Integral evaluation, semiempirical

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). 

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... 

Another school has also developed and attempted to understand the functional dependence of adsorption on heterogeneous surfaces on the vapor pressure and temperature. Various empirical or semiempirical equations were proposed [24-26] and used later to represent experimental data and to evaluate EADF by inverting Eq. (1), which belongs to the class of linear Fredholm integrals of the first kind [27]. 

Although these numbers seem large, this is still a considerable improvement over evaluating every possible integral, as would be undertaken in ab initio HF theory. Most modem semiempirical models are NDDO models. After examining the differences in their formulation, we will examine their performance characteristics in some detail in Section 5.6. 

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

The two-electron integrals (Equation 6.32) are determined from atomic experimental data in the one-center case, and are evaluated from a semiempirical multipole model in the two-center case that ensures correct classical behavior at large distances and convergence to the correct one-center limit. Interestingly, this parameterization results in damped effective electron-electron interactions at small and intermediate distances, which reflects a (however less regular) implicit partial inclusion of electron correlation (Thiel, 1998). In this respect, semiempirical methods go beyond the HF level, and may accordingly be superior to HF ab initio treatments for certain properties that have a direct or indirect connection to the parameterization procedure. 

Klopman (1964) has formulated a self-consistent semiempirical formulation. Other formalisms have been given by Pople, Santry, and Segal (1965) and Kaufman (1965). Katagiri and Sandorfy (1966) have presented also a similar formulation, with particular emphasis on the evaluation of integrals and the interpretation of ionization potentials and electronic spectra of saturated hydrocarbons. 

With ab initio calculations, the dependence of the cost of SCF calculations on the size of the basis set is considerably more prohibitive than it is with semiempirical calculations. With semiempirical methods, the evaluation of the necessary Integrals over basis set functions is very fast, so that the major portion of the computer... 

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