J. Phys. Chem. 1993,97, 2425-2434
2425
Hund's Rules, the Alternating Rule, and Symmetry Holes John D. Morgan In**+ and Werner Kutzelnigg'l* Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, Massachusetts 02138, Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138, Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, and Lehrstuhl fur Theoretische Chemie. Ruhr- Universitiit Bochum. 0-4630 Bochum, Federal Republic of Germany Received: November 24, I992
We review attempts to explain Hund's first rule, that a triplet state should lie below a singlet state of the same configuration, as a consequence of the "Fermi hole" in the triplet state electron-electron distribution function. After describing the failure of such attempts, and reviewing the types of exceptions to Hund's first rule, we examine the "alternating rule", which accounts for most of the known exceptions. We show that the alternating rule is associated with whether the parity of the two-electron wave function is natural or unnatural. We show that whereas in the case of natural parity the triplet correlation function has a broad Fermi hole and the singlet correlation function does not, in the case of unnatural parity the singlet correlation function has a quartic "symmetry hole" that is even broader than the quadratic Fermi hole of the triplet correlation function. Thus this more general perspective provides a relationship between the ordering of singlet/triplet energy levels and the breadth of their respective symmetry holes. We argue that although the breadth of symmetry holes cannot be said to cuuse the ordering of singlet/triplet energy levels, both phenomena are consequences of factors of (-1)S+L+'1+'2 in the construction of the properly antisymmetrized eigenfunctions of spin and angular variables; hence the two phenomena are closely associated, although neither is the cause of the other.
Introduction It is well-known that Dudley Herschbach likes simple relations that provide insight into complex problems. We thus hope that he will find enjoyable our explorationof the relationship between Hund's rules on the ordering of intraconfigurationalatomic energy levels and the appearance of symmetry-induced holes, of which the Fermi hole is a special case, in their electron-electron distribution functions. It is a special honor to be able to present this work in an issue of the Journal of Physical Chemistry dedicated to Dudley in celebration of his 60th birthday. Several years before Dudley was born, F. Hund formulated the famous rules which bear his name for determining from the configurationof one-electron orbitals the quantum numbers that characterize the ground state of an atom:' 1. Among many-electron states arising from the same configuration of one-electron orbitals, the ground state has the largest total spin S. 2. Among many-electron states of the same total spin arising from the same configuration of one-electron orbitals, the ground state has the largest total angular momentum L. 3a. If the shell is less than half-full, then the ground state has the smallest value of J = IL + SI. 3b. If the shell is more than half-full, then the ground state has the largest value of J = IL + SI. It is to be noted that Hund formulated his rules in the case of the ground state of an atom. However, they were very soon thereafter generalized to excited states in the following form: 1. Among many-electron states arising from the same configuration of one-electron orbitals, the energy of the state decreases as the total spin S of the state increases.
' Institute for Theoretical Atomic and Molecular Physics, HarvardSmithsonian Astrophysical Observatory, and Department of Chemistry, Harvard University, and University of Delaware. Permanent address: University of Delaware. 1 Institute for Theoretical Atomic and Molecular Physics and RuhrUniversitlt Bochum. Permanent address: Ruhr-Universitiit Bochum.
2. Among many-electron states of the same total spin arising from the same configuration, the energy of the state decreases as the total angular momentum L increases. 3a. If the shell is less than half-full, then the energy of the state decreases with J = IL SI. 3b. If the shell is more than half-full, then the energy of the state increases with J = IL + SI. When Hund found his qualitative rules for determining the quantum numbers of the ground state, he was working purely empirically. With the advent of quantum mechanics and its application to the electronic structure of atoms and molecules, Slater found a quantitative procedure for generating explicit estimates of the splittings between energy levels arising from the same configuration, which in 1929 he presented in his classic artic1e"The Theory of Complex Spectra".* In his abstract, Slater stated "It is found that Hund's rule, that terms of large[st] L and S values lie lowest, has no general significance; the present theory leads to the same results as the rule when it is obeyed experimentally, but many cases that were exceptions to that rule are in agreement with the theory". To be fair to Hund, it should be noted that of the three counterexamples to Hund's rules discussed by Slater, all involve the second rule and all involve excited states, to which Hund made no pretense of applying his rules. Indeed, as was stated in 1930 by Pauling and Goudsmit, "although these rules are well fulfilled by the lower states of an atom, the higher states generally show large deviations".3 Slater's position, in effect, was that there was no need to try to explain the origin of Hund's rules and to elucidate the reasons for their frequent violation in the cases of excited states, since his method allowed one to calculate directly at the HartreeFock level the splittings between terms of a configuration. However, we think most would disagree with Slater's contention that the ability to calculate large quantities of numbers by following an algorithm obviates the explanation of patterns frequentlyexhibited by those numbers and would agree with us that the elucidation of the factors that determine the ordering of atomic energy levels remains a worthwhile subject of study.
+
0022-3654/93/2097-2425%04.00/0 0 1993 American Chemical Society
2426 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
Hund's Rules for Singly Excited States of Helium The simplest system for which one can study the generalization to excited states of Hund's first rule consists of the singly excited states of the helium atom, or of a helium-like ion. In an independent-electron picture, the spatial part of the wave function can be written as
where the normalized single-electron spatial functions (1s) and (nl) are orthogonal to each other: ( ( l s ) l ( n O )= 0 (2) The sign yields a two-electron spatial wave function that is symmetric under particle exchange and is multiplied by an antisymmetric spin singlet function of the form
+
(3) whereas the - sign yields a two-electron spatial wave function that is antisymmetric under particle exchange and is multiplied by any member of a manifold of three symmetric spin triplet functions:
88
(4)
Thus the product of a symmetric/antisymmetric spatial function with an antisymmetric/symmetric spin function yields a twoelectron wave function that overall is antisymmetric under exchange, in accordance with the Pauli principle. The exact spatial wave function ++(rl,rz) corresponding to a singlet spin function is symmetric under exchange of the spatial coordinatesof the two electrons. Usually (e.g., for singly excited states with S = 0) it is non-zero at an electron-electroncoalescence and can be shown to obey the Kato' cusp condition4
where rlz = Ir I -rzlis the interelectronicseparation and $+denotes the average of ++ over an infinitesimally small sphere centered at rlz = 0. Since an exact or approximate spatial wave function +-(rl,rz) corresponding to a triplet spin function must be antisymmetric under exchange of the spatial coordinates of the two electrons, when rI = r2 it must equal the negative of itself
++-
Provided that is continuous, as is usually assumed to be true, and can in fact be proven with considerable mathematical effort for the exact +-,4 must vanish at an electron-electron coalescence point, where rI = rz. Further analysis4shows that vanishes linearly in the neighborhood of this point, so that in the case of a wave function that is antisymmetric under exchange of the two electrons, the Kat0 cusp condition reduces to the true but uninteresting statement '0 = 0". (Nontrivial generalizations of the Kat0 cusp condition in such a case, which avoid the angular averaging and instead compute the form of the radial coefficient of a spherical harmonic, have recently been found by HoffmannOstenhof, Hoffmann-Ostenhof,and St~mnitzer.~) Since the wave function +-vanishes linearly in the neighborhood of an electronelectron coalescence, the probability density vanishes quadratically there, giving rise to what is commonly known as a Fermi hole. The quadratic vanishing of the probability density for a triplet state in the vicinity of an electron4ectroncoalescence,as opposed
+-
Morgan and Kutzelnigg to the nonvanishing of it for a singlet state, understandably prompted an 'obvious" explanation of Hund's first rule: in a triplet state the Fermi hole keeps the electronsfurther apart from each other than in a singlet state of the same configuration. This leads to a lower expectationvalue of the electron repulsion energy ( l / r l z )in the triplet state than in the singlet and hence a lower total energy.6 The flaw in this appealing argument was noted by several workers' in the 1960's and 1970's, when it became routine to do high-precision Rayleigh-Ritz variational calculations on small atoms and thus to compute reliable expectation values of various operators: although all the above statements are true for large 2 for the 2'sand the 2% states of a helium-like ion of nuclear charge 2,for smaller 2,such as the physically interesting case of neutral helium with 2 = 2, the electron repulsion energy ( 1/ r l2 ) for the 23Sstate is actually larger than it is for the higherlying 2IS state, with an overcompensatinglowering of the energy for the triplet state due to a considerably more negative electronnucleus potential energy than for the singlet state. Another way of viewing this initially surprising result is to observe that, in the absence of the l / r l z term, the 2% and the 23Sstates have the same energy. When the 1 /r12 term is added perturbatively to the Hamiltonian,not only the total energy shifts but also the individual contributions to the kinetic energy and the potential energy must shift so that the virial theorem -E = ( T ) = V) continues to be obeyed. Relative to the 2% state, the 23Sstate wave function contracts closer to the nucleus, leading to increased values of ( T ) ,-( V), ( l / r l z ) , and -E. Similarly, in a molecular system, it has been found that for the lowest pair of II, states of Hz, the electronic repulsion energy ( 1/r12) for the lower-lying triplet state is larger than it is for the higher-lyingsinglet stateas Hence the lower energy of the triplet state cannot in general be explained in terms of a lower electronic repulsion energy arising from differences in the relative spatial distributions of the electrons. This observation led to much reinterpretation of the physical basis of Hund's first rules9 The first rule is easy to derive at the level of first-order perturbation theory in the 1/Zexpansion, and a proof at the HartreeFock level was constructed by ColpaIofor the lowest pair of singly excited states of the configuration (ls)(nl) with I1 1, but in the case of correlated electrons, no quantitativeargument has yet been constructedthat would provide a general proof, with clearly stated hypotheses that exclude the not infrequently encountered exceptionsto Hund's first rule that arise from strong mixing of configurations. Since in the presence of correlation the validity of the generalization of Hund's first rule to excited states depends on the relative magnitudes of competing effects and counterexamples abound, it would seem to be extremely difficult if not impossible to formulate simple criteria that determinewhether or not the generalization of Hund's first rule to excited states is obeyed. We shall return to this point after examining the derivation of Hund's first rule for a pair of uncorrelated electrons that belong to the same configuration. The derivation of Hund's first rule within the contexts of both the 1/Z expansion and the Hartree-Fock approximation makes use of uncorrelated singlet and triplet wave functions to evaluate the expectation value of a spin-independent Hamiltonian H of the form H=hl+hz+V12 (7) where hi is a single-particle Hamiltonian that acts nontrivially only on functions of the spatial coordinate of particle i and V Iz(rl2) is the electron4ectron interaction potential. Denoting by E+and E- the expectation values of the Hamiltonian with the
The Journal of Physical Chemistry, Vol. 97, NO.10, 1993 2421
Hund’s Rules spatially symmetric/antisymmetric wave functions and using the fact that the orthogonality of (1s) and (nl) makes vanish the cross terms
in the expectation values of the single-electronoperators hl and h2, one readily finds that
E = ((ls)Ih,l(ls)) + ((nl)Ih,l(nl)) +
-
(( 1 ~ ) ( ~ l ) l ~ I 2 l ( 1)~ ) ( (~w0 ( n W l ,I(nl)( 1s) ) ( 9 )
The difference of energy between the two states is thus
AE = E + - E - = 2((ls)(~l)lVl2I(n~)(ls))
(10) Whether the singlet lies above or below the triplet, i.e., whether hE is positive or negative, is thus in this approximation a direct consequence of whether the exchange matrix element (( 1s)(nl)lV12l(nl)(1s)) is positiveor negative. Thepositivityofexchange matrix elements of this form, which is often stated to be a matter of common experience, was proved rigorously by Roothaan and by Slater,” who made use of the explicit form of the potential 1/r12 and its property of being the Green’s function for the Laplacian. In fact, it is not hard to construct for any potential that has a positive Fourier transform a general proof for twoelectron’states expressibleas a product of one-electronfunctions, whichwepresentin thecaseofour (ls)(nl) configuration,although it holds quite generally. By writing this exchange integral as
JR,d3rI JR3d3r2[( W n l ) *I *(rl1VI2(lr1- r21)[( 1s)(nl) *I (r2) (11) one can immediately see that it has the form of an inner product of (ls)(nl)* with V12*[(ls)(nl)*], the convolution of V12with (ls)(nl)*. Since (i) the inner product of two functions is the same as the inner product of their Fourier transforms and (ii) in D dimensions the Fourier transform of the convolution of two functionsis (2a)D/* times the product of their Fourier transforms, this exchange integral can be rewritten by using the Fourier transform as V I -
where the symbol denotes the Fourier transform. One therefore sees that a sufficient conditionfor the exchangeintegral to be positive is that the Fourier transform V12(k) of VI&) be positive. Such a function is known in mathematical language as “positive definite”. Since the Fourier transform (in three dimensions) of l/r is 4u/k2, it is easy to see that the electronelectron Coulomb repulsion potential energy, as well as being positive, is also positive definite, which immediately implies the positivity of exchange integrals12 and thus, at the level of firstorder perturbation theory in the 1/Z expansion, that a singly excited singlet state lies above a singly excited triplet state arising from the same configuration. The argument for the lowest pair of singly excited singlet and triplet states at the Hartree-Fock level is only slightly more complicated. One takes the one-electron spatial orbitals (1s) and (nl) which minimize the variational energy for the singlet linear combination,and then from themconstructs a trial function of triplet symmetry, which must yield an upper bound to the true triplet energy at the Hartree-Fock level. Since the one-electron energies and the direct Coulomb repulsion terms are the same A
for both symmetries and the exchange integral for a Coulomb potential is positive, the upper bound to the true triplet energy lies below the true singlet energy,thus proving that at the Hartreb Fock level the true triplet energy lies below the true singlet energy. For details, see Colpa’s article.10 Colpa’s work is significant not only for providing a rigorous proof that at the Hartree-Fock level the lowest pair of singly excited states arising from a (ls)(nl) configuration with I L 1 obeys Hund’s rule but also for suggesting generalizations of his proof to other singly excited states at the Hartree-Fock level. If his technique can be generalized so that the nonvanishing overlap term (( ls)l(nr)) can be controlled and to deal with higher excited states, for which one must take account of their orthogonality to lower states in applying the variational principle, it would provide in the Hartree-Fock approximation a proof of the apparently true fact that all singly excited states of the helium atom obey Hund’s first rule.
Exceptions to Hund’s Rules for Excited States: The “Alternating Rule” and Natural/Unnatural Parity States Exceptionsto the generalization of Hund’s first rule to excited states fall into various classes. The formulations of proofs given above require that the singlet and the triplet wave functionsbelong to the sume configuration, which is a well-defined concept at the independent-electronor Hartree-Fock level but which becomes fuzzy when configuration interaction is present, as in the real world. Strong mixing of one or both members of a nominal pair of states with one or more nearby states can, and does, produce violations of the generalization of Hund’s first rule to excited states. One of the simplest atomic cases occurs in the (3s)(n4 series of magnesium, where the IDestates lie below the 3Destates, whereas in the (3s)(np) and the (3s)(nr) series with n 1 3 the triplet states lie below the singlet states;l3 in this case, the exceptional behavior of the D series can be attributed to the mixing of the (3s)(34 IDcstatewith the higher-lyingconfiguration (3p)(3p),which produces a term of symmetry IDcbut not of 3Dc. Moreover, in molecules such as square-planar cyclobutadiene and twisted ethylene, there is the ”dynamic spin polarization” effect, in which (i) the exchange integral is tiny because the two degenerate occupied molecular orbitals are concentrated about different atoms and thus have a small differential overlap, (ii) the core is sufficiently polarizable that there is a low-lying excited molecular orbital of the right symmetry to mix with the singlet, and (iii) the exchangeintegrals that dominate the matrix elements to singly excited configurations are relatively large. If all these conditions are satisfied, which is much more difficult in an atom than in a molecule, the singlet can lower its energy below the triplet by mixing with configurations that are inaccessibleto the triplet. For details, see the discussionsof Borden14and of Kollmar and Staemmler.I5 Another, and much broader, class of exceptionsis found in the spectrumof many transition metals and rare-earth elements. These exceptions were experimentally discovered already in 1927 by Russell and Meggers16in the case of the singly ionized scandium atom. The inequivalent pair of spatial orbitals ( 3 4 and ( 4 4 with angular momentum 1 = 2 can give rise to singlet and triplet states with five values of total L ranging from I - I = 0 to I + I = 4. Russell and Meggers found the alternating rule that whereas the states with even L obeyed the usual first rule of Hund, with triplets below singlets, for the states with odd L the rule was reversed, with singlets below triplets. Russell and Meggers noted that this phenomenon “finds a parallel in pentads of similar origin in other spectra” and concluded their discussion by saying “This remarkable relation may be commended to the attention of theoretical investigators of atomic structure”. In 1935, in their classic book The Theory of Atomic Spectra, Condon and Shortley explained thealternation in theobservance/ nonobservanceof Hund’s first rulewith L in terms of an alternation
2428
Morgan and Kutzelnigg
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
in the sign of their Go contribution to the exchange integral as the total angular momentum Lincreases.I7 They did not, however, provide an explanation of the alternation in sign of the coefficient of the GOterm. In 1980 Warner, Bartell, and Blinderla argued that the alternating rule on the ordering of singlet-triplet pairs could be explained in terms of “simpleelectronic shieldingeffects induced by wave-function antisymmetrization”. If one models a doubly excited configuration by a system with two electrons equidistant from the nucleus on a spherical shell df radius r, then the electronelectron repulsion term
-1= r12
1
(r12+ 122- 2rlr2cos
Thus under inversion
*(-A,-?)
= A*(&),
with
- for ungerade states (16) and under exchange
+ for singlet, - for triplet states
q(fi,-i) = *q(B,F), with
(17) We then expand
(13) in spherical harmonics of the angular variables associated with the center of mass and relative coordinates
reduces to simply
ii = (R,e,4) (14)
2r sin(O/2) To obtain a handle on the behavior of ( l/r12),they therefore evaluated (l/sin(B/2)) for two-electron functions of the spin and the angular variables and found that this quantity, like the experimentallyobserved energy spectra, alsoobeyed an alternating rule. Their graphs of some simple approximate wave functions lent support to the rationalization of the alternating rule in terms of different angular distributions, depending on whether 11 + 12 L was even or odd. Their examination of over 600 pairs of atomic states of the same nominal configuration showed that this alternating rule was obeyed in over 90% of the cases, whereas the straightforward application of Hund’s first rule would work in little more than 50% of the cases. Warner et al. developed a heuristic explanation of the alternating rule in terms not of holes at r12 = 0 in the spatial distribution function for the relative motion of the two electrons but of holes at B = 0 in the angular distribution function, which allows for an overall rescaling of the density so that the virial theorem is obeyed. In 1985, in a comment published in Nature on an explanation by Boydi9of Hund‘s first rule in terms of the Fermi hole, Warner and Berry20 observed that “one might expect a triplet always to have less shielding, owing to the Fermi hole, but wavefunction antisymmetrycan give rise to featuresin the singlet which resemble the triplet hole”, with a reference to the 1980 article by Warner et al.Ia They proceeded to say “Asthe charge density ~ ( 0 ~has 2) been found to obey the alternating rule, the Fermi hole argument is too simplistic” and went on to discuss the case of the (2p)(3d) configurationof carbon, which can produce both singlet and triplet states of total angular momentum P, D, and F; the ID state lies below the 3D state, in violation of Hund’s first rule but in accord with the alternating rule, and in fact the ID state lies below the 3F state, in violation of Hund’s second rule. In the course of an extensive investigation2’ of the rates of convergence of Rayleigh-Ritz variational calculations using a partial wave expansion,of the type which occurs in any expansion of a many-electron wave function as an antisymmetrized sum of products of one-electron functions, we have found that twoelectron states of total angular momentum L arising from a configuration (nlll)(n212)can be divided into four cases: natural parity singlet, natural parity triplet, unnatural parity triplet, and unnatural parity singlet, each of which corresponds to distinct behavior of the wave function under total inversion of spatial coordinatesand exchangeof electrons. Theseoperationsare 5ost easily examined by expressing the spatial wave function *(R,i) of a two-electron system in terms of center-of-mass and relative coordinates R and T. for the two electrons, where
+
ii = 1
+ i2)
i = i , - i,
+ for gerade,
(15)
i = (r,O,q)
(19)
Obviously
L = even, k = even
for singlet gerade states
L = odd, k = even
for singlet ungerade states
L = odd, k = odd
for triplet gerade states
L = even, k = odd
for triplet ungerade states (20)
I and k couple to yield the total angular momentum L. The parity of the state under inversion of the spatial coordinates is (-l)L (-l)k = (-l)f+k. If this is equal to (-l)L, Le., if L - I k is even, the state is said to have naturalparity, since the parity of a one-electronstate of angular momentum 1 is (-1)‘. If (-l)f+k is -(-l)L = (-l)L+l, Le., if L - I - k is odd, the state is said to have unnatural parity. (This concept of natural and unnatural parity is already known in electron-atom scattering, where one refers to parity favored and parity unfavored transitions, whose underlying roots in symmetry were brought to light by Fano.22) This leads to the classification natural parity singlet states:
k even, L - L = even
natural parity triplet states:
k odd, L - L = odd
unnatural parity singlet states: k even,
L - L = odd
unnatural parity triplet states: k odd, L - L = even (21) The minimum allowed even/odd value of k is 0/1, respectively; however, by the triangle inequality for adding angular momenta, k = 0 is impossible for L - L odd. Hence the leading partial waves in the relative motion are
k =0
for natural parity singlet states
k=1
for either parity triplet states
k =2
for unnatural parity singlet states
(22)
To satisfy the Schradinger equation in the vicinity of r12 r = 0,
where the coupling constant h parametrizes the electron-electron interaction, the two-electron function must behave as
*:
For natural parity singlet states, where the leading partial wave
The Journal of Physical Chemistry, Vol. 97,NO. 10, 1993 2429
Hund’s Rules 0.50
I
I
0
-1
Figure 1. Qualitative sketch (arbitrary scale) of the behavior of the probability density for a two-electron systems near an electron-dectron coalescence. The top curve represents the cusp (discontinuous first derivative in Cartesian coordinates) for a singlet state of natural parity, the middle curve represents a quadratic Fermi hole with a higher-order cusp (discontinuous third derivative) for a triplet state of either natural or unnatural parity, and the bottom curve represents a quartic symmetry hole with an even higher-order cusp (discontinuous fifth derivative) for a singlet state of unnatural parity.
of the relative motion has k = 0, eq 24 is essentially Kato’s correlation cusp condition. For triplet states, which have k = 1, eq 24 implies the generalized cusp condition of Pack and Byers Brown?3while the apparently novel result with k = 2 for unnatural parity singlet states was obtained by us in ref 21. Since the helium atom has no symmetries beyond invariance under overall rotation, inversion, and exchange of its two electrons, all higher partial waves are coupled by the Coulomb interaction l/r12to one of these with k = 0, 1, or 2,so barring miraculouscancellations these are the only three possibilities for the leading behavior of a physical wave function in the vicinity of an electron-electron coalescence. If we set A = 1 in eqs 23 and 24,we get the correlation cusp relation for the exact wave function. For A = 1/Zwe obtain the relation between the unperturbed wave function @ and the firstorder function in the 1/Zexpansion
+
Thus these four cases of natural/unnatural parity singlet/ triplet states feature threedifferent kinds of behavior in thevicinity of an electron-electron coalescence: (1) For k = 0, we have natural parity singlet states, whose wave function does not vanish as r12 0 and whose probability density hence does not vanish as rI2 0. (2) For k = 1, we have either natural or unnatural parity triplet states, whose wave function vanishes linearly as r12 0 and whose probability density hencevanishes quadratically as r12 0. (3) For k = 2,we have unnatural parity singlet states, whose wave function vanishes quadratically as r12 0 and whose probability density hence vanishes quartically as r12 0. These three types of behavior of the probability density are illustrated in Figure 1. It is readily apparent from Figure 1 that just as for small r12 the probability density for a natural parity singlet state exceeds that for a natural parity triplet state, for small r12 the probability density for an unnatural parity triplet state exceeds that for an unnatural parity singlet state. The same incorrect hand-waving argument that rationalizes Hund’s first rule, that (for natural parity states) the singlet energy lies above the triplet energy, could then be extended to predict that for unnatural parity states the relation is reversed, with the triplet energy above the singlet energy. These predictions are in perfect accord with the alternating rule, since for a given pair (I,k)of angular momentum quantum numbers for the center-of-mass and relative coordinates, or equivalently for a given pair (11,12) of angular momentum quantum
--
-
-
- -
numbers for the single-particle coordinates, the parity is natural for the extremal values of Lmin = 111 - 121 and L,,, = 11 + 12 and then alternates between unnatural and natural parity for values of L between these two extremal values. Thus although the ordering of atomic energy levels in the Hartree-Fock approximation cannot be explained in terms of the Fermi hole, it can be explained in terms of the more general symmetry holes that arise from forming properly antisymmetrized eigenfunctions of spin and angular momentum, and of which the Fermi hole is a special case. The prediction that the singlet-triplet ordering of unnatural parity two-electron states is the reverse of that for natural parity states can be tested against thevery accuratecorrelated calculation of the energy levels of doubly excited states of helium by C a l l a ~ a y who , ~ ~ obtained the energies of a large number of resonant states of the term designations I.)P, 1,3D, Iq3F, and IJG below the n = 2 threshold and between the n = 2 and n = 3 thresholds. Callaway conveniently grouped his results into two tables, with the first for parity (-l)L, Le., natural parity, and the second for parity (-l)L+l,Le., unnatural parity, which is of the greater interest to us here. For the unnatural parity PC states below the n = 2 threshold, Callaway computed the energies of the three lowest singlet states and of the four lowest triplet states. The lowest 3PC state, whose energy Callaway computed to be -1.419 98 Ry, arises from the configuration ( 2 ~ ) which ~ , gives rise to no singlet P state. Limiting our comparisons then to unnatural parity IpC and 3pC states which do arise from the same configurations of the form (2p)(3p), (2p)(4p), (2p)(5p), etc., the relevant entries in Callaway’s Table I1 are
’P -1.160 43 -1.080 06 -1,048 34
3F -1.419 -1.135 -1.071 -1.044
98 43 66 46
As we expect, in every case of unnatural parity PC states arising from the same configuration, the singlet lies below the triplet. The same expectation is born out in Callaway’s entries in his Table I1 for the four lowest Dostates, the three lowest FCstates, and the three lowest Go states, which are all that he lists below the n = 2 threshold. We next proceed to the resonant states that lie below the n = 3 threshold but above the n = 2 threshold. From the electron configurations (3p)2 and (342, it is possible to make unnatural parity 3 P e states that have no counterpart IPC states. The first two entries in Callaway’s listing of 3pC resonance energies below then = 3 thresholdare these two states. Taking them intoaccount, the relevant entries in his Table I1 become ‘P
-0.557 -0,518 -0.507 -0.488 -0.484 -0.474
96 60 28 97 45 20
3 P
-0.664 16 -0.581 17 -0.543 07 -0.506 88 -0.502 08 -0.483 7 1 -0.481 98 -0.41 1 39
In every case, the unnatural parity singlet state lies below the unnatural parity triplet state of the same configuration. As can be seen in Callaway’s Table 11, our expectations are fulfilled for the lowest pair of IDo and 3D0 unnatural parity states below the n = 3 threshold but not for all the higher states. This breakdown of the ‘anti-Hund‘s first rule” for unnatural parity states may be attributable to strong configuration mixing between unnatural parity Do states arising from the (3p)(nd) and the (34(np) configurations. To make the comparisons for the unnatural parity IF( and 3Fr states, we must first take account of the (342 ,FCstate, which
2430 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
has no counterpart IFestate. Upon doing so, the relevant entries from Callaway's Table I1 become I
Fe
-0.536 -0.521 -0.497 -0.491 -0.482 -0.419 -0.476 -0.471 -0.468
3 Fe
-0.619 83 -0.525 28 -0.516 41 -0.493 43 -0.488 69 -0.482 48 -0.411 29 -0.414 53 -0.471 07
94 84 60 31 49 26 01
01 55
In every case the unnatural parity singlet state has a lower energy than the unnatural parity triplet state of the same configuration. The same holds, as can readily be seen from Callaway's Table 11, for the unnatural parity !Goand 'Go states. A similar analysis could be carried out to verify that Hund's first rule generally holds for the natural parity states in Callaway's Table I, but this would lie beyond the scope of this paper. There seem to be many more violations of Hund's first rule, presumably because of greater configuration mixing owing to the greater number of configurations that can form natural parity states with a given L and S. Why Hund's Rules Almost Always Work for Ground States We now discuss why we find few exceptions to Hund's first rule in its original formulation to ground states. We begin with specific cases with which the reader is probably familiar, before proceeding to very general cases. Since as we have seen violations of Hund's first rule at the independent-electron or the HartreeFock level are associated with states of unnatural parity, we first observe that since unnatural parity states must have 11, 12, and L 1 1, any twoelectron state formed from a single-particle state with 1 = 0 must have natural parity and therefore should not lead to a violation of Hund's first rule, even in its generalization to excited states. Thus we next consider the situation where two electrons in an unfilled shell have the same values of n and I with 1 1 1 ; e.g.
-
( n ~ ) 'De, ~ 'Pe, ISe (nd)' IG', 'F,'De, 'Pe, ISe as can be found in Condon and Sh0rt1ey.I~ In the case of twoelectron states, one can have either singlets or triplets. For such two-electron states, the state of maximum total L cannot have maximum total S, so the state of maximum total L has natural parity, and thus the triplet states have unnatural parity and the singlet states have natural parity. Unnatural parity triplet states exhibit a Fermi hole, whereas natural parity singlet states do not. Thus one expects the unnatural parity triplet states to lie below the natural parity singlet states, in agreement with Hund's first rule. Now consider states arising from three electrons in an unfilled shell with the same n and 1 with 1 1 1, e.g.
-
'DO,'Po, 4S0 2He,2Ge,4F,'F,'De, 'De,'F,4Pe (np)'
(nd)'
Consider the first case, involving the (np))configuration. The 4S0 state, which has unnatural parity, has a spin function (e.g., aaa) that is totally symmetric under exchange of any pair of particles, so its spatial function must be totally antisymmetric under the exchange of any pair of spatial variables; in other words, every pair of electrons is in a relative triplet state. As such, it
Morgan and Kutzelnigg exhibits a Fermi hole a t each electron-electron coalescence. There is another unnatural parity state arising from this configuration, the 2Do state, which can be considered as also arising from the coupling of an (np) electron with spin I / 2 to the two-electron (np)' 'Deand ISe states, which do not have holes at electronelectron coalescences, and to the (np)2 IPe state, which does. As such, the resulting three-electron unnatural parity 2Dostate can be expected not to have Fermi holes at all electron-electron coalescences and thus should have a higher energy than the unnatural parity 4S0 state. This illustrates the fact that, when considering many-electron unnatural parity states, one cannot naively assume that as usual Hund's first rule is inverted and that the state of lowest spin multiplicity has lowest energy; one must consider the individual electron pairs that make up the manyelectron state. Similar reasoning can be applied to the 4Fe and the 4Prstates of the configuration,which can also be considered as arising from the coupling of an (nd) electron with spin to the 3Feand the 3 P states, which are spatially antisymmetric under exchange and thus exhibit a Fermi hole. Since these quartet states are totally antisymmetric in the spatial variables under exchange of any pair of electrons, they exhibit Fermi holes a t all coalescences and are thus expected to have a lower energy than the doublet states, which involve mixtures with two-electron singlet states that do not have holes. We now proceed to intershell doubly excited states, with nl # n2 and 11, 12 1 1. To take a relatively simple case, the (2p)(3d) configuration gives rise to six eigenstates of total L and S: IFo, 3F0, IDo,3Do,IP0,3Po.Inthecaseofcarbon,whichwasdiscussed by Warner and Berry,17 these states lie in the order25
'Do< 'FO < 'DO < IFo < 'Po < 'Po
(26) with the IPO state anomalously below the 3P0state because of strong configuration interaction, which presently is of less concern to us than the violation for the lower states of the generalizations to excited states of Hund's first rule and his second rule: the ground state does not have maximum spin, and the IDo state lies below the IFo state. These apparent anomalies are readily explained by realizing that this IDo state is a singlet state of unnatural parity, which thus has a wide quartic hole at r I 2= 0 in its distribution function. Accordingly, one would expect it to lie below the triplet states of both unnatural and natural parity, which have only quadratic Fermi holes, and below the singlet states of natural parity, which have not even a Fermi hole. These remarks are born out by inspection of the energy level diagrams of the corresponding states of the isoelectronic sequence from Ca I to Ni IX, which always shows the unnatural parity IDo state below the other states arising from the (4p)(3d) config~ration.~3 Similar remarks apply to other configurations of the form (nl)(n'l'). One obtains a manifold of singlet and triplet states with all possible values of total L ranging from 11 - 1'1 to 1 1'. Of these, the extrema1 values of L are natural parity states, while the 'odd" ones in between are the unnatural parity states. One expects the overall ground state to be an unnatural parity singlet state, since they alone have the extended quadratic symmetry hole in the electron distribution function. This is inconsistent with the formulation to excited states of Hund's first and second rules, whereas it accords with and indeed goes beyond the alternating rule, in that it allows for a prediction of the ordering of energy levels with different L as well as different S. We can now discuss in considerable generality why Hund's first rule works so well for ground states. For atoms whosegroundstate configuration consists of having two electrons in a single open shell with 1 1 1, the state of maximal total L = 21 cannot have maximal total S. Thus the singlet states have natural parity and the triplet states have unnatural parity. Since the unnatural parity triplet states have a Fermi hole and the natural parity
+
Hund's Rules
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2431
singletstatesdonot, thegroundstateis an unnatural parity triplet state. By Hund's second rule, the ground state is a triplet state with the maximal total L = 21 - 1. Now consider the addition of a third electron with angular momentum 1 and spin s = l/2, which couple to the angular momentum and the spin of two-electron states. It is possible to make from the unnatural parity two-electron triplet states threeelectron quartet states, which will have a Fermi hole at all electronelectron coalescences. It is also possible to make doublet threeelectron states by coupling the third electron to the natural parity singlet states, but barring miraculouscancellationsall these linear combinations will fail to have a Fermi hole at some electronelectron coalescences. Thus one expects a quartet state, with maximal spin S = j / 2 , to have the lowest energy. This argument can now be repeated until the shell is halffilled, with a ground-state with total L = 0 and total S = (21 + 1)/2 for the configuration. The subsequent addition of each extra electron results in the reduction of total S by l/2, until after theaddition of another (21 1) electronseach spatial orbital is doubly occupied, yielding a unique atomic state with total L = 0 and total S = 0 for a completely filled shell. We thus see that for atoms whose ground-state configuration includes only one partly filled shell, Hund's first rule should be obeyed. What about atoms whose ground-state configuration includes two partly filled shells? In such a case, one could obtain, at least in principle, many-electron states with a pair of electrons in an unnatural parity state, with the singlet below the triplet. In fact, this actually does happen, as we shall soon see. But first, it should immediatelybe noted that in order for a pair of electrons to be in an unnatural parity state, the angular momentum of both must be greater than 0. This observation immediately excludes as candidates for violations of Hund's first rule transition metals such as chromium, niobium, molybdenum, ruthenium, rhodium, and silver, whose ground states have a single 4s or 5s electron and an incompletely filled 3d or 4d shell. The first element whose ground-state configuration involves two partly filled shells, each with I 1 1, is cerium, with 2 = 58, whose electronic configuration is
+
(1~)~(2~)~(2p)~(3~)~(3p)~(4~)~(3d)'~ (4P)6(5s)2(4d)'0(5P)6(6s)2(4n(5d) The 4f and the 5d electrons can couple to produce two-electron singlet and triplet states with L running from 1 to 5. The quartic symmetry hole in the two unnatural parity singlet states with L = 2 and L = 4 indicates that one of them should be the ground state, and Hund's second rule suggests that it should be the one with L = 4, the 'Go state, and indeed it is.26 Thus this rare violator of Hund's first rule for a ground state turns out to be perfectly explicable in terms of the alternating rule, which as we have seen is associated with a quartic symmetry hole in a twoelectron singlet state of unnatural parity. The next case of an element whose ground-state configuration has two partly tilled shells, each with I L 1, is gadolinium, with Z = 64,whose electronic configuration is
and whose ground state has the symmetry 9D with maximal spin. Qualitatively, the seven 4f electrons minimize their repulsion energy by occupying distinct spatial orbitals to produce an *S state, and then the single 5d electroncouples to this seven-electron state to produce a 9Dstate, in which all electrons are in a relative triplet state. An attempt to put the 5d electron and a 4f electron into a relative singlet state of unnatural parity would founder on compelling at least two of the 4f electrons into a relative singlet
state of natural parity, so that the net change in energy was positive rather than negative. We defer thedetails, which involve the coupling of angular momenta for eight electrons, to a later paper. The next cases of elements whose nominal ground-state configuration has two partly filled shells, each with I 2 1, occur in the actinide series: protactinium (Z= 91), uranium (2= 92), neptunium (2= 93), and curium (2= 96). For such heavy elements, in which relativistic effects become important, one should really be doingj - j rather than L - S coupling, so we do not attempt to correlate the ordering of their energy levels in terms of the generalization of Hund's first rule to the alternating rule. In summary, the relative paucity of ground states of elements that provide exceptions to Hund's first rule-really only cerium-is understandable in terms of the facts that most elements have only one open shell and that most of the rest that have two open shells have an open s shell, which cannot give rise to unnatural parity states. Of the small remainder, those in which a single electron is coupled to a half-filled shell do not lower their energy by having pairs of electrons in singlet states of unnatural parity because of a countervailing increase in energy of one or more pairs of electrons in natural parity singlet states. Thus Hund's first rule works extremely well for ground states, for which it was indeed formulated, although there are numerous exceptions for excited states in which one or more electrons have been promoted out of filled shells into unfilled shells with I 1 1. But most of these cases can be rationalized by the alternating rule, a natural generalizationof Hund's first rule to states in which two electrons have relative unnatural parity. The Alternating Rule for Energy Levels and Symmetry Holes in Wave Functions We have seen that there is a very strong correlation between the ordering of atomic energy levels and the presence and width of symmetry holes in their two-electron distribution functions. However, we must carefully avoid falling into the trap of interpreting the presence and width of a symmetry hole as the cause of the obeyance of Hund's first rule or its generalization in the form of the alternating rule. In particular, the venerable argument that a Fermi hole causes the electrons to stay further apart and thereby lowers (1/r12) and its obvious generalization to the argument that the even wider quartic symmetry hole in unnatural parity singlet states causes the electrons in them to stay even further apart are simply not correct if pressed beyond the level of first-order perturbation theory in powers of 1/Z. And yet the naive and indeed wrong explanation works so well to predict the ordering of atomic energies.... Wesuggest here that symmetry holes are not, strictly speaking, a cause of Hund's first rule as generalized to excited states, and its further generalization to the alternating rule, any more than one would maintain that the ordering of atomic energy levels, as reflected by the alternating rule, is the cause of symmetry holes. At the level of Slater's and Condon's treatments, the ordering of energy levels is a consequence of f signs in front of exchange matrix elements. Similarly, when we examine in detail the formation of eigenfunctionsof spatial and spin angular momenta, we see that the presence of a Fermi hole, or an even wider quartic symmetry hole, is a consequenceof the same f signs that appear in the linear combinations of products of one-electron functions. Both of these statements are firmly grounded in basic theory, whereas attempts to make one phenomenon the cause of the other have never produced satisfactory results and are, we believe, destined to failure. Hence we suggest that both are consequences of the same basic cause, the same f signs in the exchange terms in matrix elements of 1/r12and in the matrix elements of 6(r12)
Morgan and Kutzelnigg
2432 The Journal of Physical Chemistry, Vol. 97,No. 10, 1993 and its derivatives, which determinewhether the electron-electron distribution function and how many of its derivatives vanish at r12 = 0. To make this point more concretely, we examine a properly antisymmetrized uncorrelated two-electron wave function
which upon insertion into the angular integrals shows that for fixed values of 11 and 12 the sum over n is finite and permits the evaluation of the angular integrals to yield2'
(34) in agreement with eq 5.6 of our long paper,,' into which the overall prefactor of (-1)S+L+/l+12was incorporated. Since expression 34 can have either sign, one cannot immediatelyconclude from the pitivityof the radial integrals (which was demonstrated by Racah12) that the sum in eq 31 m
+
cG(n)(-1)L+11+12(2111)(21,
+ 1) x
n=O
as given in eqs 5.1 and 5.2 of our lengthy article.2' In determining the behavior of such a wave function in the vicinity of r12 = 0, Le., in the vicinity of rl = r2, wl= w2, the crucial issue is whether the two terms in eq 27 add together or cancel, Le., whether the factor (-l)S+L+/i+Iz is +1 or -1 and hence whether S L lI 12 is even or odd. If the state has natural parity, so that L + lI 12 is even, then the factor is +1 for a singlet state and -1 for a triplet state. If the state has unnatural parity, so that L lI + 12 is odd, the factor is just the opposite: +1 for a triplet state and -1 for a singlet state. Thus whether the wave function and also its first derivativevanishat the electron4ectroncoalescence is determined by this factor of (-1)S+L+/l+/2, Similarly, the exchange integral for such a state is given by
+ + +
+
(1
+
+ 6,1,~1112)-1(-1)S+L+'1+'2~dr,dr, #Jdol
R;(l)R@)R2(1)RI
dw,
is positive and hence that the sign of the exchange term is determined solely by the factor (-l)S+L+/i+/2 in eq 31. Further investigation is required. Although at the time its significance may not have seemed great, we can take a major step forward by invoking Racah's proof12 that the radial integrals Gcn),as well as being positive, have the propertythat the sequence G(")/(2n 1) is monotonically decreasing. If we take the limiting case of Racah's result, namely, a sequence
+
G'") = C(2n
+ 1)
(36)
where C is a positive constant, and insert it into the sum
(2)r~~Q:M/l/2(w~,w2)QLM/2/1(wl,w2) (29)
Because each 52 is a sum of products of one-electron functions, the presence of cross terms in the exchange matrix element prevents us from merely invoking the earlier result that the exchange matrix element ~ r l ) ~ r 2 ) l r ~ ~ ~ r isl )strictly ~r2)) positive. We proceed by inserting a partial wave expansion m
we obtain the sum C(-l)L+"+'2(21,
+ 1)(21, + 1) x
By a not so well-known sum rule involving products of 3 -j and 6-jsymbols, whosediscovery in a treatiseonangular momentum theory2*we owe to Dr. Volker Termath, this expression reduces to c(-1)L+"+'2(211
+ 1)(21, + 1 ) ( i
;
;)2
(39)
where thesquare of the 3 -jsymbol obeys the well-known property that it is positive for L + 11 + 12 even (Le., for natural parity) and vanishes for L + lI 12 odd (Le., for unnatural parity). For the limiting case G(n)= C(2n + 1) of Racah's theorem that G(n)/ (2n 1) forms a monotonically decreasing positive sequence, we thus would have positive exchange integrals for natural parity states and vanishing exchangeintegrals for unnatural parity states.
+
+
+
We now consider the case where the sequence G(n)/(2n l), instead of being constant, is monotonically decreasing. Since the
Hund's Rules
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2433
sum over n in eq 37 is finite, it can be rewritten as (-1)L+"+'2(21,
+ 1)(21, + 1) x
(-1)L+"+'2(211
+ 1)(21, + 1 )
nmex
G(n)
-c,, n=nmin2n+ 1
(40)
where
For nminIn I nmoxr each G("/(2n
+ 1) can be written as
fi
--G(n) ri 2n + 1 2nmin+ lj=n,in G(nmiJ
(42)
where rj is the ratio of the jth term to its predecessor, (43)
with rnm,# = 1. Thus the sum in eq 40 can be rewritten as ($emin)
nmdi
n
c n n rj 2nmin + ln=nmjn j=n,,,
to be the same, as in the perturbative first-order 1/Z expansion, this immediately shows that the term with (-1)S+L+4+/2 = -1 lies below that with (-l)S+L+'~+'z= + l . At the Hartree-Fock level, a generalization of Colpa's variational argument l o would then show that for an atom with two equivalent electrons outside a closed shell, the lowest term of a given L with (-l)S+L+ll+/z = -1 lies below the lowest term of the same L with (-1)S+L+b+l2 = +1. Thus for two-electron states with the same L arising from the same configuration,those with (-1)S+L+Il+12= -1 should lie lower than those with (-1)S+L+/l+12= +l. Fot two-electron states with natural parity, this yields Hund's first rule, that the triplet lies below the singlet, while for two-electron states with unnatural parity, it yields just the reverse, that the singlet lies below the triplet, with both results encompassed by the alternating rule, which we express here in the form that, for two-electron states of the same configuration and the same L, the state with (-1)S+L+/l+I2 = -1 lies below the state with (-1)S+L+4+4= +1, with thelowest stateof theconfiguration characterized by having the widest symmetry hole and the largest value of L. We thus see that both the alternating rule generalization of Hund's first rule and the symmetry holes in electron4ectron distribution functionsat electron-electron coalescences are direct consequences of the common factor of (-1)S+L+Ii+12in the sum (27) for the properly antisymmetrizedtwo-electron wave function. Hence there is a close association between the two, although neither is, strictly speaking, a consequence of the other.
(44)
which in turn can be reexpressed as
as is easily verified by computing in the latter expression the coefficient of r,,m,nr,,m,l+lr,,,in+2 rn. As we have seen, the last term in eq 45 is positive for a natural parity state and vanishes for an unnatural parity state. Since each rj ranges independently from 0 to 1, the key issue is thus the sign of the partial sums
...
for nmin In Inmox- 1. We have found empirically from examinationof tables prepared by Dr. Volker Termath the'partial sum rulemthat these partial sums are positive for natural parity states and negative for unnatural parity states. This is not surprising, since the formulas for 6-j coefficients contain ubiquitous prefactors of (-1)L+/l+/zmultiplying expressions that areusually positive. Thus for natural parity states, with (-1)L+Il+/2 = +1, the weighted sum of partial sums in eq 45 and hence the sum in eq 40 is positive, while for unnatural parity states, with (-1)L+Il+/z = -1, the corresponding sums are negative. In the case where all theriareclose to 1, we wouldexpect the magnitude of the sum for the natural parity state to exceed that for the unnatural parity state because of the extra nonvanishing term in q 45, in agreement with our intuition that the energy splitting between a pair of states that have no hole and a quadratic Fermi hole should be greater than that between a pair of states that have a quadratic Fermi hole and a quartic symmetry hole.29Since for either parity the sum in q 40 is multiplied by the prefactor of (-l)L+/l+/z,which is + 1 for natural parity and -1 for unnatural parity, in the case of either parity expression 37 should be positive, which implies that the sign of the exchange matrix element is determined by the factor (-1)S+L+/1+/1 in eqs 29 and 3 1. If the radial orbitals for the pair of singlet and triplet states are assumed
Epilogue We have examined the formulation of Hund's rules and past attempts to explain them in terms of the presence or absence of a Fermi hole in the electron4ectron distribution function at a coalescence point. By examining broad classes of exceptions to Hund's first rule, as naively generalized to atomic excited states, we have found that the underappreciatedalternating ruleis directly tied to whether a pair of electrons is in a relative state of natural parity, with L + lI 12 = even, or unnatural parity, with L I , 12 = odd. We have further found a very strong association of the ordering of atomic energy levels with the presence of symmetry holes in their electron4ectron distribution functions. Just as others before us have persuasively argued that Fermi holes are not the cause of Hund's first rule, we have argued that the more general symmetry holes are not the cause of the more general alternating rule for energy levels, but rather that both are consequencesof common factors of (-1)L+S+'1+'2in theconstruction of properly antisymmetrized two-electron functions, which determine how rapidly the electron4ectron distribution function vanishes at a coalescence point and which determine the sign of exchange matrix elements of the electron-electron Coulomb potential. Although we have made much progress toward elucidating at the Hartree-Fock level the causes of Hund's first rule and of the exceptions to it encompassed by the alternating rule for atomic excited states, a fully rigorous proof of Hund's rules (as appropriately generalized), encompassing first the extension to excited states within a symmetry subspace and then the much more difficult inclusion of electron correlation, with clearly stated conditions to exclude the many counterexampleswhere correlation effects dominate exchange effects, is likely to remain a challenge into the 21st century. Can we hope that by the centenary of Hund's rules, a rigorous proof will have been discovered? If this paper is found to have pointed out a path leading upward to this lofty goal, it will have served its purpose.
+
+
+
Acknowledgment. We are grateful to Dudley Herschbach and to the many members of his research group for their hospitality, which by now goes backmany years. We are particularly grateful to Dr. Stella Sung for providing Figure 1 for us and to Dr. Volker Termath for conducting a computer study of the sum in eq 38
2434 The Journal of Physical Chemistry, Vol. 97,No. 10, 1993
and for finding in Lindner’s book on angular momentum,25the sum rule reducing eq 38 to eq 39. We are further grateful to Dr. Krzysztof Szalewicz, to Dr. William C. Martin, and to two of the Journal’s reviewers for their helpful comments on this article. J.D.M. remembers with fondness the period of his sabbatical in 1985 at the University of Chicago, where he discussed with Steve Berry and Jack Warner the explanation of Hund’s first rule. This work has been supported by NSF Grants PHY-8608155 and 9215442,by an NSF grant for the Institute for Theoretical Atomic and Molecular Physics at the HarvardSmithsonian Center for Astrophysics, and by the Deutsche Forschungsgemeinschaft and the Fonds der Chemie.
References and Notes (1) Hund, F. Z . Phys. 1925, 33, 345; Linienspektren und periodisches System der Elemenre; Springer: Berlin, 1927; p 124. (2) Slater, J. C. Phys. Reo. 1929. 34, 1293. (3) Pauling, L.; Goudsmit. S. The Structure of Line Spectra; McGrawHill: New York, 1930; pp 165-6. (4) Kato, T. Commun. Pure Appl. Math. 1955, 10, 151. (5) Hoffmann-Ostenhof. M.; Hoffmann-Ostenhof, T.; Stemnitzer, H. Phys. Rev. Lett. 1992, 68, 3857. (6) Kauzmann, W. Quantum Chemistry; Academic: New York, 1957; pp 319-20. Messmer and Birss (ref 7b) attribute this incorrect argument to Kauzmann’s book and to: Pauling, L.; Wilson, E. B. Introduction to Quantum Mechanics;McGraw-Hill: New York, 1935,without givinganypagenumbers. J.D.M. has verified the Kauzmann reference but has becn unable to find the argument in Pauling and Wilson’s book. It is a welcome side benefit to be able to relieve Dudley’s academic father and grandfather of what is apparently an undeserved slur on the reputation of their classic book. (7) (a) Davidson, E. R. J . Chem. Phys. 1965, 42,4199. (b) Messmer, R. P.; Birss, F. W. J . Phys. Chem. 1%9,73,2085. (c) Lemberger, A.; Pauncz, R. Acta Phys. Acad. Sci. Hung. 1969, 27, 169. (8) Colbourn, E. A. J . Phys. B 1973, 6, 2618. (9) Kohl, D. A. J . Chem. Phys. 1972, 56, 4236. Katriel, J. Phys. Rev. A 1972, 5, 1990. Katriel, J. Theor. Chim. Acta 1972, 23, 309. Katriel, J. Theor. Chim. Acta 1972.26, 163. Killingbeck, J. Mol. Phys. 1973,25,455. Colpa, J. P.; Islip, M. F. J. Mol. Phys. 1973, 25, 701. Colpa, J. P.; Brown, R. E. Mol. Phys. 1973, 26, 1453. Katriel, J.; Pauncz, R. Theoretical Interpretations of Hund’s Rule, In Advances in Quantum Chemistry; Academic: New York, 1977; Vol. 10, pp 143-85. Shim, I.; Dahl. J. P. Theor. Chim.Acta 1978,48,165. Kutzelnigg, W., In Theoreticalhiodelsof Chemical Bonding, Part 2, the Concept o/ the Chemical Bond; Maksic, 2. B., Ed.; Springer: Berlin, 1990; pp 1 4 3 .
Morgan and Kutzelnigg (10) Colpa, J. P. Mol. Phys. 1974, 28, 581. (11) Roothaan, C. C. J. Rev. Mod. Phys. 1951, 23, 87. Slater, J. C. Quantum Theory of AtomicStructure; McGraw-Hill: New York, 1960; Vol. I, pp 486-7. ( 1 2) That the positivity of exchange integrals is a general consequence of the positive definiteness of the Coulomb potential was stated in passing by: Lieb, E. H.;Simon,B. Commun.Marh.Phys. 1977,53,191. Racah,G.Phys. Rev. 1943, 62, 460-1 made use of the specific form of the interelectronic Coulomb potential togivea proofofthepositivityofSlater’sGAradial integrals, which implies the positivity of exchange integrals. Racah further proved that GA/(2k + 1) is a monotonically decreasing function of k, which is not immediately susceptible of proof by Fourier transform techniques. (13) Moore,C. E. Atomic Energy Levels;US.Government: Washington, DC, 1971; Vol. I, pp 106-7. Bashkin, S.; Stoner, J. A., Jr. Atomic Energy Levels and Grotrian Diagrams; North-Holland: Amsterdam, 1975; Vol. I, p 389. (14) Borden, W. T. J. Am. Chem. Soc. 1975, 97, 5968. Borden, W. T. Effects of Electron Repulsion in Diradicals. In Diradicals; Borden, W. T., Ed.; Wiley: New York, 1982; especially pp 15-7 and 41-2. (15) Kollmar, H.; Staemmler, V. Theor. Chim. Acta 1978, 48, 223. (16) Russell, H. N.; Meggers, W. F. Sci. Pap. Bur. Stand. 1927,22,364. (17) Condon, E. U.; Shortley, G. H. The Theory o/ Atomic Spectra; Cambridge University: Cambridge, 1935; pp 199-207. (18) Warner, J. W.; Bartell, L. S.; Blinder, S . M. Int. J . Quantum Chem. 1980, 18, 921. (19) Boyd, R. J. Nature 1984, 310, 480. (20) Warner, J. W.; Berry, R. S. Nature 1985, 313, 160. (21) Kutzelnigg, W.; Morgan, J. D., 111 J . Chem. Phys. 1992, 96, 4484. (22) Fano, U. Phys. Rev. 1964, 135, B863. (23) Pack, R. T.; Byers Brown, W. J. Chem. Phys. 1966, 45, 556. (24) Callaway, J. Phys. Lett. 1978, 66A, 201. (25) Moore, C. E. Atomic Energy Levels; US.Government: Washington, DC, 1971; Vol. I, p 22. Bashkin, S.; Stoner, J. A., Jr. Afomic Energy Levels and Grotrian Diagrams; North-Holland: Amsterdam, 1975; Vol. I, pp 57 and 59. (26) Martin, W. C.; Zalubas, R.; Hagan, L. Atomic Energy Leuels-The RareEarth Elements; U S . Government: Washington, DC, 1978; p 209. (27) Lindgren, I.; Morrison, J. Atomic Many-Body Theory, 2nd ed.; Springer: Berlin, 1986; p 78, eq 4.32. (28) Lindner, A. Drehimpulse in der Quantenmechanik; Teubner: Stuttgart, 1984; p 55, first formula in section 4.12, withj2 = 2 j + 1. (29) We defer to a later paper the full treatment of the signs of these sums, whose properties we discovered empirically, with the assistance of Dr. Volker Termath, several weeks past the official deadline for submissions to the Herschbach issue and just one week before this paper was actually submitted. We trust that Dudley will understand and forgive us.
