Slater polynomials

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 ... 

Because the eigenfunctions of the number operators are the Slater determinants, any polynomial of number operators will also have Slater determinant eigenfunctions. Starting with a basis set of K spin orbitals, 0] (x), > us... 

Because the Slater hull constraints are insufficient to ensure A -representability, it is important to find additional methods for constraining the off-diagonal elements of the density matrix. Obtaining constraints that supersede the Slater hull requires considering Hamiltonians with a more general form than polynomials of number operators. As discussed in Section UFA, matters are especially simple if the Hamiltonian has nonnegative eigenvalues, because then the necessary conditions for A-representability take the form... 

Among the simple linear operators that are commonly used to construct constraints, polynomials of the number operators (cf. Eq. (22)) are particularly useful. Polynomials of number operators are convenient because (i) the ground-state wavefunction of number-operator polynomials is a Slater determinant and (ii) the number-operator constraints depend only on the diagonal elements of... 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

The electron distribution around an atom can be represented in several ways. Hydrogenlike functions based on solutions of the Schrodinger equation for the hydrogen atom, polynomial functions with adjustable parameters, Slater functions (Eq. 5.95), and Gaussian functions (Eq. 5.96) have all been used [34]. Of these, Slater and Gaussian functions are mathematically the simplest, and it is these that are currently used as the basis functions in molecular calculations. Slater functions are used in semiempirical calculations, like the extended Hiickel method (Section 4.4) and other semiempirical methods (Chapter 6). Modem molecular ab initio programs employ Gaussian functions. 

The minimal STO-3G basis set was used for many years. It employs Slater orbitals (STO = Slater Type Orbital) of the form P(r)exp(— r), where P(r) is a polynomial in r. With STO functions, two-electron integrals are difficult to evaluate numerically, so they are replaced each by a sum of three Gaussian functions19 (hence the acronym STO-3G). The combination of the three Gaussians is treated as a single entity, so each AO has only one coefficient in the LCAO and the base remains minimal in character. 

For the non-relativistic case various functions have been tried, and discussions of their respective merits may be found in the literature [3]. The simplest has been to use hydrogenic functions, or suitably modified La-guerre polynomials. This may be useful for purposes of analysis in simple atomic systems, but has had little impact on the molecular field. The reason for this is the complicated form of the integrals. A somewhat more efficient choice is the Slater type orbital (STO) of the form... 

Notice the details of the form of Slater-type orbitals [STOs]. The polynomial terms of the radial functions in hydrogen. Table 1.1, are simplified to include the radius raised only to the power of the principal quantum number less one. In addition, an effective principal quantum number, n, is used. 

The Fade function has a cusp at r = 0 that can be adjusted to match the Coulomb cusp conditions by adjusting the a parameter. The Sun form also has a cusp, but approaches its asymptotic value far more quickly than the Fade function, which is useful for the linear scaling methods. An exponential form proposed by Manten and Luchow is similar to the Sun form, but shifted by a constant. By itself, the shift affects only the normalization of the Slater-Jastrow function, but has other consequences when the function is used to construct more elaborate correlation functions. The polynomial Fade function does not have a cusp, but its value goes to zero at a finite distance. 

Racah s motivation for extending the vector analysis to tensors was his need to cope with matrix elements of the Coulomb interaction in a more systematic way than that provided by Slater (1929). The definition of a Legendre polynomial yields... 

Slater derived the screening rules from empirical data and these rules work well, down to the transition metal series. The hydrogen-like 3d and 4f orbitals are described by a single term in the Laguerre polynomial, but this simple description is not favorable when penetration and screening are included. The 3d and 4f orbitals have an inner part where screening is less important than in the outer parts. At least two exponential functions are needed for a relevant description. 

The STO basis set of the DZ type ean be approximated by split polynomials of the Gaussian-type functions M-NP G. Each inner AO is replaced by M GTO orbitals, the valence 2s orbital—by AT, while the p orbital—by P GTO functions. For example, the 4-31 G basis set describes every inner (Is) orbital by four GTO s, every valence 2s AO by three GTO s and every valence p AO by one GTO. It is important to point out that whereas in the case of the minimal basis set of the NG type the accuracy level of the minimal STO basis set cannot be attained even at great values of N, the use of the split-valence GTO M-NPG basis sets allows the Slater basis set level to be exceeded. 

Looking back at the pi-electron calculations, we motivated the linear combination of atomic orbitals (LCAO) concept for electron orbitals, which spread over a whole molecule. We also have in hand the polynomial functions for the radial functions of the H-atom. In the previous example for pi-electrons, we used the variational principle to optimize the coefficients of the atomic orbitals in the linear combinations. We have also observed that the H-atom model can be modified by varying the value of the effective nuclear charge, Z, to model heavier elements. In 1930, Slater [4] proposed deleting the radial nodes and small parts of the H atom orbitals to use functions of the... 

The Slater-Koster files (SLAKO) have the electronic part which includes all integrals evaluated in a grid of interatomic distances and the polynomial describing the rep-... 

In these calculations, our trial wave function will be of Slater-Jastrow form. The Slater determinants will contain orbitals taken from density functional theory (DFT) calculations. The Jastrow factor is an explicit function of electron-electron distance, enabling a highly accurate and compact description of electron correlation. The Jastrow factor consists of polynomial expansions in electron-electron separation, electron-nucleus separation, in which the polynomial expansion coefficients are optimisable parameters [21]. These parameters were determined by minimising the VMC energy. 

The Slater-type orbitals replace the polynomial in r as in hydrogenlike orbitals with a single power in r reducing computational effort. The values for 5 and n are determined empirically by the following procedure. 

There are several deficiencies in STO s. Because STO s replace the polynomial in r for a single term, STO s do not have the proper number of nodes and do not represent the inner part of an orbital well. Care must be taken when using STO s because orbitals with the different values of n but the same values of / and mi are not orthogonal to one another. Another deficiency is that ns orbitals where n > 1 have zero amplitude at the nucleus. Values have been obtained for the effective nuclear charge for a number of atoms by fitting STO s to numerically computed wavefunctions. These values are given Table 8-2 and supersede the values obtained empirically from Slater s rules. 

We begin the present section by reviewing the properties of the Laguerre polynomials, which will be used frequently both here and in Section 6.6. After a brief discussion of the hydrogenic functions, we go on to consider the more important exponential Laguerre functions and, in particular, their nodeless counterparts - the Slater-type orbitals. 

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