Slater-type orbitals,

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. 

STO and GTF basis sets, which are the most important for practical calculations, will be treated separately in the next sections. 

For the sake of completeness we comment here on some other possibilities. 

Some time ago the one-center expansions (OCE) seemed to be prom-9 

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

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A) SLATER-TYPE ORBITALS AND GAUSSIAN-TYPE ORBITALS... 

These atomic orbitals, called Slater Type Orbitals (STOs), are a simplification of exact soil tion s of the Sch rbdin ger eq nation for the... 

These functions are universally known as Slater type orbitals (STOs) and are just the leading term in the appropriate Laguerre polynomials. The first three Slater functions are as follows ... 

Fig. 2.5 The Is Slater type orbital and the best Gaussian equivalent.

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].

Several functional forms have been investigated for the basis functions Given the vast experience of using Gaussian functions in Hartree-Fock theory it will come as no surprise to learn that such functions have also been employed in density functional theory. However, these are not the only possibility Slater type orbitals are also used, as are numerical... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

We do not know the orbitals of the electrons either. (An orbital, by the way, is not a ball of fuzz, it is a mathematical function.) We can reasonably assume that the ground-state orbitals of electrons I and 2 are similar but not identical to the Is orbital of hydrogen. The Slater-type orbitals... 

The Slater-type orbitals are a family of functions that give us an economical way of approximating various atomic orbitals (which, for atoms other than hydrogen, we don t know anyway) in a single relatively simple form. For the general case, STOs are written... 

Hoffmann introdueed the so-ealled extended Hiiekel method. He found that a value for K= 1.75 gave optimal results when using Slater-type orbitals as a basis (and for ealeulating the Sg,v)- The diagonal h elements are given, as in the eonventional Hiiekel method, in terms of valenee-state IP s and EA s. Cusaehs later proposed a variant of this parameterization of the off-diagonal elements ... 

For first- and seeond-row atoms, the Is or (2s, 2p) or (3s,3p, 3d) valenee-state ionization energies (aj s), the number of valenee eleetrons ( Elee.) as well as the orbital exponents (es, ep and ej) of Slater-type orbitals used to ealeulate the overlap matrix elements Sp y eorresponding are given below. 

In quantum ehemistry it is quite eommon to use eombinations of more familiar and easy-to-handle "basis funetions" to approximate atomie orbitals. Two eommon types of basis funetions are the Slater type orbitals (STO s) and gaussian type orbitals (GTO s). STO s have the normalized form ... 

Two Slater type orbitals, i andj, centered on the same point results in the following overlap integrals ... 

These are evaluated analogous to exereise 1, letting denote eaeh of the individual Slater Type Orbitals. 

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

Hoffmann introdueed the so-ealled extended Hiiekel method. He found that a value for K= 1.75 gave optimal results when using Slater-type orbitals as a basis (and for ealeulating the... 

The complete neglect of differential overlap (CNDO) method is the simplest of the neglect of differential overlap (NDO) methods. This method models valence orbitals only using a minimal basis set of Slater type orbitals. The CNDO method has proven useful for some hydrocarbon results but little else. CNDO is still sometimes used to generate the initial guess for ah initio calculations on hydrocarbons. 

FIGURE 10.1 Approximating a Slater-type orbital with several Gaussian-type orbitals. 

Slater type orbital (STO) mathematical function for describing the wave function of an electron in an atom, which is rigorously correct for atoms with one electron... 

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