An Alternative Formulation of the Variational Problem

The objective is to minimize the total energy as a function of the molecular orbitals, subject to the orthogonality constraint. In the above formulation, this was handled by means of Lagrange multipliers. The final Fock matrix in the MO basis is diagonal, with the diagonal elements being the orbital energies. During the iterative sequence, i.e. before the orbitals have converged to an SCF solution, the Fock matrix is not diagonal. Starting from an initial set of molecular orbitals, the problem may also be formulated as a rotation of the orbitals (unitary transformation) in order to make the operator diagonal.10 Since the operator depends on the orbitals, the procedure again becomes iterative.

The orbital rotation is given by a unitary matrix U, which can be written as an exponential transformation.

The X matrix contains the parameters describing the unitary transformation of the Mbasis orbitals, being of the size of Mbasis x Mbasis. The orthogonality is incorporated by requiring that the X matrix is antisymmetric, xij _ -xji.

UfU = (ex ) (ex) = (ex f )(ex) = (e- x )(ex) = 1 (3.62)

Normally the orbitals are real, and the unitary transformation becomes an orthogonal transformation. In the case of only two orbitals, the X matrix contains the rotation angle a, and the U matrix describes a 2 x 2 rotation. The connection between X and U is illustrated in Section 16.2 (Figure 16.3) and involves diagonalization of X (to give eigenvalues of ± ia), exponentiation (to give complex exponentials that may be written as cos a ± isin a), followed by back-transformation.

In the general case, the X matrix contains rotational angles for rotating all pairs of orbitals.

It should be noted that the unoccupied orbitals do not enter the energy expression (eq. (3.32)), and a rotation between the virtual orbitals can therefore not change the energy. A rotation between the occupied orbitals corresponds to making linear combinations of these, but this does not change the total wave function or the total energy. The occupied-occupied and virtual-virtual blocks of the X matrix can therefore be chosen as zero. The variational parameters are the elements in the X matrix that describe the mixing of the occupied and virtual orbitals, i.e. there are a total of Nocc x (Mbasis - Nocc) parameters. The goal of the iterations is to make the off-diagonal elements in the occupied-virtual block of the Fock matrix zero. Alternatively stated, the off-diagonal elements are the gradients of the energy with respect to the orbitals, and the stationary condition is that the gradient vanishes.

Using the concepts from Chapter 16, the variational problem can be considered as a rotation of the coordinate system. In the original function space, the basis functions, the Fock operator depends on all the Mbasis functions, and the corresponding Fock matrix is non-diagonal. By performing a rotation of the coordinate system to the molecular orbitals, however, the matrix can be made diagonal, i.e. in this coordinate system the Fock operator only depends on Nocc functions.

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