Magnetic Exchange Couplings in Heterodinuclear Complexes Based

Article pubs.acs.org/JCTC

Magnetic Exchange Couplings in Heterodinuclear Complexes Based on Differential Local Spin Rotations Rajendra P. Joshi, Jordan J. Phillips, and Juan E. Peralta* Department of Physics and Science of Advanced Materials, Central Michigan University, Mount Pleasant, Michigan 48859, United States S Supporting Information *

ABSTRACT: We analyze the performance of a new method for the calculation of magnetic exchange coupling parameters for the particular case of heterodinuclear transition metals complexes of Cu, Ni, and V. This method is based on a generalized perturbative approach which uses differential local spin rotations via formal Lagrange multipiers (Phillips, J. J.; Peralta, J. E. J. Chem. Phys. 2013, 138, 174115). The reliability of the calculated couplings has been assessed by comparing with results from traditional energy differences with different density functional approximations and with experimental values. Our results show that this method to calculate magnetic exchange couplings can be reliably used for heteronuclear transition metal complexes, and at the same time, that it is independent from the different mapping schemes used in energy difference methods.

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INTRODUCTION

ĤHDVV = −∑ Jij Sî ·Sĵ ⟨i , j⟩

Magnetic materials based on single molecular complexes have attracted considerable attention as potential nanoscale size magnetic units with easily tunable magnetic properties, motivating their application to spintronics,1 quantum computing,2 and magnetic data storage devices.3 In addition, their study allows us to understand the interface between classical and quantum mechanics at the nanoscale level.4 These molecular complexes consist in general of one or more transition metal centers held together typically by an organic environment which reduces intermolecular interactions.5 The metal centers have one or more unpaired d electrons and are responsible for the magnetic moment of the molecule. The role played by magnetic exchange interactions between the spins on metal centers along with the magnetic anisotropy is crucial for understanding the magnetic behavior of these molecules.6−8 Experimental and theoretical studies of exchange interactions in molecular based complexes are a topic of central interest.9−12 These interactions can be understood qualitatively and quantitatively in terms of the magnetic exchange coupling parameters, J. From experiments, J can be obtained, for example, by fitting the variation of the magnetic susceptibility with temperature and external magnetic fields using several sets of independent fitting parameters.13,14 However, for large systems with low symmetry, the assumptions made in obtaining these parameters compromise the accuracy of the obtained J values. This highlights the importance of theoretical methods that can estimate the magnetic exchange coupling free from empirical data. Assuming a pairwise interaction between spins, magnetic exchange interactions are modeled by the Heisenberg−Dirac−van-Vleck (HDVV) spin Hamiltonian15 of the form © XXXX American Chemical Society

(1)

where Sk̂ is the local spin operator on magnetic center k. This model spin Hamiltonian considers the isotropic exchange interaction between local spins associated with localized unpaired electrons. The particular interaction of these local spins produces different magnetic states of a given total spin quantum number S, with the total spin for such a system ranging from the high spin state (HS) with Smax = S1 + S2 to the low spin state (LS) with Smin = S1 − S2 (S1 ≥ S2). The exchange coupling parameter Jij is routinely determined by mapping the energy difference of these different magnetic states calculated from electronic structure methods onto the respective states of the HDVV model spin Hamiltonian. Among these electronic structure methods, wave function based methods such as multireference configuration interaction,16 multiconfigurational second-order perturbation method,17 and difference dedicated configuration interaction18 provide in general a very good approximation for the coupling parameter but are restricted to small systems. On the other hand, Kohn−Sham density functional theory (KS DFT)19,20 is based on a single Slater determinant to describe the ground state and is more computationally efficient than wave function based methods. DFT results depend strongly upon the approximate exchange correlation functional employed,13,21 and in general, more reliable results are obtained with hybrid density functionals for the particular case of magnetic exchange couplings.22−24 Due to its single-reference character, the HS state can generally be represented well in DFT calculations. However, the LS state is Received: January 29, 2016

A

DOI: 10.1021/acs.jctc.6b00112 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Article

Journal of Chemical Theory and Computation usually multireference in character, which presents a well documented spin-symmetry dilemma.25 Typically this dilemma has been circumvented by considering instead “broken symmetry” (BS) solutions, which can be represented by solutions of the KS equations that do not have good total spin quantum number and are actually a superposition of two or more pure spin states. Using this BS approach, for a general heterodinuclear system with spin S1, S2 and magnetic exchange coupling J, it has been shown26−29 that J can be calculated as J=

E BS − E HS S1S2

J=

2 1 d E DFT |θ= 0 S1S2 dθ 2

(4)

It is important to remark that eq 3 is general and valid for any S1 and S2. Furthermore, the J values calculated by this method from the HS state are free from the spin contamination problem that arises while using the BS approach. To evaluate d2EDFT/dθ2, our method uses a generalized perturbative approach to Dederichs’ constrained DFT,37 based on differential local spin rotations. Here, the relative angle θ between two local spins is used as a formal constraint via Lagrange multipliers, giving place to a torque between the local spins that serves as a small perturbation. In this method, the small rotation between the local spins si = ⟨Sî ⟩ is characterized by s × s2 θy ̂ = 1 s1s2 (5)

(2)

where BS here denotes a solution with Sz = S1 − S2. Although this approach has been widely used for studying magnetic interactions in transition metal complexes, it is not free from limitations and hence mapping the BS solution using the HDVV Hamiltonian has to be handled with care. First, while converging to HS solutions is usually straightforward with modern DFT codes, obtaining the correct BS solutions on the other hand can be difficult unless carefully prepared guesses are used. Second, for general polynuclear systems with N metal atoms there are N(N−1)/2 unique couplings, which means the number of BS solutions required grows roughly as N2. This situation only becomes more complicated for heteronuclear systems that feature different spins on each metal center. Even for relatively simple homodinuclear systems there is not always agreement on how the BS methodology should be properly applied,30−32 and different mapping schemes such as the spin projected mapping proposed by Dai and Whangbo,28 the “non projected” approach of Alvarez and Ruiz,30,33 and the approximate spin projected scheme of Yamaguchi et al.34 are regularly used. Recently, Costa et al. have shown that the BS approach can be employed in heteronuclear complexes as long as the appropriate mapping is taken into account, which involves considering the different BS states for each particular spin case.13 Mayhall and Head-Gordon have shown in a recent work a simple strategy to extract exchange coupling using single excitations.35,36 The reliable “blackbox” determination of J parameters is clearly a challenging problem, and in this direction a novel method for calculating magnetic exchange coupling parameters in transition metal complexes has been proposed by some of us recently,21 whereby the coupling is defined by the torsional response of the spin density to a torque perturbation acting between spins on metal atoms. In this methodology, the J coupling is defined not by energy differences between BS and HS configurations but by a Hessian with respect to spin rotation coordinates from the HS reference solution. Starting from the HS state |S1S2⟩ of a general heterodinuclear complex, a rotation of the quantization axis of S2 about an arbitrary direction leads to the |S1R̂ θS2⟩ state. From this state, the second derivative of ⟨Ĥ HDVV⟩ at θ = 0 is

where si are the local spins on different atoms (or group of atoms) and are determined using an electron population method. Here, without loss of generality (spin−orbit is neglected in our calculations), the differential spin rotation is chosen about the y-axis, so that s1 and s2 lie in the xz plane. Subjected to this constraint, the energy is minimized using the method of Lagrange’s undetermined multipliers, where the multiplier λ functions as the torque acting between spins. This yields a modified constrained DFT equation given by

̂ + λτ )̂ ψ = ϵiψ (FKS i i

(6)

where F̂KS = ĥ + J ̂ + V̂ XC is the standard Kohn−Sham Fockian, and τ̂ is a constraining small torque operator which, for a collinear HS solution, simplifies to

τ̂ =

σ1x σx − 2 s1 s2

(7)

σx1

(3)

where is a local spin operator. Equation 6 is solved in practice self-consistently by considering λ small and solving the resulting first-order coupled-perturbed equations.21 A related non-self-consistent method based on Green’s functions has been recently proposed by Steenbock et al.38 based on the original work of Liechtenstein et al.39 It should be remarked that though the concept of a local spin40,41 is central to this methodology, it appears that the basis set dependence or arbitrariness of the spin population definition does not undercut its predictive power.42 Hereafter, we refer to the method described in the previous paragraph as the BB method, short for blackbox. This method has been tested only for homonuclear systems such as trinuclear Mn and tetranuclear Fe complexes,21 dinuclear CuII 10 complexes,42 and FeIII 7 clusters. However, the transferability of this method to heteronuclear systems has not been demonstrated yet. This is especially important considering their asymmetrical nature and different spin on each metal center. With this motivation, in this work we assessed the performance of the BB method for the heterodinuclear system, for which reliable experimental coupling values are available.

from which the J coupling parameter can be extracted (see the Supporting Information for a derivation). Here the only assumption is that the response of ⟨Ĥ HDVV⟩ under local rotations of the magnetization direction is the same as the response of the DFT energy under such rotations, in such a way that

COMPUTATIONAL DETAILS For our assessment we utilize the six heterodinuclear transition metal complexes of Cu, Ni, and V of Costa et al.13 with diverse magnetic interactions, ranging from strong antiferromagnetic (Cu−Ni) to weakly antiferromagnetic (Ni−VO) to ferromagnetic (Cu−VO). The antiferromagnetically coupled Cu(II)−

d2 ⟨ĤHDVV ⟩|θ= 0 = JS1S2 dθ 2

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Journal of Chemical Theory and Computation

Figure 1. Crystal structures for the heterodinuclear system considered in this work. (a) PAJZAB, (b) PAJZEF, (c) WIXFOZ, (d) WIXFUF, (e) BIGFAY, and (f) PUSJOC. Carbon atoms are shown in gray, small white atoms are hydrogens, blue atoms are nitrogen, oxygen is in red, vanadium is shown in orange, purple color is for nickel, and dark brown is for copper.

Table 1. Exchange Couplings (cm−1) Obtained by Different Methods for Heterodinuclear Complexes with Different Exchange Correlation Functionals and Values from Costa et al.13 Calculated Using the Energy Difference Method along with Experimental Valuesa energy differences

a

BB

BB

ref 13

complex

HSE

M06

LSDA30

HS

BS

HSE

M06

exp.

WIXFOZ WIXFUF PAJZEF PAJZAB BIGFAY PUSJOC

−25.44 −24.41 −204.87 −152.37 115.90 78.03

−46.99 −46.68 −262.34 −195.55 123.56 88.20

−21.86 −21.04 −180.63 −135.71 123.43 88.57

−22.07 −20.56 −186.34 −138.50 124.24 88.36

−21.64 −20.94 −175.22 −132.7 122.60 88.20

−23.60 −23.30 −197.07 −148.03 104.50 72.70

−42.50 −43.00 −247.71 −186.50 115.40 83.80

−17.80 −20.00 −117.80 −96.30 118.00 85.00

Here, HS and BS represent the initial guess for the BB method as high spin and broken symmetry. BB cacluations were performd with LSDA30.

(herein called LSDA30), the M06 exchange correlation functional with 27% HF exchange,46 and the HSE screened exchange correlation functional of Heyd, Scuseria, and Ernzerhof47 with its standard parameters. The LSDA30 consists of a hybrid of LSDA (Slater exchange and Vosko, Wilk, Nusair correlation)48,49 and HF exchange given by EXC = aEXHF + (1 − a)EXLDA + ECLDA, where a = 0.3. All calculations using the BB method had been carried out using the LSDA30 functional. This choice originates in the poor convergence behavior of the first-order KS equations with noncollinear GGA kernels.21,50,51 It has been previously reported that hybrid functionals when mixed with approximately 30% of HF exchange can predict magnetic exchange couplings with good accuracy.52−54 In all cases, calculations are carried out using the 6-31G* basis set for nonmetal atoms and 6-311+G for transition metal atoms. We have also calculated the spin density on individual transition metal atoms using the Minimum Basis Set (MBS) population55 analysis to characterize the different solutions. The reliability of the BB method for calculating exchange coupling parameters in heterodinuclear complexes is assessed by comparing our results with couplings obtained from energy differences and experimentally reported values. In addition we

Ni(II) complexes with oxymato bridging ligand are shown in Figure 1a,b (CCDC name PAJZAB and PAJZEF), with an experimental coupling value of J = −96.3 cm−1 and J = −117.8 cm−1, respectively. Both systems have a total charge of +2. Figure 1c,d represents the weakly antiferromagnetic Ni(II)− VO(IV) neutral system (CCDC name WIXFOZ and WIXFUF) with experimental coupling values of J = −17.8 cm −1 and J = −20.0 cm −1 , respectively. Finally, the ferromagnetic Cu(II)−VO(IV) systems BIGFAY and PUSJOC have a total charge of 0 and +1, respectively, and have experimentally determined coupling values J = 118 cm−1 and J = 85 cm−1, shown in Figure 1e,f. Self-consistent calculations were carried out in the framework of DFT as implemented in the Gausian 09 code.43 We used both the BS energy difference approach and the BB method as discussed in Introduction for calculating magnetic exchange coupling parameters. An in-house version of Gaussian Development Version44 has been utilized for all calculations involving the BB method. The structural parameters for all the complexes were taken from ref 13. For comparison, we have used three different exchange correlation functionals, namely, the local spin density functional45 with 30% HF exchange C


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Figure 2. Percentage deviation of calculated exchange coupling values with respect to the experimental values.

couplings obtained from our BB method are close to those calculated from LSDA30 using the energy difference method and to experimental values, in line with our previous findings for homonuclear transition metal complexes.21,42 Similarly, for the second Cu(II)−Ni(II) system WIXFUF with experimental coupling value of −20.00 cm−1, J couplings of −22.07 cm−1 and −21.64 cm−1 were obtained with our BB method from the HS and BS solutions, respectively. As expected, the error in this case is similar to the error for LSDA30 (about 5%) using the BS energy differences method and smaller compared to the error of 133% and 22% for the M06 and HSE with BS energy differences, respectively. Similar trends were observed for the strongly antiferromagnetic Cu(II)−Ni(II) systems, PAJZAB and PAJZEF. However, for these systems the percentage error of the calculated J couplings with respect to experimental value is larger compared to the weakly antiferromagnetic Ni(II)− VO(IV) systems, WIXFOZ and WIXFUF. For these systems, the calculated J couplings that are closer to experimental values are those obtained again with the LSDA30 functional. Finally, for the ferromagnetic BIGFAY and PUSJOC complexes with magnetic core comprised of Cu(II)−VO(IV), we obtained exchange couplings of 124.24 and 88.20 cm−1, respectively, with small deviations of less than 5% with respect to experimental values using either the HS or BS configurations as reference. We argue that this is merely due to the choice of the exchange correlation functional and not an indication of the accuracy of the method. Importantly, couplings calculated with our BB method from either the HS or BS states are very similar, indicating that our method reliably captures the physics of local spin rotations in Heisenberg-like systems. This is in contrast to a recently proposed Green function based method38 to calculate exchange coupling from the HS state in favor of avoiding self-consistency. Unlike the energy difference method, our BB method is free from the choice of different mapping schemes. However, the nominal values of S1 and S2 need to be provided a priori, which an ideal blackbox method should not require (available in the Supporting Information). Comparison of our BB results with the previously reported SF-TD-DFT method for these systems shows that the BB method can be used to predict exchange coupling for the systems with doublet and quartet spin states (antiferromagnetic systems here) with good accuracy, while SF-TD-DFT fails.13 This has been

have compared our results with the previously reported DFT values13 obtained with M06 and HSE exchange correlation functionals.

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RESULTS AND DISCUSSION First we calculated magnetic exchange couplings starting from the HS and BS configuration using our BB method. While the intent of our methodology is to use the HS state as reference, for completeness and as a consistency check, we apply it using the BS solution as reference as well. In the case of heterodinuclear complexes with spin S1 and S2, these BS solutions can be associated with |m1 = S1m2 = −S2⟩ (and |m1 = −S1m2 = S2⟩). For the case of Cu(II) and VO(IV) complexes (S1 = S2 = 1/2), this represents the only existing BS solution, which is a mix of singlet and triplet solutions. For the case of Cu(II)−Ni(II) and VO(IV)−Ni(II) complexes (S1 = 1/2 and S2 = 1), the |m1 = S1m2 = −S2⟩ BS solution can be expressed as a combination of doublet and quartet solutions. Other BS solutions can be constructed considering an “atomic” Ni(II) BS state that mixes atomic singlet and triplet, but these solutions are not considered here since they are very hard (if not impossible) to obtain using a single-determinant self-consistent approach. We show calculated J values in Table 1 along with their experimental values and the results from BS energy differences. For comparison, we also show in Table 1 the exchange couplings obtained by Costa et al.13 using BS energy differences. For WIXFOZ, we have obtained couplings of −22.07 cm−1 and −21.64 cm−1 using our BB method on the HS and BS solutions, respectively. When compared to the experimental value of −17.80 cm−1, these couplings show relative small errors with respect to previously reported values obtained by different wave function, DFT, and spin-flip timedependent-DFT (SF-TD-DFT) based methods.13 We attribute this to the inclusion of 30% of HF exchange in the XC energy, and not necessarity to the choice of the methodology for the extraction of the exchange coupling parameters, as some of us have shown previously.54 We further compare these results with those obtained from BS energy differences for LSDA30, M06, and HSE, which for this system gives couplings of −21.86 cm−1, − 46.99 cm−1, and −25.44 cm−1, respectively. The percent error in the calculated J value with respect to the experimental reference is shown in Figure 2. This figure shows that J D


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Table 2. Calculated Spin Densities on Metal Atom for Different Heterodinuclear Systems with Different Method Useda LSDA30 HS

M06 BS

HSE

HS

BS

HS

BS

complex

M1

M2

M1

M2

M1

M2

M1

M2

M1

M2

M1

M2

WIXFOZ WIXFUF PAJZEF PAJZAB BIGFAY PUSJOC

1.75 1.75 1.77 1.73 1.13 1.25

1.17 1.17 0.63 0.64 0.70 0.74

1.75 1.75 1.76 1.73 1.12 1.25

−1.17 −1.18 −0.62 −0.64 −0.70 −0.75

1.72 1.72 1.75 1.70 1.19 1.35

1.24 1.24 0.61 0.62 0.68 0.71

1.72 1.72 1.75 1.69 1.18 1.35

−1.25 −1.24 −0.60 −0.61 −0.68 −0.71

1.75 1.75 1.77 1.73 1.11 1.23

1.15 1.15 0.63 0.64 0.70 0.74

1.75 1.75 1.77 1.73 1.11 1.23

−1.16 −1.15 −0.62 −0.64 −0.70 −0.74

a

Here, HS represent high spin, BS represent broken symmetry state, and M1 and M2 represent the metal atoms 1 and 2 in the heterodinuclear system.

Figure 3. Spin density isosurfaces for the complex PAJZEF. (a) and (b) represent the zero-order spin density (in ẑ) for the HS and BS solutions (isovalue 0.002 au) and (c) and (d) represent the first-order density (in x̂) (isovalue 2.245 au). Red and green are used for positive and negative isosurfaces, respectively.

attributed to the presence of single excitations in SF-TD-DFT that destroy the S = 1 state in the Ni atoms, mixing local singlet and triplet states.13 In contrast, exchange couplings in our BB method are defined in terms the spin density torsional response, so the number of unpaired electrons, or the precise nature of the spin state (quintet, triplet, etc.), are irrelevant. Moreover, we see that for ferromagnetic systems where SF-TDDFT gives typically large deviations from experimental values, our BB method is in good agreement with experimental values. We have also analyzed the spin density at the magnetic centers using the MBS Mulliken population method.55 Inspection of the ground state (zeroth-order) atomic spin densities (Table 2) for the HS and BS solutions reveals that these heterodinuclear complexes behave as Heisenberg-like systems in the sense that the magnetization (in absolute value) at each center remains almost constant upon inversion of one spin center. It is also interesting to see that the three hybrid functionals used here yield very close values of spin densities. Since one of the fundamental implicit assumptions of the BB method is that the electronic system remains Heisenberg-like upon local differential rotations of the spin density, we attempted to verify this assumption by inspecting the firstorder spin densities. In Figure 3 we show a comparison of zeroth- and first-order spin-densities for the HS and BS solutions for the PAJZEF complex (for the remaining complexes see Supporting Information). From Figure 3 we can immediately observe that the first-order HS and BS spindensities closely resemble the zeroth-order BS and HS spin density, respectively. This is what one would expect for Heisenberg-like systems as the spin density rotates almost rigidly at each center in opposite directions. Looking into the

details of the isodensities in Figure 3, we can see that even though the first-order HS (BS) spin density looks very similar to the zeroth-order BS (HS) spin density (each at a given isodensity), there are some subtle differences, especially at the ligands, that we attribute to self-consistent relaxation effects. This in part explains the small differences between the calculated J values with the BB method from the HS and BS solutions.

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CONCLUDING REMARKS We have analyzed the performance of a novel method based on differential rotations of the local spin density to determine magnetic exchange couplings J in heterodinuclear transition metal complexes (dubbed as the BB method). To this end, we have taken a set of six heterodinuclear transition metal complexes containing Cu, Ni, and V and compared calculated couplings using the BB method, energy differences, and experimental values. Our results show that the BB method yields similar J values as the energy differences method for all heteronuclear complexes employed here. Analyzing the firstorder spin density obtained from the BB method, we attribute its success to its ability to reproduce the mechanism of local spin rotations in Heisenberg-like systems. We also show that the LSDA30 functional (a hybrid of LSDA with 30% of HF exchange) yields slightly better exchange couplings than HSE and M06 compared to experiments.

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ASSOCIATED CONTENT

S Supporting Information *

This material is available free of charge via the Internet at http://pubs.acs.org/. The Supporting Information is available E


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free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.6b00112. Derivation of eq 3 in the main text, nominal local spin values, and zeroth- and first-order spin isodensity contours (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(J.E.P.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by NSF DMR-1206920. REFERENCES

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