J. A. McMillan and T. Halpern' Argonne National Laboratorya Argonne, Illinois 60439
I I
Nuclear Ovedllpping of S Electrons A periodic property of the elemerits
Plots of many atomic properties as a function of the atomic number Z display the periodical nature of the classification of elements. This periodicity, in many cases extended to properties of compounds as well, establishes analogies that set the basis for further classification into typical, transition, innertransition, or lanthanide and actinide elements. The theoretical basis for this classification is provided, as it is well known, by the occupation of orbitals of a particular value of the angular quantum number 1. Thus, in typical elements, the ns and the np orbitals are occupied as the atomic number increases, where the principal quantum number n stands for the highest value in the atom under consideration. The transition elements are formed by filling theavailable (n- 1)dshelland are found in three series, 3d, 4d, and 5d. The inner-transition elements, also called lanthanides or rare earths, are formed mainly by occupying the 4f shell. The situation is not so clear in the actinides where the energies of the 5f and 6d levels compete in the build-up process. This situation can be anticipated in view of the fact that some shufflling between 4f and 5d levels already takes place in the rare earths. Among the periodic properties of the atoms, probably the most sensitive one is the electronic density a t the nucleus of the s orbitals, as illustrated in Figures 1-3. But before discussing the interpretation of these plots it will be advisable to recall some elementary notions of the basic theory underlying electron configurations. The hydrogen-like wavefunction of an electron in an atom, spin aside, is given by
The electronic density a t the nucleus in the case of s electrons
Figure 1 . Eledronic d m d t y a t the nucleus of nr electrons os o function of the atomic number.
where ( l ) P ( ) is the radial wavefunction, of spherical symmetry, and 81,,(8) .@,(q) is a spherical harmonic
ignoring phase considerations. The expression in brackets is a normalization coefficient and PI,lmI (cos~')is an associated Legendre polynomial.J For s electrons, I = m = 0 and, since P0,,(cos6') = 1, eqn. (1) reduces to
'Present address: Ymhiva University, Belfer Graduate School of Science, New York, N.Y. 10033. a Work performed under the auspices of the U.S. Atomic Energy Commission. E.,AND ENDE,F., "Tahlm of Functions with FormVAHNKE, ulae and Curves," (4th ed.), Dover Publications, New York, 1945.
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Figure 2. Elwtronic density mt the nucleus of in tion of the atomic number.
- 111 electmns 0s
0
funs-
rlau.) Figure 4. Radial elostmnis densities in arbitrary vnin for various warefunctions in Ga cmd Tm. Figure 3. Electronic density ot h e nucleus of ( n tion of the atomic number.
- 21s eloctronl
as a funs-
is not zero. In atoms with more than one electron the limit of eqn. (4) has to he computed using Hartree-Fock theory, where it is referred to as A
=
1 lim - P..,(r) r
-0
(5)
and given in atomic units. The parameter A2,listed in the Table for ns, (n - 1)s and (n - 2)s electrons from H to Rn, is readily available from Hartree-Fock tables. It has the meaning of 4rr-times the electronic density "inside" the nucleus expressed in electrons per atomic unit of volume. A feeling of the orders of magnitude involved is obtained after considering that one cubic centimeter contains approximately 6.7 X lo2' atomic units of volume. The magnitude of A q s determined, for eachvalne of n, by the combined coulombian interaction with the nuclear charge and the other electrons in the atom which is the basis of the iteration procedure in the Hartree-Fock model. Therefore, although one would normally expect an increase in the density a t the nucleus as the atomic number increases, it is also reasonable to expect that this increase will, to a certain extent, depend on whether the new electron added to neutralize the increase in nuclear charge is mainly localized outside or inside the ns electron under consideration. Figures 1 to 3 show quite conclusively the correctness of the argument. Consider the atom 1sK(4s1). The next atom, obtained by increasing Z in one unit, is &a(4s2). Since the second 4s electron is expected to have the same radial function as the first 4s electron, the increase in electron density a t the nucleus is basically due to the increase in the atomic number. Next, and beginning with 21S~(3d1, 48%))come the elements of the first transition series, in which to an increase of Z there corresponds occupation of a 3d wavefunction. The radius of maximum electronic density of 3d electrons is smaller than the fourth lobe and larger than the first, second, and third lobes of the 4s probability function (Fig. 4). Therefore, a 3d electron will force the first three lobes
toward the nucleus and the fourth lobe outwards. The net effect of these opposing factors is an increase of the 4s electronic density a t the nucleus significantly smaller than the one observed after addition of a 4s electron, resulting in a screening action of the 3d electrons on the nuclear attraction. Starting with soZn(3d104s2) the next in A.U. of ns, (n- - 1)s and (n Values 0f4rr1$(0)1~ electrons for Atoms H to Rn Z 1
Atom
A
la
2s
3a
6.00
Z Atom
Z
Atom
- 2)s
38
la
5s
55 56 57 58 59 60 61 62 63 04 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
Ca
Bs La Cs Pr Nd Pm Sa
Eu Gd Tb Dy
Ho Er Tm
Yb Lu Hf Ta
W Re Da Ir Pt
Au Hg
TI Pb Bi Pa At
Rn
48 2963 3225 3469 3721 3980 4247 4524 4809 5104 5410 6724 6050 6386 6732 7090 7458 7875 8310 8767 9243 9742 10258 10797 11355 11936 12536 13161 13806 14477 15158 15884 16615
58 409 491 527 562 597 631 666 701 736 772 808 845 883 921 960 1000 1106 1217 1333 1454 1580 1712 1849 1992 2140 2293 2481 2681 2892 3116 3349 3594
68 21.0 37.3 38.9 40.3 41.8 43.3 44.8 46.2 47.7 49.1 50.6 52.1 53.7 55.2 56.8 58.4 71.8 82.9 93.0 102 111 120 128 135 143 150 205 258 312 369 427 488
Volume 47, Number 9, September 1970 / 645
atom, alGa(3d'04s24p') incorporates the additional electron in 4p, which is rmtside the 4s shell (Fig. 4) and forces this latter toward the nucleus, thereby increasing significantl$ its electronic density at the nucleus. The same pattern is observed in the Rb-Xe sequence. In the Ce-Rn sequence there is an additional feature characterizing a very small change in the electronic density a t the nucleus of the 6s electrons due to an even larger screening effect of the 4f electrons which lie deeper inside the core (Fig. 4). That this is true is clearly shown in Figure 2, where the plot of the (n - 1)s is seen to be still sensitive to the occupation of the 4f level while it is
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already blind to the occupation of the (n - 1)d levels. Finally, Figure 3 illustrates the plot corresponding to the (n - 2)s electronic density at the nucleus, in which any discrimination between occupation of f, d, or p levels is wiped out due to the fact that the (n - 2)s wavefunctions lie inside the (n - l)d and the (n - 2)f. The values used in the table and figures have been taken from Fisher's tables.* 'FISHER, C. F., "Some Hartree-Foek Results for the Atoms Helium to Radon," Department of Mathematics, University of British Columbia, Vsneouver 8, B. C., Canada, January, 1968.
