Normalized radial wave functions, for

Normalized radial wave function for the 2s atomic orbital in the hydrogen atom. ... 

Normalized Radial Wave Functions R(r) for Hydrogenlike Atoms... 

Likewise, a basis set can be improved by uncontracting some of the outer basis function primitives (individual GTO orbitals). This will always lower the total energy slightly. It will improve the accuracy of chemical predictions if the primitives being uncontracted are those describing the wave function in the middle of a chemical bond. The distance from the nucleus at which a basis function has the most significant effect on the wave function is the distance at which there is a peak in the radial distribution function for that GTO primitive. The formula for a normalized radial GTO primitive in atomic units is... 

Figure 1.3 The normalized [see Chapter 3] radial wave function, Ri(r), for the Is atomic orbital in the hydrogen atom [Z = 1 ] presented as an EXCEL graph, constructed as described in the text. Note, that since Yqo = 1 for s-orbitals, the only difference between this function and the total Is atomic orbital is the factor (l/4jrf- due to the normalization constant over the angular coordinates.

Care is needed if data are copied, especially from the early literature on basis sets, either linear combinations of Slater or Gaussian functions, since it is dangerous to assume that a particular normalization or orthogonalization condition has been imposed. The normalization constants in the expressions for the basic functions may not be included in the pre-exponential coefficients and individual preferences certainly determined whether normalization constants were defined over all the integration coordinates, or simply the radial coordinate, in the modelling of the radial wave functions. 

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 radial wave function is normalized in the unit cell, and the normalization is done approximately by integrating inside two WS spheres for the two atoms in the hep unit cell. The same is used for both and in the exchange... 

We assume that the wave functions of a set of d orbitals are each of the general form specified by 9.2-1. We shall further assume that the spin function [jj% is entirely independent of the orbital functions and shall pay no further attention to it for the present. Since the radial function R(r) involves no directional variables, it is invariant to all operations in a point group and need concern us no further. The function 0(0) depends only upon the angle 0. Therefore, if all rotations are carried out about the axis from which 0 is measured (the z axis in Fig. 8.1), (0) will also be invariant. Thus, by always choosing the axes of rotation in this way (or, in other words, always quantizing the orbitals about the axis of rotation), only the function (< ) will be altered by rotations. The explicit form of the 4>(0) function, aside from a normalizing constant, is... 

This form is similar to the one presented in equ. (7.18) for a free plane wave, except for the incorporation of the incoming spherical wave boundary condition, the separate treatment of the radial function, and the normalization of these radial functions on the energy scale. Furthermore, it should be noted that equ. (7.29a) contains a j dependence of the phases and the radial function which can be understood only within a relativistic treatment. 

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