Chemical Tuning of Magnetic Exchange Couplings Using Broken-Symmetry Density Functional Theory

Using broken-symmetry Kohn-Sham Density-Functional Theory calculations, it is demonstrated that the ferromagnetic or anti-ferromagnetic character of two prototypical binuclear copper complexes can be modified, both in the sign and in magnitude, by means of chemical substitutions operated on the bridges connecting the two magnetic centers. The level of detail provided by the magnetic exchange decomposition in terms of direct exchange, kinetic exchange and core polarization puts forward the relative importance of the different bridges. At variance with the principal bridge for which chemical substitutions modify both the direct and the kinetic exchange contributions, modifications of the secondary bridge only affect the magnitude of the antiferromagnetic kinetic exchange mechanism, ultimately allowing for a direct control of the magnetic character of the modified compound.


Introduction
Magnetic exchange coupling J is one of the most important quantum interactions between two (or more) unpaired electrons. 1 In the case of a finite-size systems, this interaction is important for designing molecular compounds with controlled magnetic properties. When J is non-zero in a molecule, it favors either the parallel alignment of the magnetic electron spin angular momenta (ferromagnetic interaction) or the anti-parallel one (antiferromagnetic interaction), eventually driving the magnetic properties of the system. Accordingly, the determination of J, either experimentally [2][3][4] or by means of quantum-chemistry calculations, [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] is of tremendous importance for understanding the behavior of magnetic open-shell molecular systems.
Different physical mechanisms may compete and determine the sign and intensity of J. In the case of two spin-1/2 magnetic centers, the most important ones are the direct exchange and the kinetic exchange between the magnetic electrons, and the polarization of the nonmagnetic electrons. While high level quantum-mechanical methods can accurately evaluate the importance of these mechanisms, eg by selectively considering classes of excitations in configuration interaction calculations, 21 their computational cost limits their applicability to small molecules. On the other hand, the Kohn-Sham density-functional theory combined with the so-called broken-symmetry approach (BS-KSDFT) is a popular and effective method which often provides semi-quantitative evaluations of J. 7,15,22 By means of selective freezing of a subset of orbitals, some of the present authors recently introduced a decomposition scheme [23][24][25] which has been essentially applied to centro-symmetric molecules until now. 26 Thanks to the recent implementation of this method in the Orca package, 27 the J decomposition is now available for molecules belonging to various symmetry groups, hence expanding its range of application to systems where the magnetic centers are bridged by chemically different moieties, eg π-conjugated or not, with electron-donating or electron-withdrawing substituents, etc. Hereafter, we report J decompositions for various non-centrosymmetric copper dinuclear compounds. All of them feature oxygen-or carboxylic-bridged cores, with 2 different chemical substitutions. In all, it is demonstrated that the deep understanding of the physical contributions to J can be used to in-silico design molecular compounds with selected magnetic properties.

Molecular models
Among all the possible, available, non-centrosymmetric copper dinuclear complexes that ex- (hereafter identified as PATFIA) and its closely related analogue [{Cu(tmen) (hereafter identified as PATFOG). Both PATFIA and PATFOG, whose synthesis have been reported in 2005, 28 feature two different bridges ( Figure 1). Moreover, inspection of the corresponding experimental crystallographic structures reveal a close contact with 1 or 2 perchlorate counterions. From experiment, the two copper ions in PATFIA have been shown  Table 1, together with the model details. In almost every reported studies, the perchlorate counterions have been ignored, while the ferrocene moiety is simply replaced by a methyl group or a hydrogen atom. Looking at Table 1 entries, it is quite obvious that the correct evaluation of PATFIA J value depends on a large number of parameters: the DFT xc-functional, the geometry, the basis set, the method (BS vs other ones) etc. While some models perform wrong, even qualitatively (ferromagnetic instead of anti-ferromagnetic), some other ones overshoot the coupling by two orders of magnitude. On the other hand, besides the original calculations performed on PATFOG, 28 we are aware of only a very recent theoretical study dealing with this system, 36 for which a large set of DFT functionals have been tested, with J values ranging from −3.6 cm −1 to 59.9 cm −1 . Notice that all but one DFT functional predict PATFOG as ferromagnetic.
In the present work, the starting structures have been retrieved from the Cambridge Structural Database. 37 Comparing PATFIA and PATFOG structures, several differences show up. First, one of the bridges is hydroxo in PATFOG, while it is methoxo in PATFIA.
In the case of PATFOG, the hydroxo moiety is oriented in such a way it can form a hydrogen bond with one perchlorate molecule. This orientation is thought to govern the magnetic coupling character of these systems. Second, external ligands are dimethylethylenediamine in PATFIA, while they are tetramethylethylenediamine in PATFOG, hence feature two more methyl groups. Accordingly, it would be interesting to assess the influence on J of these two extra chemical groups. Third, the perchlorate anions are distributed differently in PATFIA and PATFOG, the latter featuring a hydrogen bond with the hydroxo bridge. While these anions are usually removed from the theoretical models, the original work from López et al 28 insisted on the structural importance of perchlorate counterions, constraining the orientation of the hydroxo bridge, ultimately modifying PATFOG magnetic properties. Together with the evaluation of the ferrocene effect on J (which has been recently confirmed to be innocent 35 ), we intend to study how these chemical differences and substitutions influence the different physical contributions to the magnetic exchange couplings in PATFIA and PATFOG, thanks to the BS-KSDFT-based decomposition scheme, as presented in the next section.

Methods
Our for which J can be decomposed using the Heisenberg-Dirac-van Vleck model as: ie, J is the sum of three important contributions: the direct exchange, the kinetic exchange and the so-called core polarization of the non-magnetic orbitals. The first step of the procedure consists in the calculation of the magnetic orbitals. Using a high-spin state (a triplet, in the case of two spin-1/2 sites) RO solutions (T,RO), both singly-occupied molecular orbitals are localized, using one of the available localizations schemes in Orca. 27 Such schemes include the popular Pipek-Mezey 38 and Boys 39 localization schemes as well as the more recently developed method of intrinsic atomic orbitals. All schemes are combined with a robust and efficient search algorithm that is designed to avoid local minima of the localization functional to the largest extent.
The remaining steps are hereafter summarized: 1. Flip the spin of one magnetic orbital to produce the broken-symmetry M s = 0 wavefunction (BS,RO), without optimizing any orbital, and evaluate the direct exchange contribution J 0 : 2. Relax the magnetic orbitals in the M s = 0 wavefunction, in the field of the frozen non-magnetic orbitals, allowing the magnetic orbitals to delocalize from one center to the other (BS,UFC): 3. Relax the core orbitals in both high-spin (M s = 1) and broken-symmetry M s = 0 wavefunctions (respectively T,UFM and BS,UFM): 4. Total magnetic coupling is obtained with the complete relaxation of orbitals in both high-spin (M s = 1) and broken-symmetry M s = 0 wavefunctions (respectively T,U and BS,U): Freezing orbitals is achieved thanks to the Local Self-Consistent-Field method, 40 in which the relaxed orbitals are variationally optimized under the constraint of being orthogonal to the frozen ones.
Because the various orbital relaxations are performed in the unrestricted formalism, the resulting Kohn-Sham wavefunctions are spin-contaminated, ie characterized by a S 2 value different from 2 (high-spin) or 0 (low-spin). Consequently, we make use of spindecontamination schemes in all the three steps. While the Yamaguchi procedure can be applied safely to the direct and kinetic exchange contributions, it is no longer the case for core polarization, as demonstrated elsewhere. 25 The selection of a xc functional is known to be very delicate in BS-KSDFT J calculations. Even the physical interpretation of the contributions to J depends on the chosen xc functional. 41 In this work, we have followed the very recent advice by Illas and coworkers,36 recommending to use the very common B3LYP functional for obtaining qualitative (ferromagnetic vs anti-ferromagnetic) to semi-quantitative (a given order of magnitude) results.
Our choice is further motivated by the main goal of the current study, in which we will compare chemically-related compounds which differ only by one or two substitutions. Regarding the basis set, we have also followed recommendations by others, selecting a fairly large basis set for copper atoms (def2-QZVPP) while keeping reasonable ones for the other atoms

Results
We have first tested the influence of the ferrocene and solvent molecules on the J decomposition results. Similarly to what has been reported by other (references in

Effect of chemical substitutions
PATFIA and PATFOG feature two different bridges which chemical nature may modify the coupling between their two copper-centered magnetic electrons. One of them is either hydroxo or methoxo, with only one oxygen atom. Oxygen-bounded moieties will be called R1 hereafter ( Figure 2). As already pointed in reference 28, their different orientations are In the following, all results have been obtained with tight convergence thresholds: 10 −8 a.u.for energy and 10 −5 a.u.for one electron energy change. In Table 2  As previously found, 28 R1 is the main responsible for the magnetic character of these PATFIA-like models. Indeed, when R1 is hydrogen, the J value is consistent with PATFOG one, even if the PATFIA geometry is kept. The direct exchange J 0 and the kinetic exchange ∆J ke are also governed by R1 only, being reduced when hydrogen replaces methyl. However, the kinetic exchange contribution is more intense than the direct exchange when R1 is a methyl group, while this is the contrary when R1 is a hydrogen atom. Polarization of the non-magnetic electrons (∆J cp ) remains small. Regarding the nature of R2, hydrogen or methyl, it looks like all the contributions to J are rather insensitive to it. The last two columns in Table 2 show two interesting features. The magnitude of J 0 depends mainly on the chemical nature of R1, not on its position. Conversely, the R1 position/orientation affects ∆J ke .
In order to maximize the R1 and R2 substitution effects, we now report J decomposition results when either R1 or R2 is a methyl group, while the other bridge is complemented with electro-donating or electro-withdrawing chemical groups. We first consider R2 = CH 3 and some carbon-centered substituents for R1 (Table 3). ∆J cp is predicted to be negligible.
Three different kinds of R1 effects can be highlighted.
• Methyl and tertiobutyl: similar direct and kinetic exchange contributions, the latter larger than the former, resulting in an overall anti-ferromagnetic coupling.
• Trimethylformate: reduced direct and kinetic exchange contributions, the latter being still larger than the former, the total magnetic exchange remaining anti-ferromagnetic.
• Trifluoromethyl: even more reduced direct and kinetic exchange contributions, the latter smaller than the former, resulting in an overall ferromagnetic coupling.
Because these R1 groups are all positioned in equivalent orientations, roughly parallel to the CuOCu plane, we can't rely on the usual explanation that was previously used to rationalize the different magnetic couplings in PATFIA and PATFOG. 28 Indeed, the chemical nature of R1 is responsible for the variations of all contributions to J.
Now we consider R1 = CH 3 , as it is the case in PATFIA, and a more extended set of substituents as R2 (Table 4). Whatever the nature of R2, ∆J cp remains negligible with Table 4: J decomposition (in cm −1 ) for PATFIA-like models, with R1= CH 3 and various R2 groups.

Analysis and discussion
As evidenced in Table 2, the magnetic character of PATFIA-like systems is mainly determined by the nature and the position of R1: anti-ferromagnetic when the bridge is methoxo, while ferromagnetic when it is hydroxo. In the original work by Lopez, 28 but already evidenced earlier, 42 this observation was explained as resulting from the different orientation of R1, more precisely to its tilt angle with respect to the CuOCu plane. If R1 is close to this plane, the compound is anti-ferromagnetic. If the out-of-plane angle is larger than 35 • , it is ferromagnetic. In our calculations, the same change of magnetic character is due to a larger decrease of the kinetic exchange contribution (about 50%) with respect to the decrease of the direct exchange contribution (about 25%), when going from methoxo to hydroxo. On the other hand, substituting methyl (resp. hydrogen) with hydrogen (resp. methyl) in R1, keeping the same original orientation, indicates that the direct exchange does only depend on the nature of the R1 group. However, the kinetic exchange does depend on both the orientation and nature of R1.
Both contributions can be rationalized by tiny but sizable variations of atomic spin populations (Table 5).
Direct exchange (J 0 ) ferromagnetic contribution is an intrinsic property of the magnetic Table 5: Mulliken atomic spin populations on selected atoms, at various stages of the J decomposition procedure. If R1 is methyl, then the total CH 3 group spin population is reported. Values in italics correspond to non-optimized positions of R1 (out-of-plane CH 3 , in-plane H). Atom labels are given in Figure 2. contribution, Figure 3a indicates that spin density is more delocalized when R1 = CF 3 , the fluorine centers carrying a noticeable population. In other words, the F electronegative centers attract electron density outside from the bridge Cu − O − Cu, which in turns lowers J 0 . About the kinetic exchange contribution, the comparison of the two pictures in Figure   3b clearly shows that the spin density variation is reduced when R1 = CF 3 , in agreement with a smaller ∆J ke value. In both compounds, this variation is restricted to the copper centers and the atoms directly connected to them. Accordingly, the spin density attracted by the fluorine atoms is not modified by the kinetic exchange contribution.
Perfect agreement between both kinetic exchange contributions fully justifies the relevance of the t and U parameters. Whatever the CF 3 position, the U parameters keeps the same value, implying that the energy of the ionic configurations in which both magnetic electrons are on the same Cu center are not affected by the position of the substituent. Conversely, the hopping energy t increases by a factor of two between R1 and R2, explaining the corresponding increase of ∆J ke . If compared to the unsubstituted PATFIA model in which CH 3 occupies both the R1 and R2 positions, it appears that t is mostly affected, and reduced, when CF 3 is in R1 position. In other words, the presence of this electron-rich substituent at position R1 decreases a lot the probability to transfer one magnetic electron from one Cu center to the other one, while it slightly increases it when placed at position R2.

Conclusions
In this article, we have presented how the new implementation in Orca of the magnetic exchange coupling decomposition, based on broken-symmetry Kohn-Sham DFT calculations, opens the way to the computer-assisted design of substituted dinuclear non-centrosymetric compounds with either a ferro-or an anti-ferromagnetic character.
In details, we have confirmed that reduced models of PATFIA and PATFOG related systems are characterized by magnetic contributions qualitatively similar to the ones obtained with the experimental crystallographic structure. In agreement with the observation already made by López and coworkers, 28 the orientation of the hydroxo vs methoxo substituents explains the different magnetic characters of PATFIA and PATFOG. However, the decomposition has allowed us to refine this interpretation by pointing out the importance of the chemical nature of these substituents, especially in the case of the direct exchange contribution.
By means of selective chemical substitutions on each bridge, we have designed PATFIAlike models which, if one can experimentally synthesize them, would exhibit either a ferromagnetic or an anti-ferromagnetic character by tuning one or several contributions to the magnetic exchange coupling. Last, we have rationalized the effect of one particular substituent, trifluoromethyle, which has been shown to drastically reduce the kinetic exchange contribution in one particular position.

Supporting Information Available
Influence of ferrocene and solvent molecules on J decomposition. Effect of the basis set quality and influence of relativistic effects. Spin densities of some of the compounds. Coordinates of all the compounds.