Research: Science and Education
Hund’s Rule in Two-Electron Atomic Systems John E. Harriman Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin–Madison, Madison, WI 53706;
[email protected] Hund’s rule summarizes the observed fact that when two or more states are associated with the same electron configuration the one with the highest multiplicity will usually have the lowest energy. This useful generalization commonly appears in texts and instruction; the challenge is to help students understand why this is the case. In such a discussion it is essential to bear in mind Einstein’s dictum, “Things should be made as simple as possible, but no simpler.” In a recent article (1) Rioux used a simple model to examine the singlet–triplet splitting in the 1s2s states of He, Li+, and similar ions. He found, as do higher-level calculations, that the dominant contribution to the splitting is the nuclear attraction contribution to the energy, rather than the electron–electron repulsion contribution. Rioux’s model gave the wrong ordering for the He 1s2s singlet and triplet states. The purpose of this article is to explore the reasons why the model fails in some cases and the origins of the splitting. Unfortunately, Rioux’s model cannot be justified for the 1s2s singlet. It uses the variation method to choose optimum parameter values, and application of this method to an excited state requires that the trial wave function be orthogonal to those of all lower-energy states of the same symmetry. This is not the case for Rioux’s model: the singlet S function is not orthogonal to the ground-state function so the energy obtained is too low. It is tempting to replace the 2s orbital in the model with a linear combination of 1s and 2s orbitals constrained to be orthogonal to the ground-state 1s orbital (2). This does improve the results, giving an excited singlet energy that is higher than the triplet energy, but it may still be lower than it should be because the model function is still not orthogonal to the true ground-state function. The simplest way to avoid this problem, while retaining the simplicity of the model, is to treat not the 1s2s singlet and triplet S states, but instead to treat the 1s2p singlet and triplet P states. The model functions for these states are orthogonal to the ground-state function by symmetry and are the lowest-energy states of their respective symmetries. The Model The model used by Rioux consists of a two-orbital approximation based on scaled hydrogenic orbitals with different scaling allowed for 1s and 2s, replaced here by 1s and 2p. The Hamiltonian and the form of the basis functions are the same for both states; the differences are the form of the wave function, a boundary condition, and the optimum parameter values. It is instructive to examine the energy expression, not just numbers. Using z1 for the scale factor in the 1s orbital and z2 for that in the 2p orbital, the energy functions can be written as
E(z1, z2) = T(z1, z2) + Vn(Z, z1, z2) + Vee(z1, z2)
where T is the kinetic energy, Vn is the nuclear attraction contribution to the potential energy for an atom or ion with atomic
number Z, and Vee is the electron–electron contribution. For an antisymmetrized spin–orbital approximation Vee can be expressed as
Vee = Uc ± Uex
where Uc and Uex are the Coulomb and exchange energies, respectively. The + sign gives the singlet energy expression and the − sign the triplet energy expression. Since the energy expressions differ for singlet and triplet, different optimum scale parameters are found for the two states. For the simple functions used in the model, Uc and Uex can be evaluated analytically as functions of z1 and z2, and each is always positive for all parameter values. It follows that for a given set of parameter values in this model the singlet will always be higher in energy than the triplet. The two states will have different optimum parameter values, however, and when these values change, all contributions to the energy change. Origin of Splitting and Hund’s Rule If only the one-electron contributions are included, as would be appropriate for noninteracting particles,
E1(z1, z2) = T(z1, z2) + Vn(Z, z1, z2)
the energy function has a minimum that occurs at z1 = z2 = Z, with a value equal to the sum of hydrogenic energies with prinicpal quantum numbers n = 1 and 2. If Uc(z1, z2) is included the energy increases as a result of the electron–electron repulsion. If the parameters are adjusted to increase the average separation, the repulsion energy will decrease. Any change of z1 and z2 away from Z will increase the one-electron contributions, however, so a compromise is reached. Up to this point the energy expression is still the same for singlet and triplet, so there is no splitting. The function Uex (z1, z2) is positive for all parameter values, and an examination of it shows that it is larger for larger z2 values, with a lesser dependence on z1. For the singlet where Uex adds to the energy, it will favor smaller z values; for the triplet Table 1. Contributions to the 1s2p Singlet–Triplet Splitting in Li+ Term
Contribution
∆T
–0.0332
∆Vn
0.0859
∆Vee
–0.0194
∆Emod
0.0332
∆Eacc
0.0334
Note: All energies are in Hartree atomic units; ∆Vee is the contribution of Uc ± Uex , ∆Emod is the total splitting calculated in this model, and ∆Eacc is the value given by accurate calculations (3, 4).
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 3 March 2008 • Journal of Chemical Education
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Figure 1. Contour plots of energy functions for Li+ 1s2p P states. The horizontal axis is z1; the vertical axis is z2. The points show the locations of the minima. The plots A and B show the total energy functions for the 1P and 3P states, respectively. Plot C shows the energy function when the exchange term is omitted, the same for singlet or triplet. The function Uex(z1,z2) is plotted in D: see text for identification of points.
where it subtracts from the energy it will favor larger values. The z values minimizing the complete energy expression for Z = 3 are z1 = 3.0064, z2 = 1.9404 for the singlet and z1 = 2.9794, z2 = 2.1632 for the triplet. Contributions of various terms in the energy to the singlet– triplet splitting ∆E = 1E − 3E are summarized in Table 1. As is well known, the largest contribution is ∆Vn and the electron interaction contribution ∆Vee actually has the wrong sign: by itself it would put the triplet at higher energy. That is also the case for He but ∆Emod is again positive, in accord with Hund’s rule. In the cases of Be2+, B3+, and C4+, ∆Vee is positive but smaller in magnitude than ∆Vn. The totals calculated with this model are all positive and in reasonable agreement with accurate values, given the simplicity of the model.
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Figure 1 shows contour plots of four functions of z1 and z2. In the contour plot for Uex (Figure 1D) the lowest value for the function in the region shown is at the lower right while the highest value is on the upper left. The points show the values of (z1, z2) that optimize the triplet energy (upper point), the energy without exchange (central point), and the singlet energy (lower point). As discussed previously, the parameter values for the triplet are displaced from those without exchange so as to increase the value of Uex while the displacement for the singlet produces a lower value for this function. The difference in sign of this term in singlet and triplet energy expressions is responsible for the different optimum parameters and is the primary reason why, in this model, the triplet energies are lower as predicted by Hund’s rule.
In summary, although the one-electron terms, especially nuclear attraction, make a larger contribution to the singlet– triplet splitting than does the electron–electron interaction term, and the latter is of the wrong sign in some cases, the difference in optimum scale parameters that leads to the ∆Vn is driven by the exchange term. Hund’s rule describes propensities, not a natural law, and there are exceptions. The advantage of this simple model is that it allows an analysis easily explained to students of the origin of the splitting and of why the triplet energy is lower. Finally, the incorrect results for He 1s2s in Rioux’s original calculation illustrate clearly the critical requirement that for a variational calculation of an excited state the trial function must be orthogonal to any lower states of the same symmetry. Literature Cited 1. Rioux, F. J. Chem. Educ. 2007, 84, 358–360. 2. Snow, R. L.; Bills, J. L. J. Chem. Educ. 1974, 51, 585–586. 3. Accad, Y.; Pekeris, C. L.; Schiff, H. Phys. Rev. A 1971, 4, 516– 536. Accad, Y.; Pekeris, C. L.; Schiff, H. Phys. Rev. A 1975, 11, 1479–1481. 4. Duon, B.; Gu, X.-Y.; Ma, Z.-Q. Eur. Phys. J. D 2002, 19, 9–12.
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Journal of Chemical Education • Vol. 85 No. 3 March 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
