4s Sometimes Is below 3d
To the Editor: The title of the article "4s is Always Above 3d!" by F. L. Pilar [J. CHEM. EDUC., 55,2 (1978)] is incorrect. The author claims that when orbital energies are compared, 4s is above 3d for all atoms. He claims this even for potassium, in which the total energy for the state with configuration (Ar)3d is hieher ,. than the total enerev for the state with confieuration (?\rJ.lr.Nevertheless the orhital energics rhat the author introducrs as evidence are limited toatonis in which 4s and :Id ~,rhiraliare occupird simullane~~usly. For pomwium he merely argues th:~tit would not be inconsistent for rhcorder oforhitill energies to br the reverse of the order of total energies. 'I'his argument is invalid, however, lor it d e ~ e n d on i the author's incorrect claim that the repulsion energy between a 4s electron and an argon core is greater than that hetween a .7d rlrctrr,n and an argrm core, 'l'he reverse is a~tuallvthe caw. since practic~llythe entire 3d charge distrihution is buried in thewre, whereas most.of the 4s distribution is outside the core. When this error of fact is corrected, the author's own argument leads to the conclusion that the 3d orhital energy for potassium would be higher than the 4s orbital energy, and hy an amount even greater than the difference in total energies. Such arguments are not definitive, however, for they ignore possible changes in the argon core in going from one state to the other. We turn instead to the orbital energies themselves. For potassium the 4s orbital energy of the 2S state with confieuratiou (Arb is lower hv 0.21 Hartree than the 3d orI,ital &rrm of the 9state with configuration (Ar):ld. For the same t w o states of Ca' the 4s orbital encrnr is lower than the 3d by 0.07 Hartree. For neutral calcium the 4s orhital energy of the ' S state with configuration (Ar)4s2is 0.13 Hartree lower than the 3d orbital energy of the 3 F ~ t a t with e configuration (Ar)3d2. The 4s orbital energies for these comparisons are Hartree-Fock results.' The 3d orbital energies have heen extrapolated from Hartree-Fock results for the same states of isoelectronic ions with atomic numbers 21 through 27.2 In addition, nonextrapolated data are available for Set, for which the 4s orhital energy of the ' S state is 0.0324 Hartree lower than the 3d orbital energy of the 3F state.3 In the previous paragraph each comparison of 3d with 4s orbital energy involves two different states of the same atom. Whv not instead comnare orhital enereies for the same state of t i e atom, say the ground state? or p&siurn, calcium,and ions isoelectronic with them this approach would have the disadvantage of comparing the energy of an occupied orbital with the energy of an unoccupied orbital (called a virtual orbital). Energies of virtual orbitals are without physical significance. and their values are hiehlv deuendent on the basis set used in calculating the self-co&isient:field wave function.4 The comparisons in the previous paragraph involve only occupied orhitals, as d o those in Pilar's article. There is no single ordering of orhital energies that is valid throughout the Periodic Table. For orhital energies, as well as total energies, there are atoms for which 4s is below 3d, and ~~
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Clementi, E., and Roetti, C., At. DntoNucl. Data Tables, 14,177 (19741,Tables 1 and 2. Ibid., Tables 32 and 33. "bid., Table 2. Hurley, A. C., "Introduction to the Electron Theory of Small
Molecules," Academic Press, London, 1976, pp. 130 and 164.
other atoms for which the reverse is true. When only 3d or only 4s electrons are present in addition to an argon core, the greater penetration and lesser repulsion energy of the 4s orbital allows Its orhital enerevto be lower than that of the 3d. If instead an atom containsine or two 4s electrons together with one or more 3d electrons, the screening of the nucleus bv the 3d electron raises the 4s orbital energiabove that of thk 3d. T e r r y S. Carlton Oberlin College Oberlin. OH 44074
To the Editor: Professor Carlton has raised an issue which does not a u m to have a unique answer. In the case of the elements scaddium and beyond-in which both 3d and 4s orhitals are used simultaneously-there is no doubt whatsoever but that 4s is above 3d: Both theory and experiment support this contention. For ~otassiumand calcium (and ions isoelectronic with them) thesituation is unclear-in'fact, the problem of 3d and 4s enemies is difficult to formulate since onlv the 4s orhital is used i n the ground states. The orhitals which minimize the total enerw -" of an atom of n electrons each describe the movement of a single electron in the average field provided by the other n - 1electrons. Any orbital other than these which is orthogonal to the groundstate orbitals may he interpreted as describing an extraelectron moving in the average field of n electrons; such an orbital is called a uirtual orhital and mav be used to describe an anion of the atom. Thus, in discussing the energy of a 3d orhital relative to the 4s orbital of the potassium electronicconfiguration (Ar)4s, we are talking about a uirtual3d orhital. This 3d orbital will he orthogonal to all the ground-state orhitals of potassium due solely to its nngulnrpart; its radial part will determine its energy and its extent of penetration into the electron shell of the neutral atom core. Consequently, Carlton's example of the Rd orhital energy o i a n excited state uf ~otassiumh ~ i n ehieher than the 48 01 the ground state does not constitute &nchsive proof of anything. This most certainly doesn't preclude the existence of a uirtual3d orhital lower in energy than the 4s of the ground state. Carlton's claim that 3d must be more penetrating than 4s since it is huried in the core is also inconclusive. In fad. 3d can be either more penetratinx or leas penetrating thands, depending on how other factors nffert its radial Dart. Certainly in scandium and beyond, 3d must be less penetrating than 4s if one is to rationalize a lower energy for (Ar)3d"-24s2 than for (Ar)3d ". Admittedly, one can argue that it is more natural to compare the 3d and 4s orbital energies as Carlton does than to go into virtual orbitals, but this is a question of taste and cannot he resolved on a ourelv obiective basis. Mv own nersonal . judgment is that ii is far more natural to taifor the model in such a way as to eliminate disquieting discontinuties such as putting 3d above 4s for two elements and then suddenly reversine them. Since the issue admits of no uniaue solution. the real qiestion may be: Which model is the least awkwaki to rationalize? In conclusion, let us suppose-just for the sake of argument-that the 3dl4s issue does have a uniaue solution and that Carlton is correct. This does not change a single mnjor point of my paper; in particular it does not alter the very important fact that i t isn't generally possihle to predict the electronic configuration of an atom or ion simply by putting electrons into the lowest energy orhitals available.
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F r a n k L. Pilar University of New Hampshire Durham, NH 03824 Volume 56, Number 11, November 1979 1 767
