Herman-Skillman

Despite many approximations the Herman-Skillman results compare well with experimental values and the program still is a useful tool for calculations related to the electronic configurations of atoms. 

Figure 2.12 Radial integrals and phases for 2p photoionization in neon as functions of the kinetic energy of the photoelectron. The radial integrals R d2p and R s2p and the corresponding phases refer to the photoionization channels 2p - sd and 2p -> ss, respectively. Instead of the total phase 2p the individual contributions are shown (equ. (7.27)), namely the Coulomb phases <7ed2p and

Figure 5.8 Angular distribution parameter / 2p for 2p photoionization in atomic magnesium as a function of photon energy. Experimental data points with error bars. Theoretical data (adapted to the experimental threshold for 2p ionization 2s - np resonances between 94 and 97 eV photon energy omitted) full curve, RRPA results [DMa83] within uncertainties of the drawing the saipe result is obtained in the MBPT approach [Alt89] broken curve, HF(1P) results [V6188] chain curve, Herman-Skillman results [DSa73]. From [KHL92],...

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. 

F irst, consider the d-state energy, c,. The only serious discrepancy between Mattheiss s calculation and the experimental optical spectra was that Mattheiss s calculation appeared to overestimate the band gap by about three electron volts. Since this gap is dominated by the energy i — r,p, the discrepancy suggested an overestimate of this difference. In fact, his calculated bands were positioned roughly in accord with the splitting predicted by term values of Herman and Skillman (1963). Finally, the same overestimate applies to the Herman-Skillman term values in comparison to Hartree-Fock term values. This suggested, then, that values for e, should be taken from Hartree-Fock calculations, and those are what appear in the Solid State Table and therefore also in Table 19-3. C. Calandra has suggested independently (unpublished) from consideration of transition metals... 

Consideration of Harlree-Fock term values from Fischer (1972), as discussed in Appendix A, indicates that Hartree-Fock values for valence s- and p-state energies are quite similar to the Herman-Skillman values given in the Solid Slate Table. Thus a more systematic treatment would result from use of the Hartree-Fock values throughout they are more appropriate for the transition metals and it would make little difference which set of values is used for other systems. The reasons for retaining Herman-Skillman values here are largely historical it is also possible that use of Hartree-Fock values would increase dLscrepancics with existing band calculations since, as indicated in Appendix A, the approximations almost universally used in solids are the same as those used in the Herman-Skillman calculation. For. s and p states the differences are small in any case. 

The difference and the ratio are between high and low oxidation states, raiiorelativity intensity of the white line. for solution samples is dependent upon the relative concentration of Ce and Ce (Ref. 36). The calculation was done with the Herman Skillman program, for example, configurations 2p - 4f 6d - and 2p, 4f 4d° for Ce and Ce, respectively were used only the difference in the transition energy was considered. 

This particular set of calculated values (by Herman and Skillman, 1963) was chosen since the approximations used in the calculation were very similar to those used in determining the energy bands that led to the parameters in Table 2-1. The values would not have differed greatly if they were taken from Hartree-Fock calculations (such values are tabulated in Appendix A). Values based on Hartree-Fock calculations have the advantage of giving good values for d states. Therefore, though the calculations in this book are based upon the Herman-Skillman values, for some applications the Hartree-Fock values may be better suited. 

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... 

The Herman-Skillman program is bundled on the CDROM with the permi.ssion of Professor Frank Herman, Distinguished Professor of Physics, San Jose State University. San Jose, California. USA. 

Unlike most modem computer programs, the Herman-Skillman code, written in the FORTRAN language, uses formatted input and output statements. Thus the white spaces in the text lines of input count to separate each input parameter. 

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. 

Run the Herman-Skillman program for the helium atom calculation. Select hs.exe and pressing the return button, or simply double click on the file icon. The program runs in interactive mode. When asked for the input file, type he.in, press return and type c he.out when asked to name the output file. The program runs in DOS mode and returns to WINDOWS after execution is complete. ... 

It is good practice to make new directories for each kind of application, so it is best perhaps to locate a copy of hs.exe in a directory named Herman-Skillman or whatever. 

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

Table 1.2 The orbital energies, in Rydbergs, for the atoms of the first short period of the Periodic Table (1 Rydberg = 13.605 eV = 0.8000 Hartrees) returned by the Herman-Skillman program.

Run hs.exe and copy the output data for the lithium 2s radial function onto columns A and B of a new spreadsheet. This means that the Thomas-Fermi mesh will be the radial mesh for the remainder of the exercise. For simplicity delete column A the dimensionless x-mesh. Label the remaining columns of translated data from the Herman-Skillman output file, r[TF], r/ (rXi-2s) and enter the headers RDF-Li 2s and RDF-Slater in cells C 5 and D 5. 

Consider the example of the boron atom and the making of comparisons of the Herman-Skillman numerical radial functions for 1 s, 2s and 2p with the Slater functions and possible Gaussian basis set approximations. 

In the first part, comparisons are made between the Slater functions and the numerical outputs of the Herman-Skillman program. In the second, the comparisons are between the Slater functions and their Gaussian representations. Finally, the role of the Slater exponent is emphasized by the restoration of the hydrogenic results using the hydrogenic Slater exponents. 

Remember, the Herman-Skillman program outputs the numerical radial function data, rR r) and so it is convenient to construct the Slater and sto-3g) approximations in this form. Remember, too, that there is a difference in the common practice with regard to the inclusion of normalization factors. The Herman-Skillman data are normalized only over the radial coordinate and this is usual, too, for Slater functions, but it is standard to normalize Gaussian functions over all the spherical polar coordinates. 

Run the Herman-Skillman program for the boron atom using the atomic data in Table 1.3. Then transfer the output mesh and rR r) data for each atomic orbital to a new worksheet in a new spreadsheet. Paste the data, as text, into columns A and B starting from cell A 9 as in Figure 1.19. Parse the text data into individual columns of r and rR r) values. 

Figure 1.19a Detail from the worksheet for the matching of the Slater and sto-3g) approximations to the Is orbital for boron as the output of the Herman-Skillman program. Note, the deletion of entries in the calculated jSIatennl) and sto-3g nl) to make clearer the graphs in the chart in this figure and Figure 1.19b for the match to the hydrogen function.

Run the Herman-Skillman program and transfer the output data for the numerical 2s radial function in boron to column B of the new spreadsheet. 

There are two considerations in the light of these comparisons. In the first place, we do not know the accuracy of the Herman-Skillman result. Secondly, it may be possible to improve the fit to the boron data by suitable choice of a modified Slater exponent or the use of a larger linear combination of primitive Gaussian functions. 

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 results of this modification to the Herman-Skillman calculation are shown in Figure 1.24, constructed, using the outputs from the program, with the different possible choices for the a exchange parameter. As you can see in the diagram, two extra columns of Herman-Skillman output were processed to add the extra r(Rr 2p) behaviour of the 2p numerical function. 

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