The SCF MO Treatment of Polyatomic Molecules

The purely electronic nonrelativistic Hamiltonian for a polyatomic molecule is (in atomic 

How are polyatomic MOs classified As might be expected, the MOs of a polyatomic molecule show the same kinds of possible symmetry behavior as the overall electronic wave function does (Section 15.2) for the proof, see C. C. J. Roothaan, Rev. Mod. Phys., 

69 (1951). The MOs are therefore classified according to the symmetry species of the molecular point gronp. For example, the MOs of H2O have the possible symmetry species fli, fl2. bi, and 2 of (15.3). Lowercase letters are used for MO symmetry species. To distinguish MOs of the same symmetry species, we number them in order of increasing energy. Thus the lowest three MOs of water are called Ifli, 2ui, and 3ai. This nomenclature is similar to that of the third column in Table 13.1. 

For SCF MO calculations on closed-shell states of molecules and atoms, electrons paired with each other are almost always given precisely the same spatial orbital function. A Hartree-Fock wave function in which electrons whose spins are paired occupy the same spatial orbital is called a restricted Hartree-Fock (RHF) wave function. (The unmodified term Hartree-Fock wave function is understood to mean the RHF wave function.) 

Most molecular quantum-mechanical methods, whether SCF, Cl, perturbation theory (Section 16.3), coupled cluster (Section 16.4), or density functional (Section 16.5), begin the calculation with the choice of a set of basis functions Xn which are used to express the MOs (pi as = IiiCriXr [Eq. (14.33)]. (Density-functional theory uses orbitals called Kohn-Sham orbitals / P that are expressed as (pf = 1,iCriXn see Section 16.5.) The use of an adequate basis set is an essential requirement for success of the calculation. 

Most polyatomic molecules have a closed-shell ground state here, the MO wave function is a single Slater determinant. For states arising from open-shell configurations, the MO wave function may require a linear combination of a few Slater determinants. 

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