Slater-type orbitals limitations

The correct limiting radial behavior of the hydrogen-like atom orbital is as a simple exponential, as in (A.62). Orbitals based on this radial dependence are called Slater-type orbitals (STOs). Gaussian functions are rounded at the nucleus and decrease faster than desirable (Figure 2.2b). Therefore, the actual basis functions are constructed by taking fixed linear combinations of the primitive Gaussian functions in such a way as to mimic exponential behavior, that is, resemble atomic orbitals. Thus... 

Slater-type orbitals were introduced in Section 5.2 (Eq. (5.2)) as the basis functions used in extended Huckel theory. As noted in that discussion, STOs have a number of attractive features primarily associated with the degree to which they closely resemble hydrogenic atomic orbitals. In ab initio HF theory, however, they suffer from a fairly significant limitation. There is no analytical solution available for the general four-index integral (Eq. 

Unfortunately, the Slater-type orbitals become increasingly less reliable for the heavier elements, including to some extent the first transition series these limitations are described in a recent review by Craig and Nyholm (5 ). The most accurate wave functions to use in these calculations would be the SCF functions obtained by the Hartree-Fock procedure outlined above, but this method leads to purely numerical radial functions. However, Craig and Nyholm (5S) have drawn attention to relatively good fits obtained by Richardson (59) to SCF 3d functions by means of two-parameter orbitals of the type... 

Various parameterizations of NDDO have been proposed. Among these are modified neglect of diatomic overlap (MNDO),152 Austin Model 1 (AMI),153 and parametric method number 3 (PM3),154 all of which often perform better than those based on INDO. The parameterizations in these methods are based on atomic and molecular data. All three methods include only valence s and p functions, which are taken as Slater-type orbitals. The difference in the methods is in how the core-core repulsions are treated. These methods involve at least 12 parameters per atom, of which some are obtained from experimental data and others by fitting to experimental data. The AMI, MNDO, and PM3 methods have been focused on ground state properties such as enthalpies of formation and geometries. One of the limitations of these methods is that they can be used only with molecules that have s and p valence electrons, although MNDO has been extended to d electrons, as mentioned below. 

In these equations, n and v are two atomic orbitals (e.g. Slater type orbitals), is the ionisation potential of the orbital and fC is a constant, which was originally set to 1.75. The formula for the off-diagonal elements (where fj, and u are on different atoms) was originally suggested by R S Mulliken. These off-diagonal matrix elements are calculated between all pairs of valence orbitals and so extended Hiickel theory is not limited to tt systems. 

Using all the above, Cizek presents the explicit, spin orbital and spin-adapted CC doubles equations (CCD) i.e., T = T2 (then called coupled-pair many-electron theory) in terms of one- and two-electron integrals over an orthogonal basis set. Assisted by Joe Paldus with some computations, he also reports some CCD results for N2, which though limited to only ti to Ug excitations, uses ab initio integrals in an Slater-type-orbital basis. He also does the full Cl calculation to assess convergence, a tool widely used in Cizek s and Paldus work and by most of us, today. He also reports results for the minimum-basis TT-electron approximation to benzene. 

The lowest ab initio SCF energy obtained so far, = - 273.5594 a.u., has apparently been calculated [2] at the optimized geometry with a polarized double or triple zeta basis of Slater -type orbitals (STO). The Hartree-Fock (HF) limit of the total energy has been estimated [10] to be EV = -273.68 a.u. A slightly higher estimate is due to Rothenberg and Schaefer [8]. 

Two other types of basis set that have been used successfully in hfs calculations are Chipman s contracted [3s,2p] bases, and basis sets based on Slater type orbitals (STOs). The former of these is mainly used in single excitation configuration interaction (CIS) calculations, and are based on a very fortuitous cancellation of errors between method and basis set. The performance of the CIS/[3s,2p] approach lies within 20-25% of experiment. One should recall, though, that once we go to larger molecular systems, the CIS method becomes computationally very demanding, STOs have mainly been used in semiempirical INDO hfcc calculations (STO-SG) and in the density functional theory (DFT) studies of Ishii and Shimitzu (STO-6G). The number of hfcc studies using these basis sets at the ab initio or DFT levels is however to date very limited. 

The two most popular basis sets consist of either Slater-type orbitals8 (STO s) or Gaussian functions. When using STO s one or more are placed on each nucleus - the more the better. The so-called minimal basis set consists of only those STO s which correspond to the occupied a.o. s in the seperated atom limit. Instead of using Slater s rules to determine orbital exponents they may be varied in order to minimize the energy. Once this optimization has been done for a small molecule the values so established can be used in bigger problems. The basis can be improved by adding additional STO s for various nuclei, e.g. with different orbital exponents. If every minimal basis a.o. is represented by two such STO s a "double Q" set is obtained. The only restriction on the number and type of STO that can be added, seems to be computer time. 

The OBS-GMCSC method offers a practical approach to the calculation of multiconfiguration electronic wavefunctions that employ non-orthogonal orbitals. Use of simultaneously-optimized Slater-type basis functions enables high accuracy with limited-size basis sets, and ensures strict compliance with the virial theorem. OBS-GMCSC wavefunctions can yield compact and accurate descriptions of the electronic structures of atoms and molecules, while neatly solving symmetry-breaking problems, as illustrated by a brief review of previous results for the boron anion and the dilithium molecule, and by newly obtained results for BH3. 

When calculating the wavefunction it is important to make a choice of basis set J. t that is suitable for the available computing power and the accuracy desired. A straightforward early approach to basis set construction was to fit an accurate Slater-type atomic orbital (STO) with n gaussians,called STO- G. The quality of STO- G wavefunctions increases as n increases. It was determined that = 3 was a good starting point, and the STO-3G basis set has been widely used, particularly where computing resources were limited or for lai er molecules. 

Wheeler and collaborators [3], in the context of nuclear physics, showed at that time that the limit in the variational procedure potential itself was not reached. Indeed, the Rayleigh-Ritz (RR) variational scheme teaches us how to obtain the best value for a parameter in a trial function, i.e., exponents of Slater (STO) or Gaussian (GTO) type orbital, Roothaan or linear combination of atomic orbitals (LCAO) expansion coefficients and Cl coefficients. Instead, the generator coordinate method (GCM) introduces the Hill-Wheeler (HW) equation, an integral transform algorithm capable, in principle, to find the best functional form for a given trial function. We present the GCM and the HW equation in Section 2. 

In all computations presented below, we use a semiempirical (INDO/S) parametrization of the Hamil-tonicin (2.1) that was fitted to reproduce the spectra of simple molecules at the singly excited Cl level. The INDO approximation limits the basis set to valence orbitals of Slater type. Exchange terms in the two-electron interaction are permitted only among orbitals located on the same atom... 

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