NBS Monograph 115: 2. Symmetry prop. rotational energy levels

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 $C_{\infty v}$ . 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 (σ) 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 $C_{\infty v}$ , 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

eq 2.01 (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 C_∞ 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,

eq 2.02 (2.2)

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

Symmetry
operation Coordinates acted upon

x_e y_e z_e ρ_e θ_e phi _e

E x_e y_e z_e ρ_e θ_e phi _e

C_∞^ε(z) x_e cos ε - y_e sin ε y_e cos ε + x_e sin ε + z_e ρ_e + θ_e φ_e + ε

σ(xz) + x_e - y_e + z_e ρ_e + θ_e - phi _e

i - x_e - y_e - z_e ρ_e π - θ_e π + phi _e

S_∞(z) x_e cos ε - y_e sin ε y_e cos ε + x_e sin ε - z_e ρ_e π - θ_e phi _e + ε

C₂(y) - x_e + y_e - z_e ρ_e π - θ_e π - phi _e

**TABLE 3. The effect of various symmetry operations on electron coordinates**
Symmetry operation	Coordinates acted upon
x_e	y_e	z_e	ρ_e	θ_e	_e
E	x_e	y_e	z_e	ρ_e	θ_e	_e
C_∞^ε(z)	x_e cos ε - y_e sin ε	y_e cos ε + x_e sin ε	+ z_e	ρ_e	+ θ_e	φ_e + ε
σ(xz)	+ x_e	- y_e	+ z_e	ρ_e	+ θ_e	- _e
i	- x_e	- y_e	- z_e	ρ_e	π - θ_e	π + _e
S_∞(z)	x_e cos ε - y_e sin ε	y_e cos ε + x_e sin ε	- z_e	ρ_e	π - θ_e	_e + ε
C₂(y)	- x_e	+ y_e	- z_e	ρ_e	π - θ_e	π - _e

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 C_∞ 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 a_i 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

d_1x d_1y d_1z d_2x d_2y d_2z

E d_1x d_1y d_1z d_2x d_2y d_2z

C_∞(z) d_1x cos ε - d_1y sin ε d_1y cos ε + d_1x sin epsilon + d_1z d_2x cos ε - d_2y sin epsilon d_2y cos ε + d_2x sin epsilon + d_2z

σ(xz) + d_1x - d_1y + d_1z + d_2x - d_2y + d_2z

i - d_2x - d_2y - d_2z - d_1x - d_1y - d_1z

S_∞^ε(z) d_2x cos ε - d_2y sin ε d_2y cos ε + d_2x sin ε - d_2z d_1x cos ε - d_1y sin ε d_1y cos ε + d_1x sin ε - d_1z

C₂(y) - d_2x + d_2y - d_2z - d_1x + d_1y - d_1z

**TABLE 4. The effect of various symmetry operations on nuclear displacement vectors**
(Image of Table 4)
Symmetry operation	Displacement vector component acted upon
d_1x	d_1y	d_1z	d_2x	d_2y	d_2z
E	d_1x	d_1y	d_1z	d_2x	d_2y	d_2z
C_∞(z)	d_1x cos ε - d_1y sin ε	d_1y cos ε + d_1x sin	+ d_1z	d_2x cos ε - d_2y sin	d_2y cos ε + d_2x sin	+ d_2z
σ(xz)	+ d_1x	- d_1y	+ d_1z	+ d_2x	- d_2y	+ d_2z
i	- d_2x	- d_2y	- d_2z	- d_1x	- d_1y	- d_1z
S_∞^ε(z)	d_2x cos ε - d_2y sin ε	d_2y cos ε + d_2x sin ε	- d_2z	d_1x cos ε - d_1y sin ε	d_1y cos ε + d_1x sin ε	- d_1z
C₂(y)	- d_2x	+ d_2y	- d_2z	- d_1x	+ d_1y	- d_1z

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; sigma (xz) and C₂(y) have the same effect; and S_&infinity;^ε(z) and C_∞^π+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 C_∞^ε(z) on the rotational variables. Thus, this operation and S_∞^ε(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, σ 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 Papou s check ek [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 phi 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 phi by π + phi everywhere in the function. The functions of the rotational variables most often considered, of course, are the rotational basis set functions $|\Omega JM\rangle$ 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 = phi , j = J, µ′ = Ω and µ = M in the quantity $[(2j+1)/4\pi]^{1/2}\cdot{\cal D}^{(j)} (\{\alpha\beta\gamma\})_{\mu^\prime\mu}$ , where ${\cal D}^{(j)} (\{\alpha\beta\gamma\})_{\mu^\prime\mu}$ 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 phi = phi 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.