Spherical harmonic functions hydrogen atom orbitals

A is a normalization constant and T/.m 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. 

Atomic density functions are expressed in terms of the three polar coordinates r, 6, and multipole formalism, the density functions are products of r-dependent radial functions and 8- and -dependent angular functions. The angular functions are the real spherical harmonic functions ytm (8, ), but with a normalization suitable for density functions, further discussed below. The functions are well known as they describe the angular dependence of the hydrogenic s, p, d,f... orbitals. 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

Each interaction is mathematically described as a second-rank tensor, which can be graphically represented by an ellipsoid and mathematically expanded as a series of spherical harmonic functions in analogy to the atomic orbitals of the 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... 

The wave functions (6.8) are known as atomic orbitals, for / = 0, 1,2, 3, etc., they are referred to as s, p, d, f, respectively, with the value of n as a prefix, i.e. Is, 2s, 2p, 3s, 3p, 3d, etc., From the explicit forms ofthe wave functions we can calculate both the sizes and shapes of the atomic orbitals, important properties when we come to consider molecule formation and structure. It is instructive to examine the angular parts of the hydrogen atom functions (the spherical harmonics) in a polar plot but noting from (6.9) that these are complex functions, we prefer to describe the angular wave functions by real linear combinations of the complex functions, which are also acceptable solutions of the Schrodinger equation. This procedure may be illustrated by considering the 2p orbitals. From equations (6.8) and (6.9) the complex wave functions are... 

A plot of the first three s orbitals for a hydrogen atom is given in Fig. 1-2. The p orbitals have one unit of angular momentum, / = 1 there are three orbitals corresponding to m = — 1, m = 0, and m = 1. (See Fig. 1-3.) Any orbital, including those of the p series, can be written as a product of a function of radial distance from the nucleus and one of the spherical harmonics Y, which are functions of angle only (this is explained in Schiff, 1968, p. 79) ... 

The exact solutions to the separate equations, which result from this coordinate transformation of the Schrddinger equation for the hydrogen atom, are the sets of functions known as the associated Laguerre polynomials, for the radial equation, and the spherical harmonics, for the angular equation. The quantum numbers, n,l and m arise naturally in the solution of Schrddinger s equation, and so the symbolic form, for the eigenfunction solutions to the H-atom problem, known as atomic orbitals, is... 

The Is atomic orbital for the hydrogen atom results as an exact solution, for the choice of the first Laguerre polynomial (n = 1) for the radial wave function and the lowest spherical harmonic (/ = 0) Yqo, for the angular wave function. Thus, from Table 1.1, the normalized Is atomic orbital for the hydrogen atom is. 

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. Fock s projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

---