Basis set function

H is the one-electron operator and i is the Slater basis set function (2s, 2p). The diagonal elements of Htj (Hit) are approximated as the valence state ionization potentials and the off-diagonal elements Htj are estimated using the Wolfsberg-Helmholtz approximation,... 

The second shortcoming of a minimal (or split-valence) basis set... functions being centered only on atoms. . . may be addressed by providing d-type functions on main-group elements (where the valence orbitals are of s and p type), and (optionally) p-type functions on hydrogen (where the valence orbital is of s type). This allows displacement of electron distributions away from the nuclear positions. 

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... 

Up to now we have assumed in this chapter the use of Slater-type orbitals. Actually, use may be made of any type of functions which form a complete set in Hilbert space. Since for practical reasons the expansion (2,1) must be always truncated, it is preferable to choose functions with a fast convergence. This requirement is probably best satisfied just for Slater-type functions. Nevertheless there is another aspect which must be taken into account. It is the rapidity with which we are able to evaluate the integrals over the basis set functions. This is particularly topical for many-center two-electron integrals. In this respect the use of the STO basis set is rather cumbersome. The only widely used alternative is a set of Gaus-slan-type functions (GTF). The properties of Gaussian-type functions are just the opposite - integrals are computed simply and, in comparison to the STO basis set, rather rapidly, but the convergence is slow. 

Some other examples of less common basis set functions are elliptical 9-12... 

For molecules the optimization of exponents is very important for small basis sets but it can never alone absorb the deficiency due to a lack of expansion functions. The number and secondly the kind of STO basis set functions (i.e. s, p, d,. .. type) are the most important considerations. The decreasing importance of the exponent optimization as the basis set grows is observable from Table 2.4,... 

This section is devoted to Gaussian basis functions which are not placed on the atom centers but at different points in space. We note on the use of the following types of basis set functions ... 

A typical (though not always unavoidable) first step in any proce-dure for the evaluation of correlation energy is the transformation of integrals over AO s (i.e. STO s, CGTF s and other possible basis set functions) with the indices V, X, and o to integrals over MO s with the indices i,j>k and i ... 

If the number of basis set functions is n, the number of (ij kl) in-tegrals becomes n, Since the evaluation of each of the latter requires the manipulation with n integrals /U.V 7 6), the direct transformation according to eqn. (4.123) is extremely ineffective. Sutcliffe... 

We have opened a page to show the relationship between iterative and direct procedures, and repeated again simple things which had been said in previous papers, because there has been some confusion about this subject in the literature of the past years. The final message is simple, that is iterative and direct procedures are not alternative approaches in setting up the ASC continuum model and they give the same results. On the contrary, the closure formulation is an approximation. The difference lies in the computational efficiency. Here, several parameters have to be considered, related to the characteristics of the computer and to the size of the problem (number of basis set functions and of tesserae). The paper we have quoted here several times (Tomasi and Persico, 1994) has been written two years ago. Meanwhile we have improved the efficiency of the direct PCM version, which is now more efficient than the iterative one, that has not been modified. 

It came out immediately clear that the supermolecule approach cannot represent the method to be used in extensive studies of solvent effects. The computational costs increase in the ab initio versions with more than the fourth power of the number of basis set functions, at a given nuclear geometry of the supermolecule. Even more important it has been the recognition that, when the size of the solvation cluster exceeds some very low limits, the number of different nuclear conformations at an equivalent energy increases exponentially computational costs increase in parallel, and the introduction of thermal averages on these conformations becomes necessary. These facts, and some attempts to overcome them, are well summarized in a dementi s monograph (Clementi, 1976). The problem of multiple equivalent minima still plagues some discrete solvation models. 

To carry out the FSS procedure, one has to choose a convenient basis set to obtain the two lowest eigenvalues and eigenvectors of the finite Hamiltonian matrix. As basis functions for the FSS procedure, we choose the following basis set functions [104-106] ... 

In the self-consistent procedure based on a linear combination basis set functions to solve Eq. 31, the following matrix elements are evaluated numerically on the grid ... 

B3LYP Three-parameter hybrid density functional method the MCPF method is an extension of the singles and doubles configuration interaction approach D95- " - is a Dunning double-zeta plus polarization and diffuse functions quality basis set D95 is the D95- " - basis set in which basis set functions and the polarization functions on the hydrogen atoms have not been included. 

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