Research: Science & Education
Rules for Determining the Ground State of a j–j Coupled Atom Mark L. Campbell* Department of Chemistry, United States Naval Academy, Annapolis, MD 21402-5026
The coupling of the angular momenta of an atom’s valence electrons has a profound effect on the energy states in the atom (1, 2). Such coupling is normally described in two limiting representations: L–S and j–j coupling. In a previous paper we reported a simple, systematic method for determining the terms and their associated J states for atoms best described by j–j coupling (2). In this paper we report rules for determining the ground state for j–j coupled atoms analogous to Hund’s rules (3) for L–S coupled atoms. The only other treatment of this subject describes rules that are much more complex than the rules described here (4 ). L–S coupling is an appropriate description when the electron–electron Coulomb repulsion energy terms for the valence electrons are much larger than the spin-orbit terms. In L–S coupling the valence electrons’ individual orbital angular momenta couple to yield the total angular momentum L, and the spin angular momenta couple to yield the total spin angular momentum S. Thus, in L–S coupling, L and S are the good quantum numbers, and an electronic state results from the coupling of L with S to yield the total angular momentum J. The electrostatic interaction normally predominates in ground-state electron configurations for light to moderately heavy atoms. However, in ground configurations of heavy atoms and many excited configurations of light and heavy atoms, the spin-orbit energy of the valence electrons contributes to a larger fraction of the energy perturbation than the residual electrostatic energy. L–S coupling inadequately describes the observed states in these cases. In the weak field case where j–j coupling is the more appropriate coupling scheme, an individual electron’s orbital and spin angular momenta couple to give j so that the good quantum numbers are the values of j, , and s for each electron. In j–j coupling, the valence electrons’ j’s couple together to yield the total angular momentum J. Because the spin quantum number for an electron can only have the value 1/2, the possible values of j for any given electron can only be + 1/2 and | – 1/2|. Thus, for an electron configuration of the type l n there will be at most two types of electrons. Each of these two types of electrons forms a subset of equivalent electrons in which equivalent electrons are defined as having the same value of n, , s, and j. According to the Pauli exclusion principle, for each subset of equivalent electrons, only those states in which the mj values are different are allowed. We use the same notation for j–j coupled states as used in our previous paper (2). Terms for j–j coupled atoms have the form [(l-1/2) a (l+1/2)b(l ′′-1/2)c…]J where l and l ′ are the letter designations for the type of orbitals (i.e., s, p, d, etc.), and ′ are the numerical values of the orbital angular momentum quantum numbers for the electrons, and a, b, c are *Henry Dreyfus Teacher Scholar.
the number of equivalent electrons in each subset. Each term has associated with it energy states with the total angular momentum J as the rigorously good quantum number. The states for each term can be determined using the systematic method described previously (2). Table 3 from ref 2 gives all the possible J states for terms that arise from electron configurations with partially filled sn, pn, dn, or f n subshells. Once the terms and J states have been determined, the ground state can be determined using the following rules: 1. The lowest energy term is found by filling the l |-1/2| states to the maximum extent possible. 2. For atoms with electron configurations with only one partially filled subshell, the state with the maximum value of J of the ground term is the ground state.
For atoms with more than one partially filled subshell, the following rules apply: 3. For atoms with l 1l ′n configurations with a partially filled l′′-1/2 subset of equivalent electrons (less than a half-filled l ′ subshell), the ground state is determined by adding – 1/2 (where is the numerical value of the l 1 electron’s orbital angular momentum quantum number) to the value of J of the l′n ground state. 4. For atoms with l 1l ′n configurations with a filled l ′′-1/2 subset of equivalent electrons (a half-filled or greater l ′ subshell), the ground state is determined by adding 1/2 – to the value of J of the l ′n ground state.
Tables 1 and 2 list the ground states determined using the above rules. Table 2 lists the ground states for the 3d and 4d transition metals for completeness, although these atoms are better described by L–S coupling. In all cases except the Ce+ ion, these rules give the ground state value of J consistent with experiment (5–8). The only case in which these rules do not correctly predict the ground state for an atom or ion is the case of Ce+. For the Ce+ ion (5d24f 1), the ground term and J state are predicted to be [(d3/2)2(f5/2)1 ] and 9/2, respectively. The experimentally reported ground state value of J is 7/2 (4 ). Hund’s rules also predict a value of 9/2 for the ground state, so neither set of rules correctly predicts the ground state value of J for Ce+. It is not necessary to refer to Table 3 from ref 2 to determine the ground state value of J. To determine the ground state for electron configurations with only one partially filled subshell, add the maximum possible values of mj for the total number of electrons in the partially filled subset of equivalent electrons. The sum obtained is the value of J for the ground state. For example, tantalum has a 5d3 configuration for which the ground term is [(d3/2)3]. The value of J for the ground state is 3/2 + 1/2 – 1/2 = 3/2. For osmium (5d6), the ground term is [(d3/2)4(d5/2)2], and the value of J for the ground state is 5/2 + 3/2 = 4. For osmium, the values of mj do not have to be summed for the d3/2 electrons because the sum of the values of mj for a full subset of equivalent electrons is zero.
JChemEd.chem.wisc.edu • Vol. 75 No. 10 October 1998 • Journal of Chemical Education
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Research: Science and Education Table 1. Ground States Using j–j Coupling for Atoms and Ions with Filled or Only One Partially Filled Subshell s1 [(s 1/2)1]1/2
d1 [(d3/2)1]3/2
f 1 [( f5/2)1]5/2
s2 [(s 1/2)2]0
d2 [(d3/2)2]2
f 2 [( f5/2)2]4
d3 [(d3/2)3]3/2
f 3 [( f5/2)3]9/2
p1 [(p1/2)1]1/2
d4 [(d3/2)4]0
f 4 [( f5/2)4]4
p2 [(p1/2)2]0
d5 [(d3/2)4(d5/2)1]5/2
f 5 [( f5/2)5]5/2
p
3
p
4
2
1
2
2
[(p1/2) (p3/2) ]3/2 [(p1/2) (p3/2) ]2
d
6
d
7
4
2
4
3
[(d3/2) (d5/2) ]4 [(d3/2) (d5/2) ]9/2
f
6
f
7
Ground State 1 4
s d
6
[( f5/2) ]0 6
Table 2. Ground States Using j – j Coupling for Neutral Atoms with More Than One Partially Filled Subshell
1
[( f5/2) (f7/2) ]7/2
1
Atom 4
[(s1/2) (d3/2) ]1/2
Nb
s1d5 [(s1/2)1(d3/2)4(d5/2)1]3
Cr, Mo
s1d7 [(s1/2)1(d3/2)4(d5/2)3]5
Ru
s1d8 [(s1/2)1(d3/2)4(d5/2)4]9/2
Rh
1 9
s d
1
4
5
[(s1/2) (d3/2) (d5/2) ]3
Pt
p5 [(p1/2)2(p3/2)3]3/2
d8 [(d3/2)4(d5/2)4]4
f 8 [( f5/2)6(f7/2)2]6
d1f 1 [(d3/2)1(f5/2)1]4
Ce
p6 [(p1/2)2(p3/2)4]0
d 9 [(d3/2)4(d5/2)5]5/2
f 9 [( f5/2)6(f7/2)3]15/2
d1f 2 [(d3/2)1(f5/2)2]11/2
Pa
d10 [(d3/2)4(d5/2)6]0
f 10 [( f5/2)6(f7/2)4]8
d1f 3 [(d3/2)1(f5/2)3]6
U
f
11
6
5
[( f5/2) (f7/2) ]15/2
f 12 [( f5/2)6(f7/2)6]6 f 13 [( f5/2)6(f7/2)7]7/2
1 4
d f
1
4
[(d3/2) (f5/2) ]11/2
d1f 7 [(d3/2)1(f5/2)6(f7/2)1]2
Np Gd, Cm
f 14 [( f5/2)6(f7/2)8]0
For atoms or ions with more than one partially filled subshell, the procedure is slightly more complicated. If the l′′-1/2 subset of equivalent electrons is partially filled, sum the maximum possible values of m j in the l ′′-1/2 subset of equivalent electrons and add to this sum the value of – 1/2 for the l 1 electron. For atoms with a completely filled l′′-1/2 subset of equivalent electrons, add the maximum possible values of mj in the l ′′+1/2 subset of equivalent electrons and add to this sum the value of 1/2 – for the l 1 electron. Thus, for uranium (6d15f 3) the ground term is [(d3/2)1(f5/2)3], and the ground state value of J is 5/2 + 3/2 + 1/2 + 2 – 1/2 = 6. For platinum (6s15d9), the ground term is [(s1/2)1(d3/2)4(d5/2)5], and the ground state value of J is 5/2 + 3/2 + 1/2 – 1/2 – 3/2 + 1/2 – 0 = 3. For gadolinium (5d14f 7), the ground term is [(d3/2)1(f5/2)6(f7/2)1], and the ground state value of J is 7/2 + 1/2 – 2 = 2. For the gadolinium(I) ion (Gd+, 6s15d14f 7), the ground term is [(s1/2)1(d3/2)1(f5/2)6 (f7/2)1], and the ground state value of J is 7/2 + 1/2 – 0 + 1/2 – 2 = 5/2. In all cases, the ground state values of J determined using these rules are the same as those determined using Hund’s rules for L–S coupling. For atoms with only one partially filled subshell, the rules for j–j coupling are actually more straightforward than Hund’s rules. Ground states of heavy atoms have previously been determined using Hund’s rules despite the
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obvious inappropriateness of using L–S coupling for these atoms. Fortunately, Hund’s rules give the correct ground state value of J for heavy atoms because of a correlation of states (9, 10). However, for heavy atoms described by j–j coupling, it is now straightforward to determine the ground-state value of J without having to resort to using Hund’s rules. Literature Cited 1. Condon, E. U.; Shortley, G. H. The Theory of Atomic Spectra; Cambridge University Press: Cambridge, 1935. 2. Gauerke, S. J.; Campbell, M. L. J Chem. Educ. 1994, 71, 457. 3. Hund, F. Linienspektren und Periodisches System der Elemente; Springer: Berlin, 1927; p 124. 4. Walker, T. E. H.; Waber, J. T. Phys. Rev. A 1973, 47, 1218. 5. Moore, C. E. Atomic Energy Levels; NSRDS-NBS 35; U.S. Government Printing Office: Washington, DC, 1971; Vols. 1–3. 6. Martin, W. C.; Zalubas, R.; Hagan, L. Atomic Energy Levels— The Rare-Earth Elements; NSRDS-NBS 60; U.S. Government Printing Office: Washington, DC, 1978. 7. Brewer, L. J. Opt. Soc. Am. 1971, 61, 1101. 8. Brewer, L. J. Opt. Soc. Am. 1971, 61, 1666. 9. Richtmyer, F. K.; Kennard, E. H.; Cooper, J. N. Introduction to Modern Physics, 6th ed.; McGraw-Hill: New York, 1969; p 456. 10. Leighton, R. B. Principles of Modern Physics; McGraw-Hill: New York, 1959; p 270.
Journal of Chemical Education • Vol. 75 No. 10 October 1998 • JChemEd.chem.wisc.edu
