Integral approximation, semiempirical

Semiempirical approaches to quantum chemistry are thus characterized by the use of empirical parameters in a quantum mechanical framework. In this sense, many current methods contain semiempirical features. For example, some high-level at initio treatments of thermochemistry employ empirical corrections for high-order correlation effects, and several advanced density functionals include a substantial number of empirical parameters that are fitted against experimental data. We shall not cover such approaches here, but follow the conventional classification by considering only semiempirical methods that are based on molecular orbital (MO) theory and make use of integral approximations and parameters already at the MO level. 

In semiempirical approaches, the standard Hartree-Fock SCF-MO equations are simplified by integral approximations which are designed to neglect all three-center and four-center two-electron integrals. The CNDO, INDO, and NDDO schemes have been introduced for this purpose [12,31]. They are rotationally invariant generahzations of the zero-dififerential-overlap approximation from ir-electron theory to valence-electron systems. The most refined of these schemes is NDDO which assumes... 

The PPP method does not attempt to explicitly specifyH" or to calculate the integrals theoretically. Rather, the integrals and y are calculated fi-om approximate semiempirical formulas, some of which contain empirical parameters. For example, when the AOs and are on atoms R and S that are bonded to each other, may be taken as where the value of the empirical parameter k is chosen... 

The PPP method does not attempt to explicitly specify integrals theoretically. Rather, the integrals // I and are calculated from approximate semiempirical formulas, some of which contain empirical parameters. For example, when the AOs fr and fs are on atoms R and S that are bonded to each other, /f may be taken as k fr fs), where the valne of the empirical parameter A is chosen so that the predictions of the theory give good agreement with experiment the overlap integral fr fs) is calcnlated from the STOs fr and fs, and not taken as zero as in (17.62). When the two different atoms R and S are not bonded to each other, is taken as zero. (Several versions of the PPP theory... 

Semiempirical calculations have been carried out by an unparameterized SCF-MO method with integral approximations [5], various versions of the CNDO [37 to 42] and INDO [6, 38, 43 to 45] methods, the MNDO [46, 47] and MINDO [48] methods, the extended Hiickel method [3, 4, 49, 50] (presumably also [51 ]), a Pariser-Parr-Pople-type open-shell method [49] (presumably also [51]), and a simple MO approach [52]. Besides some other molecular properties, the charge distribution (atomic charges and/or overlap populations) [5, 38,40,41,43,49 to 51] and the spin density distribution (and thus, the hyperfine coupling constants, compare above and p. 241) [3 to 6, 46, 48] have been the subjects of many of these studies. 

Semiempirical calculations for j or AEf were carried out by the SCF-MO method with integral approximations [23], by various versions of the CNDO [24 to 27] and INDO [1 to 4, 28] methods, by the MNDO method [29], by the extended Huckel and Pariser-Parr-Pople methods [30], and by a simple [31] MO method [32]. 

The g-tensor elements were calculated by a modified ab initio UHF method ( Nesbet ) [11], by a semiempirical SCF-MO calculation (integral approximations) [12], and by a semiempirical procedure called the energy weighted maximum overlap model [13]. 

The bjj s were calculated by the ab initio UHF ( Nesbet ) [14] and UHF and UHFASA [15,16] methods. The largest component, b x, was obtained (but p-electron spin density bxx/bret only given, see p. 233) from semiempirical MO (integral approximations) [12] and INDO [23] calculations and by the extended Hiickel method [24, 25]. 

For the CECA scheme, several (more or less successful) improvements and refinements have been developed (see e.g., [11-14]) we are not going to discuss them here in any detail. All of them (including, of course, CECA itself) are a posteriori means of analysis, that is, they can be applied after a conventional ab initio SCF calculation has been performed, in order to elucidate the results of the latter. The aim of the present paper is to use the same integral approximation scheme in order to develop an approximate ab initio scheme of a priori calculations, in which one needs not to calculate any three- and four-center integrals. In this respect, the scheme could be put in parallel with the semiempirical quantum chemical methods. However,... 

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... 

Semiempirical calculations are set up with the same general structure as a HF calculation in that they have a Hamiltonian and a wave function. Within this framework, certain pieces of information are approximated or completely omitted. Usually, the core electrons are not included in the calculation and only a minimal basis set is used. Also, some of the two-electron integrals are omitted. In order to correct for the errors introduced by omitting part of the calculation, the method is parameterized. Parameters to estimate the omitted values are obtained by fitting the results to experimental data or ah initio calculations. Often, these parameters replace some of the integrals that are excluded. 

Semiempirical methods are widely used, based on zero differential overlap (ZDO) approximations which assume that the products of two different basis functions for the same electron, related to different atoms, are equal to zero [21]. The use of semiempirical methods, like MNDO, ZINDO, etc., reduces the calculations to about integrals. This approach, however, causes certain errors that should be compensated by assigning empirical parameters to the integrals. The limited sets of parameters available, in particular for transition metals, make the semiempirical methods of limited use. Moreover, for TM systems the self-consistent field (SCF) procedures are hardly convergent because atoms with partly filled d shells have many... 

In theory, once the activity of an electrolyte in solution is known, the activity of the solvent can be determined by the Gibbs-Duhem integration (see section 2.11). In practice, the calculation is prohibitive, because of the chemical complexity of most aqueous solutions of geochemical interest. Semiempirical approximations are therefore preferred, such as that proposed by Helgeson (1969), consisting of a simulation of the properties of the H20-NaCl system up to a solute... 

In general, the semiempirical approach has been widely adopted in inorganic systems, and the most common approximation is either to assume that Hu is directly proportional to the overlap integral Sij9 or related to it by a Wolfsberg-Helmholz type of formula (43) ... 

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