Fock matrix integrals

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. 

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 ... 

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 ... 

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... 

Some convergence problems are due to numerical accuracy problems. Many programs use reduced accuracy integrals at the beginning of the calculation to save CPU time. However, this can cause some convergence problems for difficult systems. A course DFT integration grid can also lead to accuracy problems, as can an incremental Fock matrix formation procedure. 

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... 

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

The developers of ZINDO found that the parameters required to reproduce orbital energy orderings and UV spectra are different from those required to reproduce accurate structures by geometry optimization. They introduced anew pair of parameters, called the overlap weighting factors, to account for this. These parameters are provided in HyperChem in the Semi-empirical Options dialog box. Their effect is to modify the resonance integrals for the off-diagonal elements of the Fock matrix. 

Evaluate the integrals. In a conventional algorithm, they are stored on disk and read in for each iteration. In a direct algorithm, integrals are computed a few at a time as the Fock matrix is formed. 

The S matrix contains the overlap elements between basis functions, and the F matrix contains the Fock matrix elements. Each element contains two parts from the Fock operator (eq. (3.36)), integrals involving the one-electron operators, and a sum over... 

To construct the Fock matrix, eq. (3.51), integrals over all pairs of basis functions and the one-electron operator h are needed. For M basis functions there are of the order of of such one-electron integrals. These one-integrals are also known as core integrals, they describe the interaction of an electron with the whole frame of bare nuclei. The second part of the Fock matrix involves integrals over four basis functions and the g two-electron operator. There are of the order of of these two-electron integrals. In conventional HF methods the two-electron integrals are calculated and saved before the... 

Form the Fock matrix as the core (one-electron) integrals + the density matrix times the two-electron integrals. 

The first and second derivatives of the energy with respect to the X variables ( 0) and "(O)) can be written in term of Fock matrix elements and two-electron integrals in the MO basis. For an RHF type wave function these are given as... 

The gradient of the energy is an off-diagonal element of the molecular Fock matrix, which is easily calculated from the atomic Fock matrix. The second derivative, however, involves two-electron integrals which require an AO to MO transformation (see Section 4.2.1), and is therefore computationally expensive. 

The above integral screening is even more advantageous if the Fock matrix is formed incrementally. Consider two sequential density and Fock matrices in the iterative procedure (eq. (3.52)). 

Rewriting eq. (3.51) with semi-empirical labels gives the following expres.sion for a Fock matrix element, where a two-electron integral is abbreviated as (LLv a), (eq. (3.56)). 

The Huckel methods perform the parameterization on the Fock matrix elements (eqs. (3.50) and (3.51)), and not at the integral level as do NDDO/INDO/CNDO. This means that Huckel methods are non-iterative, they only require a single diagonalization of the Fock (Huckel) matrix. The Extended Huckel Theory (EHT) or Method (EHM), developed primarily by Hoffmann again only considers the valence electrons. It makes use of Koopmans theorem (eq. (3.46)) and assigns the diagonal elements in the F... 

The neglect of all three- and four-centre two-electron integrals reduces the constructioi of the Fock matrix from a formal order of to M. However, the time required fo... 

The first two terms involve products of the density matrix with derivatives of the atomic integrals, while the two last terms can be recognized as derivatives of the density matrix times the Fock matrix (eq. (3.51)). 

An analogous equation holds for the spin-down Fock matrix. The two-electron integrals in round brackets are defined by chemist s (11122) rather than the usual physicist s (12112) notation as ... 

In step 3, a criterion of convergence may be introduced to terminate the iterations. Two other points should be mentioned instead of taking the correct Singles-CI matrix, we may resort to a simpler one, omitting single bi-electronic integrals and using only Fock-matrix elements as ... 

---