The rules were developed by John C. Slater in an attempt to construct simple analytic expressions for the atomic orbital of any electron in an atom. Specifically, for each electron in an atom, Slater wished to determine shielding constants (s) and "effective" quantum numbers (n*) such that
provides a reasonable approximation to a single-electron wave function. Slater defined n* by the rule that for n = 1, 2, 3, 4, 5, 6 respectively; n* = 1, 2, 3, 3.7, 4.0 and 4.2. This was an arbitrary adjustment to fit calculated atomic energies to experimental data.
Such a form was inspired by the known wave function spectrum of hydrogen-like atoms which have the radial component
where n is the (true) principal quantum number, l the azimuthal quantum number, and fnl(r) is an oscillatory polynomial with n - l - 1 nodes. Slater argued on the basis of previous calculations by Clarence Zener that the presence of radial nodes was not required to obtain a reasonable approximation. He also noted that in the asymptotic limit (far away from the nucleus), his approximate form coincides with the exact hydrogen-like wave function in the presence of a nuclear charge of Z-s and in the state with a principal quantum number n equal to his effective quantum number n*.
Slater then argued, again based on the work of Zener, that the total energy of a N-electron atom with a wavefunction constructed from orbitals of his form should be well approximated as
Using this expression for the total energy of an atom (or ion) as a function of the shielding constants and effective quantum numbers, Slater was able to compose rules such that spectral energies calculated agree reasonably well with experimental values for a wide range of atoms. Using the values in the iron example above, the total energy of a neutral iron atom using this method is -2497.2 Ry, while the energy of an iron cation lacking a single 1s electron is -1964.6 Ry. The difference, 532.6 Ry, can be compared to the experimental (circa 1930) K absorption limit of 524.0 Ry.
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