The General LCAO Case

In later chapters we will be concerned with the LCAO model. Suppose we have a set of n atomic orbitals i(r), X2( ) d a normalized LCAO orbital 

I will assume that the atomic orbitals x, are normalized, but not generally orthogonal. That is to say, 

By analogy with the discussion of the previous section, the electron density for a single electron in is given by 

The 2 s and the 1 are called occupation numbers. In standard molecular orbital theory, occupation numbers are 0, 1 or 2 and they tell us the occupancy of a given orbital. 

The integrals involving atomic orbitals are often collected together into a matrix called the overlap matrix S 

---

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

The classic case distinguishes between an atomic core, which is essentially unperturbed by bonding, and a valence shell whose content may be accessible to bond formation. Since we suppose this simplifying assumption to be maintained in the MO treatment, an atomic orbital belonging to the valence shell will be termed a valence atomic orbital (VAO). For the construction of MOs, we utilize the following general results of the MO/LCAO model ... 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

The generalization of these results, to the LCAO-MO-MC-SCF case, can be found in the paper of Kari and Sutcliffe.31... 

The general analysis of Rudenberg s approximation in the HF LCAO method for molecules [218] and solids [223] has shown that EHT and zero differential overlap (ZDO) approximations can be considered as particular cases of Rudenberg s integral approximation. ZDO methods, considered in the next section, were applied to a wide class of molecules and solids, from purely covalent to purely ionic systems. Therefore, they are more flexible compared to the MR approximation, which is more appropriate for ionic systems. 

If there is a translational (or more generally any periodic) symmetry in an infinite solid or polymers the infinite cyclic hypermatrix (which one obtains in any LCAO theory with periodic boundary conditions) can be brought with the aid of a simple unitary transformation into a block-diagonal form /I,10/, The order of these blocks is (in the ab initio case) equal to the number of basis functions in the unit cell. In this vi/ay the original hypermatrix equation splits into N+1 matrix equations if N+1 denotes the number of blocks (unit cells). Each such equation has an index which denotes the serial number of the matrix block to which the equation belongs. If this... 

In the more general case of several LCAOs, where P has been calculated according to the occupation numbers, we have... 

Basis set dependence is important. The results in Table 16.1 were obtained for HF-LCAO calculations on pyridine. In each case, the geometry was optimized As a general rule, ab initio HF-LCAO calculations with small basis sets tend to underestimate the dipole moment, whilst extended basis sets overestimate it A treatment of electron correlation usually brings better agreement with experiment. 

That said above does not mean that a semiempirical parameterization based on the HFR MO LCAO scheme and valid for a certain narrow class of compounds or even for a specific purpose cannot be built. It is done for example in [69] for iron(H) porphyrins. But in a more general case there is no way to arrive to any definite conclusion [76] about the validity of a semiempirical parameterization in the HFR context. On the other hand we have to mention that the semiempirical method ZINDO/1 [77] which allows for... 

The molecular Schrodinger equation can be solved exacdy for the case of Hj when VAB is simply the sum of two hydrogen ion potentials. In general, however, an exact solution is not possible. Following the well-worn tracks of MO theory we look instead for an approximate solution that is given by some linear combination of atomic orbitals (LCAO). Considering the AB dimer illustrated in Fig. 3.1 we write... 

For clarity, without loss of generality, it is easier to continue the development explicitly for the case of a two-term LCAO-MO thus 7.1-4 takes the form... 

In general, the orbitals in this method are expanded as linear combinations of a basis set of Slater functions in the same spirit as in the LCAO-MO-SCF method. However, in the present case the orbitals are essentially localized, and a description such as equation (67) is clearly equivalent to a linear combination of a great many VB structures, both covalent and ionic. Thus, in the case of methane this should provide a very good description of the ground state, particularly of the potential surfaces for such processes as... 

In general, substituents are neither purely inductive nor purely mesomeric rather they possess some inductive as well as some mesomeric character. From Equations (2.19) and (2.24) it is seen that inductive as well as mesomeric effects on the HOMO and LUMO are proportional to the square of the LCAO coefficient at the substituted center g. It is therefore difficult to differentiate between these two effects from the absorption spectra. This is possible, however, by means of the MCD spectra that depend essentially on the difference AHOMO - ALUMO. (Cf. Section 3.3.) In the case of uncharged (4N-l-2)-eIectron perimeters and especially benzene, perturbations due to purely inductive substituents always yield AHOMO = ALUMO to first order in perturbation theory. Due to the energy difference in the denominator of Equation (2.24), however, AHOMO and ALUMO can be quite different in the case of mesomeric substituents, depending on the energy... 

---