NBS Monograph 115: 2.3. The Symmetry Operation

2.3. The Symmetry Operation σ

As a further aid in understanding the geometric symmetry operation σ, let us consider its effect on several electronic wave functions. In particular, let us consider diatomic-molecule wave functions derived from atomic wave functions. The one-electron atomic configuration np gives rise to an orbital P state (L = 1), which in turn gives rise to diatomic-molecule orbital Σ(Λ = 0) and Π(Λ = ±1) states. The wave functions for these diatomic-molecule states are

(2.4)

if we use the phase conventions of Condon and Shortley [7] (p. 52). These functions transform as follows under the operation σ(xz) (see table 3)

(2.5)

which can be written as

(2.6)

Equations (2.4) and (2.5) can be used to illustrate a fundamental point concerning the relationship between the choice of phase factors for a set of wave functions and their transformation properties under symmetry operations. It can be seen that the first of the transformation equations (2.5) is unaltered if |p Σ⟩ in (2.4) is defined to be -f (ρ_e ) cosθ_e rather than +f (ρ_e ) cosθ_e. Thus the transformation property represented by this first equation is independent of the choice of phases, and is an intrinsic property of the Σ state under consideration. On the other hand, the - sign on the right-hand side of the second of the transformation equations (2.5) can be changed to a + sign, simply by defining the functions | p Π_± ⟩ in (2.4) to be + f (ρ_e ) 2^-l/2 sinθ_e ${\rm e}^{\pm i\varphi_{\rm e}}$ . The transformation property represented by this second equation is thus not an intrinsic property of the Π state, but depends on the choice of phases.

It turns out that all Σ electronic wave functions are characterized by an intrinsic transformation property under the operation σ. Furthermore, this transformation property is customarily indicated by a superscript attached to the Σ label; i.e., we define Σ⁺ and Σ^- states, such that

$\sigma_v|\Sigma^\pm\rangle = \pm |\Sigma^\pm\rangle.$ (2.7)

It further turns out that all doubly-degenerate orbital electronic states, i.e., those with |Λ| > 0, are not characterized by an intrinsic transformation property under the operation σ. Nevertheless, explicit transformation equations for such functions can be written down once a set of phases for the wave functions has been chosen. To obtain correct answers in any calculation, of course, it is necessary that the phase choice implicit in one's transformation equations be consistent with the phase choice implicit in one's expressions for the various matrix elements.

Consider now the two-election atomic configuration np n′p. This configuration also gives rise to an orbital P state, which in turn gives rise to diatomic-molecule orbital Σ and Π states. The wave functions for these diatomic-molecule states are

eq 2.08 (2.8)

if we use the phase conventions of Condon and Shortley [7] (p. 76). These wave functions transform as follows under the symmetry operation σ (xz) (see table 3)

eq 2.09 (2.9)

which can be written as

(2.10)

We note in passing that the Σ state of (2.4) is a Σ⁺ state, while that of (2.8) is a Σ^- state.

It turns out that electronic orbital wave functions |LΛ⟩ having phase factors consistent with those of Condon and Shortley [7], and arising from atomic states of even parity, all transform according to (2.10), while wave functions having such phase factors, and arising from atomic states of odd parity, all transform according to (2.6). (The parity of an atomic state is determined by its behavior under the inversion operation i in table 3.)

There are several complications which arise in deciding whether to use (2.6) or (2.10). First, wave functions of the type |LΛ⟩ are most often used in discussing Rydberg states. Under these circumstances, many of the electrons in the molecule are assigned to the "core" and are not considered explicitly. Since the core plays the role of the atomic nucleus, one must consider the parity of the atomic state which corresponds to the diatomic-molecule electronic wave function involving only electrons outside the core. Second, L is often not a good quantum number, so that no particular value of L suggests itself for use in (2.6) or (2.10). Under these circumstances one can often obtain consistent and correct results for states with |Λ| > 0 by arbitrarily giving L some value and arbitrarily choosing one of the two relations (2.6) or (2.10) to represent the transformation properties. This apparently casual choice of signs actually causes no difficulty. Of course, it does require a particular phase choice for the wave functions, which must be consistent with the phase choice implicit in any matrix element expressions used. However, when L is not a good quantum number, matrix elements involving the electronic orbital part of the wave function are not evaluated explicitly (they are left as adjustable parameters). Hence, contradictions between matrix elements and transformation properties are not introduced (see sect. 2.8). Finally, the transformation properties of Σ^± states are determined by eq (2.7). Sometimes, however, it is convenient to incorporate a Σ state into a transformation scheme utilizing eq (2.6) or (2.10). It is then necessary to correlate the choice of sign and the choice of L to obtain the proper Σ state transformation properties.

Since the transformation properties of the electronic spin basis set functions |SΣ⟩ and of the rotational basis set functions |JΩ⟩ are more difficult to illustrate with simple examples than are the transformation properties of the electronic basis set functions |LΛ⟩, we shall merely state the final results here: The functions |SΣ⟩ and |JΩ⟩, when chosen to have phase factors consistent with those of Condon and Shortley [7], transform like functions |LΛ⟩ of even parity [l5,16]. We can thus summarize the effect of the operation σ on the various basis set functions described in chapter 1 by the following equations:

(2.11a)

(2.11b)

(2.11c)

where (2.11c) could be written more precisely as σ $|\Omega JM\rangle$ = (-1) ^J-Ω $|-\Omega JM\rangle$ , and where the + or - sign must be used in (2.11a) if the electronic state of the molecule correlates with a united atom state of even or odd parity, respectively. (In homonuclear diatomic molecules, this correlation is straightforward: g and u diatomic-molecule electronic states correlate with united atom states of even and odd parity, respectively.) The transformation properties given in (2.11) are, of course, consistent with the matrix elements given in (1.13) and (1.14). The transformation properties of the complete basis set functions are seen from eqs (2.11) to be

(2.12)

It was pointed out in sect. 2.2 that when the symmetry operation σ(xz) acts on a complete basis set function (corresponding to both the nonrotational and the rotational part of the problem), then its net effect is equivalent to that obtained when the laboratory-fixed coordinates of all the particles in the molecule are replaced by their negatives. States of definite parity transform into themselves or into their negatives under this operation. It can easily be seen by application of (2.12) that functions of the form $|L\Lambda S\Sigma;\Omega JM\rangle$ ± $|L-\Lambda S-\Sigma; -\Omega JM\rangle$ have a definite parity. We next consider two examples of the determination of the parity of rotational energy levels.