Evaluation of the Giant Ferromagnetic Ï-d Interaction in Iron

Article pubs.acs.org/JPCA

Cite This: J. Phys. Chem. A 2018, 122, 1678−1690

Evaluation of the Giant Ferromagnetic π−d Interaction in IronPhthalocyanine Molecule Rocío Sánchez-de-Armas and Carmen J. Calzado* Departamento de Química Física, Universidad de Sevilla, c/Profesor García González, s/n, E-41012 Sevilla, Spain ABSTRACT: The interaction between itinerant π and localized d electrons in metal-phthalocyanines, namely, Jπd interaction, is considered as responsible for the giant negative magnetoresistance observed in several phthalocyanine-based conductors, among many other important physical properties. Despite the fundamental and technological importance of this on-site intramolecular interaction, its giant ferromagnetic nature has been only recently demonstrated by the experiments conducted by Murakawa et al. in the neutral radical [Fe(Pc)(CN)2]·2CHCl3 (Phys. Rev. B 2015, 92, 054429). In this article, we present the theoretical evaluation of this interaction combining wave function-based electronic calculations on isolated Fe(Pc)(CN)2 molecules and density functional theory-based periodic calculations on the crystal. Our calculations confirm the ferromagnetic nature of the π−d interaction, with a coupling constant as large as Jπd/kB = 570 K, in excellent agreement with the experiments, and the presence of intermolecular antiferromagnetic interactions driven by the π−π overlap of neighboring phthalocyaninato molecules. The analysis of the wave function of the ground state of the Fe(Pc)(CN)2 molecule provides the clues of the origin of this giant ferromagnetic π−d interaction. bis(ethylenedithio)-tetraselenafulvalene),26,27 among others.28 The nature of the π−d interaction Jπd in metal-phthalocyanine has been a matter of controversy for some time,17,29−31 and it has been only recently that its nature and magnitude have been experimentally established for iron-phthalocyanine by Murakawa et al.32 The strategy employed consisted in the preparation of isolated Fe(Pc)(CN)2 molecular solution, through the in situ oxidation of K2[Fe2+(Pc2−)(CN−)2] by iodine bromide. The titration with IBr introduced two unpaired electrons, one resulting from the oxidation of Fe2+ to Fe3+ and the second one on the highest occupied molecular orbital (HOMO) Pc π orbital. Hence the reaction provides neutral [Fe3+(Pc−)(CN−)2] radical units. In the isolated solution, the magnetization measurements are just due to the intramolecular interaction between the Pc π-electron spin and Fe 3d moment. The Curie constant at saturation indicated the ferromagnetic nature of this interaction. Its magnitude has been determined from magnetization measurements of the single crystals of the neutral radical [Fe(Pc)(CN)2]·2CHCl3. They reveal a giant ferromagnetic intramolecular Jπd interaction in the Fe(Pc)(CN)2 molecule, with a value as large as Jπd/kB > 500 K, much larger than the intermolecular antiferromagnetic interaction of Jinter/kB = −19.5K. Actually, the π−d interaction in metalphthalocyanine is supposed to be exceptionally large, at least 1 order larger than those among known π−d systems,28 due to

1. INTRODUCTION Metal-phthalocyanine molecules have been extensively studied for a long time due to their versatility and wide range of applications as dyes,1 gas sensors,2 solar cells,3,4 biomimetic catalysts,5−7 optical and electronic devices,8 and quantum information.9 Materials based on metal-phthalocyanine molecules are molecular conductors of different dimensionality depending on the coordination of the metal atom. They form one-dimensional face-to-face stacking conductors in absence of axial ligands as in M(Pc), M = Ni, Cu, ..., and Pc = phthalocyaninato,10−12 while they behave as multidimensional conductors when the central metal is coordinated to axial ligands as in M(Pc)(L)2, with M = Fe, Co, Cr, Mn, Ru, ..., and L = CN, Br, Cl.13−21 If the metal atom has a non-null magnetic moment, the interaction between the Pc π and metal d electrons, that is, between the itinerant π-electrons and localized magnetic moments on the metal atom, modifies the transport properties of the molecular conductor. This interaction is at the origin of many interesting phenomena with remarkable importance in the field of molecular electronics and spintronics, such as charge ordering, Kondo effect, and giant magnetoresistance. In fact, this strong π−d interaction is considered responsible for the negative magnetoresistance17,22−24 of TPP[Fe(Pc)(L)2]2, PXX[Fe(Pc)(CN)2],13 and the single-component molecular conductor25 [Mn(Pc)(CN)]2O (L = CN,Br,Cl, TPP = tetraphenylphosphonium, PXX = peri-xanthenoxanthene). Also it is related to the field-induced superconducting state of the molecular conductor λ-(BETS)2FeCl4 (BETS = © 2018 American Chemical Society

Received: November 17, 2017 Revised: December 22, 2017 Published: January 16, 2018 1678

DOI: 10.1021/acs.jpca.7b11356 J. Phys. Chem. A 2018, 122, 1678−1690

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The Journal of Physical Chemistry A the fixed geometry imposed by the Pc molecule.16 For instance, in λ-(BETS)2FeCl428 and (BDA-TTP)2FeCl4 (BDA-TTP = bis(1,3-dithian-2-ylidene)tetrathiapentalene)28,33 the interaction between the π conduction electrons and the 3d local moments located in the close FeCl4−2 counterions is of the order of Jπd/kB ≈ 10 K. It is the aim of this work to determine the nature and amplitude of this interaction by means of state-of-the-art quantum chemistry methods. Although previous attempts to theoretically establish the nature and magnitude of this interaction can be found in the literature, they assigned different nature to this interaction in the case of ironphthalocyanine. In fact, Matsuura et al.31 proposed that the π−d interaction in TPP[FePc(CN)2]2 is antiferromagnetic and as large as −100 K based on the eigenvalues of a model Hamiltonian constructed from the Fe 3d and Pc π orbitals. On the one hand, as the authors indicated, Jπd “can be either ferromagnetic or antiferromagnetic depending on model parameters because of competition among various processes”, but comparing with experimental data and taking into account the magnetic anisotropy induced by the spin−orbit effect, they were in favor of the antiferromagnetic nature of this interaction. On the other hand, CASPT2 calculations performed by Yu et al.17 on TPP[FePcL2]2 for L = CN, Cl, Br indicated that this interaction is ferromagnetic, with different amplitude depending on the axial ligands: Jπd/kB = 180, 99, and 117 K, for L = CN, Cl, and Br, respectively. It is worth noting that the theoretical value for L = CN is largely underestimated with respect to the experimental data of Murakawa et al.32 (Jπd/kB > 500 K) and that the trend of the calculated Jπd couplings (CN > Br > Cl) does not correlate with the changes observed in the magnetoresistance (CN > Cl > Br). With a combined strategy using the Difference Dedicated Configuration Interaction (DDCI) method and density functional theory (DFT)-based periodic calculations on different magnetic solutions, it is possible to accurately determine the magnitude of this interaction in [Fe(Pc)(CN)2]·2CHCl3 and also the presence of intermolecular interactions between neighboring Fe(Pc)(CN)2 molecules in the crystal. Our calculations confirm the ferromagnetic nature of the π−d interaction, with a coupling constant as large as Jπd/kB = 570 K, in good agreement with the recent experiments by Murakawa et al.32 The periodic calculations corroborate the ferromagnetic nature of the intramolecular interactions and indicate the presence of antiferromagnetic intermolecular interactions, driven by the π−π overlap of the neighboring Pc molecules. The analysis of the wave function of the Fe(Pc)(CN)2 molecules provides clues about the mechanism controlling the giant ferromagnetic π−d interaction. In the work by Murakawa et al.32 the single crystals of the neutral radical [Fe(Pc)(CN)2]·2CHCl3 were fabricated in situ by electrochemical oxidation of THA[Fe3+(Pc2−)(CN−)2] (THA = tetraheptylammonium), but no X-ray data were provided. Our calculations are then based on the available X-ray crystallographic data of the isostructural [Co(Pc)(CN)2]· 2CHCl3 system.34,35 Since the size of Fe3+ ion is almost the same as that of Co3+ (ionic radius r(Fe3+) = 0.64 Å and r(Co3+) = 0.63 Å), the geometry of the Pc unit is not affected, and both crystals are isomorphous as occurs with TPP[Fe(Pc)(CN)2]2 and TPP[Co(Pc)(CN)2]2 compounds.16 Figure 1 shows three views of the crystal unit cell. The phthalocyanine units are arranged in two-dimensional (2D) sheets in the ac plane, separated by CHCl3 molecules. Along

Figure 1. Unit cell of Fe(Pc)(CN)2·2CHCl3 crystal, along the a (top, left), b (top, right), and c (bottom, left) axis. Schematic representation of the 2D phthalocyanine sheets (bottom, right).

the a axis the contact between two neighboring molecules takes place through the π−π overlap of one benzene ring of each Pc unit (Figures 2 and 5), while two benzene ring overlap occurs along the c axis (Figures 2 and 6), which correspond to the type-B and type-A modes of the Pc π−π overlaps in ref 36. The main intermolecular distances along the a and the c axes are shown in Table 1. Chloroform molecules are placed between the 2D phthalocyanine sheets introducing a large gap of ∼10 Å along the b axis. The arrangement adopted by the phthalocyanine molecules in the 2D sheets is schematically shown in Figure 1.

2. METHODOLOGY 2.1. Periodic Calculations. The Fe(Pc)(CN)2·2CHCl3 crystal was studied within DFT using the Vienna ab initio simulation package (VASP) code,37−40 employing the generalized gradient approximation (GGA) with the Perdew−Burke− Ernzerhof exchange-correlation functional41,42 and projectoraugmented wave (PAW) potentials.43,44 An effective Hubbard correction of 4 eV was used to describe the localized Fe 3d orbitals using Dudarev’s approach.45 This value was recently proposed after a systematic analysis of the influence of electron correlation on the electronic structure and magnetism of several transition-metal phthalocyanines.46 Valence electrons are described using a plane-wave basis set with a cutoff of 400 eV, and a Γ-centered grid of k-points is used for integrations in the reciprocal space, where the smallest allowed spacing between k-points is set at 0.25 Å−1.47 The space group is P1,̅ and the calculated parameters for the crystal unit cell are a = 10.58 Å, b = 10.44 Å, c = 8.14 Å, α = 96.86°, β = 110.17°, γ = 108.27°, in good agreement with experimental values.35 The ionic relaxation was performed until the Hellmann−Feynman forces were lower than 0.005 eV/Å. The van der Waals interactions were taken into account through the DFT-D2 method of Grimme.48 These interactions have been proven to play a key role in phthalocyanine adsorption on different surfaces, and their inclusion in the calculations significantly improves the agreement with experimental data.49,50 In the most stable solution there is a ferromagnetic interaction 1679


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Figure 2. (a) 2D sheets in the ac plane, showing the two types of Pc π−π overlaps: two overlapping rings along the c axis and one overlapping ring along the a axis. Top (b) and side (c) views of a Fe(Pc)(CN)2 monomer extracted from the crystal unit, with the local coordinate axes employed in this work.

Table 1. Relevant Intermolecular Distancesa along the a and the c Axes between Atoms with Significant Contribution to the Magnetic Orbitalsb along the c axis

distance (Å) intermolecular interaction a

allowed spacing between k-points (k-spacing) was set at 0.2 and 0.1 Å−1. 2.2. Calculations on Isolated Monomer and Dimers. Evaluation of the Intramolecular and Intermolecular Magnetic Coupling Constants. Three different models were selected to inspect the electronic structure of the Fe(Pc)(CN)2 unit and evaluate the intramolecular and intermolecular interactions: a monomer (Figure 3) and two dimers, along the a axis (Figure 5) and along the c axis (Figure 6). The models are directly extracted from the X-ray crystallographic data without any additional refinement. The monomer is used to accurately evaluate the intramolecular π−d interaction, while the dimers are intended to confirm the presence of this intramolecular interaction and provide information about the nature and amplitude of the intermolecular coupling constants. On each Fe(Pc)(CN)2 unit, there are two unpaired electrons, one occupying the Fe 3d orbital and one on the Pc molecule, while the dimers have four unpaired electrons in interaction. These interactions can be evaluated assuming a Heisenberg− Dirac−Van Vleck (HDVV) Hamiltonian:

along the a axis

Fe··· Fe′

Fe··· N2′

N2··· N2′

Fe··· Fe″

Fe··· N3″

N3··· N3″

8.43 ddc

5.59 πdc

3.75 ππc

10.77 dda

9.02 πda

7.36 ππa

Extracted from the X-ray crystallographic data. bLabels as in Figure 2.

between the unpaired Pc π electron and the Fe 3d one (triplet state). From the optimized geometry, single-point calculations were done to establish the relative position of the brokensymmetry solution (with antiferromagnetic π−d interaction). For these single-point calculations, and to increase the precision, the number of k-points was increased, using Γcentered grids of k-points where the smallest allowed spacing between k-points (k-spacing) was set at 0.2 and 0.1 Å−1. To study intermolecular interactions along the a and the c axes the unit cell was duplicated in both directions. Geometry optimizations were performed for each supercell, and different magnetic solutions were explored. For these relaxations, the cell parameters were maintained fixed while the atomic positions were optimized. Once the most favorable solution was identified, single-point calculations were done to evaluate the relative energy for different magnetic solutions. For these single-point calculations the number of k-points was again increased, using Γ-centered grids of k-points, where the smallest

⎛ 1 ⎞ ĤHDVV = −∑ Jij ⎜Sî Sĵ − I ⎟̂ ⎝ 4 ⎠ i 500 K and confirm the strong intramolecular π−d interaction expected for Pc systems,16 a central result of this work. Our evaluations significantly improve the previously reported CASPT2 estimates by Yu et al.17 for TPP[Fe(Pc)(L)2]2. 3.2. Mechanism of the Ferromagnetic π−d Interaction. The two electrons in two orbitals define a valence space containing two degenerate neutral determinants, |πd̅| and |dπ̅|, where each orbital carries one electron and two ionic determinants, |ππ̅| and |dd̅|, separated by the on-site repulsion U to the neutral determinants. So Uππ (Udd) represents the difference between the Coulomb repulsion in the ionic |ππ̅| (|dd̅|) determinant and the neutral determinants. The CI matrix (CASCI) can be written as72

breaks the expected degeneracy of the Fe dxz and dyz orbitals in an ideal D2h symmetry, stabilizing the dxz orbital with respect to the dyz one. In fact, the CASSCF calculations with an active space with six electrons in four orbitals, CAS(6,4) including the singly occupied Pc π orbital and the three occupied 3d Fe orbitals (dxy, dxz, dyz), gives a triplet ground state, defined by a single electronic configuration dxy2 dxz2 dyz1 π1. The active orbitals carrying the unpaired electrons are shown in Figure 3. As previously pointed out,8,31,32 there is no hybridization between the Fe 3d and the Pc π orbitals, either for 3dxz or 3dyz orbitals. The same description can be obtained with larger active spaces including additional Pc π orbitals. This preliminary description already indicates the ferromagnetic nature of the interaction between the unpaired Pc π electron and the Fe 3d one, favoring the triplet over the singlet state. It is possible to establish the relative position of the triplet with electronic configuration dxy2 dxz1 dyz2 π1 from CI calculations on the basis of the ground triplet CASSCF(6,4) orbitals. The results are reported in Table 2 for two different CI spaces, CAS(4,3)+S including just the single excitations on the top of the active space and DDCI(4,3) space. Notice that in both sets of calculations, the dxy orbital, doubly occupied in all the states considered, was excluded from the active space. The energy difference between the ground dxy2 dxz2 dyz1 π1 and excited dxy2 dxz1 dyz2 π1 triplet states is ∼0.65 eV. This energy gap is much larger than the value of 0.01−0.02 eV estimated from MP2 calculations on the ROHF wave functions by Yu et al.17 for TPP[Fe(Pc)(CN)2]2 compound. The use of the statespecific MOs of the triplet ground state introduces a penalty on the excited-state energies, displacing the CI energies of states with different configuration to much too high energy. This explains why the energy difference between the two S = 1 states is higher than expected. If instead, state-average CASSCF (6,4) MOs of the two lowest triplet states are employed the resulting triplets are close in energy. Test calculations at both CASSCF and CASPT2 levels using extended CAS indicate that the relative distribution of the four considered states is differently affected by the electron correlation, depending on the composition of the CAS. Two ingredients converging in this hybrid system, the 3d Fe electrons and the closely packed Pc π orbitals, make difficult the selection of a balanced active space. But what is relevant for this work is that the singlet−triplet separation for each electronic configuration, the Jπd coupling, is almost independent of the MO set employed. The DDCI(4,3) calculations on the basis of the state-specific CASSCF(6,4) triplet MOs give a value of Jπd/kB = 436 K for the coupling between Pc π and Fe 3dyz, very close to the value of 423 K for the interaction with Fe 3dxz. With the state average triplet CASSCF(6,4) MOs, the π−d coupling constants present very similar values at DDCI(4,3) level, Jπd/kB = 458 and 453 K for πyz and π-xz, respectively. Then, the orbitals employed in the 1682


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Figure 4. Main excitations governing the ferromagnetic π−d coupling. Intermediates #1, #2, and #3 can be classified as LMCT, MLCT, and spin polarization excitations, respectively (see text).

Since the two active orbitals present a null overlap, the hopping integrals at this level are expected to be negligible, and the π−d interaction is then governed by the positive direct exchange Kπd. In fact, it is possible to obtain the four eigenstates of the CASCI matrix and evaluate all the parameters. The singlet− triplet energy difference at this level is Jπd = E(S) − E(T) = 288 K, equals 2Kπd, since the hopping integrals are null. The on-site repulsion values, Uππ = 12.6 eV and Udd = 8.3 eV, indicate the energetic preference for the double occupation of the Fe 3d orbital over the double occupation of the Pc π orbital. Comparing with the Ueff term employed in our periodic calculations, this Udd value is almost twice the Ueff term, but it is worth noting that (i) the Udd and Uππ values were evaluated at CASCI level, while a significant decrease of these bare values is expected by the effect of the dynamical correlation at DDCI level,75 and (ii) the Ueff term in our PBE+U-D2 calculations is not just the on-site Coulomb repulsion but the difference U − J, following Dudarev’s approach.

|dπ ̅ | |π d̅ | |dd̅ | |ππ ̅ | 0 K πd 0 tπ d

tπ d

Udd

tπ′ d

tπ′ d

K π d Uππ

where tπd and t′πd are the hopping integrals, tπd = ⟨πd̅|Ĥ |dd̅⟩, tπd′ = ⟨πd̅|Ĥ |ππ̅⟩, a measure of the coupling between the ionic and neutral forms. For simplicity, hereafter it is assumed that tπd ≈ tπd′. Kπd is the exchange integral Kπd = ⟨πd̅|r12−1|dπ̅⟩, always positive at this level. The magnetic coupling constant Jπd can be expressed in the basis of these parameters, with two contributions of opposite sign:73,74 Jπ d = 2K π d −

4tπ d 2 = JF + JAF U 1683


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and a second one from the active (π or d) to πvir orbital, that gives two formally doubly excited determinants |πocπvirπ ̅ d̅ | and |π̅ocπ̅virπd| with MS = +1 in the active space and MS = −1 in the inactive orbitals and vice versa. They relax the Pc π shell and do not change the charge of Fe or Pc. This type of excitation introduces the spin polarization effect, and its contribution can be ferromagnetic or antiferromagnetic,72,78 depending on the molecular architecture. In this system, the contribution is ferromagnetic as reveals the comparison of the Jπd values resulting from the CI calculations including just the active space (CASCI calculation), 293 K, and the CAS plus the spinpolarization forms, 642 K. Notice that the resulting coupling is even larger than the DDCI value, due to the absence of the double excitations of the DDCI space, most of them introducing antiferromagnetic contributions to the coupling.72 These three types of excitations are already present in the CAS+S space; however, they do not have a prominent weight in the description of the ground state at this level. In fact, it is the interaction with the double excitations of the DDCI space that enhances the impact of these single excitations. The numerical demonstration of this effect can be obtained by comparing the CAS(2,2)+S and DDCI(2,2) evaluations in Table 2. The interaction of the LMCT and MLCT with the double excitations favors the triplet state over the singlet state,78 and this enhances the ferromagnetic contribution of the Jπd coupling. 3.3. Intermolecular Interactions. To evaluate the interactions between neighboring Fe(Pc)(CN)2 molecules we performed calculations on dimers extracted from the crystal along the a and c axes. Figures 5 and 6 show two of the active

As it is well-known, the CASCI description is only optimal for a qualitative description, the electron correlation effects being an essential ingredient for the quantitative description of the magnetic coupling.76,77 In fact, the analysis of the DDCI wave function of the ground state sheds additional light on the mechanism governing the coupling between the Fe 3d and Pc π electrons. The excitations out of the active space with larger coefficients on the wave function at DDCI(2,2) level involve the occupied πoc and the virtual πvir orbitals shown in Figure 3. πoc is essentially concentrated on the Pc molecule, in particular, in the bonding C−N π orbitals, placed on the C−N bonds in the x direction, while πvir orbital results from the combination of the diffuse Fe 3dyz counterpart and the corresponding antibonding C−N π* orbitals. Both πoc and πvir extend on spatial regions different from those of the active Pc π orbital, the larger coefficients in πoc and πvir being in those atoms without participation in the active Pc π orbital. Three different excitations with a key role in the triplet ground state can be distinguished, schematically represented in Figure 4. The dominant one corresponds to a single excitation from πoc to Fe 3dyz orbital (intermediate #1 in Figure 4). This type of excitation can be considered as a ligand-to-metal charge transfer (LMCT) form.72,77 It gives rise to a Fe2+(Pc0)(CN−)2 intermediate, where the Fe 3dyz orbital presents a double occupation, in agreement with the relative values of the on-site Coulomb repulsions Udd and Uππ. This excitation gives rise to a local triplet in the organic unit T1 of lower energy than the corresponding local singlet: T1 =

1 |(πocπ ̅ − ππoc ̅ )dd̅ ⟩ 2

S1 =

1 |(πocπ ̅ + ππoc ̅ )dd̅ ⟩ 2

This favors the triplet over the singlet state and then favors the ferromagnetic contribution to the coupling. This effect can be evidenced by calculations where the LMCT is eliminated from the DDCI space. The resulting π−d coupling is of 448 K, instead of 571 K with the complete DDCI space, which demonstrates the ferromagnetic contribution of this set of excitations. Additionally, these excitations increase the spin density on the Pc molecule, particularly on the C and N atoms forming the 16-member ring around the Fe atom. Instead intermediate #2, with a smaller contribution on the ground-state wave function, corresponds to a single excitation from Fe 3dyz orbital to the πvir orbital. This excitation can be considered as a metal-to-ligand charge transfer (MLCT) form,72,77 leading to an Fe4+(Pc2−)(CN−)2 intermediate. Since the πvir orbital also has a certain contribution of the Fe 3dyz diffuse shell, this excitation can be also described as a d→d* excitation, which allows for the orbital relaxation of the Fe center. As in the case of intermediate #1, this excitation gives rise to a local triplet in the organic unit T2 of lower energy than the corresponding local singlet S2 that favors the triplet over the singlet state. T2 =

1 |πocπoc ̅ (ππvir ̅ − πvirπ ̅ )⟩ 2

S2 =

Figure 5. Magnetic orbitals in the dimer along the a axis: (a) top view and (b) projection on the ab plane. They represent the in-phase combination (Ag symmetry) of the Fe 3d orbitals and Pc π orbitals, respectively.

orbitals for dimers along the a and the c axis, respectively. The contact between two neighboring phthalocyanine molecules takes place through the π−π overlap of the benzene rings, only one ring in the case of dimer along the a axis, two rings for dimer along the c axis (see main intermolecular distances in Table 1). The contribution of the benzene rings to the active orbitals is rather moderate, and consequently intermolecular interactions are not expected to be strong. The DDCI calculations are extremely expensive for these systems (more than 4.5 × 109 determinants in the CI space), and the EXSCI approach was employed to perform these calculations. This approach requires imposing certain thresh-

1 |πocπoc ̅ (ππvir ̅ + πvirπ ̅ )⟩ 2

The numerical demonstration of this effect is obtained from calculations where the MLCT forms have been deleted from the DDCI space. The resulting π−d coupling is of 335 K, then these forms clearly favor the ferromagnetic coupling. Finally, intermediate #3, with a smaller contribution to the wave function than the previous ones, belongs to the one hole− one particle (1h1p) set of excitations. It consists of two coupled single excitations, one from the πoc to the active space (π or d) 1684


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more excitations) at a reasonable computational cost. The calculations indicate the presence of very weak interactions between two close Pc units, Jππ, which seems to be antiferromagnetic in nature along the a axis and ferromagnetic along the c axis (Jππa and Jππc in Table 3). The Fe 3d unpaired electrons present a null interaction, Jdd = 0 in both cases. Since the absolute values of Jππ are rather small, in the limit of accuracy of the calculations, these assignments (ferro/ antiferromagnetic) must be taken with caution, but these evaluations clearly show that the intermolecular interactions are weak, as suggested by the experiments. 3.4. Periodic Calculations. Periodic DFT calculations allow the treatment of the full [Fe(Pc)(CN)2]·2CHCl3 crystal. The unit cell (111) contains just one iron-pthalocyanine and two unpaired electrons. The ground state corresponds to a ferromagnetic (FM) alignment of the two spins (the FM solution, |d1π1|). The cell was fully optimized with k-spacing = 0.25 Å−1, and then a single-point calculation on the optimized geometry gives the energy separation with the antiferromagnetic (AFM) solution (|d1π̅1|). The AFM-FM gap is shown in Table 4 for different k-spacing values. This gap can be related to the interaction parameters by means of the expectation energy of the Heisenberg Hamiltonian for these two magnetic solutions.79,80 Hence, E(FM) − E(AFM) relates to the π−d interaction inside each pthalocyanine molecule, Jπd, plus the πd coupling between neighboring molecules, πdinter amd πdinter a c . 81 The spin density maps for these two solutions are shown in Figure 7. In both cases, the spin density is concentrated on the Fe 3d and the eight C atoms of the Pc 16-member ring, in agreement with the description provided by our CASSCF/ DDCI calculations. To study intermolecular interactions along the a and c axis the unit cell was duplicated in both directions (leading to 211 and 112 supercells, respectively). Each supercell contains two iron-phthalocyanine molecules, then four unpaired electrons distributed in two Fe 3d and two Pc π orbitals. Geometry optimizations were performed for each supercell, and different magnetic solutions were explored. Once the most favorable solution was identified, single-point calculations were done to evaluate the relative energy for several magnetic solutions using different k-spacing values. The results are shown in Table 4. For both supercells, the most stable solution is |d1d̅2π1π̅2|, where each Fe(Pc)(CN)2 molecule presents an intramolecular ferromagnetic πd interaction and neighboring molecules are coupled antiferromagnetically. This result is in agreement with the experimental data of Murakawa et al.33 for [Fe(Pc)(CN)2]·

Figure 6. Magnetic orbitals in the dimer along the c axis: (a) top view and (b) projection on the bc plane. They represent the in-phase combination (Ag symmetry) of the Fe 3d orbitals and Pc π orbitals, respectively.

olds to the exchange integrals as criteria for selecting the determinants, orbitals, and integrals to be considered in the calculations. The final size of the diagonalized CI matrices is as large as 250 × 106 determinants. The thresholds, used in previous studies on different systems (Kij ≥ 1 × 10−3), represent a compromise between accuracy and feasibility. The calculated intramolecular, Jπd, and intermolecular, Jππ and Jdd, couplings are shown in Table 3. Table 3. Magnetic Coupling Interactions J/kB (K) in the Neutral Radical Fe(Pc)(CN)2 inside the Phtalocyanine Molecule Jπd and between Neighboring Molecules, Jππ and Jdd at DDCI Level on the Minimal CASa method single molecule

dedicated DDCI dimer along the a EXSCI axis dimer along the c EXSCI axis a

Jπd/kB (K)

Jππa/kB (K)

Jππc/kB (K)

Jdd/kB (K)

571.2 339.5

−3.2

0.0

276.2

+1.3

0.0

The EXSCI approach was employed for the calculations in dimers.

The results confirm the strong ferromagnetic interaction between the Pc molecule and the Fe ion, although the Jπd values evaluated in both dimers are smaller in magnitude than the estimate obtained from the monomer. This is a direct effect of the EXSCI truncation, but as mentioned it is not possible to impose a further reduction of the thresholds (i.e., to include

Table 4. Relative Energy (in meV) of the Different Magnetic Solutions at PBE+U-D2 Level with Ueff = 4 eV for Fe 3d and Different Values for the k-Spacinga (in Å−1) 111 112

211

a

|d1π1| |d1π̅1| |d1d2π1π2| |d1d̅2π1π̅2| |d1d̅2π̅1π2| |d1d2π̅1π̅2| |d1d2π1π2| |d1d̅2π1π̅2| |d1d̅2π̅1π2| |d1d2π̅1π̅2|

k-spac = 0.25

k-spac = 0.2

k-spac = 0.1

energy expression

0 5.2 0 −21.5 −8.5 21.1 0 −13.2 −5.7 10.0

0 4.5 0 −21.1 −12.2 10.6 0 −12.7 −5.6 9.8

0 4.9 0 −21.0 −12.9 9.9 0 −12.8 −5.6 9.7

0 (Jπd/2) + πdinter + πdinter a c 0 Jddc + Jππc + 2πdinter c Jπd + Jddc + Jππc + 2πdinter a Jπd + 2πdinter + 2πdinter c a 0 Jdda + Jππa + 2πdinter a Jπd + Jdda + Jππa + 2πdinter c Jπd + 2πdinter + 2πdinter a c

The intermolecular πd interactions are represented as πdinter to alleviate the notation, subscripts a and c refer to the interaction axis. 1685


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Figure 7. Spin density maps for 111 cell (top) for the ferromagnetic (left) and antiferromagnetic (right) solutions calculated with k-spacing = 0.1 Å−1 and Ueff = 4 eV. Blue and yellow surfaces represent positive and negative values of the spin density. One (top) and four (bottom) unit cells are represented, delimited by a dashed line. View along the c axis.

Figure 8. Spin density maps for the ground state of 112 (left) and 211 (right) supercells (top), calculated with k-spacing = 0.1 Å−1 and Ueff = 4 eV. Blue and yellow surfaces represent positive and negative values of the spin density. One (top) and four (bottom) supercells are represented in each figure, delimited by a dashed line. Views along the a axis and the c axis for 112 and 211 supercells, respectively.

2CHCl3 for both interactions. The spin-density maps for this solution in both supercells are shown in Figure 8. Together with the ferromagnetic solution |d1d2π1π2|, two additional magnetic solutions have been calculated, |d1d̅2π̅1π2| and |d1d2π̅1π̅2|, where the πd coupling inside each Fe(Pc)(CN)2 unit is antiferromagnetic. They differentiate in the nature of the intermolecular interactions. The solution highest in energy for both supercells is |d1d2π̅ 1π̅ 2|, where the intermolecular couplings, except those between Pc π and Fe 3d, are ferromagnetic. Then our calculations provide a set of magnetic

solutions, which relative energies are in agreement with both the ferromagnetic nature of the πd interaction inside each Fe(Pc)(CN)2 unit and the antiferromagnetic coupling between neighboring molecules. The next step is to evaluate the amplitude of the different interaction parameters from the expectation energies of the magnetic solutions. Since the distances and overlap mode of two Fe(Pc)(CN)2 units are distinct along the a and c axis, up to along six interactions can be distinguished: ππa, dda, and πdinter a along the c axis. Notice that the a axis and ππc, ddc, and πdinter c 1686


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The Journal of Physical Chemistry A the |d1d2π̅1π̅2| solutions on the supercells are in correspondence with the |d1π̅1| solution of the 111 cell. The expectation energy of the Heisenberg Hamiltonian for the supercell is just twice that of the 111 cell, and the same factor 2 is found for the calculated relative energy with respect to the ferromagnetic solution. This is a good indication of the stability and accuracy of these calculations, since three different cells, each one optimized independently, and the subsequent single-point calculations on two particular solutions give exactly the same energy gap. In the resulting set of equations, the number of unknowns (seven) exceeds the number of independent equations (five), and then some simplifications are required. Taking into account the topology of the system, the relative distances between Fe atoms in two neighboring molecules and the results provided by the DDCI calculations on the dimers, we assumed that the intermolecular d−d interactions are negligible and evaluated the five remaining parameters. The results are shown in Table 5

4. CONCLUSIONS The electronic structure and magnetic properties of the neutral radical Fe(Pc)(CN)2·2CHCl3 have been inspected by means of extended configuration interaction calculations on isolated molecules and DFT-based periodic calculations on the crystal. The former gives quantitative estimates of the intramolecular ferromagnetic π−d interaction, in good agreement with the experimental data. The latter provides indication of the presence of antiferromagnetic intermolecular interactions between neighboring molecules. The particularities of the system require extremely accurate calculations, with very demanding convergence criteria, then calculations highly demanding in time and resources. Indeed the extraction of the interaction parameters need a careful analysis of the wave function, by means of the effective Hamiltonian theory in the case of the DDCI calculations or the expectation energies of the broken-symmetry magnetic solutions in the case of the DFT periodic results. Despite the limitations imposed by the different methods, this combined strategy provides a global picture of the electronic structure and magnetic properties of the ironphthalocyanine molecule, in particular, about the mechanism controlling the giant ferromagnetic π−d coupling. This interaction, which nature has been matter of controversy for a while, plays a key role in many technological applications, including the emerging and effervescent field of molecular spintronics, where metal-phthalocyanines are postulated as promising candidates. For all these reasons, it is expected that the results reported in this work could be also useful for the understanding of the fascinating properties of related phthalocyanine-based compounds.

Table 5. Magnetic Coupling Interactions J/kB (K) Resulting from Periodic PBE+U-D2 Calculationsa

a

Jπd/kB

Jππa/kB

Jππc/kB

πdinter a /kB

πdinter c /kB

88.8

−163.6

−253.5

7.5

4.9

k-spacing = 0.1 Å−1, Ueff = 4 eV.

for k-spacing = 0.1 Å−1. Although the periodic calculations correctly predict the nature of the interactions, their relative values do not agree with the experimental estimates. The intramolecular ferromagnetic interaction is underestimated, while the intermolecular antiferromagnetic interaction is overestimated. The results suggest that the method artificially favors the antiferromagnetic interactions over the ferromagnetic ones. This could be ascribed to the Ueff correction employed for the Fe 3d orbitals. In the Fe(Pc)(CN)2 unit, this correction penalizes the situations where two electrons are placed on the same 3d orbitals (the ionic forms described in Section 4) with respect to those with one electron on the Pc π orbital and other on the Fe 3d orbitals (the neutral forms). The larger the penalty, the smaller the antiferromagnetic contribution to the coupling. Then increasing Ueff would enhance the ferromagnetic solution over the antiferromagnetic one. This is what we observed on a set of single-point calculations on the 111 cell with Ueff = 3, 4, 5, 6, and 7 eV, where the corresponding AFMFM gaps are 4.1, 4.9, 5.4, 5.6, and 5.7 meV, in good agreement with this analysis. However, even for the larger Ueff, the corresponding Jπd value (∼130 K) is still underestimated with respect to the experimental data and our DDCI evaluations. Other cause that combines with the bias on the choice of Ueff and probably has a more significant impact on this system is the well-known over-delocalization of electrons on the GGA approximation, which favors the low-spin solutions over the high-spin ones. Since the low-spin solutions are overstabilized, the global effect is the overestimation of the antiferromagnetic interactions and underestimation of the ferromagnetic ones. Then a further improvement of our periodic estimates would require alternative correlation-exchange functionals such as the hybrid ones, avoided in this work due to their huge computational cost, in particular, for highly complex systems as this one.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Carmen J. Calzado: 0000-0003-3841-7330 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge Dr. N. Cruz Hernández for valuable comments and the technical support of the Supercomputing Team of the Centro Informático Cientı ́fico de Andalucı ́a (CICA). Financial support was provided by the Ministerio de Economiá y Competitividad (Spain) and FEDER funds through the project CTQ-2015-69019-P (MINECO/FEDER).

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REFERENCES

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