## 2.2. Permutation-Inversion Symmetry Operations

The relation between the geometric symmetry operations of sect. 2.1 and the permutation-inversion symmetry operations can be demonstrated most easily by considering an equation relating the laboratory-fixed Cartesian coordinates (XiYiZi) of the electrons and nuclei in a diatomic molecule to the molecule-fixed electronic coordinates (xyz), the equilibrium positions of the nuclei, the displacement vectors di of the nuclei, and the two rotational angles &theta"; φ [15,16].

 (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 φ, 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 re; µ = m1m2/(m1  + m2) is the reduced mass of the molecule.

Consider now the effect of the four symmetry operations E, συ(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 συ(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
συ(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.

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