Slater-type orbitals Gaussian expansion

The coefficients and the exponents are found by least-squares fitting, in which the overlap between the Slater type function and the Gaussian expansion is maximised. Thus, for the Is Slater type orbital we seek to maximise the following integral ... 

Table 2.3 Coefficients and e-xponents for best-fit Gaussian expansions for the Is Slater type orbital [Hehre et al. 1969].

Stewart, R.R Small Gaussian expansions of Slater-Type Orbitals J. Chem. Phys. 52 431-438, 1970. 

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. 

A straight forward application of approximation IV to calculate W (r) maps is quite exacting, because the calculation of the potential contribution due to the couple distributions xt li 1S time consuming when directly performed on the Slater functions. This fact clashes with the basic philosophy of semiempirical methods, which is to sacrifice some reliability to speed up the calculations. It has been shown40) that expansion of each Slater-type orbital into three Gaussian functions (3G expansion41)) gives a substantial improvement of the computational times of W (r), without an appreciable reduction in the quality of the results. 

W. J. Hehre, R. F, Stewart, and J. A. Pople, /. Chem. Phys., 51, 2657 (1969). Self-Consistent Molecular-Orbital Methods. I. Use of Gaussian Expansions of Slater-Type Orbitals. 

Spatial orbitals are typically (but not necessarily) expanded in a basis set. The choice of the latter expansion is somewhat arbitrary, but the quality of the possible choices can be judged by considering completeness of the basis set and how quickly the basis converges to eigenfunctions of the Hamiltonian. Alternatives include plane-wave basis sets. Slater-type orbitals (STO), Gaussian-type orbitals (GTO), and numerical orbitals. 

The basis functions are represented as linear combinations of Slater-type orbitals (STO) or here Gaussian-type orbitals (GTO). Expansion coefficients are found by solving the secular problem ... 

The most important reason for the great progress of quantum chemistiy in recent years is replacing the Slater-type orbitals, formerly used, by Gaussian-type orbitals as the expansion functions. 

Implementations have been realized using Gaussian functions (GTO s) ([38, 39] and Slater-type orbitals (STO s) [5, 40, 41], and numerical basis sets [42, 43, 44]. The auxiliary basis may be avoided by the use of a purely numerical representation of the potential on a grid (usually called DVM - Discrete Variational Method [45, 5]), by certain approximations for the potential (Multiple Scattering concept within the so-called mufl5n-tin approximation - [46]), the linear combination of muffin-tin orbitals [47, 3], and in connection with the pseudopotential concept the application of plane-wave basis expansions - see, e.g.. Ref. [112]. 

The Gaussian basis set can also be used without any connection with the Slater-type orbitals HF equations solutions for atoms are reached by Roothaan s expansion... 

A different approach was developed by Baerends, Ellis, and Ros (1973). In addition to adopting the Slater potential for the exchange, their approach had two distinct features. The first was an efficient numerical integration procedure, the discrete variational method (DVM), which permitted the use of any type of basis function for expansion, not only Slater-type orbitals or Gaussian-type orbitals, but also numerical atomic orbitals. The second feature was an evaluation of the Coulomb potential from... 

The HF CO method is especially efficient if the Bloch orbitals are calculated in the form of a linear combination of atomic orbitals (LCAO)1 2 since in this case the large amount of experience collected in the field of molecular quantum mechanics can be used in crystal HF studies. The atomic basis orbitals applied for the above mentioned expansion are usually optimized in atoms and molecules. They can be Slater-type exponential functions if the integrals are evaluated in momentum space3 or Gaussian orbitals if one prefers to work in configuration space. The specific computational problems arising from the infinite periodic crystal potential will be discussed later. 

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