The matrix SCF equations

Once the matrix SCF equations have been set up and transformed to an orthogonal basis, the only numericad problem in their solution is the calculation of the eigenvalues and eigenvectors of a symmetric matrix. This is a problem which occurs in many branches of science, particularly those involving optimisation of some kind and has, consequently, received much attention. 

For the purposes of the solution of the matrix SCF equations it is most convenient to ask for the simplest possible solution. This is accomplished by requiring V to be symmetrical... 

As we shall shortly see when we come to actually implement the solution of the matrix SCF equations, the technique is to solve... 

Later, it will prove possible to base a whole class of interpolation and extrapolation methods on the formal properties of the solutions to the matrix SCF equations. 

The two main areas of the application of symmetry to the calculation of molecular electronic structure— calculation of integrals and symmetry blocking— can now be combined to generate an efficient way of solving the matrix SCF equations. 

Within the Hartree-Fock approximation, calculations on molecules have almost all used the matrix SCF method, in which the HF orbitals are expanded in terms of a finite basis set of functions. Direct numerical solution of the HF equations, routine for atoms, has, however, been thought too difficult, but McCullough has shown that, for diatomic molecules, a partial numerical integration procedure will yield very good results.102 In particular, the Heg results agree well with the usual calculations, and it is claimed that the orbitals are likely to be of more nearly uniform accuracy than in the matrix HF calculations. Extensions to larger molecules should be very interesting so far, published results are available for He, Heg, and LiH. 

If the shell structure is not conserved, the number of matrix elements to be included in the energy expression will be increased considerably and yield nonequivalent + and — orbitals for open-shell configurations. When the variational procedure is applied to the energy, one obtains equations that are essentially the same as the nonrelativistic equations [Eqs. (13)—(16) of Ref. 47]. Since all the deviations from the conventional SCF equations are included in Eqs. (44)-(49), the SCF equations for the TCMSs are omitted. 

The solution of the matrix HF equations can be facilitated by the use of a basis of symmetry-orbitals obtained from the basis functions by the projection operators of the point group. When symmetry-orbitals are used the degrees of freedom available to the variation principle are exposed as the number of different orbitals of each symmetry type. The matrix which defines the symmetry orbitals can be combined with the orthogonalisation matrix in the SCF procedure so that, once formed, this transformation adds nothing to the implementation of the process. Techniques and codes for the diagonalisation of blocked matrices have already been met in Appendix 12.B. 

In order to be sure that the original SCF equation is completely equivalent to the equation involving the density matrix it is useful to derive the SCF equation for C from the equation for R. 

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

The coefficients c i are given by the solution of the corresponding SCF equation (equation 2). d s the diagonal matrix with the MO energies. For consideration of Cl we must distinguish between closed-shell systems and open-shell systems. For closed-shell systems the restricted Hartree-Fock (RHF) formalism is applied, whereas for open-shell systems one has the choice between the unrestricted Hartree-Fock (UHF) or the restricted open-shell formalism (ROHF). The Fock matrix elements were formulated on the CNDO, INDO and NDDO level by Sauer et al. ... 

This reduces, of course, to the ordinary SCF equation FT = Te when F is time-independent otherwise T will acquire a time dependence and will introduce (through the time-dependent density in the G matrix) a coupling between the density fluctuations and the effective field. When the time-dependent part of h is specified it is therefore necessary to solve (12.6.13) iteratively until the fluctuations of F and p are self-consistent. There is thus a fully coupled time-dependent analogue of the SCF perturbation theory in Section 11.9. 

The significance of the Koopmans operator, whose expectation value is given expUcitly in (8.2.29) for any kind of wavefimction (exact or approximate), is now clear. According to (8.2.30), the elements of K coincide with those of —c, the matrix of Lagrangian multipliers in the MC SCF equations and they in turn are matrix elements of a 1-electron Hamiltonian containing the effective potential felt by any electron in the presence of the others. In full,... 

Once the format of the Fock matrix is known, the semiempirical molecular problem (and it is a considerable one) is finding a way to make valid approximations to the elements in the Fock matrix so as to avoid the many integrations necessary in ab initio evaluation of equations like Fij = J 4>,F4> dx. After this has been done, the matrix equation (9-62) is solved by self-consistent methods not unlike the PPP-SCF methods we have already used. Results from a semiempirical... 

In an ab initio method, all the integrals over atomic orbital basis functions are computed and the Fock matrix of the SCF computation is formed (equation (61) on page 225) from the integrals. The Fock matrix divides into two parts the one-electron Hamiltonian matrix, H, and the two-electron matrix, G, with the matrix elements... 

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