Fock matrix construction

Given these characteristics, it is evident that large-scale semiempirical SCF-MO calculations are ideally suited for vectorization and shared-memory parallelization the dominant matrix multiplications can be performed very efficiently by BLAS library routines, and the remaining minor tasks of integral evaluation and Fock matrix construction can also be handled well on parallel vector processors with shared memory (see Ref. [43] for further details). The situation is less advantageous for massively parallel (MP) systems with distributed memory. In recent years, several groups have reported on the hne-grained parallelization of their semiempirical SCF-MO codes on MP hardware [76-79], but satisfactory overall speedups are normally obtained only for relatively small numbers of nodes (see Ref. [43] for further details). 

Foster IT, TUson JL, Wagner AF, Shepard RL, Harrison RJ, KendaU RA, Littlefield RJ (1996) Toward high-performance computational chemistry I. Scalable Fock matrix construction algorithms. J Comput Chem 17(1) 109-123. doi 10.1002/ (sici)1096-987x(19960115)17 K109 aid-jcc9>3.0.co 2-v... 

Thus, the scalar basis involves about 20% fewer real quantities than the 2-spinor basis, and therefore less work in the Fock matrix construction. This applies to an uncontracted basis set. 

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... 

This is an occupied-virtual off-diagonal element of the Fock matrix in the MO basis, and is identical to the gradient of the energy with respect to an occupied-virtual mixing parameter (except for a factor of 4), see eq. (3.67). If the determinants are constructed from optimized canonical HF MOs, the gradient is zero, and the matrix element is zero. This may also be realized by noting that the MOs are eigenfunctions of the Fock operator, eq. (3.41). 

H jj is the energy of the molecules of the microenvironment (this term does not depend on the embedded electron density and is not involved in the construction of the Fock matrix). 

After the HFW integrals have been assembled, we then move on to the self-consistent field (SCF) procedure. For the most part this is the same as the HF version (10), with the exception of constructing the Fock matrix. The Fock matrix elements for an unrestricted HFW calculation are analogous to their HF counterparts and are given by... 

This is a quantity which can be easily constructed given a set of molecular orbitals (the coefficients C ) and a precalculated set of atomic orbital integrals. At this point, the Hartree-Fock equations have been reduced to a matrix eigenvector problem, FC = SCe, but not in a computationally convenient form. Following the analysis leading to equation 84, we first define the transformed Fock matrix as... 

To construct the Fock matrix, one must already know the molecular orbitals ( ) since the electron repulsion integrals require them. For this reason, the Fock equation (A.47) must be solved iteratively. One makes an initial guess at the molecular orbitals and uses this guess to construct an approximate Fock matrix. Solution of the Fock equations will produce a set of MOs from which a better Fock matrix can be constructed. After repeating this operation a number of times, if everything goes well, a point will be reached where the MOs obtained from solution of the Fock equations are the same as were obtained from the previous cycle and used to make up the Fock matrix. When this point is reached, one is said to have reached self-consistency or to have reached a self-consistent field (SCF). In practice, solution of the Fock equations proceeds as follows. First transform the basis set / into an orthonormal set 2 by means of a unitary transformation (a rotation in n dimensions),... 

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