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
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 (σ_{υ}) 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 (C_{2} ) 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 C_{∞υ} 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) |
Symmetry operation |
Coordinates acted upon | |||||
---|---|---|---|---|---|---|
x_{e} | y_{e} | z_{e} | ρ_{e} | θ_{e} | φ_{e} | |
E | x_{e} | y_{e} | z_{e} | ρ_{e} | θ_{e} | φ_{e} |
C_{∞}^{ε}(z) | y_{e} cos ε + x_{e} sin ε | + z_{e} | ρ_{e} | + θ_{e} | φ_{e} + ε | |
σ_{υ}(xz) | + x_{e} | + z_{e} | ρ_{e} | + θ_{e} | ||
i | ρ_{e} | π + φ_{e} | ||||
S_{∞}^{ε}(z) | ρ_{e} | φ_{e} + ε | ||||
C_{2}(y) | + y_{e} | ρ_{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_{∞υ} and D_{∞h} on arbitrary displacement
vectors (d_{1}, d_{2}) 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
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_{∞}^{ε}(z) | + d_{1z} | + d_{2z} | ||||
σ_{υ}(xz) | + d_{1x} | + d_{1z} | + d_{2x} | + d_{2z} | ||
i | ||||||
S_{∞}^{ε}(z) | ||||||
C_{2}(y) | + d_{2y} | + d_{1y} |
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;
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_{υ} i, and C_{2}. Of these, we need only investigate the effect of C_{2} 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 C_{2}(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,
C_{2}(y) acting on a function of the rotational variables
replaces θ by
The effect of C_{2}(y) on the polar and azimuthal angles of the internuclear axis, as defined above, is different from the effect of C_{2}(y) on the polar and azimuthal angles of an electron, as given in table 3. This difference arises because C_{2}(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.