LCAO approach

These concepts are used in the simpler linear combination of atomic orbitals (LCAO) approach. We construct a molecular orbital by starting with the wavefunctions of isolated atoms and take a linear combination of these wavefunctions to describe the state of the electron in the molecule. Consider, for example, a diatomic molecule consisting of atoms A and B. For sufficiently large distance between them, the molecular wavefunction can be written as i/f+ = i/ a + Ab = i/ a — where iff a and iff are the electron wavefunctions 

---

In our treatment of molecular systems we first show how to determine the energy for a given iva efunction, and then demonstrate how to calculate the wavefunction for a specific nuclear geometry. In the most popular kind of quantum mechanical calculations performed on molecules each molecular spin orbital is expressed as a linear combination of atomic orhilals (the LCAO approach ). Thus each molecular orbital can be written as a summation of the following form ... 

Molecules and Clusters. The local nature of the effective Hamiltonian in the LDF equations makes it possible to solve the LDF equations for molecular systems by a numerical LCAO approach (16,17). In this approach (17), the atomic basis functions are constructed numerically for free atoms and ions and tabulated on a numerical grid. By construction, the molecular basis becomes exact as the system dissociates into its atoms. The effective potential is given on the same numerical grid as the basis functions. The matrix elements of the effective LDF Hamiltonian in the atomic basis are given by... 

Generalization of this one determinant function to linear combinations of Slater determinants, defined for example as these discussed in the previous section 5.2, is also straightforward. The interesting final result concerning m-th order density functions, constructed using Slater determinants as basis sets, appears when obtaining the general structure, which can be attached to these functions, once spinorbitals are described by means of the LCAO approach. 

The basic assumption of the MO/LCAO approach is that a wave function of the type... 

The virial ratio is, as we noted above, 1.3366 for the separate-atom AO basis MO calculation, i.e. not 1.0. Now within the confines of the linear variation method (the usual LCAO approach) there is no remaining degree of freedom to use in order to constrain the virial ratio to its formally correct value (or indeed to impose any other constraint). Thus imposing the correct virial ratio on the linear variation method is, in this case, not possible without simultaneously destroying the symmetry of the wave function. Only by optimising the non-linear parameters can we improve the virial ratio as the above results show. Even at this most elementary level, the imposition of various formally correct constraints on the wave function is seem to generate contradictions. 

However, solving such an equation for a solid is something of a tall order because exact solutions have not yet been found for small molecules and even a small crystal could well contain of the order of lO atoms. An approximation often used for smaller molecules is that combining atomic wave functions can form the molecular wave functions. This linear combination of atomic orbitals (LCAO) approach can also be applied to solids. 

The MO delocalization energy (DE) calculated using a simple LCAO approach indicated that pyran-4-one (DE, 2.868/8) possesses greater aromaticity than benzene (DE 2/8) (62CCC1242). Subsequent more refined calculations gave substantially lower values, but still indicated considerable aromaticity for pyran-4-one. Values of DE, DEsp and aromaticity indices for pyran-4-one and chromone which were obtained using various MO methods are presented in Table 9. 

To evaluate Eq. (1) in a way that is analytic, robust and variational, it is first necessary to divide the density among the atoms. That is easy in any LCAO approach, where the only problem is to how to assign centers to the cross (two-center) terms of the density. The computationally most efficient way is to multiply each atomic orbital by a /8, where a is the appropriate Xa scaling factor for that atom. Then the scaled, and thus partitioned, density may be written... 

The General Hartree-Fock Equations Separation of Space and Spin the MO-LCAO-approach... 

For solids with more localized electrons, the LCAO approach is perhaps more suitable. Here, the starting point is the isolated atoms (for which it is assumed that the electron-wave functions are already known). In this respect, the approach is the extreme opposite of the free-electron picture. A periodic solid is constructed by bringing together a large number of isolated atoms in a maimer entirely analogous to the way one builds molecules with the LCAO approximation to MO (LCAO-MO) theory. The basic assumption is that overlap between atomic orbitals is small enough that the extra potential experienced by an electron in a solid can be treated as a perturbation to the potential in an atom. The extended- (Bloch) wave function is treated as a superposition of localized orbitals, centered at each atom ... 

For lattices with more than one atom per lattice point, combinations of Bloch sums have to be considered. In general, the LCAO approach requires that the result be the same number of MOs (COs in sohds) as the number of atomic orbitals (Bloch sums in sohds) with which was started. Thus, expressing the electron-wave functions in acrystaUine sohd as linear combinations of atomic orbitals (Bloch sums) is really the same approach used in the 1930s by Hund, Mulliken, Htickel, and others to construct MOs for discrete molecules (the LCAO-MO theory). 

Most chemists are well acquainted with LCAO-MO theory. The numbers of atomic orbitals, even in large molecules, however, are miniscule compared to a nonmolecular solid, where the entire crystal can be considered one giant molecule. In a crystal there are in the order of 10 atomic orbitals, which is, for all practical purposes, an infinite number. The principle difference between applying the LCAO approach to solids, versus molecules, is the number of orbitals involved. Fortunately, periodic boundary conditions allow us to smdy solids by evaluating the bonding between atoms associated with a single lattice point. Thus, the lattice point is to the solid-state scientist, what the molecule is to the chemist. 

It is possible, by using the LCAO approach or a pseudopotential approach, to make a calculation of energy bands for the distorted lattice. There are still two atoms per primitive cell, so no serious difficulty is encountered. The sum of the... 

This does not mean that the LCAO approach of the type we have used is incorrect or not useful. Recent applications of LCAO theory, based only upon electron orbitals that are occupied in the free atom, have been made to the study of simple metals (Smith and Gay, 1975), noble-metal surfaces (Gay, Smith, and Arlinghaus, 1977), and transition metals (Rath and Callaway, 1973). In fact, the LCAO approach seems a particularly effective way to obtain self-consistent calculations. The difficulty from the point of view taken in this book is that, as with many other band-calculational techniques, LCAO theory has not provided a means for the elementary calculations of properties emphasized here, but pseudo-potentials have. 

---