′ and
Ω′ ≡ Λ′ + Σ′.
Consequently, the sum in (4.14) over all values of
Λ″ and Σ″ permitted for the given (fixed) values of
*L* and *S* can be collapsed, just as if the sum were carried out
over *all possible* basis set functions, to give for the
rotational-electronic factor

(4.18) |

Combining the vibrational factor (4.16) and the rotational-electronic factor (4.18), we rewrite the Van Vleck correction (4.13) as

(4.19) |

where *D* is a constant. The centrifugal distortion correction to the
rigid-rotor rotational energy levels for a set of states characterized by fixed
*L* and *S*, and by the fact that the
(2*L* + 1) (2*S* + 1) components of
differing Λ and Σ lie close together compared to the vibrational
spacing, can thus be determined to a good approximation by including matrix
elements of the operator

(4.20) |

in the (2*L* + 1)
(2*S* + 1) × (2*L* + 1)
(2*S* + 1) Hamiltonian matrices of fixed *L*, *S*,
*J*, *M*, and *υ* to
be diagonalized exactly. By analogy, higher order centrifugal distortion
corrections can be obtained by including matrix elements of
,
etc.

Consider as a second example of centrifugal distortion corrections a situation
in which the 2(2*S* + 1) or (2*S* + 1) states of
fixed

|Λ|, *S*, *J*, *M*, and
*υ* having wave functions of the
form |Λ*S*Σ; *υ*; Ω*J M*⟩ lie close together in energy
compared to the separation between adjacent vibrational levels. Under these
circumstances, the fourth term of (4.8) can
be written

(4.21) |

where we have approximated the energy separation between two states
|Λ*S*Σ; *υ*; Ω*J M*⟩ and
|Λ″*S*Σ″; *υ*″; Ω″*J M*⟩ by the purely
vibrational energy separation, and where the summation indices take on the
values *υ*″ ≠ *υ*; Λ″ = µ|Λ|; and
Σ″ = *S*, *S* - 1,
... , - *S*.

The expression (4.21) can be written as the product of a vibrational factor and
a rotational-electronic factor, just as (4.13) was,
giving an expression identical to (4.14) except for the
absence of the quantum number *L* and the new significance of the sum over
Λ″.

The sum over *υ*″ can again
be evaluated explicitly, yielding for the vibrational factor the same quantity
-*D _{e}* given in (4.16).

The sum over Λ″ and Σ″ this time does not include all
functions generated when the operator acts on a function of the form
|Λ′*S*Σ′; Ω′*J M*⟩.
Consequently, the summation over Λ″ and Σ″ cannot be
collapsed to give an expression like (4.18). However,
the sum over Λ″ and Σ″ does include all functions
produced when **J** and **S** act on a function of the form
|Λ′*S*Σ′; Ω′*J M*⟩.
By a series of arguments it is possible to show that the centrifugal distortion
correction to the rigid-rotor rotational energy levels for a set of states
characterized by fixed |Λ| and *S*, and by the fact that the
spin-multiplet components of this state lie close together compared to the
vibrational spacing, can be determined to a good approximation by including
matrix elements of the operator

(4.22) |

in the 2(2*S* + 1) × 2(2*S* + 1) or
(2*S* + 1) × (2*S* + 1) Hamiltonian
matrices of fixed |Λ|, *S*, *J*, *M*, and
*υ* to be diagonalized exactly.
When centrifugal distortion corrections of the order of *DJ*^{2}
are negligible, (4.22) can be replaced by

(4.23) |

There are many other situations for which centrifugal distortion corrections
could be considered, e.g., the situation corresponding to basis set functions
of the form |Ω; *υ*; Ω*J M*⟩ or the situation which
arises when one of the multiplet splittings is approximately equal to a
vibrational interval. We shall not consider these here.