In the Classroom
The Genius of Slater’s Rules James L. Reed Department of Chemistry, Clark Atlanta University, Atlanta, GA 30314;
[email protected] More than 60 years ago Slater proposed a set of very simple rules for the computation of the effective nuclear charge experienced by an electron in an atom (1). Since that time these rules have been used in applications ranging from evaluating atomic overlap integrals (2) to establishing tables of atomic radii (3). Recently they were used to provide new insights into the often elusive nature of chemical hardness (4 ). Because Slater’s procedure provides for the computation of one-electron energies rather than orbital energies, it is especially well suited for instructional purposes and has been included in a number of textbooks (5). The need for simple activities that illustrate and reinforce an intuitive understanding of atoms and their properties is needed throughout the undergraduate curriculum. However, one difficulty with Slater’s original procedure has been that it fails to yield the level of detail needed to illustrate important concepts. With relatively minor modifications, Slater’s method would provide to both students and teachers a model of the atom that is instructive as well as surprisingly simple and intuitive. The Basic Concepts Slater’s method and its implementation require only a reasonable mastery of three concepts, which are already part of the first-year curriculum. The first of these is that the electron– electron interactions can be reasonably described as shielding of the nuclear charge by each of the other electrons, or *=Z
Zi
Σ
Figure 1. A diagrammatic presentation of the modified Slater’s Rules.
all electrons
– ci = Z –
i≠ j
cij
(1)
Zi*
where is the effective nuclear charge experienced by electron i and cij is the shielding of electron i by electron j. Once this is determined, the energy of the electron may be computed, using the relationship {1
e i = {1312 kJ mol
Z i* ni
2
(2)
This relationship most often appears in first-year texts in its more complicated form as the Rydberg equation. Finally, since Slater’s procedure yields one-electron energies, the total electronic energy is the sum of the one-electron energies: E = e1 + e2 + e3 + … + eN + epairing
(3)
The pairing energy was not explicitly included in Slater’s original formulation, but its explicit consideration is necessary for a number of applications. Shielding There are potentially a very large number of shielding constants. However, Slater found that, because many were virtually equal, the number could be reduced to only three. We have refined the scheme to use only three for the lighter elements (no d electrons) and a total of eight for the elements 802
through barium (no f electrons). These rules are applicable to positive, negative, and fractionally charged atoms. The rules given below are summarized in Figure 1. The Rules for Light Elements Larger Shell. An electron is not shielded by any electron in a larger shell (c ij = 0.00). Same Shell. Both s and p electrons in the same shell shield each other: 0.3228. Next Smaller Shell. The shielding of an s electron by an s or p electron in the next smallest shell is 0.8366. The shielding of a p electron by an s or p electron in the next smallest shell is 0.9155. Inner Shells. An electron is completely shielded by electrons in shells that are smaller than the next smallest (c ij = 1.00). The pairing energy for p electrons is 1360.1/n2 kJ/mol. The Rules for Heavier Elements Same Shell. In the same shell the shielding of a d electron by s and p electrons is 0.8933, but the shielding of an s or p electron by a d electron in 0.0352. The shielding of d electrons by d electrons is 0.3228. Smaller Shells. The shielding of both s and p electrons by a d electron in the next smaller shell is 0.9143. A d electron is completely shielded by any electron in a smaller shell. The pairing energy for d electrons is 9610.7/n2 kJ/mol.
Journal of Chemical Education • Vol. 76 No. 6 June 1999 • JChemEd.chem.wisc.edu
In the Classroom
Evaluation of the Shielding Constants These shielding constants are the ones which yield the best least squares (5) agreement between the experimental (6 ) and computed first ionization energies for elements hydrogen through xenon. Thus for atom M the ionization energy was computed via IE(M) = E(M+) – E(M)
(4)
where the energy is evaluated using eqs 1–3 (see the sample calculation below for details). The first ionization energies are known very accurately for hydrogen through xenon, which are the elements having only s, p, and d electrons. For most elements the ionization of d electrons occurs only for the third and subsequent ionizations of the transition metals. These were used to determine the shielding constants for d electrons. The validity of both the model and the shielding constants has been evidenced by their ability to reproduce the experimental first ionization energies and successive ionizations of the same atom (7).
Sample Calculation A potassium atom has been selected for the sample calculation. The configuration we have selected is 1s22s22p63s23p63d1 For processes involving only valence electrons the core electron energies do not change. Thus for most exercises the number of computations is greatly reduced, and students need concern themselves with only one shell. Of course to illustrate the shell structure of an atom or core ionizations (XPS or ESCA), it would be necessary to do computations for core electrons also. The determination of the shielding of a 3s electron is illustrated diagrammatically below along with the computations for all the third-shell electrons. 1.00
1s2
0.0352
1s22p6
3s2
0.8366
3p6
3d1
0.3228
Z *3s = 19 – [(1)(0.0352) + (7)(0.3228) + (8)(0.8366) + (2)(1.00)] Z *3s = 8.0124 e3s = { 9,358.7 kJ/mol * Z 3p = 19 - [(1)(0.0352) + (7)(0.3228) + (8)(0.9155) + (2)(1.00)] Z *3p = 7.3812 e3p = { 7,942.3 kJ/mol Z *3d = 19 - [(8)(0.8933) + (8)(1.00) + (2)(1.00)] Z *3d = 1.8536 e3d = { 500.9 kJ/mol epairing = 1360.1/9 = 151.1 kJ/mol
The electronic energy (valence electrons only) is thus E[3s23p63d1]= (2)({ 9358.7)+ (6)({ 7942.3) + (1)({ 500.9) + (3)(151.1) E[3s23p 63d1 ]= { 66,418.8 kJ/mol
To allow for comparison, the effective nuclear charges and one-electron energies for the configuration 1s22s22p63s23p64s1 have been computed to yield e3s = { 9,441.1 e3p = { 8,018.2 e4s = { 436.5 E[3s23p 64s1] = { 66,974.9
Z *3s = 8.0476 Z *3p = 7.4164 Z *4s = 2.3072
This is one of the more involved computations, but it will serve to illustrate one of the subtleties involved in the aufbau process. The ordering of the energy levels of the fourth-period elements has been the subject of some controversy (8–11). The ordering of the one-electron energies that the model yields is 3s < 3p < 3d < 4s, which would seem to suggest that potassium should be the first transition metal. However, in spite of this ordering, the configuration that makes potassium an alkali metal is the lower energy configuration. The accepted configuration is favorable not because of the lower energy of the outermost electron, but rather because of the poorer shielding of the 3s and 3p electrons by the 4s electron. This lowers their energies and thus lowers the total energy. As one can see, the computations are quite simple both in concept and in execution. They are actually simpler than many of the computations that general chemistry students are expected to do routinely. Applications Slater’s rules find application throughout the undergraduate curriculum. It should be emphasized that the activities and presentation format should be tailored to the audience. For this reason, in addition to the text form of the rules, an equivalent graphical presentation has been included (Fig. 1). Furthermore the rules themselves have been divided into those for light elements and those for heavier elements with this same idea in mind. Finally, students need not be asked to do all the types of computations done in the sample calculation. The instructor may choose to give the students the effective nuclear charges or one-electron energies.
Aufbau Generating electron configurations is the first activity that students undertake after learning about quantum numbers. Because students have already been introduced to the basic concepts of thermochemistry, they can be shown or asked to carry out activities which show that the ground state configuration simply results in the maximum exothermicity for the aufbau or filling process. Because pairing energies can be computed, it can be easily shown that pairing electrons in the same orbital may be energetically unfavorable. Ionizations and Energy Level Diagrams Energy level diagrams are frequently used to facilitate learning. Students may be provided with or asked to prepare them. The energy level diagrams for sulfur, its cation, and its anion are given in Figure 2. The ionization process involves moving an electron from the highest occupied orbital to the zero-energy state. Although it would seem that the ionization energy should equal the difference in these two energies, except for the group 1 elements, the experimental ionization energies are much smaller. However, comparing the energylevel diagrams for the atom and its cation reveals that the very endothermic ionization is accompanied by an exothermic relaxation of the remaining valence electrons. Similarly, one might expect that the ionization energy and electron affinity of an atom would have similar values. The small values of the experimental electron affinities can be understood to arise from not only the increase in the energy of the remaining electrons, but also the increase in the acceptor orbital energy during the process.
JChemEd.chem.wisc.edu • Vol. 76 No. 6 June 1999 • Journal of Chemical Education
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In the Classroom
which is simply the average of the one-electron energies of the component orbitals. The promotion energy is then E(promotion) = E(ground state) – E(valence state) (6) The following are just a few examples. Although the atomization energies of sets of compounds of the type MX n suggests that the bonds become weaker as n increases, this apparent difference can be shown to arise primarily from differences in the promotion energies, and the bonds are essentially the same. In addition, it can be shown that the instability of noble gas compounds is not the result of weaker bonds, which would be counter to the periodic trend in bond strength, but rather the result of very large promotion energies. Figure 2. Energy level diagrams for sulfur, its cation, and its anion.
Spectroscopy The optical transitions among the configurations of atoms cannot be observed experimentally, but can be computed from the spectroscopic term energies (12). The energies for the transitions among the configurations computed using Slater’s method agree quite well with the experimentally computed transition energies. The ionization of core electrons by highenergy radiation has been used to probe the electronic structure of atoms, molecules, and atoms in molecules. The XPS and ESCA chemical shifts computed for core ionizations agree quite well with those observed experimentally (13). It may not, however, be immediately obvious that the XPS and ESCA chemical shifts should correlate with atomic charge (14). The XPS and ESCA energies computed for fractionally charged atoms not only yield this same type of correlation, but the exercise actually reveals that the XPS and ESCA chemical shift–atomic charge dependence arises from the differences in the relaxation energies of the valence electrons during photoionization. Periodicity and Periodic Trends Many of the properties of isolated atoms as well as atoms in molecules exhibit periodic behavior and trends. All these can be related directly or indirectly to trends in effective nuclear charge and principal quantum number. One of the strengths of Slater’s model is that the effective nuclear charge is directly interpretable in terms of the configuration, nuclear charge, and types of electron–electron interactions. Thus the model can assist in the interpretation of the periodic behavior of such properties as ionization energies, electron affinities, electronegativities, various atomic radii, bond distances, overlap integrals, and even chemical properties such as acid–base strengths and oxidation–reduction potentials. Promotion Energy The energetics of the promotion of atoms to their valence states or to hypervalent states (expanded octet) is often determinant in both structural and reaction chemistry. In the central field approximation it is easily shown that the one-electron energy (expectation value) of any hybrid orbital, spx d y, is given by 1 espxdy = e + x ep + y ed (5) 1+x +y s
804
In Closing I feel, as do many others, that most general chemistry courses attempt to include far too much diverse material. On the other hand, simple computations are currently used to enhance student understanding of topics such as stoichiometry, kinetics, and equilibrium but not atomic structure. It would seem that a few very simple computations would be appropriate for beginning students. It should also be noted that all the concepts (eqs 1–3) are already among those beginning students are expected to master. Finally, once having developed this simple skill, both students and teachers can use these “back of the envelope” calculations to enhance instruction in other topics in later courses. Acknowledgment I wish to acknowledge the support of the Army Research Office, contract DAAA15-94-K0004. Literature Cited 1. Slater, J. C. Phys. Rev. 1930, 36, 57. 2. Mulliken, R. S.; Rieke, C. A.; Orloff, H. J. Chem. Phys. 1949, 17, 1248. 3. Slater, J. C. J. Chem. Phys. 1964, 41, 3199. 4. Reed, J. L. J. Phys. Chem. 1997, 101, 7396. 5. Douglas, B. E.; McDaniel, D. H.; Alexander, J. J. Concepts and Models in Inorganic Chemistry, 3rd ed.; Wiley: New York, 1994; p 40. Sharp, A. G. Inorganic Chemistry; Longman: New York, 1981; p 70. Porterfield, W. W. Inorganic Chemistry: A Unified Approach, 2nd ed.; Academic: San Diego, 1993; p 23. 6. Nelder, J. A.; Mead, R. Comput. J. 1965, 7, 308. 7. Huheey, J. E. Inorganic Chemistry, 3rd ed.; Harper-Row: New York, 1983; p 43. 8. Carlton, T. S. J. Chem. Educ. 1978, 55, 2. 9. Carlton, T. S. J. Chem. Educ. 1979, 56, 767. 10. Pilar, F. L. J. Chem. Educ. 1979, 56, 767. 11. Melrose, M. P.; Scerri, E. R. J. Chem. Educ. 1996, 73, 498. 12. Brink, C. D. J. Chem. Educ. 1991, 68,376. 13. Tudela, D. J. Chem. Educ. 1993, 70, 956. 14. Moore, C. E. Atomic Energy Levels; National Bureau of Standards Circ. 467, Vol. I; National Bureau of Standards: Washington, DC, 1948. 15. Ghosh, P. K. Introduction to Photoelectron Spectroscopy; Wiley: New York, 1983; p 10. 16. Ibid., p 60.
Journal of Chemical Education • Vol. 76 No. 6 June 1999 • JChemEd.chem.wisc.edu
