Fock matrices

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... 

Choose th e DIIS SCF con vergen ce accelerator to poten tially speed up SCF eon vergen ee. DIIS often reduees the number of iteration s required to reach a con vergen ec limit. However, it takes memory to store the Fock rnalriees from th c previous iteration s an d this option may increase th e com pu tation a I time for individual iteration s because th e Fock m atrix h as to be calcu la ted as a lin car corn -biriation of the current Fock matrix and Fock matrices from previous iteration s. 

The extent to which this condition does not occur is a m easiire of deviance from self-con sisten cy. Th e DIIS melh od ii ses a lin ear combination of previoii s Fock matrices to predict a Fock matrix that minimizes [I - K. This new Rich matrix is then used by the SCF calculation. 

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

III an SCF calculation. many iterations may beneetled to achieve SCr con vergeiice. In each iteration all the two-electron integrals are retrieved to form a Fock matrix. Fast algorith m s to retrieve the two-cicetron s integrals arc important. 

So only the two-electron integrals wilh p. > v. and I>aand [p.v > 7.a need to he computed and stored. Dp.v.la on ly appears m Gpv, and Gvp, w hereas ih e original two-electron integrals con tribute to other matrix elemen is as well. So it is m iich easier to form ih e Fock matrix by using the siipermairix D and modified density matrix P th an the regular format of the tw O-electron in tegrals and stan dard den sity m atrix. 

The elenienis of the CXlDO/2 Fock matrix (for the RHFeasc i thus hccorn e... 

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 ... 

I he Fock matrix is a ff x ff square matrix that is symmetric if real basis functions are used. Tile Roothaan-Hall equations (2.149) can be conveniently written as a matrix equation ... 

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. 

I he Fock matrix must next be transformed to F by pre- and post-multiplying by... 

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 ... 

---