The Fock Matrix

The Fock matrix F(n) is defined as (Siegbahn et al, 1981 Jensen and Agren, 1986) 

The Fock matrix appears naturally in the calculation of the orbital part of the electronic gradient  

In the MCSCF case the undifferentiated Fock matrix is symmetric since the orbital optimization ensures that ffj = 2(FfJ — FfJ) = 0. The Fock matrix also appears in the calculation of expectation values of one-index transformed Hamiltonians (see Appendix F). 

The construction of the Fock matrix may be simplified in the MCSCF case when the orbital space is partitioned in an inactive, an active, and a secondary space. Siegbahn et al. (1981) have shown that the Fock matrix elements then become 

In these expressions we have used the Roothaan-Bagus integrals (Roothaan and Bagus, 1963) 

---

The basic self-consistent field (SCF) procedure, i.e., repeated diagonalization of the Fock matrix [26], can be viewed, if sufficiently converged, as local optimization with a fixed, approximate Hessian, i.e., as simple relaxation. To show this, let us consider the closed-shell case and restrict ourselves to real orbitals. The SCF orbital coefficients are not the... 

In order to form the Fock matrix ofan ah iniiio calculation, all the... 

If Lhc live iiul c pun lien i oiic-ceiiicr iwo-cIccLroii integrals are expressed by symbols such as Gss, Gsp, defiiietJ above, then the Fock matrix element contributions from the one-center two-elec-iron in icgrals are ... 

By replacing the superscripts a and (i by Pand tx, respectively, in th e above th ree eq u ation s. you can easily get three similar equations for the Fock matrix elements for beta orbitals. Similar expressions to the above for Fock matrix elements ol restricted Ilartree-Fock (RIIF) calculations can be generated by simply icplaeing 1- (or I P) by 1/2 P in the above equation s. 

The Fock matrix elements for a closed-shell system can be expanded as follows by substituting the expression for the Fock operator ... 

The elements of the Fock matrix can thus be written as the sum of core. Coulomb anc exchange contributions. The core contribution is ... 

When the Coulomb and exchange operators are expressed in terms of the basis functions and the orbital expansion is substituted for xu then their contributions to the Fock matrix element take the following form ... 

Lei us consider how we might solve the Roothaan-Hall equations and thereby obtain the molecular orbitals. The first point we must note is that the elements of the Fock matrix, u liich appear on the left-hand side of Equation (2.162), depend on the molecular orbital oetficients which also appear on the right-hand side of the equation. Thus an iterative pi oeedure is required to find a solution. 

In ub initio calculations all elements of the Fock matrix are calculated using Equation (2.226), ii re peifive of whether the basis functions ip, cp, formally bonded. To discuss the semi-empirical melh ids it is useful to consider the Fock matrix elements in three groups (the diagonal... 

With these approximations the Fock matrix elements for CNDO become ... 

In a closed-shell system, P = P) = P and the Fock matrix elements can be obtained by making this substitution. If a basis set containing s, p orbitals is used, then many of the one-centre integrals nominally included in INDO are equal to zero, as are the core elements Specifically, only the following one-centre, two-electron integrals are non-zero (/x/x /x/x), (pit w) and (fti/lfM/). The elements of the Fock matrix that are affected can then be written a." Uxllow s ... 

Having the Slater atomic orbitals, the linear combination approximation to molecular orbitals, and the SCF method as applied to the Fock matrix, we are in a position to calculate properties of atoms and molecules ab initio, at the Hartree-Fock level of accuracy. Before doing that, however, we shall continue in the spirit of semiempirical calculations by postponing the ab initio method to Chapter 10 and invoking a rather sophisticated set of approximations and empirical substitutions... 

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... 

Choose the DIIS SCF convergence accelerator to potentially speed up SCF convergence. DIIS often reduces the number of iterations required to reach a convergence limit. However, it takes memory to store the Fock matrices from the previous iterations and this option may increase the computational time for individual iterations because the Fock matrix has to be calculated as a linear combination of the current Fock matrix and Fock matrices from previous iterations. 

The preceding discussion means that the Matrix equations already described are correct, except that the Fock matrix, F, replaces the effective one-electron Hamiltonian matrix, and that Fdepends on the solution C ... 

The two equations couple because the alpha Fock matrix depends on both the alpha and the beta solutions, C and cP (and sim ilarly for the beta Fock matrix). The self-consistent dependence of the Fock matrix on molecular orbital coefficients is best represen ted, as before, via the den sity matrices an d pP, wh ich essen -tially state the probability of describing an electron of alpha spin, and the probability of finding one of beta spin ... 

Thus, HyperChem occasionally uses a three-point interpolation of the density matrix to accelerate the convergence of quantum mechanics calculations when the number of iterations is exactly divisible by three and certain criteria are met by the density matrices. The interpolated density matrix is then used to form the Fock matrix used by the next iteration. This method usually accelerates convergent calculations. However, interpolation with the MINDO/3, MNDO, AMI, and PM3 methods can fail on systems that have a significant charge buildup. 

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... 

In order to form the Fock matrix of an ab initio calculation, all the core-Hamiltonian matrix elements, H y, and two-electron integrals (pvIXa) have to be computed. If the total number of basis functions is m, the total number of the core Hamiltonian matrix elements is... 

So only the two-electron integrals with p > v, and X > a and [pv] > [Xa] need to be computed and stored. Dpv,Xa only appears in Gpv, and Gvp, whereas the original two-electron integrals contribute to other matrix elements as well. So it is much easier to form the Fock matrix by using the supermatrix D and modified density matrix P than the regular format of the two-electron integrals and standard density matrix. 

The Raffenetti integral format emphasizes the speed of computing the Fock matrix. 

---