## 2. Symmetry properties of the rotational energy levels

In this chapter we discuss the symmetry properties of the rotational energy levels of diatomic molecules [1] (pp. 237-240). The rotational levels of all diatomic molecules can be classified as + or - according to their parity, i.e., according to the behavior of the complete molecular wave function (apart from translation) when the laboratory-fixed Cartesian coordinates of all particles are replaced by their negatives. The rotational energy levels of homonuclear diatomic molecules can be classified in addition as s (symmetric) or a (antisymmetric) with respect to permutation of identical nuclei. The symmetry designations +, -, s, a are very important, since perturbations and optical transitions between two rotational levels are limited by selection rules on these quantities.

It turns out that the two permutation-inversion symmetry operations mentioned in the preceding paragraph, i.e., the laboratory-fixed inversion operation I, and the permutation P of identical nuclei in a homonuclear diatomic molecule, are related [14-16] to the geometric symmetry operations found in the usual group theory tables [6] (pp. 312-340), i.e., rotation, reflection, inversion. and rotation-reflection operations. It is convenient at this time to consider these two kinds of symmetry operations in more detail.

## 2.1. Geometric Symmetry operations

Heteronuclear diatomic molecules belong to the point group . This point group [6] (p. 330) contains as symmetry elements the identity (E), one infinite-fold rotation axis (C ), and an infinite number of reflection planes (συ) containing the rotation axis. Homonuclear diatomic molecules belong to the point group D∞h. This point group [6] (p. 330) contains, in addition to the symmetry elements found in , an inversion center (i), one infinite-fold rotation-reflection axis (S ), and an infinite number of twofold rotation axes (C2 ) perpendicular to the C axis.

It is relatively easy to visualize the effect of these geometric symmetry operations on an electron belonging to the diatomic molecule, and to determine in this way the precise effect of these geometric symmetry operations on the coordinates of the electron. Let

 (2.1)

be the Cartesian coordinates and spherical polar coordinates of an electron in an axis system fixed in the diatomic molecule such that the internuclear axis is the z axis. Table 3 indicates the new coordinates of this electron, after it has been subjected to representative symmetry operations of the point groups Cυ and D∞h. When one of these symmetry operations acts on a function containing the electron coordinates, its effect is to replace, everywhere in the function, each coordinate by the quantity found in table 3 at the intersection of the appropriate row and column. For example,

 (2.2)

TABLE 3. The effect of various symmetry operations on electron coordinates

Symmetry
operation
Coordinates acted upon
xe ye ze ρe θe φe
E xe ye ze ρe θe φe
Cε(z) xe cos ε - ye sin ε ye cos ε + xe sin ε ze ρe + θe φe + ε
συ(xz) xe ye ze ρe + θe φe
i xe ye ze ρe π - θe π + φe
Sε(z) xe cos ε - ye sin ε ye cos ε + xe sin ε ze ρe π - θe φe + ε
C2(y) xe ye ze ρe π - θe π - φe

These geometric symmetry operations also act on the vibrational and rotational variables of a diatomic molecule. The transformations of the vibrational variable can be obtained by considering the effect of the various symmetry operations on the vibrational displacement vectors. Pictorially speaking, these displacement vectors are subjected to the various symmetry operations while the molecular framework, i.e., the equilibrium position of each atom in the molecule, is left unchanged [6] (pp. 77-101). Table 4 indicates quantitatively the effect of representative symmetry operations of Cυ and D∞h on arbitrary displacement vectors (d1, d2) for the two nuclei of a diatomic molecule. The vibrational variable for diatomic molecules is usually taken to be the internuclear distance r, given by |(a1 + d1) - (a2 + d2)|, where a1 and a2 represent equilibrium positions for the two atoms. Since the equilibrium positions ai lie on the z axis, and since they are not changed in value by any of the symmetry operations, it is easy to show that the transformations of table 4 leave the vibrational variable r unaltered.

TABLE 4. The effect of various symmetry operations on nuclear displacement vectors
(Image of Table 4)

Symmetry
operation
Displacement vector component acted upon
d1x d1y d1z d2x d2y d2z
E d1x d1y d1z d2x d2y d2z
Cε(z) d1x cos ε - d1y sin ε d1y cos ε + d1x sin ε d1z d2x cos ε - d2y sin ε d2y cos ε + d2x sin ε d2z
συ(xz) d1x d1y d1z d2x d2y d2z
i d2x d2y d2z d1x d1y d1z
Sε(z) d2x cos ε - d2y sin ε d2y cos ε + d2x sin ε d2z d1x cos ε - d1y sin ε d1y cos ε + d1x sin ε d1z
C2(y) d2x d2y d2z d1x d1y d1z

The effect of geometric symmetry operations on rotational variables is not as obvious as the effect on electronic and vibrational variables. The rotational variables of a molecule actually represent Eulerian angles, indicating how the (right handed) molecule-fixed axis system is rotated with respect to some (right handed) laboratory-fixed axis system [6] (pp. 285-286). For every orientation of the molecule-fixed axis system there is a set of corresponding Eulerian angles. For every set of Eulerian angles there is a corresponding orientation of the molecule-fixed axis system. It thus seems intuitively obvious that a geometric symmetry operation corresponding to a pure rotation should effect a change in the Eulerian angles corresponding to that pure rotation. It is not possible, however, to represent the change from a right handed to a left-handed axis system by a set of Eulerian angles. Consequently, it is not intuitively obvious how sense-reversing symmetry operations (reflections, inversion, rotation-reflections) should affect the Eulerian angles. It turns out that a consistent and useful scheme of geometric symmetry operations can be obtained if a sense-reversing operation is defined to have the same effect on the Eulerian angles as does the pure rotation obtained from the sense-reversing operation by multiplication by the inversion [15, 16]. According to this prescription, i and E have the same effect on the rotational variables; συ(xz) and C2(y) have the same effect; and S&infinity;ε(z) and Cπ+ε(z) have the same effect.

The rotational variables of linear molecules present an additional complication [16], since they consist of two Eulerian angles rather than three [3] (pp. 6-16). Because the third Eulerian angle is missing, it is not possible to carry out the operation Cε(z) on the rotational variables. Thus, this operation and Sε(z), both of which were used in classifying electronic and vibrational levels, cannot be used in classifying rotational energy levels. The remaining symmetry operations are E, συ i, and C2. Of these, we need only investigate the effect of C2 on the rotational variables. Note added in proof: A recently published article by Bunker and Papoušek [27] presents a more sophisticated discussion of the complications associated with symmetry operations for linear molecules than does reference [16].

The rotational variables θ and φ represent for a diatomic molecule the polar and azimuthal angles of the internuclear axis in the laboratory-fixed axis system. Since C2(y) corresponds to a twofold rotation about an axis perpendicular to the internuclear axis, this symmetry operation reverses the direction of the internuclear axis. Thus, C2(y) acting on a function of the rotational variables replaces θ by π - θ and φ by π + φ everywhere in the function. The functions of the rotational variables most often considered, of course, are the rotational basis set functions themselves. Explicit expressions for these functions, corresponding precisely to the phase conventions used in this monograph, can be obtained by setting α = π/2, β = θ, γ = φ, j = J, µ′ = Ω and µ = M in the quantity , where is given in eq (15.27) of Wigner's book on group theory [17]. In this same connection, the 3 × 3 rotation matrices used in eqs (2.3) below can be obtained by setting χ = π/2, θ = θ, and φ = φ in Appendix I of [6].

The effect of C2(y) on the polar and azimuthal angles of the internuclear axis, as defined above, is different from the effect of C2(y) on the polar and azimuthal angles of an electron, as given in table 3. This difference arises because C2(y) represents a rotation about an axis which is always perpendicular to the internuclear axis, but which is not in general perpendicular to the position vector from the origin to a given electron.