Symmetry Properties of Hunds Case a Basis Functions

A detailed discussion of the symmetry properties of Hund s case (a) basis functions, InJOASE), is presented in Section 3.2.2.1 followed in Section 3.2.2.2 by a brief summary of the symmetry properties of other classes of basis functions, such as those that are expressed in ion-core Rydberg electron) form and those that specify molecule-fixed R-axis (Sr, 1r, sr, jR) rather than z-axis projection quantum numbers. 

1 Symmetry Properties of Hund s Case (a) Basis Functions 

Before using the crv operator to classify symmetrized basis functions as even versus odd, it is necessary to insert a note of caution about four distinct types of symmetry classifications, each frequently and misleadingly called parity, and each associated with the effect of the crv operator on different parts of the basis function. 

These four distinct, x -related symmetry classifications are (i) total parity (effect of crv on the complete electronic-vibration-rotation wavefunction) (ii) e//-symmetry (total parity of the complete wavefunction exclusive of a (—l) 7 or (—l)-7-1/2 rotational factor, respectively for molecules with an even or odd number of electrons) (Hi) the intrinsic even/odd symmetry under crv(xz) of the electronic wavefunction, exclusively for A = 0 states, E+ or E and (iv) the high-J (or case (b)) limiting behavior of the spatial coordinates of the electronic wavefunction under av xz) reflection in a plane that contains the internuclear 

The (Tv operator corresponds to a reflection through the molecule-fixed xz plane. If it operates on the spatial and spin coordinates of all electrons, the nuclear displacement vectors, and the rotational wavefunction (expressed in terms of Euler angles, which specify the orientation of the molecule-fixed coordinate system relative to a laboratory-fixed coordinate system), then the eigenvalues of av label the total parity, , of a rotating molecule basis function, 

---