The Hartree Fock equation can, in principle be solved by any standard method for solving integro-differential equations. For atoms, solutions of the HF equations reaching the HF limit is routine and can be carried out by numerical integration. However, for large molecules, solutions reaching the HF limit are not possible. In fact, the HF procedure leads to a complicated set of near-intractable, integro-differential equations, which can only be solved with any ease for a one-center problem.

This problem of the numerical ab initio theory was solved in 1951 when, independently, Roothaan and Hall linearized the HF problem by expanding each one-electron spin orbital (here since the spin* is treated separately, each spatial function) not as a product as a linear combination of finite, fixed basis functions, (with p = 1, 2, ..., and 2 A N, A corresponds to the number of fixed basis functions),:

As the energy is minimized, the coefficients, , will be optimized, while the basis functions remain unchanged.

Although any set of mathematically-suitable functions which spans the space of electron distribution could be used as basis functions, the concept of molecular orbitals as linear combination of atomic orbitals (LCAO-MO) have proved to be very useful. Moreover, each spin orbital can be described by more than just one basis function. This set of atomic orbitals, , is often normalized but generally not orthogonal and it is called basis set (See Chapter 7).

The matrix of the Lagrange undetermined multipliers, , can be diagonalized by a unitary transformation resulting in the Roothaan equations

,

in which the coefficients crs also were transformed (but we keep the notation crs).* These equations can be expressed in matrix form:

Fc = eSc,

where F = (Fvu) is the matrix of the Fock operator, e is a N_basis x N_basis matrix of the transformed Lagrange multipliers (orbital energies e ), S = (Svu) is the overlap matrix and c represents the vector of the coefficients.

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* A matrix U is unitary, if with (A linear operator, whose inverse is its adjoint, is called unitary; corresponds to the identity matrix). The inner product, , is in that case invariant. During a unitary transformation, , the eigenvalue of a matrix, M, will be preserved. The undetermined Lagrange multipliers, , form a Hermitian matrix, M. Therefore, this matrix can be transformed into a diagonal matrix: .

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The Roothaan equations allow the solution of the Hartree Fock equation using matrix techniques, since the these equations can be considered as a matrix representation of the Hartree Fock equation in a infinite basis. In practice, for computational efficiency, the number of basis functions is limited. Therefore, these calculations do not reach the HF limit. The Roothaan equation derived above cannot be solved like a simple matrix eigenvalue problem, because the overlap matrix, S, appears on the right hand side. This problem will be tackled in chapter 2.5.

Roothaan’s matrix procedure is, without any doubt, the most important method used for calculating electronic energies and lies at the heart of any ab initio program package. If the nucleus-nucleus repulsion energy is added, the total energy of the considered molecular system is obtained. In most cases, when referring the Hartree Fock method, it should be remembered that the Roothaan matrix method is the one applied in the HF picture.

last changes: 01.04.2008, AS
questions & comments to: axel.schulz@uni-rostock.de