An under-approximation of entropy for elemental multiconfigurational ground state electronic structures

For most of the elements in the periodic table, the number of possible electron permutations for an isolated neutral atom in its ground state is limited by the Russell Saunders (LS) coupling that gives rise to Hund's rule.¹ However, for heavier elements that exhibit jj-coupling behavior, Hund's rule gives way to greater degrees of freedom and larger spin–orbit coupling resulting in vastly more complex electronic structures and behaviors such as strong correlation effects, multiconfigurational ground states, instabilities, non-integer orbital occupancies, itinerant magnetism, heavy fermions, and superconductivity.^1–8 Among the actinides, plutonium (Pu) is arguably the most complex as its intermediate behavior between LS- and jj coupling gives rise to its varied instabilities and properties.^{1,2,5–7,9–14}

In both experimental and theoretical efforts to explain the complexity of plutonium, much emphasis is placed on the electron occupancy of the 5f orbital.^{2,4,6,7,10,15–20} For example, Table I shows 5f occupancies reported by Booth et al. for a series of Pu allotropes and compounds as determined by resonant x ray emission spectroscopy (RXES) showing non-integer occupancies below six.^7, Figure 1 shows angle integrated photoemission spectra (PES) for a variety of Pu compounds and allotropes in the order of 5f occupancies revealing a trend between the predominance of the weight of density of states at the Fermi energy (E_F) and 5f occupancy.^7,15,16,21 Specifically, no significant weight is apparent at E_F for PuO₂ with 5f⁴ occupancy, but as 5f occupancy approaches the itinerant to localized crossover between 5f⁵ and 5f⁶, the weight at E_F becomes more and more dominant in the spectrum. This weight at E_F has been described in theoretical modeling of strongly correlated metals as a partially occupied quasiparticle cut off by E_F.^22–24 It is important to note that in these quantitative and qualitative experimental methods of probing 5f occupancy, the results are representative of bulk material which are distinct from an isolated neutral atom, even in the case of the Pu metal allotropes, and particularly in the case of highly correlated electron materials.

The distinction between orbital occupancies for atomic and bulk Pu has been modeled previously by Ryzhkov et al., in which it was determined that the ground state configuration for an isolated neutral atom of Pu is expected to be [Rn]7s²5f⁶, while that of bulk Pu deviates toward a 5f⁵ occupancy.^18,19 In these calculations, the resulting 5f occupancy in all bulk cases was a non-integer value between 5f⁵ and 5f⁶, with the remainder of the eight total electrons distributed to varying non-integer occupancies in the 7s-, 6d-, and 7p-orbitals. These results are consistent with those of Booth et al. summarized in Table I for the two allotropic variants of metallic Pu. The 5f occupancies for the oxide and intermetallics being less than five are not surprising since bonding between the Pu atoms and oxygen or other ligand atoms would be expected to transfer electrons away from Pu.

Herein, a combinatorial approach to describe the electron degrees of freedom in terms of allowable permutations of quantum electronic configurations will be applied toward an entropic approach to determine orbital occupancies in plutonium. This effort expands on previously published work toward an over-approximation of entropy for multiconfigurational ground states.⁸ Specifically, the number of potential electron configurations that could conceivably contribute to the quantum mechanical superposition of a multiconfigurational ground state was previously extremely inclusive in terms of both the occupational configurations and the number of electron permutations resulting from a relaxing of Hund's rule generally (even for the case of LS-coupling). In order to establish an under-approximation of entropy, the previous combinatorics will be reworked more exclusively by enforcing Hund's rule generally within a single occupancy configuration for each element in the periodic table. This will enable a bracketing of the range of possible electronic structure entropies for multiconfigurational ground state electronic structures, which can be compared to known values. The resulting under-approximation will then be used to eliminate orbital occupancies from consideration as contributing to the multiconfigurational ground state of Pu. Finally, resulting orbital occupancies from a statistical mechanics approach will be compared to those summarized from the literature for Pu.

The number of permutations under the LS coupling constraints of Hund's rules for all fillings of s-, p-, d-, and f-orbitals are given in Tables II–V, respectively. The specific permutations for each occupancy of the s- and p-orbitals are also given in Tables II and III. Lists of all Hund's rule constrained permutations for the d- and f-orbitals are included as the supplementary material (Tables SVII and SVIII).

Having a capacity of two electrons (l = 0; m_l = 0; and m_s = ±½), the s-orbital can be either empty, half-filled, or filled such that there is no combinatorial possibility of a permutation that violates Hund's rule. Therefore, Table II is identical to the table reported previously for the s-orbital in which Hund's rule was relaxed for the possibility of jj-coupling.⁸ This is also true for empty, singly occupied, singly deficient, or fully filled p-, d-, and f- orbitals (p⁰, p¹, p⁵, p⁶, d⁰, d¹, d⁹, d¹⁰, f⁰, f¹, f¹³, and f¹⁴). In all other cases, the number of permutations is greatly reduced as Hund's rule violating configurations have been eliminated. For example, the maximum number of permutations for partially filled p-, d-, and f- orbitals are 6, 20 and 70, respectively, as compared to 20, 252, and 3432, respectively, in the case of a relaxed Hund's rule. Furthermore, for all half-filled orbitals (s¹, p³, d⁵, and f⁷), enforcement of Hund's rule limits the number of permutations to exactly two possibilities corresponding to all unpaired spin-up or spin-down states. This is in stark contrast to the maximum number of permutations occurring for the half-filled orbitals when Hund's rule is relaxed. With the exception of the s- orbital for which permutations are not affected by the application of Hund's rule, the maximum number of permutations for each orbital occurs symmetrically at the midpoints between completely empty and half-filled, and between half-filled and fully filled. Thus, the greatest number of permutations for each of the p-, d-, and f-orbitals occur at partial fillings of p¹, p², p⁴, p⁵; d², d³, d⁷, d⁸; f³, f⁴, f¹⁰, and f¹¹.

These comparisons between the number of configurational permutations for various fillings of the orbitals have significant fundamental implications for the proposed hypothesis correlating combinatorics to electronic stability and complexity. Specifically, with the degrees of freedom for an electronic configuration being represented by the number of quantum permutation configurations, it has been hypothesized that greater instability and complexity occurs for larger numbers of permutations, and greater stability occurs for lower numbers of permutations. Such an interpretation as applied to the permutations in Tables II–V tabulated under the constraint of Hund's rule provides a fundamental explanation for the greater stabilities of empty, half-filled, and fully filled orbitals.

With the number of permutations established for each orbital under the constraints of Hund's rule, the numbers of permutations for the established ground states of isolated neutral atoms for each element can be determined. The established ground state electronic structures for each element in the periodic table as listed in Table VI are consistent with those listed on the Royal Society of Chemistry website. The number of permutations for each established ground state in Table VI is determined by multiplying the number of permutations for each corresponding orbital occupancy listed from Tables II–V. For example, the number of permutations for the established ground state of U is 1 × 10 × 70 = 700, corresponding to the product of the s², f³, and d¹ permutations, respectively.

Combining the over-approximation of entropies published previously with those given herein under a generalized LS-coupling assumption, a comparison can now be made between the two approximations and known standard molar entropies, as shown in Fig. 2. The majority of standard molar entropies for each element shown in Fig. 2 were obtained from the online Periodic Table of the Elements database on the Gordon England Thermal Spray Coating Consultant's website, with others taken from the University of Wisconsin, Madison's ChemPRIME project and various other references.^25–27 Standard molar entropies are not included for astatine, francium, or any transplutonic elements.

In order to compare with the calculated over- and under-approximations, the known standard entropies were converted from units of J K⁻¹mol⁻¹ to units of eV K⁻¹ per individual atom. Ideally, the known entropies would be bracketed between the over- and under-approximations. However, the resulting comparison shows that for many of the elements, the standard molar entropies are greater than those predicted combinatorically as an over-approximation for Boltzmann's entropy. It is important to note that at standard temperature and pressure, many of the elements are gases (H, He, N, O, F, Ne, Cl, Ar, Kr, I, Xe, and Rn) and liquids (Br and Hg). Therefore, any determination of entropy at standard conditions for these fluids results in a much greater entropy due to the contribution of rotational and translational entropy than that calculated for the electronic structure alone. More generally, it should be noted that the calculations are based only on the entropy of the electronic configuration in isolated neutral atoms, whereas the standard molar entropies are determined for bulk materials which includes atomic motion. For the elements that are solid at standard conditions, this includes vibrational motion.

With the exception of technetium (Tc) and protactinium (Pa), the entropic contribution of motion explains the large entropies of all the outliers well above the calculations. The known entropies for all the remaining solid elements fall within the upper limit of the maximum over approximations reported previously. However, many of these elements do not fall within the range bracketed by the over- and under-approximations for their respective elements. In Fig. 3, the data are replotted with the s-block, p-block, d-block, and f-block elements color coded red, blue, cyan, and magenta, respectively. In doing so, it becomes apparent that the known entropies of the s- and p-block elements fall above the approximations in almost all cases with the notable exceptions of boron (B), and carbon (C), which appear below even the under-approximation; and silicon (Si) which falls within the bracketed approximations. It is interesting that the known entropies for both boron (6.115 × 10⁻⁵eV/K) and carbon (5.908 × 10⁻⁵eV/K) are very close in value to what would be expected from exactly two possible energy degenerate configurations (5.97 × 10⁻⁵eV/K). It should also be noted that the known entropy of carbon plotted in Figs. 2 and 3 is that of graphite, while the value for diamond is even lower (2.46 × 10⁻⁵eV/K). This could indicate that the electronic structure for diamond contains only one possible configuration, and the known entropy is due primarily to atomic motion. The same could be said for boron and graphite, although it is also possible that both bulk forms contain two possible energy degenerate electronic configurations.

For the f- and d-block elements, where multiconfigurational ground states are most likely to occur, the known values largely fall within the bracketed region between the under- and over-approximation calculations. Furthermore, while the over-approximation calculations would suggest that entropy increases as a half-filled state in these orbitals is approached, the known data suggest flatter, if not opposite trends. Comparing the known data to the under-approximation calculations seems to reveal better correlation with the expected values decreasing as half filling is approached, as is the case with the first 4 elements in each d-block series as shown in Table VII for Sc, Ti, V, and Cr; Y, Zr, Nb, and Mo; and Lu, Hf, Ta, and W. As half filling is exceeded, the known entropy values tend to increase as shown in Table VII for Fe, Co, Ni, Cu, and Zn; Ru, Rh, Pd, Ag, and Cd; and for Os, Ir, Pt, Au, and Hg. The only exception to this upward trend beyond half filling in the d-block appears for the transition between Co and Ni, where the flat transition is consistent with that of the under-approximation.

The under-approximation calculation for entropy would suggest that at half-filling of the d-orbital, the entropy would drop significantly. However, in all cases, the known entropies for the element corresponding to half-filling (Mn, Tc, and Re) are greater than those for the elements on either side, with the 18.8 × 10⁻⁴eV/K entropy shown for Tc being an outlier of the data possibly attributable to the difficulty in determining molar entropy for the radioactive element.

For the f-block elements, known entropy data are plotted across the lanthanide series, but are unavailable for transplutonic elements in the actinide series. This makes comparisons between the approximation calculations and the known entropies more difficult because consistent trends between the lanthanide and actinide series cannot be established, including a comparison between the established ground states for half-filling of the f-orbital expected for Eu and Am. However, comparisons can still be made between the observations of the d-block elements and the lanthanide series. For example, the known entropy for Eu shown in Table VIII is the largest value for all of the lanthanide elements, consistent with the observation in the d- block of maximum entropy for half-filling of the respective orbitals. It is also notable that the known entropies for Nd and Pm are exactly the same and that the values for Dy and Ho are very close, consistent with the predictions of the under-approximation, which is in line with the observation made for Co and Ni. For the actinide series, we see that this observation lends itself to the known values of U and Np, which are very similar as would be expected from the under-approximated values. Finally, the trend observed for decreasing/increasing entropy as half-filling of the d-orbital is approached/exceeded is comparable (although not as consistent) to the decrease in known entropies within the Pr, Nd, Pm, Sm series and the increase in known entropies within the Gd, Tb, Dy series. A similar trend is not apparent in the early actinide series, even when discounting Pa as an outlier similar to that of Tc.

These observations provide an opportunity for predictive experimentation of the transplutonic elements. For example, based on these observations, it would be expected that the entropy of Am would be larger than that of both Pu and Cm, if not all the other actinides (bar Pa), and that the entropy of Cf and Es would be very close in value as predicted by the under-approximated values, with the values for Cm and Bk increasing to that of Cf and Es. Additional comparisons and observations can be made for relative experimental entropies for element groups (or columns) in the periodic table. For example, Fig. 4 shows a general trend for all the s-block and p-block element groups for increasing entropy with each consecutive period. Table VII shows that this trend persists for all d-block element entropies with each element exhibiting greater entropy than the element directly above (with the exception of the Tc outlier). It is interesting that this trend seems to be universal for every column of elements in the s-, p-, and d-blocks of the Periodic table, but completely opposite for the available data in the f-block as shown in Table VIII. For the f-block, the entropy values of La and Ac are very close, but for every other column in Table VIII, the known entropy for the lanthanide is much greater than that of its corresponding actinide. It begs the question as to whether the entropies of the transplutonic actinides would be less than or greater than those of their lanthanide counterparts on the latter side of the itinerant to localized transition. It would not be unreasonable to hypothesize that the entropies of the transplutonic actinides would be greater than those of their corresponding lanthanides because the later actinides are known to behave more lanthanide-like and thus would be more likely to maintain the trends observed everywhere else in the Periodic table, except for the early actinides.

The underapproximation of entropy for plutonium calculated herein of 2.27 × 10⁻⁴eV/K is derived from the established ground state of 7s²5f⁶6d⁰7p⁰, which yields 14 possible electron permutations under the constraints Hund's rule (as compared to the 3003 permutations for when Hund's rule is relaxed). The overapproximated entropy of 1.31 × 10⁻³eV/K for Pu plotted in Figs. 2–4 is derived from 487 630 electron permutations for the nine occupancy configurations given in Table IX with Hund's rule relaxed.⁸ The known molar entropy of Pu (5.65 × 10⁻⁴eV/K) falls well within these bracketed over- and under-approximated values. With Hund's rule enforced, Table IX shows the reduced number of permutations and corresponding Boltzmann's entropy for each of the nine individual occupancy configurations, resulting in 4112 cumulative permutations, which corresponds to a Boltzmann's entropy of 7.17 × 10⁻⁴eV/K (still above that of the known entropy). The known entropy for bulk Pu of 5.64 × 10⁻⁴eV/K would correspond to no more than 700 energy degenerate configurations (which will be referred to as the 700-permutation limit). Therefore, of the nine occupancy configurations given in Table IX, the configuration that comes closest to the known entropy value for bulk Pu without going over is 7s²5f⁵6d¹7p⁰.

A number of total electron permutations for these 105 occupancy configurations have been recalculated under the constraints of Hund's rule and tabulated in the supplementary material (Table SIX). By eliminating all Hund's rule violating configurations, the cumulative number of permutations decreases to 212 317 possible combinations. In this section, the bounding conditions (4) for the orbital occupancies of a multiconfigurational ground state of Pu will be gradually tightened. At each step, the probabilities of finding each orbital with a specific occupancy will be calculated under an energy degeneracy assumption, and a resulting non-integer occupancy multiconfigurational ground state representation will be given based on these probabilities. In every case, if the resulting set of occupancy configurations contributing to a multiconfigurational ground state include configurations both above and below the 700-permutation limit, the calculation will be performed twice: once for all the contributing occupancy configurations, and again for only those configurations that fall below the 700-permutation limit.