The ability to predict the chemical and physical properties of a material is directly related to the structure and interactions of its electrons. For materials comprised of f-block elements (the lanthanides and actinides found in the last two rows of the periodic table), the complexity of electronic structure has presented great difficulty in understanding, modeling, and predicting material properties. The complexity of multiconfigurational ground state electronic structures is illustrated herein by the combinatorics of electron permutations within individual and cumulative occupancy configurations. A non-integer orbital occupancy representation of multiconfigurational ground states is described for superposition mixing between multiple near-energy degenerate occupancy configurations and generalized in such a way that established ground states are returned by approximation for elements with less-complex electronic structures. By considering the occupancy configurations as statistical mechanics macrostates, and the permutations of electrons as statistical mechanics microstates within those macrostates, an over-approximation of entropy for multiconfigurational elemental ground state electronic structures has been calculated.

Topics
Angular momentum coupling, Electronic structure, Electronic configuration, Entropy, Electron energy loss spectroscopy, X-ray absorption spectroscopy, Combinatorics, Quantum mechanical principles, Statistical thermodynamics, Nonequilibrium statistical mechanics

From our earliest exposure to atomic structure and the periodic table, we begin to predict material properties based on the electronic structure. For example, it is commonly taught that for non-transition metal elements, the number of valence electrons for atoms ranges from 1 to 8 depending on the column (groups 1, 2, and 13–18), and that nature prefers eight valence electrons (or 2 in the case of period 1 elements). The tendency of nature toward an octet of valence electrons is used to explain the most energetically favorable ionic states, similarities in properties of like-group elements, and the most stable compounds formed by the transfer and sharing of electrons. Although the octet rule is useful and elegant, it is also insufficient to explain the existence, structure, and properties of the transition elements.

With a more fundamental quantum-based understanding of the electronic structure, the reasons for the existence of transition metals, the octet rule, and the overall structure of the periodic table become more apparent. Specifically, the octet rule is understood as the stability of completely filled s- and p-orbitals each having electron capacities of two (l = 0; ml = 0; ms = ±½) and six (l = 1; ml = −1, 0, +1; ms = ±½), respectively, with each period, n, having ns and np orbitals (except for period 1, where no 1p orbital exists). For principal quantum number n ≥ 3, there exist nd-orbitals with a capacity of ten electrons (l = 2; ml = −2, −1, 0, +1, +2; ms = ±½). The energy of the 3d orbital is typically higher than that of the 4s orbital such that the filling of the 3d orbitals occurs between the 4s and 4p orbitals within period 4. This explains the position of the d-block transition metals between the s- and p-blocks in the periodic table, the filling of which lags behind by one period resulting in there being only eight elements in period 3 instead of 18.

Similar to the d-orbitals, the filling of f-orbitals, corresponding to l = 3 (only existing for n ≥ 4) with a capacity of 14 electrons (ml = −3, −2, −1, 0, +1, +2, +3; ms = ±½), lags behind by two periods, such that the filling of the 4f(5f) orbitals does not occur until period 6(7). The f-block of the periodic table, consisting of the lanthanides and actinides, sits between the s- and d-blocks, indicating that the filling of the (n − 2)f-orbitals typically occurs after the ns orbital and before the (n − 1)d-orbitals. Thus, the filling of orbitals for any period, n, in the periodic table proceeds approximately in consecutive order of atomic number with the two ns-states, followed by any existing (n − 2)f-states, followed by any existing (n − 1)d-states, and, finally followed by any existing np-states.

This pattern of consecutive orbital filling, referred to as the aufbau principle, holds for all s-block and p-block elements, but not necessarily for the transition metals, particularly the lanthanides and actinides. Deviations from this filling pattern are often explained by the interplay between Hund's rule and the aufbau principle where the Russell Saunders (LS) coupling dominates; the stability of empty, half-filled, and completely filled orbitals; d-orbital collapse; relativistic effects; and core level screening, among other mechanisms.1–4 As we move through the periodic table in order of atomic number, the complexity of electronic structure progressively increases as each consecutive period introduces more orbitals with greater diversity of energies, more crowded energy level profiles, and greater potential for more complex interactions and resulting ground state structures. With this complexity comes a greater predominance of jj coupling over LS-coupling as well as more complex and difficult to explain mechanisms for aufbau filling pattern deviations.3,4 Regardless of the mechanism involved, even for the most complex electronic structures, deviations from the aufbau filling pattern can be generally attributed to energetic favorability.

Kotliar and Vollhardt describe this increase in complexity as “a whole zoo of exotic ordering” resulting from the large degrees of freedom for open d- and f-electron shells.5 This is consistent with observed deviations from the aufbau filling pattern for established elemental ground states that occur exclusively within the d- and f-blocks of the periodic table. It is these degrees of freedom that give rise to multiconfigurational ground states and strongly correlated materials.5,6 Of all the elements in the periodic table, plutonium (Pu) has long been regarded as the most complex element, and, therefore, a prime example of this complexity.7–11 Of Pu, Hecker et al. wrote that, “[p]lutonium is of interest because of its nuclear properties. However, it is its electronic structure and the resulting physical and chemical properties that make plutonium the most complex element in the periodic table.”12 There are numerous ways in which this complexity manifests itself, such as its instabilities,12 its electronic structure that has been experimentally observed to exhibit a jj-skewed intermediate between LS and jj coupling,3,4,13 its non-integer 5f occupancies,7,14 its important position in the itinerant to localized transition or crossover,13,15 and the difficulty theory has had in accurately modeling these properties.16 

Janoschek et al. described the ground state of Pu as being, “governed by valence fluctuations, that is, a quantum mechanical superposition of localized and itinerant electronic configurations.”7 This description made specifically for Pu could reasonably be applied to any multiconfigurational element ground state. In this paper, the idea of electrons existing in quantum mechanical superposition between near-energy-degenerate ground state configurations is explored. Specifically, the application of quantum mechanical superposition to the energy degeneracies that exist in multiconfigurational ground states of individual neutral atoms will be represented herein with a non-integer orbital occupancy notation. This notation will be generally applicable to the ground state electronic structures of neutral atoms for every element in the periodic table. However, specific emphasis will be placed on elements with historically controversial or unique ground states as well as elements considered to possess multiconfigurational ground states such as the actinides, particularly Pu.

Bounding conditions for the number of possible occupancy states as well as the possible electron permutations within each occupancy state that could potentially contribute to superposition mixing will be considered generally at first. Once generalized combinations have been described, bounding conditions will be tightened in an attempt to approach a more physical description of multiconfiguration ground states. By considering each occupancy state as a statistical mechanics macrostate and the possible permutations of electrons within each occupancy state as statistical mechanics microstates, principles of classical statistical mechanics and thermodynamics will be applied to the combinations of quantum configurations in an attempt to describe the entropy of multiconfigurational ground states and to correlate the number of microstates to observed complexity and instability.

Progressing through the periodic table in order of atomic number, the first element that deviates from the aufbau filling pattern of ground state electron configurations is chromium (Cr). The expected electronic configuration based on the aufbau filling pattern would be an argon core with two 4s electrons and four 3d electrons: [Ar]4s23d4. However, the established ground state electronic configuration for a neutral atom of Cr is [Ar]4s13d5. This deviation from the filling pattern is indicative of a crossover of energy levels between the filling of the fifth d-state and the second s-state1 and suggests that the filling of the fifth d-state is energetically favored to the pairing of the 4s orbital.

In order to apply a non-integer occupancy interpretation to ground state electronic configurations, it is hypothesized that effective non-integer orbital occupancies can occur in the ground states of any element atoms possessing near-energy degenerate configurations. Based on basic quantum mechanical principles, it would be expected under this interpretation that deviations from integer orbital occupancies are directly related to the number of near-degenerate configurations and inversely related to the energy difference of the configurations. For demonstration purposes, this interpretation is applied to more simple systems, such as Cr which could be represented as having an effective ground state with non-integer occupancies as
(1)
where γ represents a deviation (0 < γ < 2) from integer occupancies. This would imply a superposition between two or more 4s and 3d orbital occupancies and assumes that the 4p orbital energy is sufficiently high that it does not participate. This could be described combinatorically as the ground state existing in a superposition between the [Ar]4s23d4, [Ar]4s13d5, and [Ar]3d6 configurations. Energetic considerations would likely rule out at least the latter of the three configurations. The magnitude of γ would be representative of the deviation from integer occupancy resulting from the differing probabilities of finding a Cr atom in one of the two near-energy degenerate states.

Given that the ground state electron configuration of Cr is well established as [Ar]4s13d5, the magnitude of the deviation could be considered to be γ ≈ 1, such that the ground state approximates to the established integer occupancy of [Ar]4s13d5. In other words, the probability of finding the ground state of a Cr atom in the [Ar]4s13d5 state is much greater than that of finding it in the [Ar]4s23d4 or [Ar]3d6 states, approaching 100%. Thus, this interpretation of quantum superposition for near-energy degenerate ground state electronic configuration approximates to the well-established energetically favorable ground state of Cr while allowing for consideration of quantum superposition mixing with other configurations. The quantum interpretation of this representation would be that a sufficient energy difference exists between the established ground state and the first excited state that no significant superposition mixing between the two exists.

Similar rationales and treatment can be readily applied to copper (Cu), which exhibits a ground state configuration of [Ar]4s13d10 (instead of [Ar]4s23d9), for which the complete filling of the 3d-orbital with a half-filled 4s-orbital is energetically favored to the filled 4s-orbital and a single electron deficiency of the 3d-orbital. In these two examples, the focus is on Cr and Cu because they deviate from the aufbau filling pattern. However, the ground states of these elements are well established as being lower in energy than what would be predicted by the aufbau filling pattern. A more interesting application of this quantum superposition interpretation can be applied to another period 4 transition metal that is in proximity to one of these aufbau deviants, for which the filling approaches an instability. Specifically, the ground state of nickel (Ni) has been described as having two near-energy-degenerate configurations, [Ar]4s23d8 and [Ar]4s13d9, with some controversy over which has the lower energy.17 By describing the ground state electron configuration of Ni as a superposition between these two states, the ground state can be described as
(2)
The quantum interpretation would be that the ground state of Ni exists in a superposition between the [Ar]4s23d8 and [Ar] 4s13d9 states. If the energies of these two configurations are considered to be nearly degenerate, we could consider γ as being approximately equal to 0.5, indicating a near equal probability of finding a Ni atom in one of the two configurations. This could be represented as the Ni ground state having effective non-integer occupancies of approximately 1.5 and 8.5 for the s- and d-orbitals, respectively,
(3)
Any detectable deviation of γ from a value of 0.5 would resolve the controversy over which configuration is truly lower in energy and, thus, the true ground state.

Similar to the observed deviations in the aufbau filling pattern for ground states within the period 4 transition metals, those observed for the higher period d-block transition metals all involve instabilities between the filling of their respective s- and d-orbitals with one possible notable exception for lawrencium (Lr) that will be addressed later. These deviants are highlighted in Table I.

Table I.

Abbreviated established ground state electron configurations for d-block transition metal elements with aufbau deviants highlighted.

4s23d1 Sc  4s23d2 Ti  4s23d3 V  4s13d5 Cr  4s23d5 Mn  4s23d6 Fe  4s23d7 Co  4s23d8 Ni  4s13d10 Cu  4s23d10 Zn 
5s24d1 Y  5s24d2 Zr  5s14d4 Nb  5s14d5 Mo  5s24d5 Tc  5s14d7 Ru  5s14d8 Rh  4d10 Pd  5s14d10 Ag  5s24d10 Cd 
6s25d1 Lu  6s25d2 Hf  6s25d3 Ta  6s25d4 W  6s25d5 Re  6s25d6 Os  6s25d7 Ir  6s15d9 Pt  6s15d10 Au  6s25d10 Hg 
7s27p1 Lr  7s26d2 Rf  7s26d3 Db  7s26d4 Sg  7s26d5 Bh  7s26d6 Hs  7s26d7 Mt  7s26d8 Ds  7s26d9 Rg  7s26d10 Cn 
4s23d1 Sc  4s23d2 Ti  4s23d3 V  4s13d5 Cr  4s23d5 Mn  4s23d6 Fe  4s23d7 Co  4s23d8 Ni  4s13d10 Cu  4s23d10 Zn 
5s24d1 Y  5s24d2 Zr  5s14d4 Nb  5s14d5 Mo  5s24d5 Tc  5s14d7 Ru  5s14d8 Rh  4d10 Pd  5s14d10 Ag  5s24d10 Cd 
6s25d1 Lu  6s25d2 Hf  6s25d3 Ta  6s25d4 W  6s25d5 Re  6s25d6 Os  6s25d7 Ir  6s15d9 Pt  6s15d10 Au  6s25d10 Hg 
7s27p1 Lr  7s26d2 Rf  7s26d3 Db  7s26d4 Sg  7s26d5 Bh  7s26d6 Hs  7s26d7 Mt  7s26d8 Ds  7s26d9 Rg  7s26d10 Cn 
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By generalizing the representations of the ground state configurations based on the non-integer occupancy interpretation, the configurations of the d-block transition metals can be represented, as shown in Table II. Specifically, the ground state for every element in the d-block is represented by the predicted aufbau occupancy configuration with a γ value deviation from the occupancies. The γ deviations for each element are positive values subtracted from the maximum occupancy of the s-orbitals and added to the aufbau filling pattern occupancy of each d-orbital. For a γ value of zero, the expected ground state from the aufbau filling pattern is returned, which accounts for most of the elements in Table I. With only two exceptions (Lr and Pd), the aufbau deviants can be represented with a γ value of 1 (similar to Cr and Cu) to give the established ground states, with Pd requiring a γ value of 2. The value of γ in this representation is generally bound by 0 ≤ γ ≤ 2, except for a more stringent bounding of 0 ≤ γ ≤ 1 for the second to last column, and γ = 0 for the last column of the d-block. These γ values are indicative of a maximum possible number of electron configurations, namely, s2dn, s1dn+1, and s0dn+2. In the last two columns of the d-block, the combinations are constrained by the s- and d-orbital occupancies: s2d9 and s1d10 for the second to last column, and only s2d10 for the last.

Table II.

Generalized non-integer occupancy ground state representations for d-block elements assuming only s- and d-orbital participation in superposition mixing with aufbau deviants highlighted.

4 s 2 − γ S c 3 d 1 + γ S c Sc  4 s 2 − γ T i 3 d 2 + γ T i Ti  4 s 2 − γ V 3 d 3 + γ V V  4 s 2 − γ C r 3 d 4 + γ C r Cr  4 s 2 − γ M n 3 d 5 + γ M n Mn  4 s 2 − γ F e 3 d 6 + γ F e Fe  4 s 2 − γ C o 3 d 7 + γ C o Co  4 s 2 − γ N i 3 d 8 + γ N i Ni  4 s 2 − γ C u 3 d 9 + γ C u Cu  4 s 2 − γ Z n 3 d 10 + γ Z n Zn 
5 s 2 − γ Y 4 d 1 + γ Y Y  5 s 2 − γ Z r 4 d 2 + γ Z r Zr  5 s 2 − γ N b 4 d 3 + γ N b Nb  5 s 2 − γ M o 4 d 4 + γ M o Mo  5 s 2 − γ T c 4 d 5 + γ T c Tc  5 s 2 − γ R u 4 d 6 + γ R u Ru  5 s 2 − γ R h 4 d 7 + γ R h Rh  5 s 2 − γ P d 4 d 8 + γ P d Pd  5 s 2 − γ A g 4 d 9 + γ A g Ag  5 s 2 − γ C d 4 d 10 + γ C d Cd 
6 s 2 − γ L u 5 d 1 + γ L u Lu  6 s 2 − γ H f 5 d 2 + γ H f Hf  6 s 2 − γ T a 5 d 3 +   γ T a Ta  6 s 2 − γ W 5 d 4 + γ W W  6 s 2 − γ R e 5 d 5 + γ R e Re  6 s 2 − γ O s 5 d 6 + γ O s Os  6 s 2 − γ I r 5 d 7 + γ I r Ir  6 s 2 − γ P t 5 d 8 + γ P t Pt  6 s 2 − γ A u 5 d 9 + γ A u Au  6 s 2 − γ H g 5 d 10 + γ H g Hg 
7 s 2 − γ L r 6 d 1 + γ L r Lr  7 s 2 − γ R f 6 d 2 + γ R f Rf  7 s 2 − γ D b 6 d 3 + γ D b Db  s 2 − γ S g 6 d 4 + γ S g Sg  7 s 2 − γ B h 6 d 5 + γ B h Bh  7 s 2 − γ H s 6 d 6 + γ H s Hs  7 s 2 − γ M t 6 d 7 + γ M t Mt  7 s 2 − γ D s 6 d 8 + γ D s Ds  7 s 2 − γ R g 6 d 9 + γ R g Rg  7 s 2 − γ C n 6 d 10 + γ C n Cn 
4 s 2 − γ S c 3 d 1 + γ S c Sc  4 s 2 − γ T i 3 d 2 + γ T i Ti  4 s 2 − γ V 3 d 3 + γ V V  4 s 2 − γ C r 3 d 4 + γ C r Cr  4 s 2 − γ M n 3 d 5 + γ M n Mn  4 s 2 − γ F e 3 d 6 + γ F e Fe  4 s 2 − γ C o 3 d 7 + γ C o Co  4 s 2 − γ N i 3 d 8 + γ N i Ni  4 s 2 − γ C u 3 d 9 + γ C u Cu  4 s 2 − γ Z n 3 d 10 + γ Z n Zn 
5 s 2 − γ Y 4 d 1 + γ Y Y  5 s 2 − γ Z r 4 d 2 + γ Z r Zr  5 s 2 − γ N b 4 d 3 + γ N b Nb  5 s 2 − γ M o 4 d 4 + γ M o Mo  5 s 2 − γ T c 4 d 5 + γ T c Tc  5 s 2 − γ R u 4 d 6 + γ R u Ru  5 s 2 − γ R h 4 d 7 + γ R h Rh  5 s 2 − γ P d 4 d 8 + γ P d Pd  5 s 2 − γ A g 4 d 9 + γ A g Ag  5 s 2 − γ C d 4 d 10 + γ C d Cd 
6 s 2 − γ L u 5 d 1 + γ L u Lu  6 s 2 − γ H f 5 d 2 + γ H f Hf  6 s 2 − γ T a 5 d 3 +   γ T a Ta  6 s 2 − γ W 5 d 4 + γ W W  6 s 2 − γ R e 5 d 5 + γ R e Re  6 s 2 − γ O s 5 d 6 + γ O s Os  6 s 2 − γ I r 5 d 7 + γ I r Ir  6 s 2 − γ P t 5 d 8 + γ P t Pt  6 s 2 − γ A u 5 d 9 + γ A u Au  6 s 2 − γ H g 5 d 10 + γ H g Hg 
7 s 2 − γ L r 6 d 1 + γ L r Lr  7 s 2 − γ R f 6 d 2 + γ R f Rf  7 s 2 − γ D b 6 d 3 + γ D b Db  s 2 − γ S g 6 d 4 + γ S g Sg  7 s 2 − γ B h 6 d 5 + γ B h Bh  7 s 2 − γ H s 6 d 6 + γ H s Hs  7 s 2 − γ M t 6 d 7 + γ M t Mt  7 s 2 − γ D s 6 d 8 + γ D s Ds  7 s 2 − γ R g 6 d 9 + γ R g Rg  7 s 2 − γ C n 6 d 10 + γ C n Cn 
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Combinatorically, the elements in Table II have up to three possible occupancy configurations, which could lend themselves to effective non-integer occupancy ground states resulting from the electrons existing in quantum superpositions between these configurations. However, energetically in most cases, the energy separations for these configurations are likely large enough to limit significant superposition mixing such that the established ground state configuration would be returned by approximation. Although this might seem like unnecessary tedium to represent an already established ground state, the example for Ni described previously shows that some utility in this quantum-based interpretation can be found, especially for elements with more controversial and complex electronic structure.

In all of the treatment of the d-block transition metals thus far, it has been assumed that any superposition mixing of configurations only involves the s- and d-orbitals. However, the established ground state of Lr being [Rn]7s25f147p1 would call this into question, at least for the later periods of the periodic table. The entry for Lr in Table II is the only exception for which no value of γLr yields the established ground state given in Table I. Within the framework of the superposition mixing interpretation, it would seem that at some point, the p-orbital becomes relevant in the ground state configuration superpositions. It would be expected that a greater probability of superposition mixing of configurations would occur for higher period elements than lower period elements due to there being more orbitals with greater diversity of energies, more crowded energy level profiles, and decreasing energy separations of orbitals with increasing radial extent.

Given that the period 7 d-block elements have established deviations involving the s- and d-orbitals, the representations of ground state configurations for these elements could be adjusted to account for the possibility of p-orbital participation in superposition mixing as shown in Table III (which excludes consideration of f-orbital participation). The constraints for γs, γd, and γp in Table III are based on the capacity of each orbital and the total number of electrons available, with the additional constraint of γs + γd + γp = 0 being a general statement of electron conservation. For Lr, the following γ values would return the established ground state configuration: γs,Lr = 0, γd,Lr = −1, and γp,Lr = +1. However, this more generalized representation increases the number of occupancy configuration combinations and allows for consideration of more complex superposition mixing of these available configurations. Combinatorically, the configurations available for Lr include
7 s 2 6 d 1 7 p 0 ,   7 s 2 6 d 0 7 p 1 ,   7 s 1 6 d 2 7 p 0 ,   7 s 1 6 d 1 7 p 1 ,   7 s 1 6 d 0 7 p 2 ,   7 s 0 6 d 3 7 p 0 ,   7 s 0 6 d 2 7 p 1 ,   7 s 0 6 d 1 7 p 2 ,   7 s 0 6 d 0 7 p 3 .
Table III.

Generalized non-integer occupancy ground state representations for period 7 d-block elements assuming s-, d-, and p-orbital participation in superposition mixing.

Element Configuration Constraints
γ s γ d γ p
Lr  7 s 2 + γ s , L r 6 d 1 + γ d , L r 7 p 0 + γ p , L r  −2 ≤ γ s , L r ≤ 0  −1 ≤ γ d , L r ≤ 2  0 ≤ γ p , L r ≤ 3 
Rf  7 s 2 + γ s , R f 6 d 2 + γ d , R f 7 p 0 + γ p , R f  −2 ≤ γ s , R f ≤ 0  −2 ≤ γ d , R f ≤ 2  0 ≤ γ p , R f ≤ 4 
Db  7 s 2 + γ s , D b 6 d 3 + γ d , D b 7 p 0 + γ p , D b  −2 ≤ γ s , D b ≤ 0  −3 ≤ γ d , D b ≤ 2  0 ≤ γ p , D b ≤ 5 
Sg  7 s 2 + γ s , S g 6 d 4 + γ d , S g 7 p 0 + γ p , S g  −2 ≤ γ s , S g ≤ 0  −4 ≤ γ d , S g ≤ 2  0 ≤ γ p , S g ≤ 6 
Bh  7 s 2 + γ s , B h 6 d 5 + γ d , B h 7 p 0 + γ p , B h  −2 ≤ γ s , B h ≤ 0  −5 ≤ γ d , B h ≤ 2  0 ≤ γ p , B h ≤ 6 
Hs  7 s 2 + γ s , H s 6 d 6 + γ d , H s 7 p 0 + γ p , H s  −2 ≤ γ s , H s ≤ 0  −6 ≤ γ d , H s ≤ 2  0 ≤ γ p , H s ≤ 6 
Mt  7 s 2 + γ s , M t 6 d 7 + γ d , M t 7 p 0 + γ p , M t  −2 ≤ γ s , M t ≤ 0  −6 ≤ γ d , M t ≤ 2  0 ≤ γ p , M t ≤ 6 
Ds  7 s 2 + γ s , D s 6 d 8 + γ d , D s 7 p 0 + γ p , D s  −2 ≤ γ s , D s ≤ 0  −6 ≤ γ d , D s ≤ 2  0 ≤ γ p , D s ≤ 6 
Rg  7 s 2 + γ s , R g 6 d 9 + γ d , R g 7 p 0 + γ p , R g  −2 ≤ γ s , R g ≤ 0  −6 ≤ γ d , R g ≤ 1  0 ≤ γ p , R g ≤ 6 
Cn  7 s 2 + γ s , C n 6 d 10 + γ d , C n 7 p 0 + γ p , C n  −2 ≤ γ s , C n ≤ 0  −6 ≤ γ d , C n ≤ 0  0 ≤ γ p , C n ≤ 6 
Element Configuration Constraints
γ s γ d γ p
Lr  7 s 2 + γ s , L r 6 d 1 + γ d , L r 7 p 0 + γ p , L r  −2 ≤ γ s , L r ≤ 0  −1 ≤ γ d , L r ≤ 2  0 ≤ γ p , L r ≤ 3 
Rf  7 s 2 + γ s , R f 6 d 2 + γ d , R f 7 p 0 + γ p , R f  −2 ≤ γ s , R f ≤ 0  −2 ≤ γ d , R f ≤ 2  0 ≤ γ p , R f ≤ 4 
Db  7 s 2 + γ s , D b 6 d 3 + γ d , D b 7 p 0 + γ p , D b  −2 ≤ γ s , D b ≤ 0  −3 ≤ γ d , D b ≤ 2  0 ≤ γ p , D b ≤ 5 
Sg  7 s 2 + γ s , S g 6 d 4 + γ d , S g 7 p 0 + γ p , S g  −2 ≤ γ s , S g ≤ 0  −4 ≤ γ d , S g ≤ 2  0 ≤ γ p , S g ≤ 6 
Bh  7 s 2 + γ s , B h 6 d 5 + γ d , B h 7 p 0 + γ p , B h  −2 ≤ γ s , B h ≤ 0  −5 ≤ γ d , B h ≤ 2  0 ≤ γ p , B h ≤ 6 
Hs  7 s 2 + γ s , H s 6 d 6 + γ d , H s 7 p 0 + γ p , H s  −2 ≤ γ s , H s ≤ 0  −6 ≤ γ d , H s ≤ 2  0 ≤ γ p , H s ≤ 6 
Mt  7 s 2 + γ s , M t 6 d 7 + γ d , M t 7 p 0 + γ p , M t  −2 ≤ γ s , M t ≤ 0  −6 ≤ γ d , M t ≤ 2  0 ≤ γ p , M t ≤ 6 
Ds  7 s 2 + γ s , D s 6 d 8 + γ d , D s 7 p 0 + γ p , D s  −2 ≤ γ s , D s ≤ 0  −6 ≤ γ d , D s ≤ 2  0 ≤ γ p , D s ≤ 6 
Rg  7 s 2 + γ s , R g 6 d 9 + γ d , R g 7 p 0 + γ p , R g  −2 ≤ γ s , R g ≤ 0  −6 ≤ γ d , R g ≤ 1  0 ≤ γ p , R g ≤ 6 
Cn  7 s 2 + γ s , C n 6 d 10 + γ d , C n 7 p 0 + γ p , C n  −2 ≤ γ s , C n ≤ 0  −6 ≤ γ d , C n ≤ 0  0 ≤ γ p , C n ≤ 6 
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Although the energies of many of these configurations are surely much higher than others and, therefore, probably would not participate significantly in superposition mixing, it does demonstrate how the inclusion of more combinations of orbital configurations increases the potential for near-energy-degenerate configurations that could participate in superposition mixing. In this particular case, it would likely be reasonable to (at least) consider the first two states (7s26d17p0 and 7s26d07p1) because it has been shown that the energy difference between these two configurations is small.18 Thus, a superposition between the two states might correspond to a configuration of
(4)
with γs,Lr = 0, −0.5 < γd,Lr < 0, and 0 < γp,Lr < 0.5 to represent the slight energetic favorability toward the s2p1 configuration.
Similar to Lr, rutherfordium (Rf) has been shown to have two near-energy-degenerate configurations (7s26d2 and 7s26d17p1) with only 0.3 to 0.5 eV difference in energy.19–21 Unlike Lr, Rf contains 12 possible occupancy configurations, given the extra electron in the neutral atom,
7 s 2 6 d 2 7 p 0 ,   7 s 2 6 d 1 7 p 1 ,   7 s 2 6 d 0 7 p 2 ,   7 s 1 6 d 3 7 p 0 ,   7 s 1 6 d 2 7 p 1 ,   7 s 1 6 d 1 7 p 2 ,   7 s 1 6 d 0 7 p 3 ,   7 s 0 6 d 4 7 p 0 ,   7 s 0 6 d 3 7 p 1 ,   7 s 0 6 d 2 7 p 2 ,   7 s 0 6 d 1 7 p 3 ,   7 s 0 6 d 0 7 p 4 .
By only considering the two lowest energy state configurations for superposition mixing, we can express the non-integer ground state for Rf as
(5)
with γs,Rf = 0, −1 < γd,Rf < −0.5, and 0.5 < γp,Rf < 1 representing the slightly lower energy of the 7s26d2 established ground state.
The number of occupancy combinations for the remainder of the period 7 d-block elements are given in Table IV. Those for dubnium (Db) and seaborgium (Sg) continue to increase consecutively by three with 15 occupancy configurations for Db,
7 s 2 6 d 3 7 p 0 ,   7 s 2 6 d 2 7 p 1 ,   7 s 2 6 d 1 7 p 2 ,   7 s 2 6 d 0 7 p 3 ,   7 s 1 6 d 4 7 p 0 ,   7 s 1 6 d 3 7 p 1 ,   7 s 1 6 d 2 7 p 2 ,   7 s 1 6 d 1 7 p 3 ,   7 s 1 6 d 0 7 p 4 ,   7 s 0 6 d 5 7 p 0 ,   7 s 0 6 d 4 7 p 1 ,   7 s 0 6 d 3 7 p 2 ,   7 s 0 6 d 2 7 p 3 ,   7 s 0 6 d 1 7 p 4 ,   7 s 0 6 d 0 7 p 5 ,
and 18 occupancy configurations for Sg
7 s 2 6 d 4 7 p 0 ,   7 s 2 6 d 3 7 p 1 ,   7 s 2 6 d 2 7 p 2 ,   7 s 2 6 d 1 7 p 3 ,   7 s 2 6 d 0 7 p 4 ,   7 s 1 6 d 5 7 p 0 ,   7 s 1 6 d 4 7 p 1 ,   7 s 1 6 d 3 7 p 2 ,   7 s 1 6 d 2 7 p 3 , 7 s 1 6 d 1 7 p 4 ,   7 s 1 6 d 0 7 p 5 ,   7 s 0 6 d 6 7 p 0 ,   7 s 0 6 d 5 7 p 1 ,   7 s 0 6 d 4 7 p 2 ,   7 s 0 6 d 3 7 p 3 ,   7 s 0 6 d 2 7 p 4 ,   7 s 0 6 d 1 7 p 5 ,   7 s 0 6 d 0 7 p 6 .
Table IV.

Generalized number of occupancy combinations and consecutive changes in these numbers for each period 7 d-block element assuming s-, d-, and p-orbital participation in superposition mixing.

Elements Combinations Increase
Lr  9  ⋯ 
Rf  12  3 
Db  15  3 
Sg  18  3 
Bh  20  2 
Hs  21  1 
Mt  21  0 
Ds  21  0 
Rg  20  −1 
Cn  18  −2 
Elements Combinations Increase
Lr  9  ⋯ 
Rf  12  3 
Db  15  3 
Sg  18  3 
Bh  20  2 
Hs  21  1 
Mt  21  0 
Ds  21  0 
Rg  20  −1 
Cn  18  −2 
View Large
For bohrium (Bh), the increase in occupancy combinations is suppressed by the six-electron occupancy of the p-orbital, resulting in 20 combinations,
7 s 2 6 d 5 7 p 0 ,   7 s 2 6 d 4 7 p 1 ,   7 s 2 6 d 3 7 p 2 ,   7 s 2 6 d 2 7 p 3 ,   7 s 2 6 d 1 7 p 4 ,   7 s 2 6 d 0 7 p 5 ,   7 s 1 6 d 6 7 p 0 ,   7 s 1 6 d 5 7 p 1 ,   7 s 1 6 d 4 7 p 2 , 7 s 1 6 d 3 7 p 3 ,   7 s 1 6 d 2 7 p 4 ,   7 s 1 6 d 1 7 p 5 ,   7 s 1 6 d 0 7 p 6 ,   7 s 0 6 d 7 7 p 0 ,   7 s 0 6 d 6 7 p 1 ,   7 s 0 6 d 5 7 p 2 ,   7 s 0 6 d 4 7 p 3 ,   7 s 0 6 d 3 7 p 4 , 7 s 0 6 d 2 7 p 5 ,   7 s 0 6 d 1 7 p 6 .
For hassium (Hs), meitnerium (Mt), and darmstadtium (Ds), a maximum of 21 occupancy combinations are reached. For Hs,
7 s 2 6 d 6 7 p 0 ,   7 s 2 6 d 5 7 p 1 ,   7 s 2 6 d 4 7 p 2 ,   7 s 2 6 d 3 7 p 3 ,   7 s 2 6 d 2 7 p 4 ,   7 s 2 6 d 1 7 p 5 ,   7 s 2 6 d 0 7 p 6 ,   7 s 1 6 d 7 7 p 0 ,   7 s 1 6 d 6 7 p 1 , 7 s 1 6 d 5 7 p 2 ,   7 s 1 6 d 4 7 p 3 ,   7 s 1 6 d 3 7 p 4 ,   7 s 1 6 d 2 7 p 5 ,   7 s 1 6 d 1 7 p 6 ,   7 s 0 6 d 8 7 p 0 ,   7 s 0 6 d 7 7 p 1 ,   7 s 0 6 d 6 7 p 2 ,   7 s 0 6 d 5 7 p 3 , 7 s 0 6 d 4 7 p 4 ,   7 s 0 6 d 3 7 p 5 ,   7 s 0 6 d 2 7 p 6 .
For Mt,
7 s 2 6 d 7 7 p 0 ,   7 s 2 6 d 6 7 p 1 ,   7 s 2 6 d 5 7 p 2 ,   7 s 2 6 d 4 7 p 3 ,   7 s 2 6 d 3 7 p 4 ,   7 s 2 6 d 2 7 p 5 ,   7 s 2 6 d 1 7 p 6 ,   7 s 1 6 d 8 7 p 0 ,   7 s 1 6 d 7 7 p 1 ,   7 s 1 6 d 6 7 p 2 ,   7 s 1 6 d 5 7 p 3 ,   7 s 1 6 d 4 7 p 4 ,   7 s 1 6 d 3 7 p 5 ,   7 s 1 6 d 2 7 p 6 ,   7 s 0 6 d 9 7 p 0 ,   7 s 0 6 d 8 7 p 1 ,   7 s 0 6 d 7 7 p 2 ,   7 s 0 6 d 6 7 p 3 ,   7 s 0 6 d 5 7 p 4 ,   7 s 0 6 d 4 7 p 5 ,   7 s 0 6 d 3 7 p 6 .
For Ds,
7 s 2 6 d 8 7 p 0 ,   7 s 2 6 d 7 7 p 1 ,   7 s 2 6 d 6 7 p 2 ,   7 s 2 6 d 5 7 p 3 ,   7 s 2 6 d 4 7 p 4 ,   7 s 2 6 d 3 7 p 5 ,   7 s 2 6 d 2 7 p 6 ,   7 s 1 6 d 9 7 p 0 ,   7 s 1 6 d 8 7 p 1 , 7 s 1 6 d 7 7 p 2 ,   7 s 1 6 d 6 7 p 3 ,   7 s 1 6 d 5 7 p 4 ,   7 s 1 6 d 4 7 p 5 ,   7 s 1 6 d 3 7 p 6 ,   7 s 0 6 d 10 7 p 0 ,   7 s 0 6 d 9 7 p 1 ,   7 s 0 6 d 8 7 p 2 ,   7 s 0 6 d 7 7 p 3 , 7 s 0 6 d 6 7 p 4 ,   7 s 0 6 d 5 7 p 5 ,   7 s 0 6 d 4 7 p 6 .
For roentgenium (Rg) and copernicium (Cn), the number of occupancy combinations begins to be suppressed by the filling of the d-orbital with Rg having 20 combinations,
7 s 2 6 d 9 7 p 0 , 7 s 2 6 d 8 7 p 1 ,   7 s 2 6 d 7 7 p 2 ,   7 s 2 6 d 6 7 p 3 ,   7 s 2 6 d 5 7 p 4 ,   7 s 2 6 d 4 7 p 5 ,   7 s 2 6 d 3 7 p 6 ,   7 s 1 6 d 10 7 p 0 ,   7 s 1 6 d 9 7 p 1 , 7 s 1 6 d 8 7 p 2 ,   7 s 1 6 d 7 7 p 3 ,   7 s 1 6 d 6 7 p 4 ,   7 s 1 6 d 5 7 p 5 ,   7 s 1 6 d 4 7 p 6 ,   7 s 0 6 d 10 7 p 1 ,   7 s 0 6 d 9 7 p 2 ,   7 s 0 6 d 8 7 p 3 ,   7 s 0 6 d 7 7 p 4 , 7 s 0 6 d 6 7 p 5 ,   7 s 0 6 d 5 7 p 6 ,
and Cn 18 combinations
7 s 2 6 d 10 7 p 0 ,   7 s 2 6 d 9 7 p 1 ,   7 s 2 6 d 8 7 p 2 ,   7 s 2 6 d 7 7 p 3 ,   7 s 2 6 d 6 7 p 4 ,   7 s 2 6 d 5 7 p 5 ,   7 s 2 6 d 4 7 p 6 ,   7 s 1 6 d 10 7 p 1 ,   7 s 1 6 d 9 7 p 2 , 7 s 1 6 d 8 7 p 3 ,   7 s 1 6 d 7 7 p 4 ,   7 s 1 6 d 6 7 p 5 ,   7 s 1 6 d 5 7 p 6 ,   7 s 0 6 d 10 7 p 2 ,   7 s 0 6 d 9 7 p 3 ,   7 s 0 6 d 8 7 p 4 ,   7 s 0 6 d 7 7 p 5 ,   7 s 0 6 d 6 7 p 6 .
The number of configurations, thus, derived for each of the seventh period d-block transition metals would be the same for the fourth, fifth, and sixth period d-block transition metals under the assumption that superposition mixing within those analogous series also involves the ns, (n − 1)d, and np orbitals. However, it is assumed that the potential for superposition mixing is greater for later period elements than early period elements due to lower energy separations of orbitals and greater radial extent. Although the combinatorics of this representation is illustrative of the greater potential for superposition mixing, it is not the intent to suggest that all these combinations would participate significantly. By strict definition of the ground state, all but one combination in most, if not all, cases would be considered excited state configurations. If a quantum superposition of near-energy-degenerate configurations does occur, it would need to be determined which configurations participate, and where to draw the line between participating and nonparticipating configurations. This will be discussed later, after addressing the f-block lanthanides and actinides.

For the lanthanide and actinide elements, the ground states given as simple occupancy configurations from casual online searches cease to be reliable or even meaningful, given their multiconfigurational nature. The lanthanide and actinide elements are historically considered to possess ground states occupancies that can be described multiconfigurationally as 4fn(6s5d6p)m and 5fn(7s6d7p)m, respectively, where n and m are electron occupancies.22–55 Consistent with this description, it will be assumed in this section that all four orbitals have the potential to contribute to near-energy-degenerate configurations, resulting in the generalized representation of electronic configurations shown in Table V. Constraints for the associated γ values are again determined by the number of electrons available in a neutral atom, the respective orbital occupancies, and the conservation of electrons requirement for the sum of γ values (γs + γf + γd + γp = 0). For all zero γ values, the predicted aufbau configuration would be returned.

Table V.

Generalized non-integer occupancy ground state representations for f-block elements assuming s-, f-, d-, and p-orbital participation in superposition mixing.

Element Configuration Constraints
γ s γ f γ d γ p
La  6 s 2 + γ s , L a 4 f 1 + γ f , L a 5 d 0 + γ d , L a 6 p 0 + γ p , L a  −2 ≤ γ s , L a ≤ 0  −1 ≤ γ f , L a ≤ 2  0 ≤ γ d , L a ≤ 3  0 ≤ γ p , L a ≤ 3 
Ce  6 s 2 + γ s , C e 4 f 2 + γ f , C e 5 d 0 + γ d , C e 6 p 0 + γ p , C e  −2 ≤ γ s , C e ≤ 0  −2 ≤ γ f , C e ≤ 2  0 ≤ γ d , C e ≤ 4  0 ≤ γ p , C e ≤ 4 
Pr  6 s 2 + γ s , P r 4 f 3 + γ f , P r 5 d 0 + γ d , P r 6 p 0 + γ p , P r  −2 ≤ γ s , P r ≤ 0  −3 ≤ γ f , P r ≤ 2  0 ≤ γ d , P r ≤ 5  0 ≤ γ p , P r ≤ 5 
Nd  6 s 2 + γ s , N d 4 f 4 + γ f , N d 5 d 0 + γ d , N d 6 p 0 + γ p , N d  −2 ≤ γ s , N d ≤ 0  −4 ≤ γ f , N d ≤ 2  0 ≤ γ d , N d ≤ 6  0 ≤ γ p , N d ≤ 6 
Pm  6 s 2 + γ s , P m 4 f 5 + γ f , P m 5 d 0 + γ d , P m 6 p 0 + γ p , P m  −2 ≤ γ s , P m ≤ 0  −5 ≤ γ f , P m ≤ 2  0 ≤ γ d , P m ≤ 7  0 ≤ γ p , P m ≤ 6 
Sm  6 s 2 + γ s , S m 4 f 6 + γ f , S m 5 d 0 + γ d , S m 6 p 0 + γ p , S m  −2 ≤ γ s , S m ≤ 0  −6 ≤ γ f , S m ≤ 2  0 ≤ γ d , S m ≤ 8  0 ≤ γ p , S m ≤ 6 
Eu  6 s 2 + γ s , E u 4 f 7 + γ f , E u 5 d 0 + γ d , E u 6 p 0 + γ p , E u  −2 ≤ γ s , E u ≤ 0  −7 ≤ γ f , E u ≤ 2  0 ≤ γ d , E u ≤ 9  0 ≤ γ p , E u ≤ 6 
Gd  6 s 2 + γ s , G d 4 f 8 + γ f , G d 5 d 0 + γ d , G d 6 p 0 + γ p , G d  −2 ≤ γ s , G d ≤ 0  −8 ≤ γ f , G d ≤ 2  0 ≤ γ d , G d ≤ 10  0 ≤ γ p , G d ≤ 6 
Tb  6 s 2 + γ s , T b 4 f 9 + γ f , T b 5 d 0 + γ d , T b 6 p 0 + γ p , T b  −2 ≤ γ s , T b ≤ 0  −9 ≤ γ f , T b ≤ 2  0 ≤ γ d , T b ≤ 10  0 ≤ γ p , T b ≤ 6 
Dy  6 s 2 + γ s , D y 4 f 10 + γ f , D y 5 d 0 + γ d , D y 6 p 0 + γ p , D y  −2 ≤ γ s , D y ≤ 0  −10 ≤ γ f , D y ≤ 2  0 ≤ γ d , D y ≤ 10  0 ≤ γ p , D y ≤ 6 
Ho  6 s 2 + γ s , H o 4 f 11 + γ f , H o 5 d 0 + γ d , H o 6 p 0 + γ p , H o  −2 ≤ γ s , H o ≤ 0  −11 ≤ γ f , H o ≤ 2  0 ≤ γ d , H o ≤ 10  0 ≤ γ p , H o ≤ 6 
Er  6 s 2 + γ s , E r 4 f 12 + γ f , E r 5 d 0 + γ d , E r 6 p 0 + γ p , E r  −2 ≤ γ s , E r ≤ 0  −12 ≤ γ f , E r ≤ 2  0 ≤ γ d , E r ≤ 10  0 ≤ γ p , E r ≤ 6 
Tm  6 s 2 + γ s , T m 4 f 13 + γ f , T m 5 d 0 + γ d , T m 6 p 0 + γ p , T m  −2 ≤ γ s , T m ≤ 0  −13 ≤ γ f , T m ≤ 2  0 ≤ γ d , T m ≤ 10  0 ≤ γ p , T m ≤ 6 
Yb  6 s 2 + γ s , Y b 4 f 14 + γ f , Y b 5 d 0 + γ d , Y b 6 p 0 + γ p , Y b  −2 ≤ γ s , Y b ≤ 0  −14 ≤ γ f , Y b ≤ 2  0 ≤ γ d , Y b ≤ 10  0 ≤ γ p , Y b ≤ 6 
Ac  7 s 2 + γ s , A c 5 f 1 + γ f , A c 6 d 0 + γ d , A c 7 p 0 + γ p , A c  −2 ≤ γ s , A c ≤ 0  −1 ≤ γ f , A c ≤ 2  0 ≤ γ d , A c ≤ 3  0 ≤ γ p , A c ≤ 3 
Th  7 s 2 + γ s , T h 5 f 2 + γ f , T h 6 d 0 + γ d , T h 7 p 0 + γ p , T h  −2 ≤ γ s , T h ≤ 0  −2 ≤ γ f , T h ≤ 2  0 ≤ γ d , T h ≤ 4  0 ≤ γ p , T h ≤ 4 
Pa  7 s 2 + γ s , P a 5 f 3 + γ f , P a 6 d 0 + γ d , P a 7 p 0 + γ p , P a  −2 ≤ γ s , P a ≤ 0  −3 ≤ γ f , P a ≤ 2  0 ≤ γ d , P a ≤ 5  0 ≤ γ p , P a ≤ 5 
U  7 s 2 + γ s , U 5 f 4 + γ f , U 6 d 0 + γ d , U 7 p 0 + γ p , U  −2 ≤ γ s , U ≤ 0  −4 ≤ γ f , U ≤ 2  0 ≤ γ d , U ≤ 6  0 ≤ γ p , U ≤ 6 
Np  7 s 2 + γ s , N p 5 f 5 + γ f , N p 6 d 0 + γ d , N p 7 p 0 + γ p , N p  −2 ≤ γ s , N p ≤ 0  −5 ≤ γ f , N p ≤ 2  0 ≤ γ d , N p ≤ 7  0 ≤ γ p , N p ≤ 6 
Pu  7 s 2 + γ s , P u 5 f 6 + γ f , P u 6 d 0 + γ d , P u 7 p 0 + γ p , P u  −2 ≤ γ s , P u ≤ 0  −6 ≤ γ f , P u ≤ 2  0 ≤ γ d , P u ≤ 8  0 ≤ γ p , P u ≤ 6 
Am  7 s 2 + γ s , A m 5 f 7 + γ f , A m 6 d 0 + γ d , A m 7 p 0 + γ p , A m  −2 ≤ γ s , A m ≤ 0  −7 ≤ γ f , A m ≤ 2  0 ≤ γ d , A m ≤ 9  0 ≤ γ p , A m ≤ 6 
Cm  7 s 2 + γ s , C m 5 f 8 + γ f , C m 6 d 0 + γ d , C m 7 p 0 + γ p , C m  −2 ≤ γ s , C m ≤ 0  −8 ≤ γ f , C m ≤ 2  0 ≤ γ d , C m ≤ 10  0 ≤ γ p , C m ≤ 6 
Bk  7 s 2 + γ s , B k 5 f 9 + γ f , B k 6 d 0 + γ d , B k 7 p 0 + γ p , B k  −2 ≤ γ s , B k ≤ 0  −9 ≤ γ f , B k ≤ 2  0 ≤ γ d , B k ≤ 10  0 ≤ γ p , B k ≤ 6 
Cf  7 s 2 + γ s , C f 5 f 10 + γ f , C f 6 d 0 + γ d , C f 7 p 0 + γ p , C f  −2 ≤ γ s , C f ≤ 0  −10 ≤ γ f , C f ≤ 2  0 ≤ γ d , C f ≤ 10  0 ≤ γ p , C f ≤ 6 
Es  7 s 2 + γ s , E s 5 f 11 + γ f , E s 6 d 0 + γ d , E s 7 p 0 + γ p , E s  −2 ≤ γ s , E s ≤ 0  −11 ≤ γ f , E s ≤ 2  0 ≤ γ d , E s ≤ 10  0 ≤ γ p , E s ≤ 6 
Fm  7 s 2 + γ s , F m 5 f 12 + γ f , F m 6 d 0 + γ d , F m 7 p 0 + γ p , F m  −2 ≤ γ s , F m ≤ 0  −12 ≤ γ f , F m ≤ 2  0 ≤ γ d , F m ≤ 10  0 ≤ γ p , F m ≤ 6 
Md  7 s 2 + γ s , M d 5 f 13 + γ f , M d 6 d 0 + γ d , M d 7 p 0 + γ p , M d  −2 ≤ γ s , M d ≤ 0  −13 ≤ γ f , M d ≤ 2  0 ≤ γ d , M d ≤ 10  0 ≤ γ p , M d ≤ 6 
No  7 s 2 + γ s , N o 5 f 14 + γ f , N o 6 d 0 + γ d , N o 7 p 0 + γ p , N o  −2 ≤ γ s , N o ≤ 0  −14 ≤ γ f , N o ≤ 2  0 ≤ γ d , N o ≤ 10  0 ≤ γ p , N o ≤ 6 
Element Configuration Constraints
γ s γ f γ d γ p
La  6 s 2 + γ s , L a 4 f 1 + γ f , L a 5 d 0 + γ d , L a 6 p 0 + γ p , L a  −2 ≤ γ s , L a ≤ 0  −1 ≤ γ f , L a ≤ 2  0 ≤ γ d , L a ≤ 3  0 ≤ γ p , L a ≤ 3 
Ce  6 s 2 + γ s , C e 4 f 2 + γ f , C e 5 d 0 + γ d , C e 6 p 0 + γ p , C e  −2 ≤ γ s , C e ≤ 0  −2 ≤ γ f , C e ≤ 2  0 ≤ γ d , C e ≤ 4  0 ≤ γ p , C e ≤ 4 
Pr  6 s 2 + γ s , P r 4 f 3 + γ f , P r 5 d 0 + γ d , P r 6 p 0 + γ p , P r  −2 ≤ γ s , P r ≤ 0  −3 ≤ γ f , P r ≤ 2  0 ≤ γ d , P r ≤ 5  0 ≤ γ p , P r ≤ 5 
Nd  6 s 2 + γ s , N d 4 f 4 + γ f , N d 5 d 0 + γ d , N d 6 p 0 + γ p , N d  −2 ≤ γ s , N d ≤ 0  −4 ≤ γ f , N d ≤ 2  0 ≤ γ d , N d ≤ 6  0 ≤ γ p , N d ≤ 6 
Pm  6 s 2 + γ s , P m 4 f 5 + γ f , P m 5 d 0 + γ d , P m 6 p 0 + γ p , P m  −2 ≤ γ s , P m ≤ 0  −5 ≤ γ f , P m ≤ 2  0 ≤ γ d , P m ≤ 7  0 ≤ γ p , P m ≤ 6 
Sm  6 s 2 + γ s , S m 4 f 6 + γ f , S m 5 d 0 + γ d , S m 6 p 0 + γ p , S m  −2 ≤ γ s , S m ≤ 0  −6 ≤ γ f , S m ≤ 2  0 ≤ γ d , S m ≤ 8  0 ≤ γ p , S m ≤ 6 
Eu  6 s 2 + γ s , E u 4 f 7 + γ f , E u 5 d 0 + γ d , E u 6 p 0 + γ p , E u  −2 ≤ γ s , E u ≤ 0  −7 ≤ γ f , E u ≤ 2  0 ≤ γ d , E u ≤ 9  0 ≤ γ p , E u ≤ 6 
Gd  6 s 2 + γ s , G d 4 f 8 + γ f , G d 5 d 0 + γ d , G d 6 p 0 + γ p , G d  −2 ≤ γ s , G d ≤ 0  −8 ≤ γ f , G d ≤ 2  0 ≤ γ d , G d ≤ 10  0 ≤ γ p , G d ≤ 6 
Tb  6 s 2 + γ s , T b 4 f 9 + γ f , T b 5 d 0 + γ d , T b 6 p 0 + γ p , T b  −2 ≤ γ s , T b ≤ 0  −9 ≤ γ f , T b ≤ 2  0 ≤ γ d , T b ≤ 10  0 ≤ γ p , T b ≤ 6 
Dy  6 s 2 + γ s , D y 4 f 10 + γ f , D y 5 d 0 + γ d , D y 6 p 0 + γ p , D y  −2 ≤ γ s , D y ≤ 0  −10 ≤ γ f , D y ≤ 2  0 ≤ γ d , D y ≤ 10  0 ≤ γ p , D y ≤ 6 
Ho  6 s 2 + γ s , H o 4 f 11 + γ f , H o 5 d 0 + γ d , H o 6 p 0 + γ p , H o  −2 ≤ γ s , H o ≤ 0  −11 ≤ γ f , H o ≤ 2  0 ≤ γ d , H o ≤ 10  0 ≤ γ p , H o ≤ 6 
Er  6 s 2 + γ s , E r 4 f 12 + γ f , E r 5 d 0 + γ d , E r 6 p 0 + γ p , E r  −2 ≤ γ s , E r ≤ 0  −12 ≤ γ f , E r ≤ 2  0 ≤ γ d , E r ≤ 10  0 ≤ γ p , E r ≤ 6 
Tm  6 s 2 + γ s , T m 4 f 13 + γ f , T m 5 d 0 + γ d , T m 6 p 0 + γ p , T m  −2 ≤ γ s , T m ≤ 0  −13 ≤ γ f , T m ≤ 2  0 ≤ γ d , T m ≤ 10  0 ≤ γ p , T m ≤ 6 
Yb  6 s 2 + γ s , Y b 4 f 14 + γ f , Y b 5 d 0 + γ d , Y b 6 p 0 + γ p , Y b  −2 ≤ γ s , Y b ≤ 0  −14 ≤ γ f , Y b ≤ 2  0 ≤ γ d , Y b ≤ 10  0 ≤ γ p , Y b ≤ 6 
Ac  7 s 2 + γ s , A c 5 f 1 + γ f , A c 6 d 0 + γ d , A c 7 p 0 + γ p , A c  −2 ≤ γ s , A c ≤ 0  −1 ≤ γ f , A c ≤ 2  0 ≤ γ d , A c ≤ 3  0 ≤ γ p , A c ≤ 3 
Th  7 s 2 + γ s , T h 5 f 2 + γ f , T h 6 d 0 + γ d , T h 7 p 0 + γ p , T h  −2 ≤ γ s , T h ≤ 0  −2 ≤ γ f , T h ≤ 2  0 ≤ γ d , T h ≤ 4  0 ≤ γ p , T h ≤ 4 
Pa  7 s 2 + γ s , P a 5 f 3 + γ f , P a 6 d 0 + γ d , P a 7 p 0 + γ p , P a  −2 ≤ γ s , P a ≤ 0  −3 ≤ γ f , P a ≤ 2  0 ≤ γ d , P a ≤ 5  0 ≤ γ p , P a ≤ 5 
U  7 s 2 + γ s , U 5 f 4 + γ f , U 6 d 0 + γ d , U 7 p 0 + γ p , U  −2 ≤ γ s , U ≤ 0  −4 ≤ γ f , U ≤ 2  0 ≤ γ d , U ≤ 6  0 ≤ γ p , U ≤ 6 
Np  7 s 2 + γ s , N p 5 f 5 + γ f , N p 6 d 0 + γ d , N p 7 p 0 + γ p , N p  −2 ≤ γ s , N p ≤ 0  −5 ≤ γ f , N p ≤ 2  0 ≤ γ d , N p ≤ 7  0 ≤ γ p , N p ≤ 6 
Pu  7 s 2 + γ s , P u 5 f 6 + γ f , P u 6 d 0 + γ d , P u 7 p 0 + γ p , P u  −2 ≤ γ s , P u ≤ 0  −6 ≤ γ f , P u ≤ 2  0 ≤ γ d , P u ≤ 8  0 ≤ γ p , P u ≤ 6 
Am  7 s 2 + γ s , A m 5 f 7 + γ f , A m 6 d 0 + γ d , A m 7 p 0 + γ p , A m  −2 ≤ γ s , A m ≤ 0  −7 ≤ γ f , A m ≤ 2  0 ≤ γ d , A m ≤ 9  0 ≤ γ p , A m ≤ 6 
Cm  7 s 2 + γ s , C m 5 f 8 + γ f , C m 6 d 0 + γ d , C m 7 p 0 + γ p , C m  −2 ≤ γ s , C m ≤ 0  −8 ≤ γ f , C m ≤ 2  0 ≤ γ d , C m ≤ 10  0 ≤ γ p , C m ≤ 6 
Bk  7 s 2 + γ s , B k 5 f 9 + γ f , B k 6 d 0 + γ d , B k 7 p 0 + γ p , B k  −2 ≤ γ s , B k ≤ 0  −9 ≤ γ f , B k ≤ 2  0 ≤ γ d , B k ≤ 10  0 ≤ γ p , B k ≤ 6 
Cf  7 s 2 + γ s , C f 5 f 10 + γ f , C f 6 d 0 + γ d , C f 7 p 0 + γ p , C f  −2 ≤ γ s , C f ≤ 0  −10 ≤ γ f , C f ≤ 2  0 ≤ γ d , C f ≤ 10  0 ≤ γ p , C f ≤ 6 
Es  7 s 2 + γ s , E s 5 f 11 + γ f , E s 6 d 0 + γ d , E s 7 p 0 + γ p , E s  −2 ≤ γ s , E s ≤ 0  −11 ≤ γ f , E s ≤ 2  0 ≤ γ d , E s ≤ 10  0 ≤ γ p , E s ≤ 6 
Fm  7 s 2 + γ s , F m 5 f 12 + γ f , F m 6 d 0 + γ d , F m 7 p 0 + γ p , F m  −2 ≤ γ s , F m ≤ 0  −12 ≤ γ f , F m ≤ 2  0 ≤ γ d , F m ≤ 10  0 ≤ γ p , F m ≤ 6 
Md  7 s 2 + γ s , M d 5 f 13 + γ f , M d 6 d 0 + γ d , M d 7 p 0 + γ p , M d  −2 ≤ γ s , M d ≤ 0  −13 ≤ γ f , M d ≤ 2  0 ≤ γ d , M d ≤ 10  0 ≤ γ p , M d ≤ 6 
No  7 s 2 + γ s , N o 5 f 14 + γ f , N o 6 d 0 + γ d , N o 7 p 0 + γ p , N o  −2 ≤ γ s , N o ≤ 0  −14 ≤ γ f , N o ≤ 2  0 ≤ γ d , N o ≤ 10  0 ≤ γ p , N o ≤ 6 
View Large

In the case of the d-block transition metals, the assumption that only s- and d-orbital superposition mixing occurs limited the occupancy combinations to a maximum of three possibilities. By considering the possibility of p-orbital participation, the possible combinations increased to a maximum of 21. For the f-block lanthanides and actinides, the number of occupancy combinations is shown in Table VI, ranging from 19 combinations for lanthanum (La) and actinium (Ac), to as many as 223 combinations for ytterbium (Yb) and nobelium (No). Table VI also shows how the number of combinations increases with consecutive atomic number to the end of the lanthanide/actinide series, unlike the d-block where the number of combinations dropped off. Listings of the specific occupancy combinations for each actinide are provided as supplementary material. Comparing the maximum number of combinations for the f-block (223) to the maximum number for the d-block (21) illustrates a greater likelihood of over an order of magnitude for superposition mixing in the f-block.

Table VI.

Generalized number of occupancy combinations, consecutive changes in these numbers, and the total number of possible permutations of electrons for each f-block element assuming s-, f-, d-, and p-orbital participation in superposition mixing.

Elements Occupancy combinations Increase Permutations
La, Ac  19  ⋯  4 960 
Ce, Th  31  12  35 960 
Pr, Pa  46  15  201 376 
Nd, U  64  18  906 192 
Pm, Np  84  20  3 365 856 
Sm, Pu  105  21  10 518 300 
Eu, Am  126  21  28 048 800 
Gd, Cm  147  21  64 512 240 
Tb, Bk  167  20  129 024 480 
Dy, Cf  185  18  225 792 840 
Ho, Es  200  15  347 373 600 
Er, Fm  212  12  471 435 600 
Tm, Md  220  8  565 722 720 
Yb, No  223  3  601 080 390 
Elements Occupancy combinations Increase Permutations
La, Ac  19  ⋯  4 960 
Ce, Th  31  12  35 960 
Pr, Pa  46  15  201 376 
Nd, U  64  18  906 192 
Pm, Np  84  20  3 365 856 
Sm, Pu  105  21  10 518 300 
Eu, Am  126  21  28 048 800 
Gd, Cm  147  21  64 512 240 
Tb, Bk  167  20  129 024 480 
Dy, Cf  185  18  225 792 840 
Ho, Es  200  15  347 373 600 
Er, Fm  212  12  471 435 600 
Tm, Md  220  8  565 722 720 
Yb, No  223  3  601 080 390 
View Large

By only considering the combinatorics based on the assumptions made, the maximum number of possible occupancy combinations increases greatly from a maximum of three for the early d-block elements, to a maximum of 21 for the later d-block elements, to as many as 223 for the lanthanides and actinides. It is not the intent to suggest that every single one of these combinations contributes to superposition mixing, but rather to illustrate how the probability of finding two or more near-energy-degenerate configurations increases drastically as more orbitals are introduced with increasingly crowded energy separations. However, from a purely combinatorial perspective, the occupancy configurations addressed thus far is incomplete. It is not sufficient for multiconfigurational ground states to only consider the combinations of occupancy configurations. The permutations of electrons within each orbital must also be considered.

Specifically, there is only one way for an orbital to be completely empty or completely filled. However, there is more than one way for any orbital to be partially filled. For example, an s-orbital having a capacity of two electrons (l = 0; ml = 0; ms = ±½) can be either empty, half-filled, or filled. For a half-filled s-orbital with only a single electron (s1), there are two possible permutations for the electron (spin up or spin down) as shown in Table VII. As the orbital capacity increases for p-, d-, and f-orbitals, so too does the number of possible permutations as shown in Table VIII, Table IX, and Table X, respectively. The maximum number of permutations for each orbital type occurs at half capacity corresponding to maximum degrees of freedom for the electrons. Specifically, 20 electron permutations exist for a half-filled p3 orbital, 252 electron permutations in a half-filled d5 orbital, and 3432 permutations for a half-filled f7 orbital. This becomes critically important in terms of determining how these partially filled orbitals combine to potentially increase the number of permutations for a given occupancy configuration. Lists of all permutations for d- and f-orbitals are included as the supplementary information.

Table VII.

Possible permutations for all possible s-orbital occupancies.

s-occupancy Number of permutations List of permutations
s0  1  [ ] 
s1  2  [u ] 
[d ] 
s2  1  [ud] 
s-occupancy Number of permutations List of permutations
s0  1  [ ] 
s1  2  [u ] 
[d ] 
s2  1  [ud] 
View Large
Table VIII.

Possible permutations for all p-orbital occupancies.

p-occupancy Number of permutations List of permutations
p0  1  [ ][ ][ ] 
p1  6  [u ][ ][ ] 
[d ][ ][ ] 
[ ][u ][ ] 
[ ][d ][ ] 
[ ][ ][u ] 
[ ][ ][d ] 
p2  15  [ud][ ][ ] 
[u ][u ][ ] 
[u ][d ][ ] 
[u ][ ][u ] 
[u ][ ][d ] 
[d ][u ][ ] 
[d ][d ][ ] 
[d ][ ][u ] 
[d ][ ][d ] 
[ ][ud][ ] 
[ ][u ][u ] 
[ ][u ][d ] 
[ ][d ][u ] 
[ ][d ][d ] 
[ ][ ][ud] 
p3  20  [ud][u ][ ] 
[ud][d ][ ] 
[ud][ ][u ] 
[ud][ ][d ] 
[u ][ud][ ] 
[u ][u ][u ] 
[u ][u ][d ] 
[u ][d ][u ] 
[u ][d ][d ] 
[u ][ ][ud] 
[d ][ud][ ] 
[d ][u ][u ] 
[d ][u ][d ] 
[d ][d ][u ] 
[d ][d ][d ] 
[d ][ ][ud] 
[ ][ud][u ] 
[ ][ud][d ] 
[ ][u ][ud] 
[ ][d ][ud] 
p4  15  [ud][ud][ ] 
[ud][u ][u ] 
[ud][u ][d ] 
[ud][d ][u ] 
[ud][d ][d ] 
[ud][ ][ud] 
[u ][ud][u ] 
[u ][ud][d ] 
[u ][u ][ud] 
[u ][d ][ud] 
[d ][ud][u ] 
[d ][ud][d ] 
[d ][u ][ud] 
[d ][d ][ud] 
[ ][ud][ud] 
p5  6  [ud][ud][u ] 
[ud][ud][d ] 
[ud][u ][ud] 
[ud][d ][ud] 
[u ][ud][ud] 
[d ][ud][ud] 
p6  1  [ud][ud][ud] 
p-occupancy Number of permutations List of permutations
p0  1  [ ][ ][ ] 
p1  6  [u ][ ][ ] 
[d ][ ][ ] 
[ ][u ][ ] 
[ ][d ][ ] 
[ ][ ][u ] 
[ ][ ][d ] 
p2  15  [ud][ ][ ] 
[u ][u ][ ] 
[u ][d ][ ] 
[u ][ ][u ] 
[u ][ ][d ] 
[d ][u ][ ] 
[d ][d ][ ] 
[d ][ ][u ] 
[d ][ ][d ] 
[ ][ud][ ] 
[ ][u ][u ] 
[ ][u ][d ] 
[ ][d ][u ] 
[ ][d ][d ] 
[ ][ ][ud] 
p3  20  [ud][u ][ ] 
[ud][d ][ ] 
[ud][ ][u ] 
[ud][ ][d ] 
[u ][ud][ ] 
[u ][u ][u ] 
[u ][u ][d ] 
[u ][d ][u ] 
[u ][d ][d ] 
[u ][ ][ud] 
[d ][ud][ ] 
[d ][u ][u ] 
[d ][u ][d ] 
[d ][d ][u ] 
[d ][d ][d ] 
[d ][ ][ud] 
[ ][ud][u ] 
[ ][ud][d ] 
[ ][u ][ud] 
[ ][d ][ud] 
p4  15  [ud][ud][ ] 
[ud][u ][u ] 
[ud][u ][d ] 
[ud][d ][u ] 
[ud][d ][d ] 
[ud][ ][ud] 
[u ][ud][u ] 
[u ][ud][d ] 
[u ][u ][ud] 
[u ][d ][ud] 
[d ][ud][u ] 
[d ][ud][d ] 
[d ][u ][ud] 
[d ][d ][ud] 
[ ][ud][ud] 
p5  6  [ud][ud][u ] 
[ud][ud][d ] 
[ud][u ][ud] 
[ud][d ][ud] 
[u ][ud][ud] 
[d ][ud][ud] 
p6  1  [ud][ud][ud] 
View Large
Table IX.

Number of permutations for all possible d-orbital occupancies.

d-occupancy Permutations
d0  1 
d1  10 
d2  45 
d3  120 
d4  210 
d5  252 
d6  210 
d7  120 
d8  45 
d9  10 
d10  1 
d-occupancy Permutations
d0  1 
d1  10 
d2  45 
d3  120 
d4  210 
d5  252 
d6  210 
d7  120 
d8  45 
d9  10 
d10  1 
View Large
Table X.

Number of permutations for all possible f-orbital occupancies.

f-occupancy Permutations
f0  1 
f1  14 
f2  91 
f3  364 
f4  1001 
f5  2002 
f6  3003 
f7  3432 
f8  3003 
f9  2002 
f10  1001 
f11  364 
f12  91 
f13  14 
f14  1 
f-occupancy Permutations
f0  1 
f1  14 
f2  91 
f3  364 
f4  1001 
f5  2002 
f6  3003 
f7  3432 
f8  3003 
f9  2002 
f10  1001 
f11  364 
f12  91 
f13  14 
f14  1 
View Large
Determining the number of possible permutations for a given occupancy configuration simply requires multiplying the numbers of permutations for each individual orbital occupancy (Tables VII–X). Using Pu as an example, we can consider the total number of possible permutations for any or all of the 105 occupancy configurations available (Table VI):
7 s 2 5 f 6 6 d 0 7 p 0 ,   7 s 2 5 f 5 6 d 1 7 p 0 ,   7 s 2 5 f 5 6 d 0 7 p 1 ,   7 s 2 5 f 4 6 d 2 7 p 0 ,   7 s 2 5 f 4 6 d 1 7 p 1 ,   7 s 2 5 f 4 6 d 0 7 p 2 ,   7 s 2 5 f 3 6 d 3 7 p 0 , 7 s 2 5 f 3 6 d 2 7 p 1 ,   7 s 2 5 f 3 6 d 1 7 p 2 ,   7 s 2 5 f 3 6 d 0 7 p 3 ,   7 s 2 5 f 2 6 d 4 7 p 0 ,   7 s 2 5 f 2 6 d 3 7 p 1 ,   7 s 2 5 f 2 6 d 2 7 p 2 ,   7 s 2 5 f 2 6 d 1 7 p 3 , 7 s 2 5 f 2 6 d 0 7 p 4 ,   7 s 2 5 f 1 6 d 5 7 p 0 ,   7 s 2 5 f 1 6 d 4 7 p 1 ,   7 s 2 5 f 1 6 d 3 7 p 2 ,   7 s 2 5 f 1 6 d 2 7 p 3 ,   7 s 2 5 f 1 6 d 1 7 p 4 ,   7 s 2 5 f 1 6 d 0 7 p 5 , 7 s 2 5 f 0 6 d 6 7 p 0 ,   7 s 2 5 f 0 6 d 5 7 p 1 ,   7 s 2 5 f 0 6 d 4 7 p 2 ,   7 s 2 5 f 0 6 d 3 7 p 3 ,   7 s 2 5 f 0 6 d 2 7 p 4 ,   7 s 2 5 f 0 6 d 1 7 p 5 ,   7 s 2 5 f 0 6 d 0 7 p 6 , 7 s 1 5 f 7 6 d 0 7 p 0 ,   7 s 1 5 f 6 6 d 1 7 p 0 ,   7 s 1 5 f 6 6 d 0 7 p 1 ,   7 s 1 5 f 5 6 d 2 7 p 0 ,   7 s 1 5 f 5 6 d 1 7 p 1 ,   7 s 1 5 f 5 6 d 0 7 p 2 ,   7 s 1 5 f 4 6 d 3 7 p 0 , 7 s 1 5 f 4 6 d 2 7 p 1 ,   7 s 1 5 f 4 6 d 1 7 p 2 ,   7 s 1 5 f 4 6 d 0 7 p 3 ,   7 s 1 5 f 3 6 d 4 7 p 0 ,   7 s 1 5 f 3 6 d 3 7 p 1 ,   7 s 1 5 f 3 6 d 2 7 p 2 ,   7 s 1 5 f 3 6 d 1 7 p 3 , 7 s 1 5 f 3 6 d 0 7 p 4 ,   7 s 1 5 f 2 6 d 5 7 p 0 ,   7 s 1 5 f 2 6 d 4 7 p 1 ,   7 s 1 5 f 2 6 d 3 7 p 2 ,   7 s 1 5 f 2 6 d 2 7 p 3 ,   7 s 1 5 f 2 6 d 1 7 p 4 ,   7 s 1 5 f 2 6 d 0 7 p 5 , 7 s 1 5 f 1 6 d 6 7 p 0 ,   7 s 1 5 f 1 6 d 5 7 p 1 ,   7 s 1 5 f 1 6 d 4 7 p 2 ,   7 s 1 5 f 1 6 d 3 7 p 3 ,   7 s 1 5 f 1 6 d 2 7 p 4 ,   7 s 1 5 f 1 6 d 1 7 p 5 ,   7 s 1 5 f 1 6 d 0 7 p 6 , 7 s 1 5 f 0 6 d 7 7 p 0 ,   7 s 1 5 f 0 6 d 6 7 p 1 ,   7 s 1 5 f 0 6 d 5 7 p 2 ,   7 s 1 5 f 0 6 d 4 7 p 3 ,   7 s 1 5 f 0 6 d 3 7 p 4 ,   7 s 1 5 f 0 6 d 2 7 p 5 ,   7 s 1 5 f 0 6 d 1 7 p 6 , 7 s 0 5 f 8 6 d 0 7 p 0 ,   7 s 0 5 f 7 6 d 1 7 p 0 ,   7 s 0 5 f 7 6 d 0 7 p 1 ,   7 s 0 5 f 6 6 d 2 7 p 0 ,   7 s 0 5 f 6 6 d 1 7 p 1 ,   7 s 0 5 f 6 6 d 0 7 p 2 ,   7 s 0 5 f 5 6 d 3 7 p 0 , 7 s 0 5 f 5 6 d 2 7 p 1 ,   7 s 0 5 f 5 6 d 1 7 p 2 ,   7 s 0 5 f 5 6 d 0 7 p 3 ,   7 s 0 5 f 4 6 d 4 7 p 0 ,   7 s 0 5 f 4 6 d 3 7 p 1 ,   7 s 0 5 f 4 6 d 2 7 p 2 ,   7 s 0 5 f 4 6 d 1 7 p 3 , 7 s 0 5 f 4 6 d 0 7 p 4 ,   7 s 0 5 f 3 6 d 5 7 p 0 ,   7 s 0 5 f 3 6 d 4 7 p 1 ,   7 s 0 5 f 3 6 d 3 7 p 2 ,   7 s 0 5 f 3 6 d 2 7 p 3 ,   7 s 0 5 f 3 6 d 1 7 p 4 ,   7 s 0 5 f 3 6 d 0 7 p 5 , 7 s 0 5 f 2 6 d 6 7 p 0 ,   7 s 0 5 f 2 6 d 5 7 p 1 ,   7 s 0 5 f 2 6 d 4 7 p 2 ,   7 s 0 5 f 2 6 d 3 7 p 3 ,   7 s 0 5 f 2 6 d 2 7 p 4 ,   7 s 0 5 f 2 6 d 1 7 p 5 ,   7 s 0 5 f 2 6 d 0 7 p 6 , 7 s 0 5 f 1 6 d 7 7 p 0 ,   7 s 0 5 f 1 6 d 6 7 p 1 ,   7 s 0 5 f 1 6 d 5 7 p 2 ,   7 s 0 5 f 1 6 d 4 7 p 3 ,   7 s 0 5 f 1 6 d 3 7 p 4 ,   7 s 0 5 f 1 6 d 2 7 p 5 ,   7 s 0 5 f 1 6 d 1 7 p 6 , 7 s 0 5 f 0 6 d 8 7 p 0 ,   7 s 0 5 f 0 6 d 7 7 p 1 ,   7 s 0 5 f 0 6 d 6 7 p 2 ,   7 s 0 5 f 0 6 d 5 7 p 3 ,   7 s 0 5 f 0 6 d 4 7 p 4 ,   7 s 0 5 f 0 6 d 3 7 p 5 ,   7 s 0 5 f 0 6 d 2 7 p 6 .
Given that the number of permutations within a single orbital is greatest for half filling, and that the f-orbital contains the greatest number of permutations, it would be tempting to assume that the greatest number of electron permutations occurs for the occupancy configurations with seven f-electrons. However, this is not the case. The greatest number of permutations actually occurs for the 7s05f46d37p1 occupancy configuration (720 720 permutations total), corresponding to the product of the number of permutations for each orbital (⁠ 1 × 1001 × 120 × 6⁠). The least number of permutations (one) occurs for the 7s25f06d07p6 occupancy configuration (⁠ 1 × 1 × 1 × 1⁠) since there is only one way for the s- and p-orbitals to be completely filled, and only one way for the f- and d-orbitals to be completely empty. The sum of all permutations for all of the occupancy configurations listed for Pu is 10 518 300 in total. A similar determination of the permutations for each of the actinides (which applies the same to the lanthanides) is included in Table VI.

The assumptions and combinatorial reasonings thus far are intentionally broad in order to not exclude configurations that could be close in energy to the ground state, but not obviously so. Many of the configurations that can be listed combinatorically likely represent excited states with energies sufficiently high that they would never significantly contribute to superposition mixing. In fact, many of the combinatorically listed configurations likely represent energies that would surpass the filling of even higher order orbitals such as the 8s- or 5g-orbitals in the case of the actinides. If the objective is non-exclusion, perhaps this framework for actinides should be expanded to include possible participation by these higher order orbitals, which would increase the combinations even more drastically.

It is more reasonable to try to reduce the number of configurations that could practically participate in superposition mixing than to further expand it. In order to illustrate how the number of combinations could be cut down to a more reasonable and practical number, plutonium (Pu) will again be used as an example. Figure 1 summarizes the assumptions made thus far about which orbitals participate in superposition mixing for various sections of the periodic table. However, the combinatorial treatment thus far gives no indication for why greater complexity and instabilities would exist for Pu than for other lanthanides and actinides, and no clues as to why the itinerant to localized transition exists. Furthermore, if we assume the complexity and instabilities observed correlate to the number of orbital occupancies and permutations, then the treatment thus far would suggest consecutively increasing complexity for each actinide element beyond Pu as well as comparable complexity for each lanthanide in column with its corresponding actinide.

Fig. 1.
Summary of orbital participation assumptions made thus far in the development of a generalized representation of non-integer occupancy mutliconfigurational ground states.
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Summary of orbital participation assumptions made thus far in the development of a generalized representation of non-integer occupancy mutliconfigurational ground states.

Fig. 1.
Summary of orbital participation assumptions made thus far in the development of a generalized representation of non-integer occupancy mutliconfigurational ground states.
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Summary of orbital participation assumptions made thus far in the development of a generalized representation of non-integer occupancy mutliconfigurational ground states.

Close modal

One often cited specific example of Pu's complexity is the number of solid-state allotropes. Pu possesses more solid-state allotropes than any other element with six allotropes at ambient pressure and a seventh allotrope at modestly increased pressure.11 The volume changes between these various allotropes of Pu can be quite significant with the largest being a 25% difference in volume between α-phase and δ-phase Pu allotropes. Another example that explains why Pu is chosen as an example among the other lanthanides and actinides involves a comparison of the radial extent of the electrons (the Wigner–Seitz radius) of actinides, lanthanides, and transition metals.12 When marching through the actinides, the Wigner–Seitz radii exhibit a trend very similar to that of the period 6 d-block transition metals up to Pu; however, from Am on, the trend becomes more lanthanide-like. Finally, it has been experimentally observed that Pu exhibits spin–orbit coupling behavior described as a jj-skewed intermediate between LS- and jj-coupling behavior.3,4 This all demonstrates the important position of Pu related to the itinerant to localized transition or crossover.13,15 It was once believed that this transition occurs between Pu and Am,56 but it is now understood to occur within the allotropes of Pu, with δ-Pu being very near the apex of this transition.

Given that Pu does seem to exhibit these greater instabilities, and that the actinides beyond Pu appear to behave more lanthanide-like, the assumptions in Fig. 1 should be reconsidered. Specifically, based on the observed properties of the early actinides (Ac, Th, Pa, U, Np, and Pu) as compared to the transplutonic actinides, it will be assumed that the orbital superposition mixing for the transplutonic actinides should resemble those of the lanthanides, while the orbital mixing for the early actinides should somehow be distinct. In terms of coupling behavior, LS-coupling has been shown to represent the lanthanides, while the actinides surrounding the itinerant to localized transition exhibit behavior better described by jj-skewed intermediate coupling.3,4 Thus, the established ground state configurations shown in Table I along with the various considerations related to the itinerant to localized transition described earlier will be used to alter the assumptions summarized in Fig. 1. Specifically, the assumption for s- and d-orbital participation for the early d-block elements will remain. However, for the established ground states for the period 7 d-block elements, there is no example of an s-orbital occupancy less than two. Combined with the apparent p-orbital participation evident for Lr, any quantum superposition associated with a multiconfigurational ground state for the period 7 d-block elements will be assumed to involve only the d- and p-orbitals.

In an attempt to represent the conformity between the lanthanides and transplutonic actinides as being distinct from the early actinides (i.e., to represent the itinerant to localized transition), it will be assumed that the s-orbital participates for the early actinides, but not for the transplutonic actinides or lanthanides. Furthermore, it will be assumed that the p-orbital does not participate for any of the f-block elements. Finally, all occupancy configurations for f-block elements with d-occupancy greater than two will be excluded.

Based on these revised assumptions, summarized in Fig. 2, the number of occupancy configurations for Pu is reduced to only nine,
7 s 2 5 f 6 6 d 0 7 p 0 ,   7 s 2 5 f 5 6 d 1 7 p 0 ,   7 s 2 5 f 4 6 d 2 7 p 0 ,   7 s 1 5 f 7 6 d 0 7 p 0 ,   7 s 1 5 f 6 6 d 1 7 p 0 ,   7 s 1 5 f 5 6 d 2 7 p 0 ,   7 s 0 5 f 8 6 d 0 7 p 0 , 7 s 0 5 f 7 6 d 1 7 p 0 ,   7 s 0 5 f 6 6 d 2 7 p 0 .
Fig. 2.
Summary of revised orbital participation assumptions for superposition mixing in multiconfigurational ground states.
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Summary of revised orbital participation assumptions for superposition mixing in multiconfigurational ground states.

Fig. 2.
Summary of revised orbital participation assumptions for superposition mixing in multiconfigurational ground states.
View largeDownload slide

Summary of revised orbital participation assumptions for superposition mixing in multiconfigurational ground states.

Close modal
These occupancies for the 5f orbitals encompass the range of those observed experimentally and reported for PuO2, PuSb2, PuCoIn5, PuCoGa5, δ-phase Pu, and α-phase Pu from x-ray absorption spectroscopy (XAS), high energy-electron energy-loss spectroscopy (HE-EELS), resonant x-ray emission spectroscopy (RXES), and photoemission spectroscopy (PES)3,4,14,57–60 as well as those determined computationally.61,62 The generalized multiconfigurational ground state representation of Pu from Table V remains unchanged
(6)
as does the electron conservation condition
(7)
However, the bounding conditions for Pu from Table V 
(8)
(9)
(10)
(11)
must be revised along with the assumptions as follows:
(12)
(13)
(14)
(15)

Table XI summarizes the number of occupation configurations and permutations for all the elements in the periodic table based on the revised assumptions summarized in Fig. 2. The constraints made to the other elements result from similar tightening of their respective bounding conditions as described earlier for Pu. Comparing Table XI to Table IV and Table VI, it is apparent that the number of possible combinations is greatly reduced for the period 7 d-block and f-block elements. Furthermore, consideration of the permutations available under these assumptions demonstrates much greater conformity to the itinerant to localized transition and provides better distinction between the lanthanides and early actinides. Specifically, the greatest number of permutations (487 630) now occurs for Pu corresponding to the greater complexity and instabilities observed. Unlike what was observed previously, the number of permutations for the later actinides no longer runs away but, rather, decreases toward the end of the actinide series as the degrees of freedom are constrained.

Table XI.

Reduced number of occupancy combinations, and the total number of possible permutations of electrons for each actinide element assuming orbital mixing as summarized in Fig. 2 as well as d-orbital occupancy restricted to a maximum of 2 for f-block elements.

Atomic number Element Occupancy combinations Permutations
1  H  1  2 
2  He  1  1 
3  Li  1  2 
4  Be  1  1 
5  B  1  6 
6  C  1  15 
7  N  1  20 
8  O  1  15 
9  F  1  6 
10  Ne  1  1 
11  Na  1  2 
12  Mg  1  1 
13  Al  1  6 
14  Si  1  15 
15  P  1  20 
16  Si  1  15 
17  Cl  1  6 
18  Ar  1  1 
19  K  1  2 
20  Ca  1  1 
21  Sc  3  220 
22  Ti  3  495 
23  V  3  792 
24  Cr  3  924 
25  Mn  3  792 
26  Fe  3  495 
27  Co  3  220 
28  Ni  3  66 
29  Cu  2  12 
30  Zn  1  1 
31  Ga  1  6 
32  Ge  1  15 
33  As  1  20 
34  Se  1  15 
35  Br  1  6 
36  Kr  1  1 
37  Rb  1  2 
38  Sr  1  1 
39  Y  3  220 
40  Zr  3  495 
41  Nb  3  792 
42  Mo  3  924 
43  Tc  3  792 
44  Ru  3  495 
45  Rh  3  220 
46  Pd  3  66 
47  Ag  2  12 
48  Cd  1  1 
49  In  1  6 
50  Sn  1  15 
51  Sb  1  20 
52  Te  1  15 
53  In  1  6 
54  Xe  1  1 
55  Cs  1  2 
56  Ba  1  1 
57  La  2  24 
58  Ce  3  276 
59  Pr  3  1 904 
60  Nd  3  8 736 
61  Pm  3  28 392 
62  Sm  3  68 068 
63  Eu  3  123 552 
64  Gd  3  172 458 
65  Tb  3  186 472 
66  Dy  3  156 156 
67  Ho  3  100 464 
68  Er  3  48 776 
69  Tm  3  17 304 
70  Yb  3  4 236 
71  Lu  3  220 
72  Hf  3  495 
73  Ta  3  792 
74  W  3  924 
75  Re  3  792 
76  Os  3  495 
77  Ir  3  220 
78  Pt  3  66 
79  Au  2  12 
80  Hg  1  1 
81  Tl  1  6 
82  Pb  1  15 
83  Bi  1  20 
84  Po  1  15 
85  At  1  6 
86  Rn  1  1 
87  Fr  1  2 
88  Ra  1  1 
89  Ac  8  2 480 
90  Th  9  12 820 
91  Pa  9  47 768 
92  U  9  133 588 
93  Np  9  288 080 
94  Pu  9  487 630 
95  Am  3  123 552 
96  Cm  3  172 458 
97  Bk  3  186 472 
98  Cf  3  156 156 
99  Es  3  100 464 
100  Fm  3  48 776 
101  Md  3  17 304 
102  No  3  4 236 
103  Lr  2  16 
104  Rf  3  120 
105  Db  4  560 
106  Sg  5  1 820 
107  Bh  6  4 368 
108  Hs  7  8 008 
109  Mt  7  11 440 
110  Ds  7  12 870 
111  Rg  7  11 440 
112  Cn  7  8008 
113  Nh  1  6 
114  Fl  1  15 
115  Mc  1  20 
116  Lv  1  15 
117  Ts  1  6 
118  Og  1  1 
Atomic number Element Occupancy combinations Permutations
1  H  1  2 
2  He  1  1 
3  Li  1  2 
4  Be  1  1 
5  B  1  6 
6  C  1  15 
7  N  1  20 
8  O  1  15 
9  F  1  6 
10  Ne  1  1 
11  Na  1  2 
12  Mg  1  1 
13  Al  1  6 
14  Si  1  15 
15  P  1  20 
16  Si  1  15 
17  Cl  1  6 
18  Ar  1  1 
19  K  1  2 
20  Ca  1  1 
21  Sc  3  220 
22  Ti  3  495 
23  V  3  792 
24  Cr  3  924 
25  Mn  3  792 
26  Fe  3  495 
27  Co  3  220 
28  Ni  3  66 
29  Cu  2  12 
30  Zn  1  1 
31  Ga  1  6 
32  Ge  1  15 
33  As  1  20 
34  Se  1  15 
35  Br  1  6 
36  Kr  1  1 
37  Rb  1  2 
38  Sr  1  1 
39  Y  3  220 
40  Zr  3  495 
41  Nb  3  792 
42  Mo  3  924 
43  Tc  3  792 
44  Ru  3  495 
45  Rh  3  220 
46  Pd  3  66 
47  Ag  2  12 
48  Cd  1  1 
49  In  1  6 
50  Sn  1  15 
51  Sb  1  20 
52  Te  1  15 
53  In  1  6 
54  Xe  1  1 
55  Cs  1  2 
56  Ba  1  1 
57  La  2  24 
58  Ce  3  276 
59  Pr  3  1 904 
60  Nd  3  8 736 
61  Pm  3  28 392 
62  Sm  3  68 068 
63  Eu  3  123 552 
64  Gd  3  172 458 
65  Tb  3  186 472 
66  Dy  3  156 156 
67  Ho  3  100 464 
68  Er  3  48 776 
69  Tm  3  17 304 
70  Yb  3  4 236 
71  Lu  3  220 
72  Hf  3  495 
73  Ta  3  792 
74  W  3  924 
75  Re  3  792 
76  Os  3  495 
77  Ir  3  220 
78  Pt  3  66 
79  Au  2  12 
80  Hg  1  1 
81  Tl  1  6 
82  Pb  1  15 
83  Bi  1  20 
84  Po  1  15 
85  At  1  6 
86  Rn  1  1 
87  Fr  1  2 
88  Ra  1  1 
89  Ac  8  2 480 
90  Th  9  12 820 
91  Pa  9  47 768 
92  U  9  133 588 
93  Np  9  288 080 
94  Pu  9  487 630 
95  Am  3  123 552 
96  Cm  3  172 458 
97  Bk  3  186 472 
98  Cf  3  156 156 
99  Es  3  100 464 
100  Fm  3  48 776 
101  Md  3  17 304 
102  No  3  4 236 
103  Lr  2  16 
104  Rf  3  120 
105  Db  4  560 
106  Sg  5  1 820 
107  Bh  6  4 368 
108  Hs  7  8 008 
109  Mt  7  11 440 
110  Ds  7  12 870 
111  Rg  7  11 440 
112  Cn  7  8008 
113  Nh  1  6 
114  Fl  1  15 
115  Mc  1  20 
116  Lv  1  15 
117  Ts  1  6 
118  Og  1  1 
View Large
By considering the possible permutations of electrons as statistical mechanics microstates within their respective occupancy configuration macrostates, principles of statistical and thermodynamics can be applied to the electronic structures of elemental ground states. For example, classical Boltzmann's entropy S is defined by
(16)
where kB is the Boltzmann constant and W is the number of statistical mechanics microstates in a system, which correspond to the number of permutations listed in Table XI for electronic structures. Boltzmann's entropies for every element in the periodic table are calculated for each element and plotted in Fig. 3. It is important to note that Boltzmann's entropy as described earlier assumes energy degeneracy between all the microstates used. In some cases, the entropies of electronic structure shown in Fig. 3 could be taken as exact values. For example, for all the group 1 elements (H, Li, Na, K, Rb, Cs, and Fr), the two microstates available correspond to the two possible configurations of a single electron in an s-orbital (spin up and spin down), which are truly energy degenerate. However, in most cases, the microstates are not energy degenerate, in which case, the values plotted in Table XI represent an over-approximation of entropy. Even with the reduced number of microstates arrived at with the revised assumptions, the number of macrostates included for each element is still potentially overestimated, especially for the elements that are not considered to possess a multiconfigurational ground state.
Fig. 3.
Boltzmann entropy of the electronic structures for each element in the periodic table based on the permutation microstates given in Table XI.
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Boltzmann entropy of the electronic structures for each element in the periodic table based on the permutation microstates given in Table XI.

Fig. 3.
Boltzmann entropy of the electronic structures for each element in the periodic table based on the permutation microstates given in Table XI.
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Boltzmann entropy of the electronic structures for each element in the periodic table based on the permutation microstates given in Table XI.

Close modal

Consideration of the Gibbs entropy would result in a better representation of the entropies of multiconfigurational ground state electronic structures. However, the determination of Gibbs entropy requires knowing the probabilities of finding the electrons in a specific configuration, which, in turn, would require knowledge about the relative energy distributions of all the configurations. Given the large number of permutations, it would be difficult to identify the energies of each and every configuration relative to the others. However, the generalized representation of non-integer orbital occupancy multiconfigurational ground state electronic structures could provide a starting point for considering this possibility, especially when applied to lower atomic number elements with much simpler ground state structures and well-known energy distributions (i.e., those that are not considered to have multiconfiguration ground states). Furthermore, defining macrostates beyond those described herein as occupational configurations (such as macrostates defined by LS-coupling or jj-coupling) could serve to help inform measurable quantities such as energy and heat capacity from thermodynamic principles that can be used for verification by experiment and other computational methods.3,4,63

In the example given for Ni, the effective non-integer occupancy ground state of [Ar]4s1.53d8.5 assumes perfect energy degeneracy between the [Ar]4s23d8 and [Ar]4s13d9 configurations with the quantum interpretation being a 50% probability of finding the ground state in either of these two states. It is more likely that the two configurations are only near-energy degenerate (if at all), and that the effective non-integer occupancy representation would skew toward either an [Ar]sx<1.5dy>8.5 or a [Ar]sx>1.5dy<8.5 state. Regardless, [Ar]4s03d10 is not even considered as a contributing configuration in the superposition mixing, being considered an excited state configuration with an energy sufficiently high so as to make the probability of finding the ground state in this configuration very close (if not equal) to 0% by approximation. However, the three occupancy configuration macrostates for Ni along with the 66 microstate permutations given in Table XI (upon which the Boltzmann's entropy for Ni in Fig. 3 is based) make no such exclusions for the ground state. Thus, the entropy for the ground state of Ni represented in Fig. 3 is certainly an over-approximation, as is the case for most of the early period d-block elements.

In the cases of Lr and Rf discussed previously, only the two lowest energy configurations for each were considered for superposition mixing, resulting in effective non-integer occupancy configuration representations of [Rn]7s26d0<n<0.57p0.5<m<1 and [Rn]7s26d1<n<1.57p0.5<m<1, respectively. For configurations with higher energy, the probability of finding the electrons in that configuration is lower from a quantum perspective. Therefore, as consecutively higher energy configurations are included as contributing to superposition mixing, the influence on orbital occupancies would be less and less with each inclusion. It is worth noting that the s-orbital occupancies in both the representations for Lr and Rf are exactly two since both of the lowest two energy configurations have this s-orbital occupancy. The inclusion of any additional configuration with an s-orbital occupancy of one or zero would result in an effective non-integer s-orbital occupancy of less than two. Any potential experimental determination of an integer orbital occupancy ground state would serve to eliminate the number of occupancy configurations that need be considered, which is one potential example of the utility of the generalized non-integer occupancy ground state representation.

For the lanthanides and actinides where as many as 487 630 configuration permutations can be listed, the ability to exclude those that violate Hund's rule becomes more difficult because of the tendency of higher mass elements to exhibit jj-coupling behavior over LS-coupling.3,4 Given that the LS-coupling describes the lanthanides, and that the transplutonic actinides tend to behave more like the lanthanides, it could be reasoned that the transplutonic elements also exhibit LS-coupling behavior. If this is the case, then perhaps Hund's rule violating configurations could be used to reduce the number of microstates for the lanthanides and transplutonic actinides, while the jj-skewed coupling for the actinides surrounding the itinerant to localized transition would still maintain the larger numbers of microstates and, thus, higher entropy. This would support the statistical dynamics interpretation of correlating complexity and instability in these elements to the number of near-energy degenerate ways in which the electrons can arrange themselves.

A determination of which configurations to include or exclude from contribution to a multiconfigurational ground state can be made by considering the relative energies of the various configurations. Temperature would play a significant role in such an effort since any contributing configurations could not have an energy beyond what would be thermodynamically accessible at a given temperature. In most cases, this would likely eliminate a great deal of configurations for most of the elements in Fig. 3 for moderate temperatures. By the strictest definition of the ground state, only truly energy degenerate configurations would be relevant at absolute zero, while additional configurations would contribute as temperature progressively increases. While the computational cost in determining relative energies for all the configurations is potentially very expensive for elements like Pu, it may not be insurmountable and is certainly achievable for lower mass elements with less-complex electronic structures, or even higher mass elements with greater constraints on their degrees of freedom.61,62

It is difficult to point to experimental methods capable of determining the ground state orbital occupancies of an isolated neutral atom for a specific element. However, there exists a wealth of published data addressing the f occupancies for a series of lanthanide and actinide allotropes and compounds, including RXES, resonant photoemission spectroscopy (RPES), XAS, and HE-EELS, which can be used to inform which configurations should be included and excluded.3,4,14,22–55,57–60,64,65 The orbital occupancies that can be coaxed from these methods do not represent the ground state of a single isolated neutral atom, but rather the average occupancies associated with atoms under chemical bonding. Even in the case of the metallic allotropes of Pu, the results reveal the occupancy for atoms chemically boded to their nearest neighbors. However, from these results, the γ values for individual element atoms can at least be further constrained.

A method that approaches the probing of single isolated atoms is scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS).66 STS based methods have already proven to be powerful tools in the study of emergent phenomena in strongly correlated f-electron electron systems and provide a means of not just probing but also manipulating atoms.67–91 The combination of STS, which can seamlessly probe both the occupied and unoccupied density of states for individual atoms, combined with the ability to manipulate and isolate individual atoms comes close to providing a direct observation of the electronic structure of a single isolated atom. However, even then, the ground state would be perturbed by interactions with the substrate on which the atom is manipulated.

Regardless of these methods, the concept of a ground state electron configuration for a neutral isolated atom remains in large part theoretical. Furthermore, the assumption that a correct understanding of multiconfigurational ground states of f-electron elements would lead to an explanation for complex strongly correlated electron behavior might be naively influenced by the octet rule, which serves to so elegantly explain bonding for simple s-block and p-block elements. However, given the vast combinations of electron permutations illustrated herein, some value and utility might be found in applying a more statistical and thermodynamics approach to the quantum structure of electrons in these multiconfigurational ground states. By analogy, representing the positions and velocity of every gas molecule in a room can be computationally extremely expensive. However, the application of statistical and thermodynamic principles eliminates the need for such brute force computation. Similarly, it is possible that a similar treatment of the instabilities and complexities that arise from vast quantum degrees of freedoms for electrons in f-electron systems could provide a fresh perspective on the f-electron challenge.

A generalized representation for multiconfigurational ground states has been described in such a way that effective non-integer orbital occupancies arising from quantum superpositions of possible electron configurations can be considered. It has been demonstrated that the non-integer orbital occupancy representation can be applied generally across the elements of the periodic table, returning established ground states for elements with less-complex ground states by approximation. Bounding conditions for the representations have been described and tightened in such a way to simulate the itinerant to localized transition observed within the actinide series. A combinatorial approach to electron degrees of freedom has been worked to provide a framework for a statistical and thermodynamics treatment of multiconfigurational ground states. An over-approximation of entropy based on the combinatorics for elemental ground state electron structures has been plotted as an initial step toward a more informed entropy determination.

See the supplementary material for comprehensive lists of the specific occupancy combinations for each actinide and tables listing all permutations for d- and f-orbitals.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Heavy Element Chemistry Program under Early Career FWP No. EC2021LANL05. The author would like to thank Krzysztof Gofryk, Ladislav Havela, and James G. Tobin for valuable discussions.

The author has no conflicts to disclose.

Miles F. Beaux: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Writing – original draft (lead).

The data that support the findings of this study are available within the article and its supplementary material.

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