Molecular orbitals are composed of linear combinations of atomic orbitals. The shape of the atomic orbitals are precisely those of the hydrogen atom. However, the radial extent of the electrons in the atoms of the periodic table are determined independently by study of each atom. A collection of parameterized radial functions is known as a basis set. With a basis set in hand, one has the input needed to construct the linear combinations of atomic orbials (LCAO). Just as in the approximate treatment of atoms using the Hartree approximation, the MO approach uses a mean field approach. This means that each atom is treated separately and the Schrodinger equation is solved for the motion of each atom in the field of all of the other atoms. This approach reduces the many-electron problem to a collection of one-electron equations. Once linearized in this way the equations can be solved by a standard method known as matrix diagonalization.
The secular detrminant
The energy of the molecular orbitals can be calculated by inserting the LCAOs into the average energy expression.
The expression can be expanded in terms that represent the self-energies (Coulomb integrals) and interaction energies (resonance integrals). The overlaps are also calculated in the denominator. The variational principle is applied by taking derivatives with respect to the coefficients of the LCAOs. The result is a set of linear equations of terms with the form Hij - ESij. When these terms are written in matrix form as equations for the coefficients of the LCAOs we constructed a determinant, which must be equal to zero if the set of equations for the coefficients has a solution. THis determinant is called the secular determinant. The general form is discussed below.
Properties of molecular orbitals
The matrix diagonalization procedure results in a set of orbitals that are orthogonal to one another. The electrons can be placed in these orbitals in pairs or according to Hund's rule if there are degenerate levels. There are as many MOs in a solution as AOs used in the basis set. The number of occupied MOs is determined by the number of electrons. The remaining MOs are not occupied. The most basic implementation of MO theory is the Hartree-Fock approach. This is basically what has been outlined up to this point using a one-electron approximation and the assumption that each electron can be treated as a separate quantity in the field of all of the other electrons. There is not a unique solution of such a system. Therefore, the Hartree-Fock approach includes a self-consistent field (SCF) method that searches systematically for the lowest energy solution to the approximate set of equations. The variational theorem states that any approximate energy calculation for an approximate wave function will always be greater than the true energy. This theorem means that the HF method will always approach the true enegy from above. It also provides a means to solve for the coefficients on each iteraction of the SCF. The procedure required is to take the derivative of the LCAO with respect to the individual coefficients of the AOs. These derivatives provide a set of coefficients that minimizes the energy. On each iteration of the self-consistent field one needs to take a derivative to obtain the coefficients. Then one uses those coefficients to calculate the energy by the matrix diagonalization. Then one adjusts the coefficients and recalculates the energy. When the energy changes by less than one part in 10-6 the calculation is considered to be converged and the energy is reported.
