Double-zeta STO-exponents

Table 13 The optimized double-zeta STO-exponents obtained by dementi from accurate Hartree-Fock calculations on the atoms from He to Ne.

Because of the complexity of the PHF function, only very small electronic systems were initially considered. As first example, the electronic energy of some four electron atomic systems was calculated using the Brillouin procedure [8]. For this purpose, a short double zeta STO basis set. Is, Is , 2s and 2s , with optimized exponents was used. The energy values obtained are given in Table 1. In the same table, the RHF energy values calculated with the same basis are gathered for comparison. It is seen that the PHF model introduces some electronic correlation in the wave-function. Because of the nature of the basis set formed by only s-type orbitals, only radial correlation is included which account for about 30% of the electronic correlation energy. 

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. 

In order to have a better basis set, we can replace each STO of a minimal basis set by two STOs with different orbital exponent f (zeta). This is known as a double zeta basis set. In this basis, there is a linear combination of a contracted function (with a larger zeta) and a diffuse function (with a smaller zeta) and the coefficients of these combinations are optimized by the SCF procedure. Using H2O as an example, a double zeta set has two Is STOs on each H, two Is STOs, two 2s STOs, two 2px, two 2pv and two 2pz STOs on oxygen, making a total of 14 basis functions. 

Many calculations for atoms have led to the development of a number of recipes for deciding the best values of and n. A further important issue is the size of the basis set. A minimal basis set of STOs for an atom would include one function for each SCF occupied orbital with different n and / quantum numbers in equation (6.56) for the chlorine atom, therefore, the minimal basis set would include s, 2s, 2p, 3s and 3p functions, each with an optimised Slater orbital exponent . A higher order of approximation would be to double the number of STOs (the double zeta basis set), with orbital exponents optimised ultimately the Hartree-Fock limit is reached, as it has been for all atoms from He to Xe [13]. 

The split-basis (46), outlined in Chapter 1, the separating of the Huzinaga sto-4g) basis set, Table 1.7, into the Gaussian, with the smallest exponent and so the most diffuse function, as one component and the other three primitives as the second component in the contraction, is a simple Gaussian equivalent of dementi s double-zeta Slater functions. It provides a good example of the double-zeta approach applied in Gaussian basis set theory. 

A double-zeta (DZ) basis set is obtained by replacing each STO of a minimal basis set by two STOs that differ in their orbital exponents ((zeta). (Recall that a single STO is not an accurate representation of an AO use of two STOs gives substantial improvement.) For example, for QHa a double-zeta set consists of two Is STOs on each H, two Is STOs, two 2s STOs, two 2p two 2py, and two 2p STOs on each carbon, for a total of 24 basis functions this is a (4s /2s) basis set. (Recall that we did a double-zeta SCF calculation on He in Section 13.16.) Since each basis function Xr in < i = 2, CriXr has its own independently determined variational coefficient c , the number of variational parameters in a double-zeta-basis-set wave function is twice that in a minimal-basis-set wave function. A triple-zeta (TZ) basis set replaces each STO of a minimal basis set by three STOs that differ in their orbital exponents. 

The next improvement of the basis sets is a doubling of all basis functions, producing a Double Zeta (DZ) type basis. The term zeta stems from the fact that the exponent of STO basis functions is often denoted by the Greek letter A DZ basis thus employs two x-functions for hydrogen (Is and Is ), four x-functions (Is, Is, 2s and 2s ) and two sets of p-functions (2p and 2p ) for first row elements, and six y-functions and four sets of p-functions for second row elements. The importance of a DZ over a minimum basis can be illustrated by considering the bonding in the HCN molecule. The H—C bond will primarily consist of the hydrogen s-orbital and the p -orbital on C. 

A minimal basis set has rather limited variational flexibility particularly if exponents are not optimized. The first step in improving upon the minimal basis set involves using two functions for each of the minimal basis functions—a double zeta basis set. The best orbital exponents of the two functions are commonly slightly above and slightly below the optimal exponent of the minimal basis function. This allows effective expansion or contraction of the basis functions by variation of linear parameters rather than nonlinear exponents. The SCF procedure will weight either the coefficient of the dense or diffuse component according to whether the molecular environment requires the effective orbital to be expanded or contracted. In addition, an extra degree of anisotropy is allowed relative to an STO-3G basis since, for example, p orbitals in different directions can have effectively different sizes. 

Thus, we have a linear variation procedure that, in effect, allows for AO expansion and contraction. It is akin to optimizing an orbital exponent, but it does not require nonlinear variation. Of course, one still has to choose the values of the larger and smaller of Fig. 11-1. This is normally done by optimizing the fit to very accurate atomic wavefunctions or by a nonlinear variation on atoms. A basis set in which every minimal basis AO is represented by an inner-outer pair of STOs is often referred to as a double-zeta basis set. 

For accurate calculations, we need to go beyond the STO minimal representation. An obvious extension would be to introduce more functions of the same type as those already present in order to improve the radial flexibility. Thus, in the double-zeta basis sets, we use two STOs for each occupied AO. The exponents of such sets may, for example, be obtained from atomic Hartree-Fock calculations, optimized to give the lowest energy. 

---