Slater functions

Once the least-squares fits to Slater functions with orbital exponents e = 1.0 are available, fits to Slater function s with oth er orbital expon cn ts can be obtained by siin ply m ii Itiplyin g th e cc s in th e above three equations by It remains to be determined what Slater orbital exponents to use in electronic structure calculation s. The two possibilities may be to use the "best atom" exponents (e = 1. f) for II. for exam pie) or to opiim i/e exponents in each calculation. The "best atom expon en ts m igh t be a rather poor ch oicc for mo lecular en viron men ts, and optirn i/.at ion of non linear exponents is not practical for large molecules, where the dimension of the space to be searched is very large.. 4 com prom isc is to use a set of standard exponents where the average values of expon en ts are optirn i/ed for a set of sin all rn olecules, fh e recom -mended STO-3G exponents are... 

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 ... 

One approach, using a local density approximation for each part, has E - = Es -1- Evwn, where Eg is a Slater functional and Evwn is a correlation functional from Vosko, Wilk, and Nusair (1980). Both functionals in this treatment assume a homogeneous election density. The result is unsatisfactory, leading to enors of more than 50 kcal mol for simple hydrocarbons. 

The various MO calculations use different basis sets and have different ways of calculating multicenter coulomb and exchange integrals. The current trend in MO is to expand as a linear combination of atomic orbitals (LCAO). The atomic orbitals are represented by Slater functions with expansion in gaussian functions, taking advantage of the additive rule. When the calculation is performed in this... 

Here position vectors ra. The overlap between these functions is given by S. After an FT and integrating over momentum coordinates of one particle, the EMD of H2 molecule within VB and MO theory are derived as... 

Approximate linear dependence of AO-based sets is always a numerical problem, especially in 3D extended systems. Slater functions are no exceptions. We studied and recommended the use of mixed Slater/plane-wave (AO-PW) basis sets [15]. It offers a good compromise of local accuracy (nuclear cusps can be correctly described), global flexibility (nodes in /ik) outside primitive unit cell can be correct) and reduced PW expansion lengths. It seems also beneficial for GW calculations that need low-lying excited bands (not available with AO bases), yet limited numbers of PWs. Computationally the AOs and PWs mix perfectly mixed AO-PW matrix elements are even easier to calculate than those involving AO-AO combinations. 

International Tables for Crystallography 1992). The function <]/> for Slater-type radial functions can be expressed in terms of a hypergeometric series (Stewart 1980), or in closed form (Avery and Watson 1977, Su and Coppens 1990). The latter are listed in appendix G. As an example, for a first-row atom quadrupolar function (/ = 2) with n, = 2, the integral over the nonnormalized Slater function is... 

In general, the Slater function is not an exact solution of any Schrodinger equation (except the Is- wavefunction, which is the exact solution for the hydrogen-atom problem). Nevertheless, asymptotically, the orbital exponent C is directly related to the energy eigenvalue of that state. Actually, at large distances from the center of the atom, the potential is zero. Schrodinger s equation for the radial function R(r) is... 

At r—all the terms with r and r - are negligible. To take derivatives with respect to r, it is obvious that the derivative from the exponential factor of the Slater function, Eq. (6.2), is much larger than from the algebraic factor. Therefore, the Schrodinger equation implies... 

The spirit of the Slater atomic wavefunction is to use the term with the highest power r" to represent the entire algebraic factor in the hydrogenlike wavefunctions. This approach is particularly suitable for treating STM-related problems. In the processes pertinent to STM, only the values of atomic wavefunctions a few Angstroms away from the nucleus of the atom are relevant, not the values near the core. Within the same spirit, we can derive all the Slater functions from a single function... 

Tables of overlap integrals (55-57) involving Slater functions are available, and most of those required in calculations on organometallic molecules may be obtained by interpolation. Details of the calculation of both individual and group overlap integrals are given in Sec. IV.

A similar SCF calculation of ferrocene has been made by Shustorovich and Dyatkina (73) in which Slater functions were used for the iron orbitals. These calculations gave an exactly opposite charge distribution to that of Dahl and Ballhausen, owing to the more contracted metal orbitals used by the latter authors. Because values of overlap integrals of the type S (2pa3da) and S(2p7T3d7T) calculated by the latter authors are almost identical with those calculated directly from the Watson functions (74), it seems that the charge distribution calculated by Dahl and Ballhausen is the correct one. 

A. Radial Functions and Atomic Orbital Energies.—Self-consistent field (SCF) radial functions for vanadium 3d and 4s orbitals were taken from Watson s report.16 Watson gives no 4p function, so it is estimated as having approximately the same radial dependence as the 4s function. Analytic 2s and 2p oxygen SCF radial functions were obtained by fitting the numerical functions given by Hartree17 with a linear combination of Slater functions. These radial functions are summarized... 

The simplest basis sets are those used in the simple Hiickel and the extended Hiickel methods (SHM and EHM, Chapter 4). As applied to conjugated organic compounds (its usual domain), the simple Hiickel basis set consists of just p atomic orbitals (or geometrically p-type atomic orbitals, like a lone-pair orbital which can be considered not to interact with the er framework). The extended Hiickel basis set consists of only the atomic valence orbitals. In the SHM we don t worry about the mathematical form of the basis functions, reducing the interactions between them to 0 or —1 in the SHM Fock matrix (e.g. Eqs. 4.64 and 4.65). In the EHM the valence atomic orbitals are represented as Slater functions (Sections 4.4.1.2 and 4.4.2). 

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. 

Fig. 5.12 Comparison of Slater, STO-1G and STO-3G functions for hydrogen. The Slater function shown is the most appropriate one for hydrogen in a molecular environment, and the Gaussians are the best 1-G and 3-G fits to this Slater function. Slater and Gaussian functions are usually characterized by parameters designated f (zeta) and a, respectively, as shown [31]...

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