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On The Density Functional Theory Treatment of Lanthanide Coordination Compounds: A Comparative Study in a Series of Cu−Ln (Ln = Gd, Tb, Lu) Binuclear Complexes Marilena Ferbinteanu,† Alessandro Stroppa,‡,§ Marco Scarrozza,‡ Ionel Humelnicu,∥ Dan Maftei,∥ Bogdan Frecus,⊥ and Fanica Cimpoesu*,⊥ †

Department of Inorganic Chemistry, University of Bucharest, Dumbrava Rosie 23, Bucharest 020462, Romania SPIN Institute of Consiglio Nazionale delle Ricerche, L’Aquila 67100, Italy § International Centre for Quantum and Molecular Structures, and Physics Department, Shanghai University, 99 Shangda Road, Shanghai, 200444 China ∥ Physical and Theoretical Chemistry Department, Alexandru Ioan Cuza University, 11 Bd. Carol I, 700506 Iasi, Romania ⊥ Institute of Physical Chemistry, Splaiul Independentei 202, Bucharest 060021, Romania ‡

S Supporting Information *

ABSTRACT: The nontrivial aspects of electron structure in lanthanide complexes, considering ligand ﬁeld (LF) and exchange coupling eﬀects, have been investigated by means of density functional theory (DFT) calculations, taking as a prototypic case study a series of binuclear complexes [LCu(O2COMe)Ln(thd)2], where L2− = N,N′-2,2-dimethylpropylene-di(3-methoxy-salicylidene-iminato) and Ln = Tb, Lu, and Gd. Particular attention has been devoted to the Cu− Tb complex, which shows a quasi-degenerate nonrelativistic ground state. Challenging the limits of density functional theory (DFT), we devised a practical route to obtain diﬀerent convergent solutions, permuting the starting guess orbitals in a manner resembling the run of the β electron formally originating from the f8 conﬁguration of the Tb(III) over seven molecular orbitals (MOs) with predominant f-type character. Although the obtained states cannot be claimed as the DFT computed split of the 7F multiplet, the results are yet interesting numeric experiments, relevant for the ligand ﬁeld eﬀects. We also performed broken symmetry (BS) DFT estimation of exchange coupling in the Cu−Gd system, using diﬀerent settings, with Gaussian-type and plane-wave bases, ﬁnding a good match with the coupling parameter from experimental data. We also caught BS-type states for each of the mentioned series of diﬀerent states emulated for the Cu−Tb complex, ﬁnding almost equal exchange coupling parameters throughout the seven LF-like conﬁgurations, the magnitude of the J parameter being comparable with those of the Cu−Gd system.

1. INTRODUCTION

Since the wave functions needed in molecular magnetism problems are, in general, multiconﬁgurational, a natural computational approach is the framework of complete active space self consistent ﬁeld (CASSCF)7,8 calculations and related techniques, such as the subsequent second-order perturbation increments.9,10 A frequent instance of DFT11 as the method of choice in molecular magnetism is the use of the broken symmetry (BS) method for estimation of exchange coupling parameters.12,13 The computation of d-transition metal complexes (mononuclears or d−d polynuclears) does not imply large diﬃculties, although special attention should be devoted to achieve proper d-type orbital schemes. The quantum chemistry of lanthanides is more complicated, because of the non-auf bau conﬁguration

The study proposed in this paper originates from interests related with the molecular magnetism based on the f-type ions, but the general relevance of it is larger, regarding the bonding regime in lanthanide complexes. The electronic structure of ftype ions raises methodological concerns, such as the use of density functional theory (DFT) theory in situations nearby its validity limits. The chemistry of lanthanide complexes is drawing continuous interest,1 especially due to the large intrinsic magnetic anisotropy of most f-ions.2 The magnetic anisotropy is a key parameter for building single molecule magnets (SMM)3 and related systems. Molecular magnetism4 was developed in the past decades with direct interest in structure−property correlations,5,6 using computational chemistry and theoretical methods as valuable complements to the experimental ﬁndings and inquires. © 2017 American Chemical Society

Received: March 5, 2017 Published: August 7, 2017 9474

DOI: 10.1021/acs.inorgchem.7b00587 Inorg. Chem. 2017, 56, 9474−9485

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Inorganic Chemistry and weakly interacting features of the f shell, as we will illustrate in the discussion. Namely, in orbital energy schemes, the f shell can be buried below many ligand doubly occupied levels or, in the case of d−f complexes, below the d-type orbitals of the transition metal ion. The singly occupied molecular orbitals (SOMOs) belonging to the transition metal ions are, usually, frontier orbitals. Earlier,14 we realized pioneering breakthroughs in the multiconﬁguration treatment of lanthanides, avoiding the convergence problems by obtaining preliminary good orbitals for initializing the CASSCF calculations. Namely, we devised a strategy for going directly to the CASSCF treatments of lanthanide complexes, using a wave function assembled from fragments (the free lanthanide ions plus the rest of the complex). Several signiﬁcant applications of this methodology, e.g., the account of magnetic anisotropy, were illustrated.15,16 Initially,14 we claimed that the DFT methods cannot be used for the treatment of f complexes, because of their limitation to single determinant wave functions and given the alleged nonaufbau issues. However, later on, calculations of several Cu−Gd complexes were presented17,18 proving that in certain cases the DFT can be used on lanthanide compounds. The present study can be considered as an extension of the above cited works, discussing the limitations of the DFT in these circumstances. Recently, an alternative treatment on series of Cu−Gd complexes was reported by means of plane-waves basis set calculations.19 For the situation of a lanthanide ion with the nondegenerate f7 ground term (such as GdIII or EuII), the unrestricted single-determinant approaches seem to oﬀer a reasonable electronic structure solution, which can be obtained with common computational packages. While the use of DFT is relatively clear for the half-ﬁlled f7 conﬁguration, the other fn cases show degenerate ground states, where the DFT faces conceptual and technical problems. In this work, we study such cases, considering compounds of terbium ions as one of the simplest cases. We selected a prototypical Cu−Tb binuclear synthesized by Costes et al.20 showing single molecule magnet features. Before considering the Cu−Tb system, we will take the simpler Cu−Gd congener,21 as well as the hypothetical Cu−Lu system, to reveal aspects related to the f shell. In real compounds, with low symmetry, the degeneracy of the multiplets is usually raised by the ligand ﬁeld eﬀects. In addition, the true electronic structure of lanthanide sites is decisively inﬂuenced by the spin−orbit coupling, which leads to other systems of multiplets, accounted by the J quantum number instead of the pure orbital pattern driven by the L quantum numbers. On the other hand, since the two eﬀects are originating from diﬀerent sources, namely, the ligand ﬁeld from outside the ionic radius, while the spin−orbit is intrinsically from the inner atomic part, these can be analyzed by separate computational experiments. In fact, in the ligand ﬁeld phenomenology, the two-electron and spin−orbit parts can be taken as in the free ion (eventually adjusted to new values of parameters, to meet a particular system). A methodology to tackle the ligand ﬁeld (LF) problems inside of the DFT is referred as LFDFT, from the work of Daul and Atanasov, initially tailored for d-type complexes22 and later on the f-type ones.23 A recent work by Aravena et al. revisited by advanced multireference multiconﬁguration methods the systems treated in the pioneering applications of LFDFT on lanthanides,23 concluding the conceptual and technical superiority of CASSCF-like methods over the DFT strategies, in the approach of electron structure of lanthanide complexes.24

The DFT attempt to deal with the LF problems remains however an interesting ﬁeld for methodological concerns. One of us contributed also to the extended LFDFT approach, unifying in the algorithm the LF scheme with the two-electron and spin−orbit parameters on the d and f shells, taken together.25 The two-open-shell LFDFT was used to account for the 4f−5d transitions in lanthanide ions, in various environments, to engineer by ﬁrst principle approaches the lanthanide-based phosphors for turning the emission of blue diodes toward white light.25 The LFDFT is a combination of nonstandard DFT procedures and postcomputational analyses. Exploiting the fact that the conceptual DFT allows fractional orbital occupations,26 and that some packages, such as the ADF (Amsterdam Density Functional) code,27 have this feature implemented by proper keywords, the LFDFT uses the idea of average of conﬁgurations (AOC) as background for the reference Kohn−Sham orbitals of the system. Namely, a transition metal complex described as dn conﬁguration or a lanthanide mononuclear with fn occupation are treated by a set of ﬁve or seven Kohn−Sham orbitals, having imposed the fractional n/5 or n/7 occupation numbers. A successful AOC calculation is obtained if these orbitals are preponderantly made of the corresponding d or f type atomic orbitals (AOs). The AOC is a technical realization of the barycenter state heuristically conceived in ligand ﬁeld pictures as representing the perturbation smeared on a sphere, around the active site. In LFDFT, the self-consistent AOC orbitals are submitted to a series of computational experiments, generating the conﬁgurations from the combinatorial approach of n electrons in the ﬁve d-type or seven f-type AOs (now with integer electron populations). These conﬁgurations are not real states, but useful conventional objects, needed to extract the LF and twoelectron parameters, to set a post-DFT ligand ﬁeld modeling. Since most of the regularly used DFT programs do not have the facility to run the AOC procedure and to generate diﬀerent Slater-determinant-like conﬁgurations, the LFDFT strategy cannot be generaly approached by any general computer package. Other interesting analyses aiming for LF information from DFT calculations are due to Schäﬀer et al.28 We are also interested in alternative procedures, focusing on lanthanide compounds. As mentioned previously, there are already several DFT calculations reported on Cu−Gd binuclear models.17−19 Here, we go further, presenting the DFT treatment on Cu−Tb congeners.

2. COMPUTATION DETAILS The calculations were done with diﬀerent codes. For computations based on Gaussian-type orbitals, we used the GAMESS code,29 with the SBKJC30 eﬀective core potentials and basis sets for Gd, Tb, and Lu, while we used 6-311G* for the Cu, N, or O and 6-31G for the C or H atoms. The molecular structure is taken for the experimental Cu−Tb system, assuming the same skeleton for the Cu−Gd and Cu− Lu systems. The density functional theory (DFT) calculations were performed, comparatively with B3LYP, BP86, and SVWN31 functionals. Another series of calculations was done with the VASP package,32 which makes use of plane-wave basis sets, using the PBE33 functional. Further modeling was done with the Amsterdam density functional (ADF) code,27,28 using the TZP (triple-ζ with polarization) basis set for all the atoms. The ADF allows a fractional occupation number and imposing non-auf bau occupation schemes. A particular keyword, called “SlaterDeterminants,” enables numeric experiments with diﬀerent conﬁgurations realized on the grounds of frozen orbitals, 9475

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Inorganic Chemistry previously prepared and stored in binary ﬁles (TAPE21 type) loaded at the initialization of the calculation.

The second ligand room, open to host the large lanthanide ion and the other terminal ligands, is sketched by the phenolate groups (which are establishing a bridge with the transition metal ion) and the rather distant ether oxygen atoms from CH3O groups attached to the benzene rings of the ligand frame. Because of the two phenolate groups, the large ligand is negatively charged, so that the complex formed only with the transition metal, [LCu], can be regarded as a neutral ligand, able to approach the lanthanide trivalent ion. The three smaller ligands are carrying, each, mononegative charges, the whole [LCu(O2COMe)Ln(thd)2] complex being neutral. The molecule is close to the Cs point group, having the methyl-carbonate anion, (O2COMe)−, approximately sitting in the quasi-symmetry plane, perpendicularly to the mean plane of the {N2O2O2} coordination sequence of the main ligand. The molecular frames shown on the bottom part of Figure 1 are viewed perpendicularly to the Cs quasi-symmetry plane (left inset) and parallel to it, orthogonal to the {N2O2O2} coordination frame of the Schiﬀ base ligand (right side inset). The diketonate thd− anions are coordinating as chelate arches, one incorporated in the approximate symmetry plane, while the other roughly perpendicular to it. In total, the coordination number of the lanthanide ion is nine, while the transition metal is placed near the basis of a square pyramid. The Cu−Ln axis is approximately incorporated in the {N2O2O2} coordination mean plane of the binucleating ligand. 3.2. The Cu−Lu Congener of the Cu−Tb Complex. The non-aufbau nature of the lanthanide complexes is illustrated considering the lutetium congener, where the f shell is closed, with the f14 conﬁguration. The orbitals having main f content, shown in Figure 2, are located very low in the energy scheme (see the left side panel). The SOMO corresponding to the Cu(II) site is the highest occupied level, with many doubly occupied orbitals existing between d and f levels. Aside from the “bar code” representation of all the MO energies, the distribution f atomic orbitals (AOs) are shown as a vertical histogram. One ﬁnds that the f-type AOs are concentrated in seven MOs having population lines reaching almost the +2 margin. Although the orbital energies are not identical with the ligand ﬁeld scheme, their splitting can be taken as a qualitative measure of the LF strength. Thus, in cm−1, the relative MO energies are in the following sequence, {0, 0, 110, 241, 263, 439, 505}, with a couple of accidentally degenerate lowest levels. One ﬁnds that the gaps are in the range expected for a LF split on the f shell in usual complexes.36 The middle inset from Figure 2 shows magniﬁed the split of the f-type MOs. Because of the above-noted accidental degeneracy, only six lines are visible. The right side of Figure 2 shows the shapes of the discussed MOs, observing their nature as almost pure f AOs (mutually remixed, in comparison to the standard functions, because of the LF action). Assuming that the same pattern is realized in the case of other lanthanide ions, with a partly ﬁlled f shell, the unpaired electrons will be placed on inner orbitals, resulting then in a non-auf bau status, in the framework of spin restricted approach. Such a spin restricted non-auf bau conﬁguration is very diﬃcult to converge, because the electrons from upper doubly occupied levels ﬂow in the f area, during the iterations. Also, one may face a nonphysical mixing with distant ligand functions having, accidentally, energies close to the f orbitals in the molecule.

3. RESULTS AND DISCUSSION 3.1. The Selected Case Study. We considered a series of Cu−Ln isostructural complexes with binucleating ligands belonging to the chemical outcome of Costes et al.,20,21 considering the prototypic features of this class of compounds encountered in both early34 and recent reports.35 Among the many available Costes-type compounds, we selected a pattern with rich coordination at the lanthanide site, made of two thd = tetra-methyl-heptanedionato chelating ligands (analogues of acetyl-acetonate with tert-butyl instead of methyl groups), a methyl-carbonate ligand bridging the d and f centers, aside from the general encapsulation of both metal ions ensured by the large frame of L2− = N, N′-2, 2-dimethyl-propylenedi(3methoxy-salicylideneiminato). The general formula reads [LCu(O2COMe)Ln(thd)2]. Particularly, interesting is the Cu−Gd system,21 oﬀering the resolution of exchange eﬀects, and the Cu−Tb congener,20 which, showing single molecule magnet (SMM) behavior, is relevant for anisotropic properties. The Cu−Gd and Cu−Tb experimental molecular geometries are closely similar. For full comparability, we took the geometry of the Cu−Tb for all the Cu−Ln cases. As an extreme simpliﬁcation, the Ln center can be replaced with the closed-shell Lu(III) ion. For other numeric experiments, the copper center can be replaced by Zn(II), taking, e.g., binuclear Zn−Tb, conﬁning it then to numeric experiments regarding the ligand ﬁeld regime of the lanthanide ion. The structure of the complexes is shown in Figure 1. The large ligand has two rooms, the smaller one, having an equatorial plane realized by two imine nitrogen atoms and two phenolate oxygens, occupied preferentially by the transition metal ion.

Figure 1. Scheme of the modeled complex (top part) and two orientations of experimental molecular geometries of the Cu−Tb complex (bottom part). The hydrogen atoms are not represented. 9476

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Figure 2. Orbital scheme of the Cu−Lu hypothetical congener of the considered Cu−Tb system resulting from restricted B3LYP calculations. Left panel: the bar diagram representation of MO energies. The abscissa corresponds to the representation of the f AO content in the MO, with the concentration of the ligand ﬁeld sequence in the −0.560 to −0.562 hartree energy interval. This zone is magniﬁed in the middle panel. The MOs can be visualized on the right side, concluding their AO-like nature.

3.3. The Cu−Gd Congener of the Cu−Tb Complex. Although a restricted open shell DFT treatment of lanthanide complexes is diﬃcult (while potentially useful, if it is wanted to set a LF postcomputational interpretation), a convergent wave function can be obtained working in the unrestricted scheme. As will be clariﬁed in the following, this solution occurs due to the large energy separation between fα and fβ sets, or between the occupied versus empty f-type MOs. The simplest situation is in the case of Gd(III) complexes, where the unrestricted occupation concerns the seven fα-type MOs. Then, a useful preamble to the Cu−Tb system, where we shall face the quasidegeneracy problem, is to consider the Cu−Gd analogue, having a lanthanide with a nondegenerate ionic ground state. The Cu−Gd complex is realized experimentally,21 with a structure similar to those of the Cu−Tb one, as shown in Figure 1. For comparability, we will consider the Cu−Gd system at the experimental geometry of the Cu−Tb one. In the Gd(III) case, the f-type MOs belong to the α occupied set, while the virtual set is entirely of β nature. From Figure 3, one notes a large gap between the two subsets of f-type MOs. This fact is in sharp contrast with the ligand ﬁeld phenomenological ideas, where a unique orbital set is assumed. For LF aims, although aware of relative independence of the two spin sets in unrestricted calculations, one may expect not very large α vs β energy separations. Such a situation happens in the case of the d-type complexes, where the occupied functions with preponderant metal-ion character fall in the HOMOs zone, while the virtuals related with empty d-type components are expected near the frontier, among LUMOs. The f-type complexes show a diﬀerent scheme, not only with occupied α functions much below the HOMO energy but also with β virtuals rather high above the LUMO line. This aspect was not previously noticed by computational chemists, raising here the question of its physical meaning. A simple answer is to see the situation as a technical necessity, taking only the total molecular energy as a signiﬁcant outcome, rather than the individual MOs. If Figure 3 is compared with the related

Figure 3. Orbital diagram of the Cu−Gd system, resulting from unrestricted B3LYP calculations. The content of f AOs into MOs is shown by population histograms. The α orbitals are shown on the right side, while the β populations are drawn on the left.

diagram from Figure 2, one may say that the upward shift of the β virtuals occurs to avoid the non-auf bau situation, appearing if the empty functions would have remained around the occupied ones. However, the energy of the fα type MOs remains approximately in the same position of the scale as described previously for the completely ﬁlled f14 shell of the Cu−Lu complex. The f content of the relevant sequence of α orbitals is close to the unity, as illustrated by the bars on the right-bottom part of Figure 3. Conversely, the f content in the β virtuals is smeared, as visible in the upper-left corner of the ﬁgure, having almost a Gaussian distribution, with a half-width of about 0.2 hartree. This spreading is in formal contradiction with the idea 9477

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are similar in absolute contours but that the BS has the spin polarization reverted around the Cu atom. A part of the spin density in the d-type coordination sphere is delocalized over the surrounding atoms. The density on the f center is almost perfectly spherical, due to the overall quasi-symmetry imposed by the half-ﬁlled shell. For comparison, we considered the BS-DFT problem in plane-waves frame, using the VASP code32 and a DFT+U approach in the Dudarev formalism.38 One obtains then a J = 2.65 cm −1 coupling parameter when the imposed U parameters19 are Uf(Gd) = 6 eV and Ud(Cu) = 6 eV. The coupling strength decreases to J = 0.85 cm−1 when the DFT+U is switched-oﬀ to simple DFT (Uf(Gd) = 0 and Ud(Cu) = 0). Then, the caught range is comparable with the experimental value and the previous result with Gaussian bases. 3.4. The Emulation of Lanthanide Multiplets in the Cu−Tb Complex. We will explore in the following the limits of the DFT approach, taking the Cu−Tb complex. The free terbium(III) ion has a degenerate groundstate, 7F, but the degeneracy is lifted in a low symmetry environment. Formally, under this circumstance, one may use DFT, being in the limits of the theorem requesting nondegenerate ground states. However, the gaps between the states of the split 7F multiplet are expected to be small, in the range of tens and hundreds of cm−1 units, facing a challenging methodological inquiry. The f8 conﬁguration of the Tb(III) can be described as consisting of seven α electrons accomplishing a half-ﬁlling of the f shell, plus one β electron. Equivalently, in a restricted scheme, it can be regarded as containing one doubly occupied f orbital and six unpaired electrons. The components of the 7F term can be thought of as resulting from the successive placement of the β electron on the seven f orbitals. From a restricted-type perspective, the 7F term can be presented as moving the doubly occupied orbital over the seven available orbitals. In a phenomenological sense, the conﬁguration interaction (CI) matrix describing the split of the 7F term in a given molecular environment is the same as the ligand ﬁeld (LF) matrix, because of the one-electron eﬀective nature of the problem. The CI matrix of the many-electron problem diﬀers from the LF one by a shift in the diagonal part, experiencing the same two-electron amount for all seven conﬁgurations, in the limits of LF theoretical premises. A clear account of the LF problem is realized on a computational level by complete active space self consistent ﬁeld (CASSCF).7,8 As pointed out in the Introduction,14−16 we initialize the multiconﬁguration procedure with orbitals produced merging those of the free Tb3+ ion and the MOs of the [LCu(O2COMe)(thd)2](3−) molecular complex. With the above initialization, the CASSCF gives straightforwardly the energy levels outlined in Table 2. The copper center is accounted for by one electron, located in one active orbital. Then, aside from the eight electrons in seven orbitals corresponding to the f8 conﬁguration of Tb(III), the multiconﬁguration calculations are of a CASSCF(9,8) setting (i.e., nine electrons in eight orbitals). The calculation produces two series of spin states, one for high spin S = 7/2 (labeled HS) and one for low spin S = 5/2 spin (labeled LS), resulting from the coupling of the STb = 3 and SCu = 1/2 metal ion spins. The canonical orbitals are produced by an average over seven states, each spin series revealing the LF split originating from the ground term of Tb(III). The comparison of the two series of CASSCF states

of the f shell conserving its quasi-AO identity in the MO diagrams. However, in the strict sense, the virtuals are not contributing to the properties of the system, being just a computational leftover. To check whether this electronic conﬁguration accounts for the physical features of the system, we take as a test the simulation of the d−f exchange coupling parameter by the broken symmetry (BS) approach.12 For a Cu−Gd complex, the BS method consists of two unrestricted calculations for the conﬁgurations named high spin (HS) and broken symmetry (BS), with Sz(HS) = 7/2 + 1/2 = 4 and Sz(BS) = 7/2 − 1/2 = 3 projections. The exchange coupling parameter is obtained picking from each output the energy, E, and the spin square expectation value, ⟨S2⟩, putting them in the following formula:37 J=

E BS − E HS ⟨S2⟩HS − ⟨S2⟩BS

(1)

The exchange parameter estimated with B3LYP functional, JCuGd = +1.85 cm−1 is in agreement to the experimental one,21 JCuGdexp = +2.1 cm−1, if we consider the parameters related to the following form of the Heisenberg exchange Hamiltonian:

Ĥ = −2JS1̂ ·S2̂

(2)

The Mulliken spin populations from Table 1 show that the computation respected the idea of BS treatment, having values Table 1. Details of the BS-DFT Calculations on the Cu−Gd System, with B3LYP Functional Mulliken spin populations state

energy (atomic units)

⟨S2⟩

Gd

Cu

HS: {Gd(7α), Cu(α)} BS: {Gd(7α), Cu(β)}

−4440.30370943 −4440.30365016

20.013 13.012

7.030 7.028

0.727 −0.725

close to the {Gd(7α), Cu(α)} vs {Gd(7α), Cu(β)} conﬁgurations for HS vs BS states. The +0.7 and −0.7 spin populations on the copper site can be taken as representative for the respective α and β conﬁgurations of the d ion. The departure from unity is due to the fact that d-type magnetic orbitals contain ligand contributions, usually amounting to a few tens of percent. In turn, the eﬀective contribution of the Gd(III) comprises almost seven electrons, in line with the quasi-atomic nature of the half-ﬁlled f shell. The BS result is also well illustrated in Figure 4, showing the spin density. One observes that the HS and BS density maps

Figure 4. Spin density maps from the B3LYP calculations on the Cu− Gd complex: (a) the HS case, (b) the BS conﬁguration. The surface is drawn at 0.001e/Å3, with the α spin density in blue and the β spin density in yellow. 9478

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Table 2. CASSCF Calculation of the Levels Associated with the Ligand Field Split and the 7F Term of the Tb(III) Site in the Cu−Tb Complexa 1 2 3 4 5 6 7

EHS (Hartree)

ΔEHS (cm−1)

⟨S2⟩HS

ELS Hartree)

ΔELS (cm−1)

⟨S2⟩LS

J (cm−1)

−4438.774720 −4438.774666 −4438.773758 −4438.773478 −4438.771996 −4438.771866 −4438.771012

0.0 11.7 211.0 272.5 597.7 626.3 813.8

15.75 15.75 15.75 15.75 15.75 15.75 15.75

−4438.774696 −4438.774643 −4438.773735 −4438.773455 −4438.771974 −4438.771839 −4438.770985

5.3 16.7 216.1 277.5 602.6 632.2 819.6

8.75 8.75 8.75 8.75 8.75 8.75 8.75

0.76 0.72 0.73 0.72 0.70 0.85 0.83

a

The calculations are performed on S = 7/2 and S = 5/2 spin states corresponding to the respective parallel and anti-parallel coupling between the STb = 3 and SCu = 1/2 metal ion spins.

Figure 5. Density diﬀerence maps obtained subtracting the total density of the Cu−Gd system from the corresponding series of diﬀerent orbital conﬁgurations of the Cu−Tb congener. The shape of the surfaces (contours with six or eight lobes) suggests the accommodation of the β electron in f-type MOs, when comparing the f8 and f7 systems. The surfaces are drawn at 0.01 e/Å3 values.

Table 3. DFT Emulation of the Levels Associated with the Ligand Field Split and the 7F Term of the Tb(III) Site, by Initiating Calculation with Permuted MOs from the Cu−Gd Complex 1 2 3 4 5 6 7

EHS (Hartree)

ΔEHS (cm−1)

⟨S2⟩HS

EBS (Hartree)

ΔEBS (cm−1)

⟨S2⟩BS

J (cm−1)

−4458.057022 −4458.056245 −4458.055981 −4458.055841 −4458.055368 −4458.050106 −4458.046252

0.0 170.6 228.5 259.4 363.1 1517.9 2363.9

15.762 15.762 15.762 15.762 15.761 15.761 15.762

−4458.056979 −4458.056201 −4458.055937 −4458.055790 −4458.055322 −4458.050046 −4458.046201

9.5 180.4 238.2 270.4 373.3 1531.1 2375.1

9.762 9.762 9.761 9.761 9.761 9.761 9.762

1.59 1.62 1.61 1.84 1.70 2.21 1.87

states: 0.0, 0.2, 102.9, 103.5, 207.3, 219.5, 288.1, 292.5, 319.0, 440.2, 441.1, 618.6, and 618.8 (all values in cm−1). This series corresponds to the LF splitting of the J = 6 multiplet of the Tb(III) ion. One notes a series of quasi-degenerate couples corresponding to non-null ± Jz projections. The value at 319.0, having no quasi-degenerate neighbor, corresponds to the Jz = 0 case. Particularly, the ground state quasi-doublet is a prerequisite for the SMM features of the Cu−Tb complex.20 Although the CASSCF-SO calculations seem to describe very well the LF regime and subsequent properties of the system, we will not advance further along this line, since the multiconﬁguration treatment was taken only for comparative purposes. We will attempt to produce diﬀerent components originating from the Ligand Field (LF) splitting of the 7F multiplet, changing by permutations the initial orbital guess. Namely, taking the natural orbitals generated by the Cu−Gd calculations (i.e., postcomputationally handled functions having a restrictedlike occupation pattern, with occupation numbers close to 1), we prepared the initialization of the input in several ways, targeting the orbital where the β electron of the f8 conﬁguration of Tb(III) will be placed (see the Supporting Information).

reveals the d−f spin coupling ported on each component of the 7 F multiplet. Thus, for each LF level, one may assign a J coupling parameter, practically computed with a formula similar to eq 1, except that now the ⟨S2⟩ expectation values refer to the S(S + 1) exact eigenvalues. It is very interesting to see that the J parameters are almost equal for all seven orbital states, around the J = 0.76 cm−1 average value. The total spin quantum number does not inﬂuence the distribution of the states assigned to the LF splitting. In addition, repeating the calculation for a Zn−Tb hypothetical complex, where there is no spin coupling and the molecular spin is that of the Tb(III), S = 3, the series of CASSCF states shows almost the same relative energy values as for the S = 7/2 and S = 5/2 spin multiplets of the Cu−Tb molecule. The total splitting (see Table 2), about 800 cm−1, and the successive gaps are in the range expected for the ligand ﬁeld eﬀects on the f shell. Table 2 shows that the ﬁrst and second LF states are quite close in energy. This can be qualitatively related with the accidental degeneracy of the lowest two orbital energies noticed for the Cu−Lu complex. The spin−orbit (SO) calculations based on CASSCF states of the Zn−Tb complex yield the following sequence of lowest 9479

DOI: 10.1021/acs.inorgchem.7b00587 Inorg. Chem. 2017, 56, 9474−9485

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Inorganic Chemistry

Table 4. AO Analysis of the Restricted Natural Orbitals with Almost Double Occupation and Predominant f Character, in the Seven Diﬀerent SCF Solutions of the Cu−Tb Complex Assimilated with the F Ground Term of the f8 Conﬁgurationa y(3x2−y2) f1 f2 f3 f4 f5 f6 f7 column norm.

0.000 −0.002 0.129 0.532 0.455 −0.002 −0.420 0.826

xyz −0.500 0.970 −0.121 0.282 −0.319 −0.308 −0.454 1.299

yz2

z3 −0.001 0.012 −0.428 0.009 −0.374 −0.768 0.139 0.966

0.002 −0.002 0.465 −0.685 0.047 −0.001 −0.372 0.909

xz2 0.000 −0.001 −0.282 0.007 −0.516 0.000 0.169 0.612

z(x2−y2) 0.860 −0.009 0.006 −0.002 −0.375 −0.523 0.392 1.143

x(x2−3y2) 0.000 −0.003 0.550 −0.012 0.177 0.000 0.427 0.719

row norm. 0.995 0.970 0.902 0.912 0.943 0.979 0.951

The MOs, labelled f1−f7 are given in rows. The f-AOs given as heads of columns are ascribed in the following molecular alignment: the Cu−Tb on the x axis and the z axis perpendicular on the mean ligand plane (see Figure 1). a

sketched by Kahn39,40 and conﬁrmed by us, in the paper, claimed as the ﬁrst multiconﬁgurational ab initio account of a d−f complex.14 The revealed situation suggests that the mechanism outlined for frequent Cu−Gd ferromagnetism is general also for any Cu−Ln system. It is only the more complex situation of other lanthanides, driven by ligand ﬁeld and spin−orbit factors, that precludes a clear resolution of the exchange parameters as a basis of discussion for this eﬀect. The wave functions of the unrestricted calculations taken as surrogates of the diﬀerent LF states can be characterized in an analytical manner, decomposing the natural orbitals having main f content and an occupation number close to the value of 2. This corresponds to the above-discussed rationalization of the Slater determinants serving as a basis of the 7F term of Tb(III), characterized by the doubly occupied component running over the f8 conﬁgurations. Equivalently, these can be presented as the eﬀective orbitals carrying the β density of the f8 conﬁgurations, whose density contours are caught in Figure 5. Table 3 shows the decomposition of these MOs in f-type AOs, in the considered conventional orientation (see caption of Table 3). A global measure is realized taking the norms of lines and columns (the square root from the sum of squared coeﬃcients, for the corresponding vectors). The norms along the lines are measuring the f-type AO content of each natural orbital. The values tabulated at the end of f1−f7 lines, ranging between 0.90 and 0.99, illustrate that the corresponding natural orbitals are almost pure f atomic functions. The norms of the columns should ideally also have values close to unity, meaning that each standard AO occurs once in the set, retrieved from its smearing among all the MOs. One notes that certain AOs, such as xz2 and x(x2−3y2), having rather low norms, seem to be not represented well in the given MO set. Conversely, the xyz and z(x2−y2) functions, with supra-unitary norms, are encountered “in excess.” This suggests that the given set is not a complete basis, neither as a transformation of approximate f AOs nor in the many-electron sense, if the represented doubly occupied natural orbitals are taken as signatures for the Slater determinants of the f8 ground multiplet. It is expected that the natural MOs forming the Slater determinant and having occupations close to the unity are representing the complement of the doubly occupied function. If we denote by C the 7 × 7 matrix contained in Table 4, a measure of the completeness is given by the eigenvalues of the C·CT product. Under ideal conditions, the eigenvalues should be unity. In our situation, we have the following eigenvalues: 0.029, 0.233, 0.408, 0.635, 0.865, 2.035, and 2.123. A couple of

In this manner, if the states with diﬀerent starting orbitals converge to diﬀerent solutions, one may say that, at least empirically, a series of conﬁgurations similar to the split of the F multiplet was mimicked. The fact that we obtained diﬀerent conﬁgurations in the SCF processes is proven in Figure 5, taking the diﬀerence density maps with respect to the Cu−Gd reference. This is a global characterization, since the excess electron (from f8 vs f7 conﬁguration) is smeared among several MOs. It is clear that the drawn proﬁles are of f AO type, having six- or eight-lobe shapes. As the orientation of the lobes varies, one certiﬁes that the calculations caught diﬀerent results, in the desired sense, getting LF conﬁgurations from the f8 set. One may see from Table 3 that a part of the simulated states show energies in the range expected for a LF split, comparable also with the CASSCF results from Table 2. However, the last two states are deﬁnitely too high. At the same time, it must clearly be seen that the computed conﬁgurations cannot be claimed as physical states. Or, in other words, the resulted energies cannot be conceived as obeying, all together, the phenomenology of a conﬁguration interaction. In spite of such indeterminacies, the numerical experiment is worth analysis. The situation is somewhat similar to the BSDFT approach. The BS spin conﬁgurations are not real states but computed experiments which can yield the parameters of the Heisenberg spin Hamiltonian. Similarly, the diﬀerent orbital conﬁgurations generated previously are not claimed as LF states but as energy expectation values relevant for a LF eﬀective Hamiltonian. For each of the seven conﬁgurations taken as surrogates of the F-type levels, we considered the HS and BS calculations, tuned by the spin coupling with the copper center. The gap between HS and BS energies at a ﬁxed conﬁguration yields the exchange coupling parameter. It is interesting to see that the magnitude of the coupling is almost the same on all seven LFtype states. This feature was observed also in the CASSCF calculations. The J values estimated by DFT are larger than the CASSCF ones, by factors ranging from about 2 to 3. At the same time, the average parameter for Cu−Tb exchange, J ∼ 1.78 cm−1, is similar to those estimated by BS-DFT for the Cu−Gd complex. This suggests a relative independence of the exchange mechanism from the conﬁguration of the lanthanide center, concluding then that a spin polarization driving force acts similarly under all circumstances. In terms of the full conﬁguration interaction, the mechanism implies, aside from the d and f unpaired electrons, ligand bridge orbitals and 5d virtuals of the lanthanide site. This is the qualitative mechanism 9480

DOI: 10.1021/acs.inorgchem.7b00587 Inorg. Chem. 2017, 56, 9474−9485

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Inorganic Chemistry

in Figure 6 only the f1 case (the lowest energy), the other situations showing a similar pattern. Note that the density

0 and 2 eigenvalues represent two nonindependent functions, having to take only one of the involved eigenvectors. The above list shows two low eigenvalues and two others close to the ∼2 magnitude. This means that, instead of having seven independent f-type functions, we collected only ﬁve independent functions, while the other two are merely accidents. This explains why ﬁve of the energy expectation values from Table 3 are in the range of LF split (hundreds of reciprocal centimeters), while the remaining two are outside this scale. In accordance with the well-known limits of DFT, one may not expect to account self-consistently states diﬀerent from the ground level. However, when symmetry exists, one may consider the possibility of diﬀerent variational lowest energy solutions for each irreducible representation, provided that these do not correspond to degenerate situations. In our case, there is no rigorous symmetry, but we mentioned a good closeness to the Cs point group. In addition, there is a hidden supplementary symmetry contained in an eﬀective LF modeling. Namely, the so-called holohedrization eﬀect41,42 concerns the fact that the Hamiltonian set on a single-shell basis, d or f, is expanded only in even spherical harmonic functions. If another version of ligand ﬁeld modeling is taken (e.g., the angular overlap model, AOM),43−45 avoiding the explicit expansion of the Hamiltonian with spherical harmonic operators, the eﬀective artiﬁcial symmetry persists. Since the odd symmetry components cannot be accounted for in a LF Hamiltonian, this means that the actual symmetry is always raised up, as if an inversion operation acts. Thus, the action of a ligand occurs not as the asymmetric M-A perturbation, but as if the ligand is smeared along an inversion axis (A/2)−M−(A/2). When a given electron structure problem is set with MOs eﬀectively behaving as an atomic shell, namely f in our discussion, then the holohedrization is also tacitly imposed on the collected outcome. In our case, a combination of an approximate symmetry plane and a hidden inversion, would bring the eﬀective symmetry of the lanthanide site to C2h. In addition, if it is assumed that the mean plane of the binucleating ligand can be perceived as a symmetry element, one may arrive at the D2h point group, where the f shell behaves as the au + 2b1u + 2b2u + 2b3u representation. In this case, there are four symmetry channels to be spanned by the f shell and therefore the hope for diﬀerent “ground states” corresponding to the au, b1u, b2u, and b3u representations. We got ﬁve states with a reasonable LF spacing, i.e., accidentally a little better than one may expect from pseudosymmetry reasons. The total LF spacing or certain successive gaps are overestimated. Probably the cause is the fact that, in an unrestricted frame, the separation recorded for α vs β orbital energies is not in line with the LF modeling paradigm. However, the outcome of distinct DFT calculations assignable to LF conﬁgurations answers the challenge of advanced exploitation of the DFT frame, up to its limits of interpretation. If we denote by ε the diagonal matrix made from energies listed in Table 3 (the ΔEHS values, shifted with the barycenter in zero) and ignoring that the C matrix taken from Table 4 is rather imperfect, in principle, one may produce the LF matrix by the following product VLF = CT·ε·C. However, since the DFT seems not able to produce a correct range of ε eigenvalues, then the LF parameters are faulted. In the following, we will analyze the energy diagrams of the converged unrestricted MOs of the Cu−Tb system, illustrating

Figure 6. Orbital diagram of the unrestricted B3LYP calculation on the lowest energy conﬁguration of the Cu−Tb system (the f1 case). The patterns of the other six SCF solutions look similar. The content of ftype AOs into MOs is shown by histograms.

corresponding to the β electron of the f8 conﬁguration spreads among few orbitals, with energy placed between the seven α occupied functions and the six β virtuals. The set of α f-type MOs seems collaterally perturbed, one level being shifted distinctly up by about 0.05 hartree, while the remaining six are kept in a narrow interval. Furthermore, the virtual orbitals are visibly split into two groups, as a byproduct of spin polarization. The situation contradicts the usual intuition and premises of the ligand ﬁeld phenomenology. On the other hand, it can be speculated as a consequence of a non-auf bau-like tendency, since the occupied α and β f-type orbitals are placed much below the frontier energy. With such a pattern, there is no chance to take orbital energies as an approximation of the LF scheme, although in the previous cases of closed-shell Lu(III) and a half-ﬁlled shell of Gd(III) complex, this convention could be adopted, in semiquantitative respects. With the BP86 functional, taken as an example of the generalized gradient approximation (GGA), the energies of the computed states are 0, 122, 163, 1618, 1697, 2713, and 2755 cm−1. Here, only two gaps are in the LF proper range, the other levels being quite overestimated. Within the local density approximation (LDA), taking the SVWN functional, the simulated LF states are 0, 5, 7, 60, 72, 920, and 920 (in cm−1). Although, here, apparently, all the states are in the range of desired small energies, the very small spacing between the lowest three states suggests that the results are not reasonable. The locality premises (i.e., borrowing a functional designed from homogeneous electron gas) are not well satisﬁed if the f shell is concerned, because of its steep radial variation. Then, one may say that the hybrid functional performs better than the LDA or GGA ones, because the fraction of exchange accounted for by exact Hartree−Fock integrals helps in providing a realistic trend. 9481

DOI: 10.1021/acs.inorgchem.7b00587 Inorg. Chem. 2017, 56, 9474−9485

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Inorganic Chemistry 3.5. Checking the Ligand Field Density Functional Theory (LFDFT) Methodology. To complete the perspective on the account of the LF bonding regime via DFT, we will turn to the algorithm of computation and analysis called LFDFT, designed by Daul and co-workers.22,23 As pointed out in the Introduction, the key step is the average of conﬁgurations calculation (AOC), which implies a canonicalization of restricted-type. For the Cu−Tb complex, the MO assignable to the d-type unpaired electron of the copper atom should receive occupation 1, while the MOs presumed to be of f type should have 8/7 − 1.143 for a series of seven levels. The previous considerations pointed toward a non-auf bau situation. However, this situation is practically impossible to catch, since it is hard to ﬁgure the position of a deep f-type orbital window and to have it stabilized along the iterations. On the other hand, one may try to impose now the aufbau scheme. We have not succeeded in the convergence for the Cu−Tb complex, probably because of the arbitrary mix of the Cu and Tb orbitals with the 1 and 1.143 occupation numbers, while they are enforced together in the frontier part. However, we got the aufbau solution for the Zn−Tb analogue, discarding the problem with the occupation of the d-type MOs, introducing the d10 site. The B3LYP calculation reached only moderate convergence, which was not improved tightening the integration threshold or by the available iteration controls. Under these conditions, the computed orbital energies and simulated LF conﬁgurations resulted in much overestimated gaps. In this circumstance, we will discuss only the results based on the BP86 functional. Thus, the orbital energies, computed with BP86 under the canonicalization of restricted type with the 8/7 fractional occupation of the f-type seven MOs, are 0, 113, 597, 710, 1742, 2428, and 2783 (in cm−1). Using these orbitals, the ADF code27,28 can generate conﬁgurations with a keyword called “SlaterDeterminants.” Considering the given problem, we generated the seven conﬁgurations moving the β electron of the f8 conﬁgurations along the LF-type MOs. The energies are 0, 465, 476, 926, 1260, 2330, and 2336 (in cm−1). For the numeric experiments with LF-like conﬁgurations, we used the unrestricted scheme. The “SlaterDeterminants” calculation does not iterate over orbitals, keeping them as obtained from restricted-type self-consistency. Both the MO energies and also those of the emulated conﬁgurations are showing four levels in the acceptable range, while three others are clearly overestimated. The situation seems a bit worse than in the previous approach with permuted orbitals as a tentative “synthesis” of LF-like states. Probably this is the price paid for the enforced aufbau scheme, which brings some undesired artiﬁcial mixing with the orbitals from the large ligand. Note that there is no clear correlation between the MO energies and those of the conﬁgurations. Actually, the above given ordering is realized with the β electron placed in the following sequence of LF-type MOs: 6, 7, 3, 2, 1, 5, and 4. The situation illustrates the intricacies and somewhat nonsystematic trends of various DFT treatments attempted as LF computation experiments. The physics of ligand ﬁeld states is, no doubt, formally better accounted for in the frame of multiconﬁguration methods, such as CASSCF. However, considering that CASSCF does not contain nondynamical correlation, it is yet tempting to explore the limits of the DFT, as the LFDFT philosophy does. In previous works conﬁned to the ionic small systems,24,25 beneﬁting yet from the advent of high local symmetry, the

LFDFT methodology, in ADF code, seemed to work very well. However, as revealed by the actual case study, it may be a bit problematic in the case of large systems, with organic ligands and low symmetry. It must be recalled that we do not see the generated DFT conﬁgurations as true LF terms. These are expectation values of imposed conﬁguration, which can reveal certain LF parameters. Actually, the LFDFT algorithm does not break the DFT limitations to nondegenerate ground states, in spite of the fact that it tentatively approaches multiplets. Thus, judging heuristically, one must realize that the ban of degenerate ground states from the DFT capabilities comes from the fact that, in the case of equivalent states, one may deform the electron cloud without energy costs. Then, the density cannot act as the unique driver of the system. If, however, one imposes the AOC condition on a set of degenerate or quasi-degenerate orbitals, one approaches in fact a unique state of the system, equivalent with the state average in CASSCF procedures. Thus, the AOC orbitals are optimized in limits obeying the DFT requirements. Then, in the “SlaterDeterminants” step, since no iteration is done, no variational rule is claimed to be at work. The method emulates the diagonal elements of an eﬀective Hamiltonian but must clearly realize that it cannot oﬀer any information about the nondiagonal elements.

4. CONCLUSIONS Due to the electronic structure particularities of lanthanide ions, the treatment of magnetic f, d−f, and f−f complexes is a subtle deal, particularly in the frame of DFT, considering its limitations to nongenerate groundstates. We assigned the problems to a non-auf bau pattern of the lanthanide ions in the molecule, analyzing this issue on a series of Cu−Lu, Cu−Gd, and Cu−Tb complexes. While in the solid-state methods, based on plane waves, the DFT+U treatment is tailored to cope with a part of lanthanide ion speciﬁcs,19,38 the molecular approaches are yet an open ﬁeld of debate. In the molecular DFT methods based on Gaussian atomic orbitals, the Gd(III) ion is relatively easily approachable because of its nondegenerate ground state.17,18 We advanced more on this line, analyzing the Cu− Tb case, to probe to what extent the ions with a degenerate ground state are treatable by DFT. The ligand ﬁeld eﬀects are removing the degeneracy of the ion in the molecule, but the small gaps keep the systems in quasi-degenerate situations, challenging the limits of DFT applicability. We found that a series of diﬀerent self-consistent solutions can be emulated by permuting the starting orbitals. A part of the resulted states are in the range expected for ligand ﬁeld splitting, while the remaining others are strongly overestimated. Although these cannot be fully assigned to ligand ﬁeld states, one may value them as relevant numeric experiments, in a manner similar to broken symmetry treatments, which do not render physical spin eigenvectors but enable the extraction of exchange coupling parameters. The assessment dedicated to the account of ligand ﬁeld has larger importance, considering the temptation of DFT-based modeling in various corners of structural chemistry and materials sciences, including the felements. We answered the technical and conceptual issues raised by such problems, leaving also challenging questions open for further debate. 9482

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.7b00587. Used molecular geometry and computational methodologies illustrated in sample input ﬁles and commented keywords, further numeric experiments and considerations related with the plane-wave DFT+U calculations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] ORCID

Fanica Cimpoesu: 0000-0003-0036-304X Notes

The authors declare no competing ﬁnancial interest.

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ACKNOWLEDGMENTS We are thankful for the ﬁnancial support of UEFISCDI, project PNIII-P4-IDEI PCE 108/2017. A.S. and F.C. are thankful for travel grants convened by the Bilateral Agreement CNR-Italy/ Romanian Academy (Joint Projects 2014-2016) N.0006436 (27.01.2014).

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REFERENCES

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DOI: 10.1021/acs.inorgchem.7b00587 Inorg. Chem. 2017, 56, 9474−9485

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DOI: 10.1021/acs.inorgchem.7b00587 Inorg. Chem. 2017, 56, 9474−9485