The d Orbital Multi Pattern Occupancy in a Partially Filled d Shell: The KFeF3 Perovskite as a Test Case

The occupancy of the d shell in KFeF3 is t2g4eg2, with five α and one β electrons. The Jahn–Teller lift of degeneracy in the t2g sub-shell produces a tetragonal relaxation of the unit cell (4.09 vs. 4.22 Å, B3LYP result) not observed experimentally. In order to understand the origin of this apparent contradiction, we explored, with a 2 × 2 × 2 supercell (40 atoms per cell), all possible local structures in which contiguous Fe atoms have a different occupancy of the t2g orbitals with the minority spin electron. A total of 6561 configurations (with occupancies from (8,0,0) to (3,2,2) of the 3 t2g orbitals of the 8 Fe atoms) have been explored, with energies in many cases lower (by up to 1550 μEh per 2 Fe atoms) than the one of the fully ordered case, both for the ferromagnetic and the anti-ferromagnetic solutions. The results confirm that the orientation of the β d electron of Fe influences the electrostatics (more efficient relative orientation of the Fe quadrupoles of the d shell) of the system, but not the magnetic interactions. Three hybrid functionals, B3LYP, PBE0, and HSE06, provide very similar results.


Introduction
Only in a few cases, the ABX 3 perovskites maintain the cubic structure. There are at least three mechanisms that produce a symmetry lowering, and, possibly, an increase of the size of the unit cell, from 1 to 2 or 4 or 8 formula units (f.u.). The most common one has been described about 50 years ago by Glazer [1,2], who investigated all possible structures that can be obtained from the aristotype cubic structure by tilting the BX 6 octahedra with respect to one, two, or three Cartesian directions, and assuming periods not longer than two octahedra. In general, the octahedra remain regular during the tilting. KMnF 3 is an example of compound belonging to this class of perovskites; the many phase transitions it is passing through as a function of the temperature are still a matter of debate [3].
In the second case the transition metal (TM) at the center of the octahedron moves (say vertically), breaking the symmetry (the inversion center is lost), and this activates the ferroelectricity of the compound. A prototype of this family is KNbO 3 .
The third case involves the Jahn-Teller effect [4][5][6] (for a very recent discussion of the effect, see Reference [7], and the many references therein) , that is active when the t 2g or e g subshells are partially occupied. Reducing the symmetry allows to lift the degeneracy of the t 2g or e g subshells of the aristotype cubic structure, with a consequent energy gain. It is this third mechanism that will be discussed in the present study.
When looking at the series of the 11 KMF 3 perovskites, from Ca to Zn, we would expect that Ca, V, Mn, Ni, and Zn might be involved in the first class (octahedron rotation). Sc, Ti, Fe, and Co are expected to undergo a Jahn-Teller deformation, as the t 2g subshell is as S222, all symmetry independent configurations (they are 6561) or patterns corresponding to all possible compositions are built and optimized, and the energies compared with the one of the fully ordered cell.
These compositions are labelled using integer triplets (n yz , n xz , n xy ), where n uv indicates in how many Fe atoms the β electron occupies the d uv orbital. Obviously, n yz + n xz + n xy = 8. The energies are compared with the one of the (0,0,8) configuration, referred to as the fully ordered one. The full set of calculations is repeated for the ferromagnetic (FM) and the anti-ferromagnetic (AFM) solutions.
The paper is structured as follows: in Section 2 the computational conditions are defined. In Section 3, the results obtained with the B3LYP functional are presented, followed by some conclusions. In the Appendix A, tables and figures, complementary to the ones reported in the main text of the manuscript, are reported: (i) PBE0 and HSE06 results, very close to the B3LYP ones.
(ii) Hartree-Fock (HF) results, qualitatively similar to the B3LYP ones. (iii) Technical details and results of many inner checks of the accuracy of the obtained results: as 6561 total energies are compared, spanning a relatively narrow energy range, and obtained by optimizing a cell containing 40 atoms, it is essential to show that each one of these energies is determined with an accuracy such that the differences remain meaningful.

Materials and Methods
The FM and AFM solutions of KFeF 3 have been evaluated by using the full range B3LYP [15,16] and PBE0 [17], and the range separated HSE06 [18,19] hybrid functionals, as well as the Hartree-Fock Hamiltonian. An all electron Gaussian type basis set and the CRYSTAL code [20][21][22] have been adopted. The triple zeta type 7-311G, 8-6-411(41d)G and 8-6-511G contractions, consisting of 13, 27, and 17 Atomic Orbitals (AO) for F, Fe, and K, respectively, are similar to the ones used in our previous study on this compound [23]. Exponents and coefficients of the contractions have, however, been optimized; the basis sets are given in Tables A1 and A2.
The Coulomb and Hartree-Fock exchange series are controlled by five parameters [24] that were set to T 1 = T 2 = T 3 = T 4 and T 5 = 2·T 1 , with T 1 = 10 (for a complete description of the role of these parameters, see also Refs. [25,26]); these values are required for an accurate evaluation of the small differences (in energy and geometry) between the various configurations of the system, ranging between 10 −3 and 10 −5 E h . As regards the DFT exchangecorrelation contribution to the Fock matrix, it was evaluated by numerical integration over the unit cell volume. Radial and angular points for the integration grid were generated through a Gauss-Legendre radial quadrature and Lebedev two-dimensional angular point distributions. In the present work, a pruned grid with 99 radial and 1454 angular points was used (see XXLGRID keyword in the CRYSTAL manual [24]).
In the geometry optimization, the BFSG scheme [27][28][29][30][31] has been adopted. The process stops when the root-mean-square (RMS) and the absolute value of the largest component of both the gradients and the estimated displacements are smaller than the corresponding thresholds. The default values for the four parameters controlling the process are: When these four conditions are simultaneously satisfied, the optimization stops. For the present investigation, in view of the small energy differences between the various configurations, the default values reported above have been divided by 10.

Considered Cells
Two cells were considered. The first one, referred to as the reference case, or fully ordered case (REF), contains one formula unit with a single transition metal atom, and then 5 atoms overall. It corresponds to the FM state with the cubic system Pm3m (221). The single β d electron is localized in the d xy orbital, that is then doubly occupied. The two other t 2g d orbitals are singly occupied. Obviously, d xy can be replaced by d xz or d yz . The Jahn-Teller effect implies that REF is tetragonal. For the AFM solution, REF is doubled. Combining various d occupancies in contiguous lattice positions requires larger cells. A 40-atoms supercell, obtained by expanding the unit cell of the aristotype by 2 × 2 × 2 has been used. It contains 8 metal atoms (see Figure 1) and is referred to as S222 with the orthorhombic space group Pmmm (47). For simplicity, most of the presented results will refer to the FM solution only. For the B3LYP case, however, also the AFM results will be presented and compared with the FM ones.
The AFM solutions will be shown to follow the same path as the FM ones, as the energy differences between the various configurations are due to electrostatics, that is not altered by the FM-AFM transition. In the following, energies and volumes will refer to 2 f.u. (10 atoms).
So the determination of the number of SICs can reduce the resource requirement because only one configuration must be calculated in the M set. The determination of the number of SICs, and their enumeration, has been discussed by several authors (see Refs. [33][34][35][36][37][38][39]) in the case of solid solutions, where various atomic species occupy a given set of positions (D) within the cell. In that case, atoms considered as colors are isotropic, so they do not possess directional properties as the d orbitals do. This makes the present case different with respect to the solid solution case. Obviously these differences apply to cases with partial f or g orbital occupancy as well. Mathematical details of the method will be given elsewhere [40] (for additional information, see also Ref. [33]). The list of the 162 SICs and some of their properties are summarized in Tables A3 and A4. The results of this analysis is that the 6561 configurations belong to 162 SICs. Then, for the purposes of the present study, in which we are interested mainly in the total energy and volume of each configuration, SCF calculations and geometry optimizations can be limited to the 162 SICs.
Each representative of a SIC will be identified in the following by the 3 integers, (n xy ,n xz ,n yz ), with the condition that n xy +n xz +n yz =8.

Results
KFeF 3 is a very ionic compound, with net charges, as obtained from a Mulliken analysis, very close to the formal +1, +2, and −1 charges on K, Fe, and F, respectively, as Table 1 shows. For the four functionals here considered, two columns are reported: the first one (REF, for reference case) provides the net charges of the primitive cell containing 5 atoms. The second column (SOL) shows the maximum absolute difference for this net charge among the 8 positions of all the S222 explored configurations. The SOL columns document that in the various configurations the net charges are extremely close to the ones of the REF case. For Fe, the oscillation is always smaller than 0.012 |e|; for F and K it is even smaller. So we can conclude that the net charges remain nearly constant in all sites of the full set of explored configurations.
The occupancy of the d shell on Fe is t 4 2g e 2 g , with five α electrons and only one β electron in the t 2g sub-shell, a typical case in which the Jahn-Teller theorem applies. Table 1 reports (line 5) the number of electrons in the doubly occupied t 2g orbital, that in our REF convention is d xy . The PBE0, B3LYP, and HSE06 populations are very similar, at 1.99 |e|, and the UHF one is only slightly higher at 2.00 |e|. In the various S222 configurations, the doubly occupied orbital can also be d xz or d yz ; the second columns show that the maximum deviation from the REF value is as small as 0.001 |e|. The same is true for the other four d orbitals, whose population is not shown, and is close to one. The lowest part of the table reports the non-null components of the quadrupole tensor (evaluated through a Mulliken partition of the charge density) of the atoms of the unit cell. It should be underlined that the dipole moments on all atoms are null for symmetry reasons. The quadrupole of the Fe atom is by far larger than the ones of the F and K atoms. The five α d electrons on Fe generate a (nearly) spherical charge distribution (and then null quadrupole components in the spherical harmonic basis here used), that is broken by the β electron. The only component of the quadrupole on Fe is then xy, xz, or yz according to the double occupancy on the various Fe atoms, as illustrated by Figure 2. It should be noticed that in the spin density maps, see Figure 3, referring to the REF case (β electron in d xy ), the excess of α electrons due to d xz and d yz appears, whereas for electostatics the doubly occupied orbital, d xy , is providing the dominant contribution.

Fe Fe
Fe Fe  The energy difference between the cubic structure in which the β electron on Fe is uniformly distributed on the three t 2g orbitals and the tetragonal one in which a single t 2g orbital is doubly occupied can be decomposed in a Jahn-Teller contribution (or electronic relaxation) evaluated at the cubic geometry reducing the symmetry from cubic (Pm3m) to tetragonal (P 4 m bm) and shifting one of the t 2g (see the EIGSHIFT option in the CRYSTAL manual), and then imposing the (1,0,0) instead of the (1/3,1/3,1/3) occupancy of the three t 2g orbitals, and the change in energy from the metrically cubic tetragonal solution and its geometrically optimized analog. It turns out that the Jahn-Teller energy contribution is three orders of magnitude larger than the geometry contribution: 1.1 E h compared to just 0.95 mE h (or 25 meV). As a consequence of relaxation, the c lattice parameter becomes shorter than the a and b ones: 4.09 and 4.22 Å at the B3LYP level (PBE0 and HSE06 are very similar), and 4.12 Å versus 4.24 Å at the UHF level. The Fe-F distances are just half the lattice parameters, then 2.05 and 2.12 Å in B3LYP (two and four neighbors, respectively).

The Orientation of the Doubly Occupied d Orbital
The B3LYP energies, cell parameters, volumes, kind of d-d interaction for the full set of S222 configurations are shown in Tables A3-A5.
The spanned energy range for the FM solutions is illustrated in Figure 5, where the multiplicity and then the symmetry of the SICs is also provided (see top left of the figure, and colors). It should be noticed that the low symmetry SICs (largest multiplicity) cluster in the central part of the energy interval and that the most and less stable configurations are characterized by high symmetry. The central part of the energy interval is densely populated, while the edges are less populated. This is a consequence of the relatively small size of the supercell.
Only 21 of the 162 SICs are less stable than REF. At n xy constant, the most stable configuration of (0, n xz = 0, n xy ) is always more stable than any of the (n yz = 0, n xz = 0, n xy ). This suggests that a few relative orientations of the d orbitals are stabilizing. Figure 2 illustrates the four possible relative orientations of the d orbitals (and then quadrupoles) on first neighbors Fe atoms. Interestingly, the two most and the less stable configurations of the full set (see Figure 1, and labels 1, 2, and 162 in Table A3) belong to (0,4,4).
Looking in Figure 1 at the relative orientations of the doubly occupied d orbitals along the Cartesian axis a (or x), b (or y) and c (or z) between two first neighbor iron atoms, the stablest configurations are characterized by only two (out of four) types of relative orientations, the most unstable by three.
In order to test the influence of the Hamiltonian on the presented results, calculations have been performed on the (2,3,3) SICs using PBE0, HSE06, and UHF. The (2,3,3) composition has been chosen because it corresponds to the largest number of configurations (1680) and SICs (39). The results are shown in Figures A1 and A2 The above numbers provide a clear evidence that the fully ordered case is less stable than many "mixed" configurations, but not of all of them. From data in Table A5, containing the optimized lattice parameters (obtained at B3LYP level) of the S222 cells for each SIC, the cell distortion from cubic metric, δ = max(a,b,c)-min(a,b,c), strongly depends on the composition. The REF cell shows the maximum distortion (δ = 0.24 Å) that reduces rapidly as the composition allows even patterns: δ < 0.10 Å in (1,3,4), (2,2,4), and (2,3,3) configurations (more than two-thirds of the full set) and δ ≈ 0.03 Å in the (2,3,3) patterns. The cell appears, however, as cubic, after averaging over the SICs and the possible orientations of the configurations (for example 0,4,4), (4,0,4), and (4,4,0)).
We can now try to identify the origin of the relative stability of many "mixed" configurations with respect to the fully ordered one. In principle, there are three variables that can stabilize one configuration with respect to another: (a) A different charge distribution. We already commented that the net charges and the quadrupole on Fe are extremely similar in all configurations; so we can conclude that this variable has essentially no effect on energy differences. This statement obviously does not include the different orientations of the quadrupole on the Fe ions. (b) A different geometry. We already commented that the cell volume of the configurations shows extremely small variations suggesting very similar geometries. The lattice parameters of the most stable and the less stable SICs are very close, as Table A5 shows. The small volume variations hide however dissimilar local situations around iron atoms. A tetragonally deformed octahedron with two apical (2.047 Å) and four equatorial (2.108 Å) distances characterizes REF. The difference between the Fe-F distances is even larger in the most stable configuration (2.122/2.109/2.032 Å); such a large difference is, however, not maintained in all configurations, for example, it reduces to 0.03 Å in the less stable configuration (Figure 1). It is not easy, however, to correlate many small distance differences to energy variations. However, certainly many small differences in the Fe-F and Fe-Fe distances can contribute with a non-negligible percentage to the energy stabilization/destabilization. (c) The different quadrupolar interaction in the various configurations. As already anticipated, the quadrupoles on the Fe atoms is expected to play the most important role in the stabilization of some configurations with respect to others. This effect is, however, not easily evaluated, because, as discussed at point b, when the occupancy of the t 2g d orbital is changing, also the Fe-F distances are changing. So there is certainly a coupled geometry-electrostatics effect in the stabilization. The FM-AFM energy difference is more than twice the energy difference between the most stable and the REF FM solutions (1549 µE h , see caption of Figure 1, and Table A3). However the FM-AFM energy differences of the 162 SICs are very similar, with a mean value of −3443 µE h and a standard deviation of 45 µE h .The orbital disorder is slightly less efficient for the AFM than for the FM solutions, by about 180 µE h , a small fraction of the energy range associated to disorder. As shown by the top panel of Figure 6, the disorder stabilisation effects for FM and AFM are strongly and linearly related, and the stability order of the 162 SICs is not significantly modified. For example, the relative stability of five most stable and the 10 least stable configurations is the same for FM and AFM. In the densely populated central energy interval, where SICs differ by few µE h , the stability order of the 162 AFM SICs is slightly different from the FM one. Note, in particular, that the REF configuration, that in the FM list (see Table A3) is number 141, in the AFM list (Table A4) is number 113, due to its large stabilization shown in Figure 6, and becomes more stable than the columnar [14] (see below) configuration, that is now at number 123.
In summary, the small variations between the FM and AFM cases, on the one hand, do not alter the analysis performed in this section on the basis of the FM solutions; on the other hand, confirm that the energy differences among the 162 SICs are mostly dictated by electrostatics, and are nearly independent from the magnetic interactions determining the FM-AFM energy differences. The maximum FM-AFM energy difference is observed for REF, that realizes the maximum Pauli pression of the majority spin electrons of F due to the d electrons of Fe; it is minimum for the most stable configurations that, with a more effective packing, also minimize the Pauli pression on F.   This suggests that the structural changes are very small. The volume/energy figure defines a broad band. At first, considering lowest and highest energy points, volume and energy seem to be anti-correlated. However, looking at SICs belonging to a given composition, a second trend appears, orthogonal to the first one, as shown in the figure.
In a recent paper Varignon et al. [14] investigated at the FM level three configurations of a KFeF 3 supercell containing 4 Fe atoms that they call single, columnar, and 3D checkerboard. The first one coincides with the REF configuration, and the third one with the most stable configuration shown in Figure 1. The columnar configuration is shown in Figure A3; it has the same basal plane as the most stable one of Figure 1; all planes along the c lattice parameter are the same as the basal plane. This configuration is not the second-most stable one; actually it is at number 124 in the list of the 162 FM SICs shown in Table A4. A quantitative comparison with the present results is not easy, because, (i) in Varignon et al. the geometry is not fully optimized; (ii) They use a plane-wave basis set, with pseudopotentials, whereas here we use an all electron Gaussian type basis set. (iii) The Hamiltonian in their case is LDA+U. We are, then, unable to establish a quantitative correspondence with the B3LYP, PBE0, HSE06, and UHF functionals used here. Their 3D checkerboard and columnar energies (see their Table III) are 1984 and 1286 µE h per 2 f.u. lower than single; the present numbers are 1549 and 99, 1359 and 83, and 1390 and 98 µE h for B3LYP, PBE0, and HSE06, respectively.

Simple Models for Describing the Stability Order
We can look for a simple model for describing the energy of the various configurations. We suppose that the interactions are additive and that the interaction energy of each Fe with the other Fe atoms is the sum of the individual interactions with the first Fe neighbors, and that farther neighbors are irrelevant. Variation of the Fe-Fe and Fe-F distances within a given configuration, and for the various configurations is not taken into account. Four types of interactions are rationalized on the basis of the relative orientation of the d shells on Fe, as shown in Figure 2. If, on the two Fe atoms, unlike d orbitals are doubly occupied, the interaction is J, if the atoms are connected along the direction (x, y, or z) appearing in both d orbitals (i.e., d xz -d yz along z); in the other cases the interaction is S (i.e., d xz -d yz along x or y). If the doubly occupied orbital on the two Fe atoms is the same, the interaction is I, if the atoms are aligned along one of the two directions appearing in the d orbitals (i.e., d xy -d xy along x or y); it is K in the other cases (d xy -d xy along z).
Looking at the Fe-F-Fe path, we observe that I, J and K are symmetric with respect to the two Fe atoms, so that F is expected to be midway. The S interaction, on the contrary, is not symmetric, and F is expected to be closer to one of the Fe atoms. For example, if one of the d orbitals is, say xz (on Fe 1 ), and the other xy (on Fe 2 ), and the direction of the bond is x, then F will be closer to Fe 2 . This local point of view must, however, take into account the various constraints deriving from the 3D structure. The maximum difference between the shortest and the longest Fe-F bond length, in the set of the 162 SICs, is observed for S (0.14 Å); for the other three interactions it is ≈0.06 Å. The mean short and long bond for I, J, and K interactions are very close, 2.086/2.098, 2.087/2.098 and 2.075/2.088 Å, respectively, but differ significantly for S: 2.049/2.122 Å.
S222 What about I and J? Consider the two most stable configurations, both belonging to the (0,4,4) composition. They both are characterized by 16 S interactions. The most stable, however, has 8 J, whereas in the other there are 8 I. Then exchanging J with I destabilizes the configuration. Looking now at Figure 7, where the two energies appear to the left, top (red diamonds), we observe that the same exchange increases the volume (the second trend indicated in the figure). Such a simple model, based on the S, I, J, and K interactions, is unable to reproduce the computed energies. Its major deficiency is its failure to approach the zero energy of the reference configuration. A more sophisticated model has then been explored, taking simultaneously into account the three interactions associated to each Fe site. A three-letter string (i.e., ISS or IJK) is associated to each site, where the letters refer to the four Fe-Fe interactions previously defined. Such model keeps partial track of the connectivity between the Fe-Fe couples sharing a common Fe. Starting from the cubic symmetry of the aristotype structure of perovskite, and in order to keep the number of parameters as small as possible, anagrams are merged in a unique parameter (this means that we consider equivalent, for example, IJK, IKJ, JIK, JKI, KIJ, and KJI). The model contains 12 parameters. Its ability to mimic the dataset is significantly improved with respect to the model based simply on the four interactions S, I, J, and K. It produces 143 independent energies for the 162 SICs. The model is as follows: where n i αβγ and C αβγ are the number of Fe sites involved in simultaneously α, β, and γ interactions and the corresponding effective coefficients of interaction (ECI), respectively. Obviously, for S222, ∑ n i αβγ = 8. The AFM ECIs as resulting from the best fit are: −744.  Figure A4. The energy of the (0,0,8) REF configuration is very well reproduced: it is characterized by 8 IIK interactions whose ECI is 0.0 and -0.8 µE h for the AFM and the FM cases, respectively. It is interesting to note that the stabilizing character of these three fold interactions decreases as their "S content" decreases, the exception being KSS (+355.2 µE h ).
These ECI call for some comments: (i) Only configurations containing two or three different t 2g d orbitals doubly occupied can be more stable than REF characterized by a single type of doubly occupied d orbital; (ii) It is impossible to build a configuration containing only S interactions, so the stablest configuration found in this work is the stablest for any supercell within the present approximations; (iii) The high stability of the configurations with a large number of S interactions can be related to the orientation of the quadrupole on Fe atoms; the S arrangement corresponds to the herringbone structure reported for molecular crystals characterized by a dominant contribution from the molecular quadrupole, such as benzene [41]; (iv) The flexibility of the Fe-F bonds allowed by the local asymmetry of the S interaction is expected to contribute to the stability of the configurations.

Conclusions and Perspectives
Many transition metal compounds of the first row, with the 3d shell partially occupied, undergo a Jahn-Teller deformation, when the number of degenerate orbitals in a given sub-shell is larger than the number of α or β electrons to be allocated.
In the perovskite compounds KMF 3 , this is the case of 6 (Sc, d 1 ; Ti, d 2 ; Cr, d 4 ; Fe, d 6 ; Co, d 7 ; and Cu, d 9 ) out of 10 systems of the family starting from KScF 3 and ending with KZnF 3 . The cubic symmetry of the ideal perovskite structure, with two d sub-shells containing three t 2g and two e g orbitals, reduces to tetragonal as a consequence of the split of (say) the d xy from the d xz and d yz (Sc, Ti, Fe, Co), or of the d z 2 from the d x 2 −y 2 orbital (Cr, Cu). The deformation of the unit cell in the cases of the Fe compound, explored here, is of the order of 0.13 Å (the lattice parameter c is shorter than a and b); the energy difference due to this tetragonal relaxation is as small as 1 mE h .
There is, however, no reason for having the double occupancy in d xy rather than in d xz or d yz ; it is shown here that if a supercell (2 × 2 × 2) is built in which the three t 2g orbitals are occupied on contiguous sites, in most of the cases an energy is reached which is lower than the one of the fully ordered case (all cells in the infinite lattice with the d xy orbital doubly occupied).
In particular, it is shown here that: It should be underlined that the present calculations require the preparation of a large number of input files including the definition of the initial guess for the 8 Fe ions. All these manipulations are, however, performed automatically through scripts available to the reader. The present calculations exploiting the symmetry both to catalog the d patterns and to simplify the ab initio calculations, performed with an all electron scheme and (full range) hybrid functionals, are relatively cheap. The tools implemented for the present analysis are general, so that in the near future extensions are foreseen to the following sets of compounds:

Conflicts of Interest:
The authors declare no conflicts of interest.

Abbreviations
The following abbreviations are used in this manuscript:   Figure 5 ((2,3,3) panel) in the main text.    [14] in the main text), labelled (i), (ii), and (iii) in their Table III. They appear at L SIC = 1, 124 and 141.
L SIC n S n I n J n K ∆E ∆V L SIC n S n I n J n K ∆E ∆V L SIC n S n I n J n K ∆E ∆V       Table III of Reference [14] of the main text.  Figure A4. Comparison between the B3LYP relative energies (∆E calc ) and the relative energies predicted according to the 12 interaction model described in the text (∆E pred ) for the 162 SICs. Energies refer to 2 f.u. Top: E pred as a function of E calc . The even distribution with respect to the perfect agreement straight line (dashed line) confirms the good quality of the model. Bottom: Deviation (∆E pred -∆E calc ) as a function of ∆E calc . The two panels share the horizontal scale. The energy differences are a small fraction of the energy interval covered by the set of SICs; only 4 exceed 10% of the energy interval.  Figure A5. As in previous Figure A4, for the AFM solutions.