2.2. Permutation-Inversion Symmetry Operations

(2.3)

The left-hand sides of eqs (2.03) contain the Cartesian coordinates of an electron and the two nuclei in an axis system parallel to that fixed in the laboratory, but located at the center of mass of the equilibrium configuration of the nuclei. The right-hand sides of eqs (2.3) all contain a 3 × 3 rotation matrix, the direction cosine matrix, which transforms vector components from a molecule-fixed axis system to a laboratory-fixed axis system. This matrix is a function of the rotational variables θ and , specifying the direction of the internuclear axis of the diatomic molecule in the laboratory-fixed axis system. The column vector on the far right in the first of eqs (2.3) contains the molecule-fixed coordinates of an electron. The second and third column vectors on the far right contain the positions of the two nuclei in the molecule-fixed axis system. At equilibrium (d₁ = d₂ = 0) both nuclei lie on the z axis, with the center of mass at the origin, and with internuclear distance r_e; µ = m₁m₂/(m₁  + m₂) is the reduced mass of the molecule.

Consider now the effect of the four symmetry operations E, σ(xz), i, and C₂(y) on the coordinates in eqs (2.3). The transformations of the coordinates on the right side of (2.3) can be obtained from table 3 and table 4 and from the text of sect. 2.1. It is fairly easy to show from (2.3) that these operations give rise to the transformations of laboratory-fixed coordinates shown in table 5. From table 5 we see that the geometric symmetry operation σ(xz), when it is applied to the electronic, vibrational. and rotational (i.e., to all) variables, is equivalent to the laboratory-fixed inversion operation I; and that the geometric symmetry operation C₂(y), when it is applied to the electronic, vibrational, and rotational variables, is equivalent to the permutation P of the two (identical) nuclei in the molecule. From table 5 we also see that the geometric symmetry operation i is equivalent to the product P · I, i.e., to the combined permutation andlaboratory-fixed inversion operation.

TABLE 5. The effect of various symmetry operations on laboratory-fixed Cartesian coordinates of the electrons and of the two nuclei

Symmetry operation	Coordinates acted upon
	Xe	Ye	Ze	X1	Y1	Z1	X2	Y2	Z2
E	Xe	Ye	Ze	X1	Y1	Z1	X2	Y2	Z2
σ(xz)	- Xe	- Ye	- Ze	- X1	- Y1	- Z1	- X2	- Y2	- Z2
i	- Xe	- Ye	- Ze	- X2	- Y2	- Z2	- X1	- Y1	- Z1
C2(y)	+ Xe	+ Ye	+ Ze	+ X2	+ Y2	+ Z2	+ X1	+ Y1	+ Z1

It is evidently necessary to distinguish clearly between the "molecule-fixed" inversion operation i, and the "laboratory-fixed" inversion operation I, since these two operations are not equivalent. In particular, i exists only if the diatomic molecule is homonuclear, whereas I exists for all diatomic molecules. The precise difference between these two inversion operations can only be understood after some study [14,15,16].

Rotational energy levels are said to be of even (+) parity if the corresponding complete molecular wave functions are invariant to the laboratory-fixed inversion operation I; they are of odd (-) parity if the wave functions transform into their negatives. Rotational energy levels of homonuclear diatomic molecules are said to be symmetric (s) if the corresponding complete molecular wave functions are invariant to the exchange of identical nuclei P; they are antisymmetric (a) if the wave functions transform into their negatives.

\