Irreducible representations total molecular wave function

In addition, group theory can be used to assess when transition dipole moments must be zero. The product of the irreducible representations of the two wave functions and the dipole moment operator within the molecular point group symmetry must contain the totally symmetric representation for the matrix element to be non-zero (note that, if the molecule does not contain an inversion center, the operator r does not belong to any single irrep, except for the trivial case of Ci symmetry see Appendix B for more details). A consequence of this consideration is that, for instance, electronic transitions between states of the same symmetry are forbidden in molecules possessing inversion centers. 

Depending on the problem, i i, and may be atomic orbitals used to construct molecular orbitals, or they may represent two different electronic states of the same atom or molecule, etc. The energy, then, expresses the extent of interaction between the two wave functions i[<. and As was shown in Chapter 4, an integral will have a nonzero value only if the integrand is invariant to the symmetry operations of the point group, i.e., belongs to the totally symmetric irreducible representation. 

The exact electronic Hamiltonian H usually commutes with a number of additional operators Ap and the exact wave functions can then be chosen to be simultaneous eigenfunctions of H and Ap. For example, the commutation of H with the spin operators and 5 implies that the eigenftmctions of H can be chosen as eigenfunctions of the total and projected spins, and the commutation of H with the symmetry operators of the molecular point group implies that we can classify the wave functions according to the irreducible representations of this point group. 

In unrestricted Hartree-Fock (UHF) theory, the wave function is represented by a single Slater determinant - no restrictions are enforced on the total spin of the system and the wave function is not required to transform as an irreducible representation of the molecular point group. Since the alpha and beta spin orbitals are separately optimized, they will in general have different spatial forms. The resulting UHF wave function may therefore yield an energy lower than that of RHF theory see the examples in Section 5.4. Although we are concerned mainly with RHF theory, much of the theory presented in this chapter carries over to unrestricted wave functions with little or no modification. The relationship between RHF and UHF theories is discussed in more detail in Section 10.10. 

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