The wave function for the two-electron atom is assumed to be a product of two identical one-electron orbital functions. The system is assumed to be a spin-singlet so that only the spatial functions need to be considered here. The SCF procedure involves two repeated steps. First, using a guess for the orbital function, an effective potential is generated. Second, with this effective potential, the differential equation for the orbital function is solved. The new orbital function is used to generate a new effective potential, which is then used to generate a newer orbital function. The procedure is repeated until successive orbital functions are considered to be close enough to each other.

A Self-Consistent Field Calculations Spreadsheet file contains two spreadsheets. The first is the one described in the associated article (1). It performs the calculation with relatively simple approximations and numerical methods, and serves to illustrate the SCF procedure for the student. The second presents a more sophisticated calculation that may be of interest to more advanced students.

Figure 1. Portion of Excel spreadsheet showing the parameters used in the SCF calculations.

The first spreadsheet is initially set up so that many of the columns are hidden. They may be unhidden if the user wishes to study the calculations in more detail. The spreadsheet also contains a graph with plots of both of the orbital functions versus r. All the columns, parameters, and the graph should be visible on a single screen.

This spreadsheet was initially prepared so that the trial function is set equal to 0 at all grid points and the orbital eigenvalue is ∈ = –1.5 au (Figure 2). The kink in the radial function is obvious in the graph. The first question posed is how should ∈ be varied to make this function smooth. An understanding of the relation between energy eigenvalue and curvature of the eigenfunction should lead to the conclusion that the eigenvalue should be lowered. The eigenvalue can be changed manually on the spreadsheet and the effect on the orbital function observed in the graph. Through trial and error, the kink can be removed at a value of about ∈ = –2.0. However, The optimum eigenvalue is more efficiently found with the use of Excel’s Solver.

Next, iterations are performed by copy and paste. Once a paste is done, all the columns change and the previous and new functions can be compared in the graph. They do not agree well at this stage, and it is necessary to remove the kink in the new function by repeating the use of Solver. Once this is done, these steps are repeated. The process continues until the results have converged.

Figure 2. Initial radial function with an eigenvalue of ∈ = –1.5 au.

Figure 3. Comparison of radial functions after the first iteration.

The second spreadsheet contains a calculation on the same system using more sophisticated techniques: The radial grid is changed to a logarithmic one, with r = from 0 to infinity; the numerical integration is performed with the Noumerov (2) method; and the total energy at convergence is equal to the exact HF energy (3).

Literature Cited

1. Hoffman, G. G. J. Chem. Educ. 2005, 82, 1418–1422.
2. Noumerov, B. V. Monthly Notices Roy. Astron. Soc. 1924, 84, 592–601.
3. A total energy of –2.8617 au was reported in Clementi, R.; Roetti, C. At. Data Nucl. Data Tables 1974, 14, 177 according to McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, 1997.