Function radial

The radial functions for atoms through argon tabulated by dementiu) are particularly useful for simple MO calculations. Other sources are Watson42 and Richardson et al.(3)for transition metal atoms, and Slater 4) for simple single-exponential-term functions. 

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Normalized radial functions for a hydrogenlike atom are given in Table A 1.1 and plotted graphically in Fig. A 1.1 for the first ten combinations of n and /. It will be seen that the radial functions for Is, 2p, 3d, and 4f orbitals have no nodes and are everywhere of... 

Table Al.l Normalized radial functions / ,/(r) for hydrogen-like atoms...

Figure Al.l Radial functions for a hydrogen atom. (Note that the horizontal scale is the same in each graph but the vertical scale varies by as much as a factor of 100. The Bohr radius Oq = 52.9 pm.)...

Notice that 1 haven t made any mention of the LCAO procedure Hartree produced numerical tables of radial functions. The atomic problem is quite different from the molecular one because of the high symmetry of atoms. The theory of atomic structure is simplified (or complicated, according to your viewpoint) by angular momentum considerations. The Hartree-Fock limit can be easily reached by numerical integration of the HF equations, and it is not necessary to invoke the LCAO method. 

The radial functions, R depend only upon the distance, r, of the electron from the nucleus while the angular functions, (6,(p) called spherical harmonics, depend only upon the polar coordinates, 6 and Examples of these purely angular functions are shown in Fig. 3-11. 

The index n is the radial degree and the index m is the azimuthal frequency. The function R r) is a radial function given by... 

Stener and co-workers [59] used an alternative B-spline LCAO density functional theory (DFT) method in their PECD investigations [53, 57, 60-63]. In this approach a normal LCAO basis set is adapted for the continuum by the addition of B-spline radial functions. A large single center expansion of such... 

Thus, the operator Hi is hermitian and the radial functions REi(r) constitute an orthonormal set with a weighting function w(r) equal to r ... 

Radial functions in terms of associated Laguerre polynomials... 

The radial functions Sni p) and R i(r) may be expressed in terms of the associated Laguerre polynomials L p), whose definition and mathematical properties are discussed in Appendix F. One method for establishing the relationship between Sniip) and L p) is to relate Sni p) in equation (6.50) to the polynomial L p) in equation (F.15). That process, however, is long and tedious. Instead, we show that both quantities are solutions of the same differential equation. 

Table 6.1. Radial functions R i for the hydrogen-like atom for n = to 6. The variable p is given by p = 2 Zrj na ...

The normalized radial functions Rniir) may be expressed in terms of the associated Laguerre polynomials by combining equations (6.22), (6.23), and (6.54)... 

The energy levels E and the radial functions depend on the reduced... 

The radial functions Rniir) for the Is, 2s, 2p, 3s, 3p, and 3d atomic orbitals are shown in Figure 6.4. For states with / 0, the radial functions vanish at the origin. For states with no angular momentum (/ = 0), however, the radial function Rno r) has a non-zero value at the origin. The function Rniir) has ( — /— ) nodes between 0 and oo, i.e., the function crosses the r-axis in — I — ) times, not counting the origin. 

Figure 6.4 The radial functions for the hydrogen-like atom.

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