Molecular orbital method secular equations

In his valuable paper Molecular Energy Levels and Valence Bonds Slater developed a method of formulating approximate wave functions for molecules and constructing the corresponding secular equations.1 Let a,b, repreamt atomic orbitals, each occupied by one valence electron, and a and 0 represent the electron spin functions for spin orientation -f i and — J, respectively. Slater showed that the following function corresponds to a valence-bond structure with bonds a-----b, c---d, and so forth ... 

We have already seen examples of semiempirical methods, in Chapter 4 the simple Hiickel method (SHM, Erich Hiickel, ca. 1931) and the extended Hiickel method (EHM, Roald Hoffmann, 1963). These are semiempirical ( semi-experimental ) because they combine physical theory with experiment. Both methods start with the Schrodinger equation (theory) and derive from this a set of secular equations which may be solved for energy levels and molecular orbital coefficients (most efficiently... 

We carried out non-spin-polarized molecular orbital calculations using a DV-Xa code called SCAT (7). In the DV-Xa method, matrix elements in the secular equation are derived from the weighted sum of integrand values at sampling points. Hence numerical atomic orbitals can be used as basis sets. These atomic orbitals were generated on each iteration in atomic-like potentials derived from spherically averaging the molecular potential around each nucleus (7). The numerical wave functions thus obtained can be used efficiently for the molecular orbitals and have a practical character making them suitable for... 

The most useful method for background studies is based on the MNDO model [18], which is a valence-electron self-consistent-field (SCF) MO treatment. It takes up a minimal basis of atomic orbitals (AOs) and the NDDO integral estimation. The molecular orbitals, ([), , and the corresponding orbital energies, s , are obtained from the linear combination of the AO base functions, cjaM, and the solution of the secular equations with Suv ... 

As an example let us consider the system of four equivalent univalent atoms at the corners of a square, discussed in the previous section by the valence-bond method. The secular equation for a one-electron wave function (molecular orbital), expressed as a, linear combination of the four atomic orbitals ua, ub, uc, and ud, is... 

The LCAO method can be used to obtain molecular orbitals for any diatomic molecule from the appropriate atomic orbitals. Minimizing the energy leads to an equation known as the secular determinant, from which the best energies for the molecular orbitals can be calculated. 

We now turn our attention to a method of deriving molecular orbitals that does not require evaluation of two-electron integrals. This is the method used with the approximate techniques known as extended Hiickel and Hiickel theory. It is also used in perturbational molecular orbital theory, which we present later in this chapter. This method takes advantage of secular determinants, a way in which to represent the Schrbdinger equation as a di-agonalizable matrix. 

Let us therefore suppose that we have set up an approximate potential field V for the molecule, in which an electron is to move. This field might, for example, be that obtained by the superposition of the Hartree fields of the component atoms. Let us also suppose that we are given a set of functions atomic orbitals of the various atoms of the molecule, although any set of independent functions could be used. The approximate orbitals can then be found by the method of trial eigenfunctions, using zero-order functions. The approximate energies of the molecular orbitals will therefore be the roots of the secular equation... 

---