Numerical radial wave functions

Exercise 1.4. Generation of the numerical radial wave function for the helium atom Is atomic orbital. 

This exercise involves the calculation of the numerical radial wave function for the helium atom in the form rR(r) output by the Herman-Skillman program for the helium atom and its processing into the equivalent of a radial wave function. 

Note, that the Herman-Skillman output, the numerical radial wave functions, rR r), are normalized in the form of equation 1.9. 

The Gaussian representation, of the numerical radial wave functions in atoms, is at the heart of modem molecular orbital theory. The different formulations are known as basis sets and the most common and proven basis sets are included in the various packages for molecular orbital theory calculations. In this book, the symbol sto-ng) identifies a linear combination of n Gaussian orbitals used, in the present context, to model the corresponding Slater function. 

Figure 1.24 Comparisons of the numerical radial wave function for boron with the Slater and sto-3g) approximations. The extra boron data were obtained by running the Herman-Skillman program using the Schwarz values for the Xa exchange term.

The new results in Figure 1.24 show, at least, that it is possible to generate different numerical radial wave functions for the boron 2p orbital [and, of course, any other orbital obtained by approximate solution of the Schrodinger equation]. But, for a light atom like boron, it is unlikely that this is the major reason for any discrepancy. 

Fig. 2.3 Schematic drawing of a 3s numerical radial wave function (full line) of a Na atom, its Slater-type approximation (STO, dashed line), and a combination of three Gaussian-type functions (GTO, dotted line).

The radial wave function has (n — l+l) nodes, where n and l are the quantum numbers. To solve the radial atomic wave equation above, the Herman-Skillman method [12] is usually used. The equation above may be rewritten in a logarithmic coordinate of radius. The radial wave equation is first expressed in terms of low-power polynomials near the origin at the nucleus [13]. With the help of the derived polynomials near the origin, the equation is then numerically solved step by step outward from the origin to satisfy the required node number. At the same time, the radial wave equation is solved numerically from a point far away from the origin, where the radial wave function decays exponentially. The inner and outer solutions are required to be connected smoothly including derivative at a connecting point. 

However, the numerical treatment of such explicitly time-dependent basis states would be time consuming compared to the treatment of bound states. Thus we search for a further simplification of by investigating the asymptotic behavior of Coulomb wave functions [42]. For rAe tt the radial wave function z/j. / is nearly independent of e and may be considered constant for integration. For et tt the exponential function in equation (18) is nearly independent of e. In both cases in equation (18) may be replaced by... 

Fig. 5. Schematic representation (after [13]) of the numerical calculation of the spatial part of the matrix element Mspace in the p + p—> d c u, reaction. The top part shows the potential well of depth Vo and nuclear radius R of deuterium with binding energy of —2.22 MeV. The next part shows the radius dependence of the deuterium radial wave function Xd(r)- The wave-function extends far outside the nuclear radius with appreciable amplitude due to the loose binding of deuterium ground state. The p-p wave-function XppM which comprise the U = 0 initial state has small amplitude inside the final nuclear radius. The radial part of the integrand entering into the calculation of Mspac is a convolution of both Xd and Xpp in the second and third panels and is given with the hatched shading in the bottom panel. It has the major contribution far outside the nuclear radius...

The primary wave function output data from the Herman-Skillman program are the products rR r), which are known as the numerical radial functions. The radial wave function itself can be recovered on division by the radial distance, r, and approximately near the origin by extrapolating to avoid the infinity. There is one other detail. For the purposes of the numerical integration procedure in the Herman-Skillman procedure the radial data are defined on a non-uniform grid, x, known as the Thomas-Fermi mesh (4). These are converted, in the output from hs.exe, to radial arrays specific to each atom, with... 

Figure 1.10 Conversion of the numerical output (a) of the Hetman-Skillman program to (b) Is and 2s radial wave functions and (c) Is and 2s radial distribution functions for lithium following the instructions in Exercise 1.5.

Figure 1.12 The better agreement in the valence region of the atom with the numerical data for the Li s radial wave function, which follows for the Clementi double-zeta basis sets for Li s and Liis mutually orthogonal.

The spherical harmonics are ordinarily used for the angular part of the orbitals, and the Hartree-Fock equations are solved to obtain the radial wave functions numerically. The Hartree-Fock wave functions are the best radial wave functions that can be obtained in the form of the Slater determinant 2.61. For... 

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... 

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