Molecular fields Gaussian functions

The formal analysis of the mathematics required incorporating the linear combination of atomic orbitals molecular orbital approximation into the self-consistent field method was a major step in the development of modem Hartree-Fock-Slater theory. Independently, Hall (57) and Roothaan (58) worked out the appropriate equations in 1951. Then, Clement (8,9,63) applied the procedure to calculate the electronic structures of many of the atoms in the Periodic Table using linear combinations of Slater orbitals. Nowadays linear combinations of Gaussian functions are the standard approximations in modem LCAO-MO theory, but the Clement atomic calculations for helium are recognized to be very instructive examples to illustrate the fundamentals of this theory (67-69). 

In addition to the set of adjustable coefficients and b contained in Eq. (13.7), the method CMF also requires ealeulation of a certain nmnber of adjustable parameters. Among them are the parameter v for the support vector regression method v-S VR and the ridge parameter y for KRR. Their values should be optimized with the aim to improve the predictive eapability of the model constracted. In addition, for each molecular field one can adjust the values of up to two parameters (attenuation factor, which is related to the width of the Gaussian function) and hj (mixing coefficient, which has the meaning of the relative contribution of molecular field of the ythtype). 

Note also that, due to the lack of discrete grids, molecular fields are treated in CMF as continuous functions with respect to spatial coordinates and therefore can be differentiated and integrated (even analytically, due to the use of Gaussian... 

The CMF approach is not confined to the simplest approximation scheme introduced by Eq. (13.5). Ar r mrmber of Gaussian functions, (both isotropic, i.e. spherically symmetrical, and non-isotropic) as well as any other set of basic functions (such as splines, wavelets, etc.) can be used for approximating continuous molecular fields. This provides the ability to work with complex types of molecular fields, including those derived from the electron density function. 

Extended basis set SCF and configuration interaction (CISD) calculations for CO have been carried out by Amos [359] with die aim to assess the effect of electron correlation on calculated molecular polarizability and polarizability derivatives. The basis set employed has been of the type (5s4p2d) contracted Gaussian function on each atom. The configuration interaction function consisted of single and double excitations of the valence shell orbitals. Polarizabilities have been calculated by the finite field method [179]. Polarizability derivatives are evaluated by numerical differentiation. The results are given in Table 10.5. 

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