of the bonding of water with
C2v symmetry. An initial assumption is that the number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion. In a sense,
n atomic orbitals combine to form
n molecular orbitals, which can be numbered
i = 1 to
n and which may not all be the same. The expression (linear expansion) for the
i th molecular orbital would be: : \ \phi_i = c_{1i} \chi_1 + c_{2i} \chi_2 + c_{3i} \chi_3 + \cdots +c_{ni} \chi_n or : \ \phi_i = \sum_{r} c_{ri} \chi_r where \ \phi_i is a molecular orbital represented as the sum of
n atomic orbitals \ \chi_r , each multiplied by a corresponding coefficient \ c_{ri} , and
r (numbered 1 to
n) represents which atomic orbital is combined in the term. The coefficients are the weights of the contributions of the n atomic orbitals to the molecular orbital. The
Hartree–Fock method is used to obtain the coefficients of the expansion. The orbitals are thus expressed as
linear combinations of
basis functions, and the basis functions are single-
electron functions which may or may not be centered on the
nuclei of the component
atoms of the
molecule. In either case the basis functions are usually also referred to as atomic orbitals (even though only in the former case this name seems to be adequate). The atomic orbitals used are typically those of
hydrogen-like atoms since these are known analytically i.e.
Slater-type orbitals but other choices are possible such as the
Gaussian functions from
standard basis sets or the pseudo-atomic orbitals from plane-wave pseudopotentials. By minimizing the total
energy of the system, an appropriate set of
coefficients of the linear combinations is determined. This quantitative approach is now known as the Hartree–Fock method. However, since the development of
computational chemistry, the LCAO method often refers not to an actual optimization of the wave function but to a qualitative discussion which is very useful for predicting and rationalizing results obtained via more modern methods. In this case, the shape of the molecular orbitals and their respective energies are deduced approximately from comparing the energies of the atomic orbitals of the individual atoms (or molecular fragments) and applying some recipes known as
level repulsion and the like. The graphs that are plotted to make this discussion clearer are called correlation diagrams. The required atomic orbital energies can come from calculations or directly from experiment via
Koopmans' theorem. This is done by using the symmetry of the molecules and orbitals involved in bonding, and thus is sometimes called
symmetry adapted linear combination (SALC). The first step in this process is assigning a
point group to the molecule. Each operation in the point group is performed upon the molecule. The number of bonds that are unmoved is the character of that operation. This reducible representation is decomposed into the sum of irreducible representations. These irreducible representations correspond to the symmetry of the orbitals involved.
Molecular orbital diagrams provide simple qualitative LCAO treatment. The
Hückel method, the
extended Hückel method and the
Pariser–Parr–Pople method, provide some quantitative theories. ==See also==