Key Role of Deep Orbitals in the dx2–y2–d3z2–r2 Gap in Tetragonal Complexes and 10Dq

Using first-principles calculations, we show that the origin of the intrinsic a1g(∼3z2 – r2)–b1g(∼x2 – y2) splitting, Δint, in tetragonal transition-metal complexes and the variations of the cubic field splitting, 10Dq, with the metal–ligand distance, R, are much more subtle than commonly thought. As a main novelty, the key role played by covalent bonding with deep valence ligand levels and thus the inadequacy of too simple models often used for the present goal is stressed. Taking as a guide the isolated D4h CuF64– complex, it is proved that Δint essentially arises from bonding with deep 2s(F) orbitals despite them lying ∼23 eV below 2p(F) orbitals. This conclusion, although surprising, is also supported by results on octahedral fluoride complexes where the contribution to 10Dq splitting from bonding with 2s(F) orbitals is behind its strong R dependence, stressing that explanations based on the crystal-field approach are simply meaningless.


INTRODUCTION
A great deal of research is currently focused on transition metal (TM) compounds due to their potential technological interest, witnessed in lasers 1 based on Al 2 O 3 :Ti 3+ or BeAl 2 O 4 :Cr 3+ or devices using manganites. 2 Among insulating TM materials, particular attention is paid to those containing Cu 2+ ions. Aside from the interest on La 2 CuO 4 , the parent compound of high-T c superconducting cuprates, 3,4 much work is done on Cu 2+ hybrid perovskites 5−9 currently employed in several devices and on Tutton salts 10−13 containing Cu 2+ due to their potential application in the study of enzymes. In addition, some Cu 2+ compounds are responsible for the color of historical pigments 14 or the stained glasses of medieval gothic architecture. 15 In insulating TM compounds, active electrons are essentially confined in the MX N complex formed by the TM cation, M, and the N ligands. For this reason, a deep insight into the covalent bonding inside the MX N unit is crucial for understanding the actual origin of optical and magnetic properties of such compounds following the way started by the pioneering work by Sugano and Shulman. 16,17 The present work is just addressed to prove that subtleties in chemical bonding can play a crucial role for reaching such a goal. Efforts are particularly focused to explain the origin of the dependence on the metal−ligand distances of two relevant splittings of the antibonding orbitals with a mainly d character in MX 6 complexes: (1) the splitting 10Dq between e g (∼3z 2 − r 2 , x 2 − y 2 ) and t 2g (∼xy, xz, yz) levels in octahedral O h complexes and (2) the splitting Δ between a 1g (∼3z 2 − r 2 ) and b 1g (∼x 2 − y 2 ) levels in tetragonal D 4h MX 6 units.
For reaching these objectives, the analysis of first-principles calculations and available experimental data is crucial. Indeed, models that use fitting parameters hardly allow one to know the actual microscopic origin of phenomena. 18−20 For this reason, rough approximations such as the superposition 21 or the angular overlap 22 models together with those based on the crystal field (CF) approach are meaningless for the present goal.
In a first step, the present work explores the influence of covalent bonding upon the splitting Δ in tetragonal MX 6 units.
Positive Δ values mean in this work that a 1g (∼3z 2 − r 2 ) has a higher energy than b 1g (∼x 2 − y 2 ). For clarifying the main ideas, the tetragonal CuF 6 4− complex is taken as a guide throughout the present work as Δ has been determined for several compounds containing such a complex. It should be noted here that optical excitations do also depend on the internal electric field induced by the rest of the lattice ions upon the electrons confined in the complex 23−26 and thus there is a contribution to Δ not related to the chemical bonding in the complex.
It is worth noting now that the gap between 2p(F) and 2s(F) valence orbitals of free F atom 27−29 is about 23 eV. Accordingly, it could be expected that Δ is much more influenced by the covalent bonding with shallow 2p(F) than with deep 2s(F) orbitals. We prove in this work that such a guess is not correct as the reality is certainly more subtle.
Tetragonal complexes are observed for Cu 2+ -doped cubic lattices 30−33 as a result of the so-called static Jahn−Teller effect, 34−36 a phenomenon ultimately due to the unavoidable presence of random strains in any real crystal. 34 Tetragonal CuF 6 4− units are also formed in Cu 2+ -doped K 2 ZnF 4 or Ba 2 ZnF 6 -layered perovskites though there is no Jahn−Teller effect as the a 1g (∼3z 2 − r 2 ) and b 1g (∼x 2 − y 2 ) levels are not degenerate following the tetragonal symmetry of the host lattice. 24 Accordingly, the theory describing the Jahn−Teller effect 34,35 cannot, in general, be transferred 4,9,24 to understand pure layered compounds such as the orthorhombic K 2 CuF 4 or Cs 2 AgF 4 . 37,38 Nevertheless, the Jahn−Teller framework is still surprisingly applied 39−41 to d 9 ions under tetragonal or lower symmetries.
Compressed tetragonal CuF 6 4− units are formed in KAlCuF 6 or CuFAsF 6 pure compounds 42−45 in addition to Cu 2+ -doped crystals. 24,46−48 By contrast, in CuF 2 or A 2 CuF 4 (A = K, Na), the tetragonally compressed CuF 6 4− units undergo an additional orthorhombic distortion, favored by the existence of adjacent complexes which share F − ligands. 9,48,49 An orthorhombic instability also takes place in copper Tutton salts 12 and in NH 4 Cl:CuCl 4 (H 2 O) 2 2− . 50,51 The interest in the Δ gap relies on the fact that it is often the lowest optical excitation of compounds 52 with tetragonal MX 6 units (M = Cu, Ag). Also, in superconductor oxocuprates, the transition temperature, T c , has been related to the magnitude of the Δ splitting. 53 In a second step, the present work is devoted to clarify quantitatively the origin of the sensitivity of 10Dq to variations of the metal−ligand distance, R, in octahedral complexes. Experimentally, it has been found that 10Dq depends on R −t , where the exponent t is often found to be close to 5. 54 By this reason, it is still claimed that the exponent t mainly comes from the CF contribution 40,55 despite this approach leading to 10Dq values much smaller than experimental ones. 54 Seeking to shed light on these issues, in addition to investigating the relation between covalent bonding and the splitting Δ, we have carried out first-principles calculations on tetragonal CuF 6 4− units at different values of axial (R ax ) and equatorial (R eq ) metal−ligand distances. In this analysis, particular attention is paid to explore how the charge on ligands is modified by varying the R ax and R eq distances. In a further step, we analyze in octahedral complexes how the variations of chemical bonding with the metal−ligand distance, R, are quantitatively related to the exponent t.
This work is organized as follows. A brief account of computational tools is given in Section 2 while Section 3 first deals with the two contributions to optical transitions for a TM complex in an insulating compound: the intrinsic one associated with the isolated complex and the extrinsic one due to the internal electric field created by the rest of the lattice ions. 23,24 That section also deals with the relation between the splitting, Δ, and the variation of charge on ligands. The main results of this work are discussed in Section 4. Special attention is paid in that section to clarify the different influences of bonding with 2p(F) and 2s(F) orbitals upon the splitting Δ (Section 4.1) of isolated CuF 6 4− units and also the origin of the dependence of 10Dq on the metal−ligand distance for octahedral complexes (Section 4.2). For the sake of completeness, the reasons behind the similarities and differences between 2p(F) and 2s(F) orbitals are discussed in Section 4.3. Finally, the applicability of the present ideas to complexes involving Cl − , Br − , or O 2− as ligands is briefly dealt with in the last section.

COMPUTATIONAL METHODS
Ab initio density functional theory (DFT) calculations on isolated CuF 6 4− complexes have been performed at fixed metal−ligand distances by means of the 2017.03 version of the Amsterdam density functional code. 69 By means of this kind of calculations, we can already explore the dependence of the intrinsic contribution Δ int to the Δ gap upon metal−ligand distances. In these DFT calculations, we have used the popular B3LYP hybrid functional (including 25% of Hartree−Fock exchange 70 ) in the spin-restricted and spin-unrestricted Kohn−Sham formalism of the DFT and high-quality allelectron basis sets of triple-ζ plus polarization type. We have verified that similar results are obtained using other hybrid functionals such as the nonempirical PBE0 one. 71

INFLUENCE OF COVALENT BONDING UPON THE Δ
SPLITTING IN TETRAGONAL COMPLEXES: A GENERAL VIEW Although in insulating compounds containing TM cations, active electrons are localized in the MX N complex, the optical properties cannot, in general, be explained considering only that isolated unit. Indeed, the localized electrons lying in the MX N complex are also subject to the electric field, , created by the rest of the lattice ions, which usually has a perturbative character. 23 By this reason, the energy, E, of an electronic transition can be divided in two contributions 24 where E int is the intrinsic contribution to the isolated MX N complex at equilibrium geometry while the extrinsic one, E ext , accounts for the effects of the internal electric field, , All energies are given in eV units. When R ax < R eq , the ground state has a hole in a 1g (∼3z 2 − r 2 ) and the extrinsic contribution tends to enhance the value of the intrinsic one. By contrast, for Cu 2+ doped into the cubic perovskite KZnF 3 , displaying an elongated equilibrium geometry, the hole in the ground state lies in b 1g (∼x 2 − y 2 ) and thus the sign of Δ int is negative. The total calculated gap, Δ = Δ int + Δ ext , is compared to available experimental data. The values of R ax and R eq metal−ligand distances are taken from experimental data for pure compounds and from calculations for systems where Cu 2+ enters as an impurity. upon the confined electrons. As an example, the intrinsic contribution for 10Dq in ruby and emerald 23,56 is the same (10Dq int = 2 eV), reflecting the identical Cr 3+ −O 2− distance (1.97 Å) in both gemstones. 23,57,58 Thus, the difference between the red ruby and the green emerald simply arises from the distinct shape of the extrinsic field in the two gemstones, leading to the small corrections 10Dq ext = 0.24 eV for ruby and 10Dq ext = −0.05 eV for emerald. 23,56 In the same vein, the color of the Egyptian blue pigment 14 is just the result of a 0.90 eV shift induced by on the highest d−d transition of the square-planar CuO 4 6− chromophore in CaCuSi 4 O 10 .
An insight into the Δ gap between a 1g (∼3z 2 − r 2 ) and b 1g (∼x 2 − y 2 ) levels of tetragonal complexes thus requires taking into account both the intrinsic, Δ int , and extrinsic, Δ ext , contributions. For the sake of clarity, the values of both contributions derived for systems with tetragonal CuF 6 4− units are displayed in Table 1. It is worth noting that the existence of an internal electric field allows one to understand why the hole in KAlCuF 6 45 and K 2 ZnF 4 :Cu 2+52 is in the a 1g (∼3z 2 − r 2 ) orbital in contrast to Cu 2+ doped into the cubic perovskites KZnF 3 or CsCdF 3 where it resides in b 1g (∼x 2 − y 2 ) due to a static Jahn−Teller effect. 30,32 Indeed, for d 9 ions under an initial octahedral symmetry, the existence of a static Jahn− Teller effect usually leads to elongated complexes 34,35 with the exception of CaO:Ni + , a matter dealt with in refs 59 and 60. The extrinsic contribution arising from a tetragonal internal field in K 2 ZnF 4 :Cu 2+ and Ba 2 ZnF 4 :Cu 2+ also explains why |Δ| has been detected 46,52 in the 0.6−1 eV region, whereas for KZnF 3 :Cu 2+ it should be below 0.5 eV, 31,52 a fact that concurs with the cubic symmetry of the host lattice. Bearing these facts in mind, the chemical bonding inside the complex essentially influences the intrinsic component, Δ int , of the total gap.
Let us consider an isolated MX 6 unit with a small tetragonal distortion depicted in Figure 1. This condition just implies that, if R ax and R eq are the two metal−ligand distances, with a mean value R m = (R ax + 2R eq )/3, it must be verified that where η reflects the tetragonal distortion from an octahedral MX 6 complex with a metal−ligand distance equal to the mean value R m .
The condition given by eq 1 is well followed by all systems of Table 1 involving CuF 6 4− units, where η < 0.16. As Δ int should always be zero when R ax = R eq ; then it is function of R ax − R eq and can be written in a good first approximation as 45,49,52 where the β(R m ) quantity only depends on the R m value. Previous studies on systems 45,49,52 with tetragonal CuF 6 4− units lead to a value β ≅ 2.7 eV/Å when R m ≅ 2.05 Å. It should be remarked that eq 3 is valid for systems displaying a static Jahn−Teller effect (like KZnF 3 :Cu 2+ ) as well as for those whose ground state is determined by the internal electric field, 45 such as it happens for CuFAsF 6 or K 2 ZnF 4 :Cu 2+ . Therefore, the β quantity is common to all systems with tetragonal CuF 6 4− units provided R m ≅ 2.05 Å. In the rough CF approach, where ligands are treated as point charges, the electrostatic potential due to ligands, V M , around the central cation (placed at r = 0) involves two contributions 61 Here, V M 0 is independent of the electronic coordinate, r, but plays a key role for placing the energy of 3d levels of Cu 2+ above that of 2p(F) ligand levels. 61 Thus, within the CF framework, the non-constant contribution V M NC (r) is the only one responsible for the splitting Δ int when R eq ≠ R ax . Accordingly, the gap, Δ int (CF), for an isolated CuF 6 4− unit, derived from the simple CF approach, is given by 17 where ⟨r d 2 ⟩ = 1.044 a.u. and ⟨r d 4 ⟩ = 2.674 a.u. correspond to f ree Cu 2+ ions 62 and Z L is the ligand charge. Using these values, R m = 2.05 Å, and even fully neglecting covalency, assuming Z L = −1, we obtain from eqs 3 and 5 that β(CF) = 0.90 eV/Å, which is three times smaller than the value β ≅ 2.7 eV/Å corresponding to CuF 6 4− at R m = 2.05 Å. This comparison strongly suggests that the intrinsic contribution, Δ int , is greatly due to the covalent bonding inside the CuF 6 4− unit. In the same way, the experimental 10Dq value of octahedral TM complexes is much higher than that calculated under the CF approach provided the right ⟨r d 4 ⟩ quantity is employed. 54, 61 For understanding the role played by chemical bonding upon Δ int , it is useful to explore how the energy levels are modified as far as we increase the size of the basis set, following an approach first proposed by Loẅdin. 63,64 In the present case, let us start with a basis set which includes only the two purely d-wavefunctions of the central cation that are degenerate under an octahedral symmetry |d a ⟩ = |3z 2 − r 2 ⟩ and |d b ⟩ = |x 2 − y 2 ⟩. Although in this first step there is no chemical bonding in the two a 1g (∼3z 2 − r 2 ) and b 1g (∼x 2 − y 2 ) levels, their energy is   The Journal of Physical Chemistry A pubs.acs.org/JPCA Article significantly raised by the repulsive interaction of electrons with the negatively charged ligands involved in the V M 0 term of eq 4. Also, in this step, the associated energies, E a and E b , of a 1g (∼3z 2 − r 2 ) and b 1g (∼x 2 − y 2 ) levels can be written as where E d corresponds to the octahedral situation (R ax = R eq ), while the corrections ε a 1 and ε b 1 are not strictly equal due to small CF effects under tetragonal symmetry. Indeed, in this first step Δ int ≅ ε b 1 − ε a 1 whose expression is just given by eq 5. In a second step, the ligand 2p and 2s wavefunctions are introduced in the basis set and then there is a change of both energy and shape of wavefunctions following the allowed 3d(TM)−2p(F) and 3d(TM)−2s(F) admixtures and the formation of antibonding orbitals. The linear combinations of 2p σ and 2s wavefunctions involving axial and equatorial ligands and transforming like a 1g and b 1g are shown on Table 2. In the case of the a 1g irreducible representation, there are two contributions termed as χ j eq (a) and χ j ax (a) (j = pσ, s), which can be mixed with d a = 3z 2 − r 2 , while d b = x 2 − y 2 can only be hybridized with the linear combinations χ j eq (b) (j = pσ, s) of equatorial ligands.
Accordingly, in this second step, the normalized |a 1g (∼3z 2 − r 2 )⟩ and |b 1g (∼x 2 − y 2 )⟩ wavefunctions have the form Although the λ j eq (a), λ j ax (a), and λ j eq (b) quantities are independent under D 4h symmetry (R eq ≠ R ax ), this is no longer true in the octahedral limit as they are related by the conditions It is worth noting that although wavefunctions such as |x 2 − y 2 ⟩ and |χ j eq (b)⟩ (j = pσ, s) are not orthogonal, the associated overlap integrals S pσ = ⟨x 2 − y 2 |χ pσ eq (b)⟩ and S s = ⟨x 2 − y 2 |χ s eq (b)⟩ are both only of the order of 0.1 at equilibrium 65 for MF 6 4− complexes (M = Cu, Ni, Co, Fe). For this reason, the total electronic charges q pσ eq (b) and q s eq (b) transferred from the central cation to 2pσ and 2s orbitals of equatorial ligands in the antibonding |b 1g (∼x 2 − y 2 )⟩ orbital are reasonably given by Similarly, the charges, q j ax (a) and q j eq (a) (j = pσ, s) transferred to axial and equatorial ligands in the antibonding |a 1g (∼3z 2 − r 2 )⟩ orbital can be approximated by In the present step, the values of orbital energies and the 3d−2p and 3d−2s admixtures come from the solution of the secular equation If we now work in second-order perturbations, the energy variations, ε a 2 and ε b 2 , induced by chemical bonding upon the | a 1g ⟩ and |b 1g ⟩ orbitals can be approximated by Here, E d − E p and E d − E s stand for the separation between the 3d levels of the central cation and the 2p and 2s levels of ligands in the complex. From the present calculations for Thus, if Δ int is mainly governed by the different chemical bonding in |a 1g ⟩ and |b 1g ⟩ orbitals, then In the same vein, within the second-order perturbation approach, the covalency parameters λ j eq (a), λ j ax (a), and λ j eq (b) are given by Thus, bearing eqs. 1, 7, 8, and 10−12 in mind, Δ int can finally be related to the charges transferred to 2pσ and 2s ligand orbitals as follows Therefore, according to eq 15, there are two contributions to the gap Δ int reflecting the bonding with 2pσ and 2s ligand The Journal of Physical Chemistry A pubs.acs.org/JPCA Article orbitals. That gap should be zero when the CuF 6 4− unit is perfectly octahedral (R eq = R ax ) and in fact eq 15, in conjunction with eqs 8−10, leads to Δ int = 0 in such a limiting case. Furthermore, eq 15 stresses that the gap, Δ int , is associated with variations of ligand charges on passing from an octahedral situation (η = 0) to a tetragonal one where R eq ≠ R ax , a view consistent with the general Hohenberg−Kohn theorem. 66 Indeed, the change of octahedral to tetragonal symmetry implies modifications of the electron−nuclei interactions (the so-called external potential in DFT 66 ) and necessarily of the associated electronic density. This change in the electronic density is then reflected on variations of ligand charges.
As the present analysis is based on a second-order perturbation approach, its validity requires that in a level like |b 1g (∼x 2 − y 2 )⟩ the charges q pσ eq (b) and q s eq (b) transferred to ligands are clearly smaller that the unity. The condition q pσ eq (b) ≪ 1 is better accomplished for fluoride than chloride or bromide complexes due to the higher electronegativity of fluorine (3.9) when compared to that of Cl (3.0) or Br (2.8). By contrast, the condition q s eq (b) ≪ 1 is much better fulfilled for all kinds of complexes due to the deep character of 2s(F), 3s(Cl), or 4s(Br) levels of free atoms. 61 For instance, the present calculations for CuF 6 4− units, discussed in the next section, give q pσ eq (b) < 0.23, while a much lower value, q s eq (b) < 0.06, is obtained for the charge transferred to 2s orbitals. This relevant fact also stresses the perturbative character of the 3d− 2s admixture.
The present approach focused on the Δ int gap of tetragonal units has also been employed for understanding the intrinsic and dominant contribution to the 10Dq value of octahedral complexes and its dependence upon the metal−ligand distance. 67 Interestingly, in the case of octahedral CrX 6 3− complexes (X = F, Cl, Br, I), it has been found 67 that the charge transferred to the valence ns level of ligands (n = 2, 3, 4, and 5 for F, Cl, Br, and I, respectively) in the antibonding e g (σ) orbital is always smaller than 0.1.
For the sake of clarity, when η ≠ 0 the Cu-wavefunction of the |a 1g ⟩ orbital is not a purely 3z 2 − r 2 orbital as it involves a small admixture (∼1%) of 4s(Cu). For elongated complexes, that admixture tends to enhance the electronic density of axial ligands, a matter discussed in ref 68. For obtaining such a 3d(Cu)−4s(Cu) hybridization in the present scheme, it is however necessary to go beyond the second-order approach. , varying the equatorial and axial metal−ligand distances but maintaining the mean distance R m = 2.05 Å. This allows one to calculate the Δ int gap simply by means of the Janak theorem 66 and to determine the charges transferred to 2pσ and 2s ligand orbitals for both a 1g and b 1g levels. Indeed, the use of the average a 1g 1.5 b 1g 1.5 configuration allows one to establish a reasonable link with the analysis carried out in Section 3 based on orbitals associated with a given electronic configuration.
The main results are collected in Table 3. The calculated Δ int values in Table 3 are consistent with the law embodied in eq 3 showing, in particular, that Δ int just changes sign on passing from R eq − R ax = 0.15 Å to R eq − R ax = −0.15 Å. A value β = 2.8 eV/Å for R m = 2.05 Å is derived from the present calculations.
As it is shown in Table 3, the charges transferred to 2pσ orbitals are, as expected, higher than those corresponding to 2s orbitals. However, when the tetragonality increases, the relative variation of q pσ eq (b) or q pσ eq (a) quantities is much smaller than that of q s eq (b) or q s eq (a) associated with 2s(F) orbitals. For instance, on passing from the octahedral situation (R eq = R ax = 2.05 Å) to R eq = 1.95 Å and R ax = 2.25 Å, q s eq (b) increases by 55% while q pσ eq (b) changes only by 2% and thus it remains nearly constant.
The quantities q s eq (b) and q pσ eq (b) are deeply related to the isotropic (A s ) and anisotropic (A p ) superhyperfine constants, respectively, for elongated CuF 6 4− units formed in Cu 2+ -doped fluoroperovskites as a result of a static Jahn−Teller effect. 30,32 Low-temperature electron paramagnetic resonance data indicate that whereas for CsCdF 3 :Cu 2+ A s = 160 (5) MHz, 32 it clearly increases up to A s = 183 (5) MHz 30 for KZnF 3 :Cu 2+ . By contrast, the measured values A p = 76 (5) MHz for CsCdF 3 :Cu 2+ and A p = 68 (5) MHz for KZnF 3 :Cu 2+ are coincident within experimental uncertainties. This fact is consistent with results for elongated NiF 6 5− species in different fluoroperovskites 72,73 involving the 3d 9 ion Ni + , where A s and q s eq (b) are highly sensitive to the actual value of R eq but not A p or q pσ eq (b). Indeed, whereas A p changes only by 3% along the series of fluoroperovskites, the variation of A s is 1 order of magnitude higher (30%).
Results are reported for both a 1g (∼3z 2 − r 2 ) and b 1g (∼x 2 − y 2 ) orbitals. The contributions Δ int (pσ) and Δ int (s) to the energy gap Δ int , derived from the Q pσ (a) − Q pσ (b) and Q s (a) − Q s (b) quantities, are also shown. It can be noted that the value of Δ int (pσ) + Δ int (s) is close to the gap, Δ int , obtained in a DFT calculation for every value of R eq and R ax . These facts already suggest that, according to eq 15, the gap Δ int is mainly due to the Δ int (s) contribution reflecting changes in the 3d−2s admixture when the tetragonality increases. This idea is certainly reinforced looking at results of present calculations embodied in Table 3. Indeed, such results prove that the obtained Δ int (s) contribution essentially accounts for the calculated gap Δ int at different values of R eq and R ax distances. For instance, for R eq = 1.95 Å and R ax = 2.25 Å, the results of Table 3 give Δ int (pσ) = −0.10 eV and Δ int (s) = −0.86 eV. Therefore, comparing these values with the figure Δ int = −0.84 eV derived from DFT calculations, we can conclude that such a gap is greatly the result of variations of the 3d−2s admixture with the tetragonality. Although this conclusion may be surprising, we can note that, from results of Table 3 for R eq = 1.95 Å and R ax = 2.25 Å, it is verified that  (17) just implying that Thus, the coupling of the 2s-ligand wavefunction, χ s eq (b), with |d b ⟩ = |x 2 − y 2 ⟩ is a little stronger than that for the 2pwavefunction, χ p eq (b). This conclusion is qualitatively consistent with the Wolfsberg−Helmholz guess 80 used before the arrival of ab initio calculations.
Bearing eq 3 in mind, we have also explored the dependence of Δ int and the β quantity upon the average value of the metal− ligand distance, R m . Varying R m in the range 1.95−2.05 Å we have found that Δ int and β are sensitive to the value of R m according to the law We have verified that the increase of Δ int and β when R m decreases is also followed by an increase of the Q s (a) − Q s (b) quantity while the contribution of Q pσ (a) − Q pσ (b) is again much less sensitive to the change of R m . This situation is thus akin to that described in Table 3.
4.2. Variation of 10Dq with the Metal−Ligand Distance for Octahedral Complexes. Bearing the present results and those previously obtained 67 on O h complexes in mind, we want now to explain quantitatively the origin of the dependence of 10Dq on the metal−ligand distance, R.
Experimental values for a variety of octahedral complexes 54,61,67,79 lead to an R dependence of the intrinsic contribution to 10Dq, (10Dq) int , given by 54 where the exponent t usually lies in the 4−6 range and thus it is close to the value t = 5 provided by CF theory. 17 According to previous results, (10Dq) int can reasonably be approximated by 67  (21) where the ratio α = (10Dq) int (s)/(10Dq) int has been found to be around 0.65 67 for the series of CrX 6 3− units (X = F, Cl, Br, I). Similarly to results of Section 4.1, the changes of (10Dq) int due to R variations are essentially driven by the (10Dq) int (s) contribution, reflecting the dependence of the q s charge of 2s, 3s, 4s, or 5s ligand orbitals on the metal−ligand distance. Thus, writing and considering small R variations (δR ≪ R), the following quantitative relation among t, t s , and α comes out α = t t s (23) Values of the exponent t s in the 6.5−8.5 range have been derived for doped cubic fluorides 54,67,76,79 and are responsible for the high sensitivity of the isotropic superhyperfine constant, A s , to R variations well observed experimentally. 54, 74−79 For Mn 2+ -doped cubic fluoroperovskites, a value t s = 8 has been obtained, 76 while from the parallel study of optical spectra, 81 t = 4.7 is found. These values are thus consistent with eq 23 and α ≈ 0.6.
Despite this fact and the early work by Sugano and Shulman, 16,17 proving that (10Dq) int essentially reflects the different covalent bonding in e g (σ) and t 2g (π) levels, experimental values of the exponent t close to 5 are still taken as a support to the validity of CF theory. 40 The sensitivity of 10Dq to R variations has a useful application for changing the shape of the fluorescence band in fluorides doped with Cr 3+ . Indeed, while the emission spectrum at ambient pressure of both Cr 3+ -doped KZnF 3 and K 2 NaGaF 6 lattices is a broad band arising from the 4 T 2 excited state, a sharp ruby-like spectrum coming from a 2 E first excited state is detected for pressures smaller than 15 GPa. 82,83 4.3. Key Role of Deep 2s(F) Orbitals in Chemical Bonding and the 2p(F)−2s(F) Gap: Microscopic Origin. The big separation, ε(2p) − ε(2s) ≅ 23 eV, between 2p and 2s levels of both free fluorine atom and the negative F − ion, 27−29 cannot be ascribed to a different extent of such orbitals. If we denote by R 2p (r) and R 2s (r) the radial functions of 2p and 2s orbitals and by R 1s (r) that of the inner 1s orbital, in Figure 2 are depicted the radial probability densities P j (r) = r 2 R j 2 (r) (j = 1s, 2s, 2p) corresponding to free fluorine atom. While the maximum of P 1s (r) is reached for r 1s = 0.06 Å, those for 2p and 2s orbitals both appear at higher distances but are very close. Indeed, as shown in Figure 2, the maximum of P 2p (r) is at r 2p = 0.36 Å and that of P 2s (r) at r 2s = 0.40 Å. It should be noted however that, when r > 0.75 Å, P 2p (r) is always a bit higher Figure 2. Radial probability densities P j (r) = r 2 R j 2 (r) (j = 1s, 2s, 2p) corresponding to free fluorine atom calculated by means of the atomic wavefunctions of Bunge et al. 29 The Journal of Physical Chemistry A pubs.acs.org/JPCA Article than P 2s (r). This situation is consistent with the calculated overlap integrals S pσ and S s for a series of octahedral MF 6 complexes 65 (M = Ni 2+ , Co 2+ , Mn 2+ , Fe 3+ , Cr 3+ , Mn 4+ ). Indeed, at equilibrium distances, S pσ and S s are both around 0.1 although S pσ is a little higher than S s . Nevertheless, due to the longer tail of the 2p orbital when compared to the 2s wavefunction (Figure 2), the dependence of S s upon the metal−ligand distance is stronger 73 than that of S pσ . The behavior of radial 2p and 2s wavefunctions depicted in Figure 2 is also consistent with the fact that the matrix elements ⟨d b |h − E d |χ pσ eq ⟩ and ⟨d b |h − E d |χ s eq ⟩ involved in eq 18 are comparable. Indeed, they just reflect that both functions look rather similar when r > 0.5 Å.
Thus, if the extent of 2p and 2s orbitals is comparable, it is now necessary to understand why ε(2p) − ε(2s) ≅ 23 eV for fluorine, while that gap is strictly equal to zero for the hydrogen atom and hydrogenic ions such as He + or Li 2+ .
It should first be stressed that the degeneracy between 2p and 2s orbitals in hydrogen is far from being accidental. In fact, it is the result of an invariant quantity, which appears however only when the potential energy, U(r), seen by the electron is strictly Coulombian 84 and thus has the form = − U r Ze r ( ) / 2 (26) at every distance r from the nucleus. When this condition is fulfilled, in addition to the angular momentum, L, the so-called Runge−Lenz operator, A, also commutes with the Hamiltonian of the problem. 84 The expression of that operator is given by where μ means the reduced mass of the hydrogen atom. The A operator connects the radial R 2s (r) and R 2p (r) wavefunctions and thus implies that the corresponding levels are to be degenerate. This operator was used by Pauli 85 for solving the energy spectrum of the hydrogen atom in the framework of the matrix quantum mechanics by Heisenberg and Born. The Runge−Lenz vector also plays a relevant role in studying the motion of planets around the sun. Its invariance implies that the position of the perihelion remains constant in time. 86 In an atom different from H or ions such as He + or Li 2+ , the self-consistent potential felt by a valence electron is not described by eq 26 in the whole range of distances to the nucleus as the net charge, Ze, seen by the electron depends on the r value. Therefore, for the fluorine atom, when r ≪ r 1s , Z ≅ 9, whereas when r 1s < r < r 2s , Z would be around 7 due to the screening by two inner electrons.
Bearing these facts in mind, the origin of the big separation between 2p and 2s levels in fluorine atom stems from the different behavior of the wavefunctions in the internal r < 0.1 Å region. As shown in Figure 2, P 2s (r) has a small maximum at r M = 0.04 Å with P 2s (r M ) = 0.4 Å −1 . By contrast, P 2p (r) is essentially zero in the 0 < r < 0.1 Å region as a result of the l(l + 1)/r 2 term in the radial equation making that R 2p (0) = 0 but R 2s (0) ≠ 0. Thus, the 2s charge P 2s (r M )Δr = 0.02e for Δr = 0.05 Å implies an energy gain for the 2s orbital with respect to the 2p one in that internal region, which can be estimated to be ∼20 eV using the virial theorem and Z = 7. It is worth noting that, if the 2p−2s separation in F mainly arises from the distinct behavior of both wavefunctions in the internal region, it is also consistent with a ε(2p) − ε(2s) value for F − that is only 5% higher than for the fluorine atom. 27,61 In the same vein, the value of ε(np) − ε(ns) for Cl − (n = 3) and Br − (n = 4) ions is only 2% higher than that for the corresponding free atom. 27,61 These considerations thus account for the big ε(2p) − ε(2s) value for fluorine and explain the fact that q s eq (b) ≪ q pσ eq (b). Moreover, due to the similar extent of the radial 2p and 2s wavefunctions when r > 0.5 Å, we can understand that the bonding with deeper 2s(F) orbitals is not negligible.
Nevertheless, it is surprising that the value of the Δ int gap essentially arises from the 3d−2s admixture rather than from the 3d−2p one despite q s eq (b) ≪ q pσ eq (b). However, from eq 15 and the results embodied in Table 3, this surprising conclusion is fully consistent with the near independence of q pσ eq (b) charges on the R ax − R eq value describing the tetragonal distortion. By contrast, q s eq (b) increases by 55% on passing from R eq = R ax = 2.05 Å to the D 4h geometry corresponding to R eq = 1.95 Å and R ax = 2.25 Å (Table 3).
This situation is thus akin to that encountered for the antibonding e g orbitals of octahedral complexes. 54,79 Therefore, for the e g (x 2 − y 2 ) orbital, q pσ eq is again found to be higher than q s eq but the dependence of q s eq on the metal ligand distance, R, is much stronger than that of q pσ eq . This important result has been explained 54,73,79 considering that λ pσ eq (e g ) depends on the ratio ⟨d(x 2 − y 2 )|h − E d |χ pσ eq ⟩/(E d − E p ). Accordingly, when R is reduced, the quantity |⟨d(x 2 − y 2 )|h − E d |χ pσ eq ⟩| increases roughly following the corresponding overlap integral S pσ . However, this increase is compensated by the additional rise of the charge-transfer excitation E d − E p due to the lessening of the metal−ligand distance on an isolated 3d complex. 54,73,87 By contrast, in the case of the admixture with the deeper 2s(F) orbital, the variation of q s eq with the distance essentially reflects that of [S s (R)] 2 . Examples of this behavior are shown in refs. 73,76,79

FINAL REMARKS
The present work highlights that the relation between spectroscopic data of TM compounds with the chemical bonding can be very subtle.
When in an isolated CuF 6 4− complex, we move from an initial octahedral situation (η = 0) to a tetragonal one with η ≠ 0 the energy of eigenstates and thus Δ int are modified. There are two sources of that change: (a) the dependence on η of the Hamiltonian and (b) the additional dependence on the distortion of the associated wavefunctions. The present analysis supports that the main contribution to Δ int arises from the variations undergone by b 1g (∼x 2 − y 2 ) and a 1g (∼3z 2 − r 2 ) wavefunctions when η is modified and thus the center of the gravity theorem 88 cannot be applied. Furthermore, Δ int is essentially associated with the variations experienced by the 2s(F) charge with η because the 2pσ(F) charge is nearly independent of the tetragonal distortion.
The present ideas can also be useful for understanding 3d complexes where fluorine is replaced by other halides or oxygen as ligand. Indeed, for these ligands, the ε(np) − ε(ns) gap is also significant and lies in the 14−18 eV range. 61 Taking as a guide the case of CdCl 2 :Cu 2+ , the tetragonal splitting, Δ int , has been measured 89 to be equal to −0.79 eV as a result of a static Jahn−Teller effect, leading to an elongated octahedral geometry. As there are no available data on the equilibrium geometry of the CuCl 6 4− unit in CdCl 2 , we have derived it through first-principles calculations giving R eq = 2.33 Å and R ax = 2.63 Å. On this basis, we obtain for CuCl 6 4− in CdCl 2 a value The Journal of Physical Chemistry A pubs.acs.org/JPCA Article β = 2.6 eV/Å that is comparable to that reported in Section 4.1 for the isolated CuF 6 4− unit. Furthermore, using these calculated R eq and R ax values in the CF expression for Δ int given in eq 5, we obtain for CdCl 2 :Cu 2+ a value Δ int (CF) = −0.12 eV, thus stressing the inadequacy of the CF approach.
From results of Section 4.1 the gap, Δ int , increases significantly upon applied pressures. This fact can be of interest in the realm of superconductor oxocuprates where the transition temperature, T c , is related 53 to the value of such a gap.
Although the ground state of MnF 6 4− and CrF 6 3− units in cubic lattices is orbitally singlet, this is no longer true for T 1g and T 2g excited states 90 where there is a coupling with the Jahn−Teller mode, e g , well seen through the progressions in luminescence spectra. 91 From calculations carried out on MnF 6 4− , Δ int = 0.147 eV for R ax − R eq = −0.06 Å was obtained, 90 thus implying β = 2.46 eV/Å. This figure is thus similar to that derived for the ground state of tetragonal CuF 6 4− units that also involves a divalent cation. Electronic levels lying far from the HOMO play also an important role in the realm of structural instabilities. 92 Therefore, due to the admixture of the ground with excited states via the electron-vibration coupling, the NH 3 molecule is non-planar 93−95 and an orthorhombic distortion appears in A 2 CuF 4 (A = K, Na) 4,9,49 and NH 4 Cl:CuCl 4 (H 2 O) 2 2−51 but not in NH 4 Cl:CuCl 4 (NH 3 ) 2 2− . 96 In the same vein, isolated Mn + ions in KCl:Mn +97 or Cu 2+ in SrCl 2 :Cu 2+54 move spontaneously from the cubic site to an off-center position, a situation not found in BaTiO 3 for an isolated Ti 4+ ion, stressing that ferroelectricity involves a cooperative distortion of all Ti 4+ ions. 97 Interestingly, the value of the excitation involved in the instability of the ammonia molecule goes up to 12 eV 94,95 although it is usually smaller for other systems 51