STOs with variable exponents

Although it is reassuring to know that the set of STOs constitute a complete set of functions for a fixed exponent, our motivation for introducing the STOs was to pave the way for functions with variable exponents. Let us examine the radial extent of the STOs. The expectation value of r and the maximum in the radial distribution curve r are given by (see Exercise 6.7) 

The question as to what sequences of STOs with variable exponents constitute a complete set is a difficult one, which has been analysed in some detail. It has been possible to establish certain conditions that ensure completeness in the limit of infinitely many functions [17]. In particular, it has been shown that the set 

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We now have at our disposal two distinct techniques for generating complete sets of STOs. We may choose to work with a single, fixed exponent according to (6.5.26), describing the radial space by functions with different n. Such a basis will contain the orbitals (Is, 2s,...), (2p, 3p,...), and so on - aU with the same exponent Alternatively, we may describe the radial space by functions with variable exponents according to (6.5.33). For each angular momentum /, we then employ only the functions of the lowest pincipal quantum number n, jdelding a basis of the type (l5( u), l5( 2s), - - -), (2p( ip), 2p(X2p. ..), and so on. 

A double-zeta (DZ) basis in which twice as many STOs or CGTOs are used as there are core and valence atomic orbitals. The use of more basis functions is motivated by a desire to provide additional variational flexibility to the LCAO-MO process. This flexibility allows the LCAO-MO process to generate molecular orbitals of variable diffuseness as the local electronegativity of the atom varies. Typically, double-zeta bases include pairs of functions with one member of each pair having a smaller exponent (C, or a value) than in the minimal basis and the other member having a larger exponent. 

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