Hydrogen atom using Slater orbitals

So called Ilydrogenic atomic orbitals (exact solutions for the hydrogen atom) h ave radial nodes (values of th e distance r where the orbital s value goes to zero) that make them somewhat inconvenient for computation. Results are n ot sensitive to these nodes and most simple calculation s use Slater atom ic orbitals ofthe form... 

The orbitals used for methane, for example, are four Is Slater orbitals of hydrogen and one 2s and three 2p Slater orbitals of carbon, leading to an 8 x 8 secular matrix. Slater orbitals are systematic approximations to atomic orbitals that are widely used in computer applications. We will investigate Slater orbitals in more detail in later chapters. 

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

These atomic orbitals, called Slater Type Orbitals (STOs), are a simplification of exact solutions of the Schrodinger equation for the hydrogen atom (or any one-electron atom, such as Li" ). Hyper-Chem uses Slater atomic orbitals to construct semi-empirical molecular orbitals. The complete set of Slater atomic orbitals is called the basis set. Core orbitals are assumed to be chemically inactive and are not treated explicitly. Core orbitals and the atomic nucleus form the atomic core. 

Ab initio calculations are based on first principles using molecular orbital (MO) calculations based on Gaussian functions. Combinations of Gaussian functions yield Slater-type orbitals (STOs), also called Slater determinants. STOs are mathematical functions closely related to exact solutions for the hydrogen atom. In their ultimate applications, ab initio methods would use Gaussian-type wave functions rather than STOs. The ab initio method assumes that from the point of view of the electrons the nuclei are stationary, whereas... 

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. 

The radial functions f (r) will be different for different atoms. Only for the hydrogen atom is the exact analytical form of the i2((r) s known. For other atoms the f (r) s will be approximate and their form will depend on the method used to find them. They might be analytical functions (e.g. Slater orbitals) or tabulated sets of numbers (e.g. numerical Hartree-Fock orbitals). 

In view of the large size of the valine molecule, our calculations (Kaplan et al., 1983) were carried out on a minimal basis of 51 Slater orbitals in the MO LCAO approximation using the method described in Sections II, C, and III, B, 1. We have taken into account 608 singly excited states of the ion (valine-He)+. The results of calculations for valine on a minimal basis were corrected with regard to the results of calculation of the influence of the basis length on the excitation probabilities of the fragments that are shown enclosed in boxes in Fig. 9 (the rest of the molecule was replaced by a hydrogen atom). 

Orbitals (GTO). Slater type orbitals have the functional form e, if) = NYi, d, e- -- (5.1) is a normalization constant and T are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. centre of a bond. 5.2 Classification of Basis Sets Having decided on the type of function (STO/GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smallest number of functions... 

All the DV-DFS calculations were performed with the Slater exchange parameter Ctof 0.7 and with 20000 DV sample points. The basis ftmctions for the actinide atoms were used up to the 7p orbital, while those for the oxygen and nitrogen atoms up to the 2p orbital and Is for the hydrogen atom. TTie calculations were carried out self-consistently untO the difference in orbital populations between the initial and final stages of the iteration was less than 0.01. 

The first use of the Fourier transformation technique for atoms or molecules was made by Boris Podolsky and Linus Pauling [17] for the hydrogen atom. Coulson [18] noted that if the r-space wave function is constmcted from one-electron functions, then there is an isomorphism between I and wave function P can be written in terms of spin-orbitals (/ as a single Slater determinant. 

We will illustrate the stages involved in the Roothaan-Hall approach using the helium hydrogen molecular ion, HeH, as an example. This is a two-electron system. Our objective here is to show how the Roothaan-Hall method can be used to derive the wavefunction, for a fixed internuclear distance of 1 A. We use HeH rather than H2 as our system as the lack of symmetry in HeH makes the procedure more informative. There are two basis functions, Isa (centred on the helium atom) and Isg (on the hydrogen). The numerical values of the integrals that we shall use in our calculation were obtained using a Gaussian series approximation to the Slater orbitals (the STO-3G basis set, which is described in Section 2.6). This detail need not concern us here. Each wavefunction is expressed as a linear combination of the two Is atomic orbitals centred on the nuclei A and B ... 

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