Gaussian Expansion of Slater-Type Orbitals

so that the integrals are evaluated actually over Gaussians. 

Such a possibility was already discussed by several authors  

As shown by Huzinaga, all s-type STO s are best expanded in terms of Is-GTF s (n = 1), p-type STO s are best expanded in terms of 2p GTF s (rig B 2). The adequacy of using GTF s with lower values is very helpful to reduce complications in molecular integral calculations. 

Alternatively, instead of the variational procedure a least squares 

Note that the same number of Gaussians, N, is used for each STO and that common Gaussian exponents are shared between ns and np or- 

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

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. 

Hehre, W. J., Stewart, R. F., and Pople, J. A., Self-consistent molecular-orbital methods. 1. Use of Gaussian expansions of Slater-type atomic orbitals, J. Chem. Phys. 51, 2657-2664 (1969). 

Hall G G 1951 The Molecular Orbital Theory of Chemical Valency VIII A Method for Calculating lomsation Potentials. Proceedings of the Royal Society (London) A205 541-552 Hehre W J, R F Stewart and J A Pople 1%9 Self-Consistent Molecular-Orbital Methods I Use of Gaussian Expansions of Slater-Type Atomic Orbitals. Journal of Chemical Physics 51 2657-2664 Hehre W J, L Radom, P v R Schleyer and J A Pople 1986 Ah initio Molecular Orbital Theory New York, John Wiley Sons. 

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. 

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 ... 

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. 

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 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]. 

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... 

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