Herman-Skillman program output

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. 

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... 

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. ... 

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. 

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. 

Figure 3.3 Demonstration of the good fit between the Slater 2s radial approximation, when rendered orthogonal to the Is function, and the numerical radial function as output of the Herman-Skillman program. The other details of the spreadsheet are as in Figures 3.1 and 3.2.

Figure 3.4 Demonstration that the internal normalization checks can be changed at will [ncheck = yes/no ] in the spreadsheet fig3-l.xls without affecting the mateh to the numerical radial function for the lithium 2s orbital, output by the Herman-Skillman program. But, note that, as we might expect, the results in F 9 and G 9 are different.

The Slater exponents are parameters in any calculations involving these Gaussians sets, which we can vary to achieve a particular result. In the present case, the desired result is a fit to the numerical radial data for lithium 2s. For the valence shell regions of the lithium 2s orbital the fit in Figure 3.7 is almost complete for the choices G = 2.7 and G.5 = 0.675. However, that, of course, does not mean that we have discovered a universal set of Slater exponents for lithium. All we have done in this analysis is set the criterion for the choice of Slater exponents, the match to the output of the Herman-Skillman program... 

Finally, finish the design with the chart for the comparison of the double-zeta orthonormal 2s function and the numerical output of the Herman-Skillman program for the lithium 2s orbital. 

Figure 3.10 Comparisons of the spreadsheet calculated Slater DZ radial function as rR(r) with the same function as outputs from the Herman-Skillman program and Mitroy s RHF atomic program.

Figure 3.12 Demonstration of the scalability of the split-basis results. In this diagram, the matches are to the output for the Herman-Skillman program for lithium Is and 2s radial functions. Close agreement is obtained simply by changing the Slater exponents to the Table 1.3 values for the case of lithium, compare Figure 3.11.

Do not be confused by the symbol. P(r). here. It is not the radial distribution, rather rR(r). the numerical radial function as in the output of the Herman-Skillman program, hs.exe. P(r) is used. here, to stay with Hartree s original derivation. 

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. 

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. 

Now, on worksheet Is , follow Figure 3.11 and enter the Herman-Skillman output with two empty columns of cells after the Thomas-Fermi mesh to allow for the recovery of the actual lithium Is and 2s radial function over the radial array, as in Chapter 1, with, as before, the irregular behaviour at the origin avoided using the INTERCEPT function of the EXCEL program, for example for the Is function... 

There are no analytical forms for the radial functions, / ni(r), as solutions of the radial wave equation. Hartree, in 1928, developed the standard solution procedure, the self-consistent field method for the helium atom by using the simple product forms of equation 1.10 to represent the two-electron wave function. Herman and Skillman (4) programmed a very useful approximate form of the Hartree method in the early 1960s for atomic structure calculations on all the atoms in the Periodic Table. An executable version of this program, based on their FORTRAN code, modified to output data for use on a spreadsheet is included with the material on the CDROM as hs.exe. 

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