4.7. Centrifugal Distortion Corrections to Rotational Energy Levels

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 m₁m₂/(m₁ + m₂) of the diatomic molecule and that B(Q) has the dimensions of energy, i.e., ML²T^-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  = H₀ + H₁, such that H₀ =  and H₁ = . 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;  ΩJ M⟩, |ΛSΣ;  υ;  ΩJ M⟩,  or |LΛSΣ; υ; ΩJ 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, ΛSΣ, or LΛSΣ, respectively.

Consider now centrifugal distortion corrections for a situation in which the (2L + 1)(2S + 1) states of fixed L, S, J, M, and υ having wave functions of the form |LΛSΣ; υ; ΩJ 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 |LΛSΣ; υ; ΩJ M⟩ and |LΛ″SΣ″; υ″; Ω″J M⟩ has been approximated by the purely vibrational energy separation E_υ - E_υ″ ≡ (υ-υ″)hν, 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, B_e, and hν are all in units of energy, i.e., ML²T ^-2. D_e can be expressed in its more common units of cm^-1 by (mentally) changing the units of B_e to cm^-1 and replacing the vibrational energy interval hν by its equivalent in cm^-1. If at the same time we drop the subscript e, we find [1] (p. 103)

(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 |LΛ′SΣ′; 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 (2L + 1) (2S + 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 (2L + 1) (2S + 1) × (2L + 1) (2S + 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(2S + 1) or (2S + 1) states of fixed
|Λ|, S, J, M, and υ having wave functions of the form |ΛSΣ; υ; ΩJ 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 |ΛSΣ; υ; ΩJ M⟩ and |Λ″SΣ″; 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′SΣ′; 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′SΣ′; 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(2S + 1) × 2(2S + 1) or (2S + 1) × (2S + 1) Hamiltonian matrices of fixed |Λ|, S, J, M, and υ to be diagonalized exactly. When centrifugal distortion corrections of the order of DJ² 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; ΩJ 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.