Noncollinear Two-Component Quasirelativistic Description of Spin

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Noncollinear two-component quasirelativistic description of spin interactions in exchange coupled systems. Mapping generalized broken-symmetry states to effective spin Hamiltonians Artur Wody#ski, and Martin Kaupp J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01067 • Publication Date (Web): 27 Jan 2018 Downloaded from http://pubs.acs.org on February 3, 2018

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Journal of Chemical Theory and Computation

Noncollinear two-component quasirelativistic description of spin interactions in exchange coupled systems. Mapping generalized broken-symmetry states to effective spin Hamiltonians Artur Wodyński∗,†,‡ and Martin Kaupp∗,† †Technische Universität Berlin, Institut f¨ ur Chemie, Theoretische Chemie/Quantenchemie, Sekr. C7, Straße des 17. Juni 135, D-10623, Berlin, Germany ´ ‡National Centre for Nuclear Research, Andrzeja Soltana 7, 05-400 Otwock-Swierk, Poland E-mail: [email protected]; [email protected]

Abstract We provide a consistent mapping of non-collinear two-component quasirelativistic DFT energies with appropriate orientations of localized spinor quantization axes for multinuclear exchange-coupled transition-metal complexes onto an uncoupled anisotropic effective spin Hamiltonian. This provides access to the full exchange interaction tensor between the centers of spin-coupled systems in a consistent way. The proposed methodology may be best viewed as a generalized broken-symmetry density-functional theory approach (gBS-DFT). While the calculations provided are limited to trinuclear systems ([M3 O(OOCH)6 (H2 O)3 ]+ where M=Cr(III), Mn(III), Fe(III)) with C3 symmetry, the method provides a general framework that is extendable to arbitrary systems. It offers

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an alternative to previous approaches to single-ion zero-field splittings, and it provides access to the less often examined antisymmetric Dzyaloshinskii-Moriya exchange interaction. Spin-orbit coupling is included self-consistently. This will be of particular importance for complexes involving 4d or 5d transition-metal centers or possibly also for f-block elements, where a perturbational treatment of spin-orbit coupling may not be valid anymore. While a comparison with experimental data was indirect due to simplifications in the chosen model structures, the agreement obtained indicates the essential soundness of the presented approach.

1

Introduction

Multi-nuclear exchange-coupled systems are the focus of intense research due to their importance, e.g., in metalloproteins (iron-sulfur clusters or the manganese-oxo-based oxygenevolving complex in photosystem II), in catalysis, or in molecule-based magnetism. A proper description of the electronic structure of exchange-coupled clusters is challenging, since most of their energetic states (except for the one with highest spin) are not adequately represented even qualitatively by a single Slater determinant. Application of multi-reference wave-function methods, which provide the most accurate description, is computationally very demanding and so far restricted only to relatively small clusters, even though methodological developments to deal with large active orbital spaces, such as the density-matrix renormalization group (DMRG) approach, bring us closer to this goal. 1–3 The much more cost-effective application of density functional theory (DFT) does not provide spin-adapted wave functions but necessarily relies on applications of broken-symmetry (BS) DFT approaches. 4 These require inevitably the application of spin-projection procedures to extract the energies of the true spin-adapted states or spin-dependent properties. 5,6 These procedures rely on a mapping of the BS-DFT energies onto suitable effective spin Hamiltonians of the Heisenberg-Dirac-van-Vleck (HDvV) or Ising types. Within a nonrelativistic or scalar (spin-free) relativistic framework, this provides direct access only to 2


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the leading (not always largest) term of the exchange-coupled interaction, the so-called isotropic or Heisenberg exchange coupling. In particular in the field of molecular magnetism, the remaining, anisotropic contributions are also of substantial interest. 7,8 Sometimes these anisotropic terms, i.e. anisotropic exchange, antisymmetric exchange or single-ion zero-field splitting (siZFS), can be even larger than isotropic exchange, for example for lanthanide compounds. 9,10 Some of the anisotropic contributions can be accessed a posteriori using perturbation theory, e.g. ZFS 11–13 or antisymmetric exchange. 14 However, estimating siZFS in multi-nuclear systems requires additional tricks, e.g. one-center approximations for spin-orbit (SO) integrals, 15,16 replacement of paramagnetic by diamagnetic ions 17 or use of localized orbitals. 18,19 As the anisotropic terms reflect SO interactions, variational inclusion of SO effects in a relativistic 4-component or quasirelativistic 2-component framework is an obvious extension that comes to mind. So far limited attempts in this direction have been made, in particular by using 2-component approaches to obtain ZFS. The variational treatment of magnetic anisotropy in solids in a DFT framework has been discussed early on, 20 and particular interest in the solid-state community has focused on the so-called antisymmetric (Dzyaloshinskii-Moriya, DM) exchange. 21 This is mostly due to the involvement of the DM interaction in two types of important magnetic structures: weak ferromagnets and relativistic spirals (see, e.g., discussion in ref. 22). The solid-state character is then accounted for by a Bloch function 22 or by a super-cell approach. 22,23 The spin-orbit coupling has been treated perturbationally 24–26 or self-consistently. 27,28 The DM parameters have been recovered by analytical derivatives of the total energy with respect to the infinitesimal deviations of magnetic moments from collinearity 24,27–29 or just by finding and comparing two chiral states (like in the present work). 22 Further pointers to the solidstate literature can be found in these papers. Inevitably, the computations on solids tend to be more complicated than for molecules, and a number of approximations are usually applied, e.g. frequently the local spin density approximation and consideration of SO effects

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only within atomic spheres. On the molecular side, explicit 2-component computations were made initially for small triplet radicals, 30 more recently for multinuclear transition metal complexes. 31 Four-component Dirac-Kohn-Sham calculations have so far been limited to transition-metal dimers. 32 Antisymmetric DM exchange for molecules has been studied even less. A first estimate of the DM interaction in some simple main-group molecules has been performed recently, 33 showing that use of a noncollinear two-component description opens new possibilities in describing the interactions of exchange-coupled systems. Here we build on the works of van W¨ ullen 31 and Takeda and coworkers 33 and provide a consistent mapping of non-collinear two-component quasirelativistic DFT energies with appropriate orientations of localized spinor quantization axes onto an uncoupled anisotropic effective spin Hamiltonian. The proposed methodology may be best viewed as a generalized BS-DFT approach (gBS-DFT). The present work will be limited to trinuclear systems ([M3 O(OOCH)6 (H2 O)3 ]+ , M=Cr(III), Mn(III), Fe(III); Figure 1) with C3 symmetry, but in principle the method can be extended to systems with any symmetry. The trinuclear systems have been chosen as a) their description is not trivial methodologically (each state mixes many parameters) but b) they are sufficiently small to be suitable test systems for the proposed approach. The paper is organized as follows. The theory part presents some basic ideas, related to the total magnetization in two-component quasirelativistic DFT (2.1), showing how the non-collinear spin states are built. Then the effective spin Hamiltonian used in the mapping procedure is established (2.2). Computational details are subsequently discussed (3) before presenting the results (4) and conclusions (5).

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Figure 1: Structures of the [M3 O(OOCH)6 (H2 O)3 ]+ model complexes (M = Cr, Mn, Fe)

2

Theory

2.1

Magnetization in a two-component framework

Before discussing magnetization in a two-component framework, some general aspects of spin-magnetization in a one-component (nonrelativistic/scalar relativistic) picture have to be recalled. At one-component level, spatial molecular orbitals are multiplied by α or β spin functions to provide spin orbitals. The spatial parts can be different (unrestricted approach) or the same (restricted approach) for spin orbitals with α and β spin, but in both cases spin is described just by projection onto a predefined axis (by convention the z-axis). In principle even for a nonrelativistic Hamiltonian spin-magnetization should be described using complex two-component spinors. Then the quantization axis could be oriented in any direction. However, in the absence of an external magnetic field and of SO coupling, the energy of the state does not depend on the direction of the spin magnetization, and a one-component framework is sufficient to describe the electron configuration. When SO coupling is introduced in self-consistently, the total magnetization (the vector sum of spin- and orbital-magnetization J = S + L) should be discussed, instead of the

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spin-magnetization only. In contrast to the nonrelativistic or scalar relativistic picture, even in the absence of an external magnetic field, the energy of the electron configuration depends now on the orientation of the total magnetization (see discussion in ref. 34). This dependence is observed experimentally at least for S > 1/2 and is one of the contributions to the zero-field interaction. The procedure proposed here computes energy differences between states with different total magnetization and maps them onto effective spin Hamiltonians (where only the magnetization of an effective spin is taken into account). In this work we have used a two-component quasirelativistic description (including scalar and SO effects self-consistently), but in principle the approach can be expanded straightforwardly to fourcomponent relativistic Hamiltonians. From a technical point of view only the local spin magnetizations are modified to construct the guess wavefunction of a specific state. The orientation of the total magnetization is ensured by the subsequent two-component SCF process. Below we will focus only on the spin magnetization since manipulation of this parameter is crucial to build different states and to perform the mapping. In the two-component picture the three spin operators (Sx , Sy , Sz ) are described by 2x2 matrices       ~ 1 0  ~ ~ 0 1 ~ ~ 0 −i ~ Sz = σz =   , S x = σx =   , Sy = σ y =   2 2 0 −1 2 2 1 0 2 2 i 0

(1)

where σi are the Pauli matrices. The quantum state of the spin has a two-component complex form (spinor):   φa  Ψ =  . φb

(2)

Two of many possible spinors (eigenspinors of the Sz operator)     0 1 Φ+ =   = sz = + 12 , Φ− =   = sz = − 21 1 0

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(3)

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correspond to the spin functions in the one-component framework. All other eigenspinors, with quantization axes in different directions, can be written as linear combinations of these two eigenspinors. It is also possible to define rotation matrices operating on two-component spinors but rotating spin magnetization in 3-dimensional space,

U (α, β, γ) = e−iαSz e−iβSy e−iγSz ,

(4)

where α, β, γ are Euler angles. In the two-component picture it is easy to find the spin-vector density matrix (see, e.g., ref. 35 for details) m(r) = [mx (r), my (r), mz (r)],

(5)

which yields the expectation value of the spin magnetization 1 2

Z

m(r)d3 r = hSi .

(6)

In this picture the vector density at each point in space can be oriented differently. However, additional efforts are required to converge the DFT computation to a state where the local spins are not organized collinearly. The exchange-correlation functional should depend on all components of the vector density instead of only the z-component as in standard onecomponent DFT. This noncollinear DFT generalization (see, e.g., ref. 36) is used in all two-component computations in this work.

2.2

The effective spin Hamiltonian

The interaction of the two spins separated in space can be described by the effective spin Hamiltonian: ˆ = E0 + SD ˆ Sˆ H

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(7)


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where E0 is a constant that shifts the energy scale to adjust the overall energy of the spin Hamiltonian to the true microscopic one. Sˆ is the total spin operator

Sˆ = Sˆ1 + Sˆ2 ,

(8)

where Sˆ1 and Sˆ2 are local spin operators, and D is the tensor describing the interaction between the spins. Different models for the effective spin Hamiltonian of exchange-coupled systems have been proposed (see, e.g., ref. 37), which differ in the separation of (and approximations to) the various contributions. The Cartesian components of the interaction tensor are mixed by rotational transformations. One possibility is to decompose the reducible tensor into terms having different rotational transformation properties. 38 For a two-spin system this results in

ˆ =H ˆ 0 + J Sˆ1 · Sˆ2 + Sˆ1 Daniso Sˆ2 + d · (Sˆ1 × Sˆ2 ) + Sˆ1 D1siZFS Sˆ1 + Sˆ2 D2siZFS Sˆ2 . H

(9)

The scalar J represents the isotropic exchange coupling. Different notations for the sign and prefactor of J are are found in the literature. Here, the sign is chosen such that a positive J indicates antiferromagnetic coupling. This is consistent with the convention used for the remaining parameters. The Daniso parameter is the symmetric, traceless, anisotropic part of the exchange tensor. The pseudovector d describes the antisymmetric (DM) part. Finally the tensors D1siZFS and D2siZFS describe site (single-ion) zero-field splitting at centers 1 and 2, respectively. The siZFS tensors have a symmetric, traceless form. We use the prefactors given by van W¨ ullen: 31 i ˆ siZF ˆ i ˆ −−→ H S = Si DsiZFS (Si − 1/2i ),

(10)

−→ where 1/2 is a vector of length 1/2, collinear to the local spin Si . The Daniso and DisiZFS tensors can each be diagonalized and described by two scalar

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parameters D and E. The siZFS energy can be written as

i EsiZF S

1 2 2 2 = Di Siz − Si (Si + 1) + Ei (Six − Siy ) , 3

(11)

the anisotropic exchange energy as

ij Eaniso

= Dij

1 Siz Sjz − Si · Sj 3

+ Eij (Six Sjx − Siy Sjy ) .

(12)

In both cases D = 23 Dz and E = 21 (Dx − Dy ). This picture (eq. 9) can be easily extended to the symmetric trinuclear systems described in this work: ˆ = H

3 X i,j>i

3 X

J Sî · Sˆj +

3 X

~1 Sî · (R(θi )† · DsiZFS · R(θi )) · (Sî − )+ 2 i,j>i

Sî · (R(θi )† · Daniso · R(θi )) · Sˆj +

i,j>i

3 X

(13)

~ · (Sî × Sˆj ). (R(θi ) · d)

i,j>i

In case of C3 symmetry, the tensors are interconnected by rotations representable by rotation matrices (R(θi ); θi = 0, 120, 240 [deg]) . If the discussion is limited to isotropic exchange coupling, the estimation of this parameter for the given symmetric trinuclear systems can make use of Noodleman’s spin projection, 39

J=

(EHS − EBS ) , 4S 2

(14)

where S is the length of the local spin vector, or using the somewhat more general equation of Yamaguchi, 40

J=

2(EHS − EBS ) , hS 2 iHS − hS 2 iBS

(15)

where hS 2 i is the total spin expectation value. The HS and BS states correspond to the

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leftmost entries in Figure 2 below.

2.2.1

The isotropic and anisotropic exchange coupling

Turning now to the mapping procedure for our specific trinuclear case with C3 symmetry, we require a number of generalized high-spin (gHSa) and broken-spin (gBSa) states with the effective spin oriented in six different directions (a = z, x, y, xy, xz, yz). The spin configurations are shown in Figure 2. To derive the parameters of the effective spin Hamiltonian of eq. 13, the length of the local spins in the gHS and gBS states has to be assumed invariant. All equations presented here are obtained as energy differences between two spin configurations calculated using eq. 13. In the following we will give the results of these mappings and defer the detailed derivations to Supporting Information.

Figure 2: Selected spin configurations used in the calculation of isotropic exchange and the anisotropic exchange tensor The final mapping equation for Dzz has the very simple form zz Daniso +J =

EgHSz − EgBSz , 4S 2

(16)

where S is the length of the local spin. For the remaining tensor elements, the equations

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become more complicated. For example,

xx Daniso

√ EgHSx − EgBSx − (3Dyy + 2 3Dxy )S 2 +J = . 5S 2

(17)

Fortunately, use of the same procedure in six directions provides a system of six equations for six parameters. The solution is given in Supporting Information (Equations S8-S12). The isotropic exchange coupling is obtained as the trace of the (Daniso + J · I3x3 ) tensor. 2.2.2

Antisymmetric exchange

Because of symmetry reasons, 41 only the dz component is nonzero for the D3h case. Takeda et. al. have given a simple formula for antisymmetric exchange in D3h symmetry, 33

dz =

∆E 33/2 S 2

,

(18)

where ∆Ex = EH1x − EH2x is the energy difference between two helical spin states (denoted H1 and H2, see Figure 3). Unfortunately, this equation is valid only for S = 1/2 (in the absence of siZFS interactions), assuming that anisotropic exchange is negligible. For systems with S 6= 1/2 and without assumptions about the anisotropic part, the formula takes the more complicated form √ xy yy yy xx xx ∆Ex (6S 2 − 3S)(DsiZF Daniso − Daniso + 2 3Daniso S + DsiZF S ) √ + . dz = 3/2 2 − 3 S 4 · 33/2 S 2 4 3

(19)

Daniso is already known from our previous considerations (see above), but we have to find yy yy xx zz xx the DsiZF S + DsiZF S part. Fortunately, DsiZF S + DsiZF S = −DsiZF S can be estimated from

two collinear configurations (see below). In case of C3 symmetry the remaining components of the pseudovector are nonzero and formulas for them are given in Supporting Information (Equations S15 and S16).

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Figure 3: Selected spin configurations used in the calculation of antisymmetric exchange and siZFS 2.2.3

Single-ion zero-field splitting

zz The DsiZF S element can be easily calculated from two gHS states (gHSz and gHSx, see

Figure 2), zz DsiZF S =

zz 4(EgHSz − EgHSx ) − 18Daniso S2 . 18S 2 − 9S

(20)

The remaining parameters have to be estimated from helical configurations (see Figure 3) with the effective spin rotated in one of the selected directions (z,xy,yz,xz). For example, xz the DsiZF S element is given by

xz DsiZF S = −

yz xz ∆Ez + (33/2 Daniso − 3Daniso ) , 2 6S − 3S

(21)

where ∆Ez is the energy difference between two helical states with effective spin oriented in z-direction. The solution for the full tensor is given in Supporting Information (Equations S17-S22).

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3 3.1

Computational details Structure optimization

Structural parameters of the three complexes (cf. Figure 1) have been optimized for the high-spin state at the hybrid DFT level with the B3LYP 42 functional using the Gaussian 09 package. 43 To take into account scalar relativistic effects, Stuttgart-type quasirelativistic effective core potentials (MDF10) 44 with associated valence basis sets 44,45 were used for the metal centers, employing standard Pople-style polarized valence triple-zeta 6-311G** basis sets 46 for the ligand atoms. Harmonic vibrational frequency analyses were used to check if the structures are local minima. The structures were chosen to have quasi C3 symmetry. Particularly for the manganese complex, this is not the case, due to Jahn-Teller distortions of the MnIII sites. This may affect our comparisons with experimental data. However, as in this work we are mainly interested in the principal aspects of the approach, we accept this approximation. Further considerations and a comparison with the true minimum structure for the manganese complex will be given in section 4.1.

3.2

Two-component energy calculations

The two-component energies of all states have been calculated (including scalar and SO terms) at the all-electron Douglas-Kroll-Hess level at fourth order (DKH4) using the efficient local DLU 47 approximation in Turbomole 6.6. 48 Noncollinear DFT has been applied, i.e. the exchange-correlation functional depends on all vector components of the spin. The B3LYP, PBE0, 49,50 BHLYP, 51 and TPSSh 52,53 functionals and uncontracted def2-TZVP 54,55 basis sets have been used. The multipole-accelerated resolution-of-the-identity approach has been used for computing the electronic Coulomb interaction (MARI-J 56 ). All calculations have been converged to 10−8 Hartree (0.002 cm−1 ). We concentrate on hybrid functionals, as use of pure GGA (generalized gradient approximation) functionals led to small HOMO-LUMO gaps and consequently to stability problems for the collinear (gBS and gHS) configurations 13



in the trinuclear systems studied here. Computations of siZFS for complexes, where some of the ions have been replaced by diamagnetic ones, were not affected as much by this problem, and so we also give GGA results.

3.2.1

Isotropic exchange and anisotropic interaction

The gHS and gBS states needed to obtain the anisotropic exchange tensor were evaluated as follows: a) converge the one-component DKH4 HS state; b) build the one-component DKH4 BS state from the converged HS state using the spin-flip tools available in Turbomole; c) use the HS and BS states as initial guess for two-component DKH computations, providing HS and BS states with effective spin oriented in z-direction (gHSz and gBSz states); d) rotate all occupied Kohn-Sham spinors of the gHSz and gBSz states by a specific angle using a locally developed code. Read in these rotated states as initial guess for subsequent SCF procedures to obtain the series of gHSa and gBSa states (a = x, y, xy, xz, yz), see Figure 2. Our treatment of anisotropic exchange may be viewed as a simplification of a more complete treatment, where computations on further states provide an overdetermined system of equations that can be fitted to also provide error estimates. We will come back to this least-squares fitting in the Results section.

3.2.2

Antisymmetric exchange and siZFS

Here we need states with noncollinear configurations of local spins (see above), built as follows: a) localize selected KS spinors for the gBSz state by the Pipek-Mezey localization procedure 57 in our external program, using the adjoint matrix diagonalization proposed by Ciupka et al. 58 (incorporating code written by Cardoso 59 ); b) rotate the localized KS spinors independently to build an appropriate noncollinear initial guess. A subsequent SCF procedure in Turbomole gives one of two helical configurations (H1z) with effective spin oriented in z-direction; c) the complementary helical state (H2z) is built by spin reflection and a subsequent SCF procedure; d) the other helical states with different directions of

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effective spin are obtained using the same routines and procedure on the gHSa and gBSa states, see Figure 3. Results for siZFS were alternatively obtained using a diamagnetic replacement procedure, in which two of the three open-shell ions were replaced by diamagnetic GaIII ions (without reoptimization of the structure, using a def2-TZVP Ga basis set) so only the isolated remaining effective spin vector was rotated. These two-component results were additionally compared against a second-order perturbational treatment of the SO contribution to local ZFS using the Pederson-Khanna 12 approach within the ORCA package. 60 Both twocomponent and perturbational results were corrected by van W¨ ullen’s prefactors. 31 Due to an instability of the coupled-perturbed KS approach in ORCA for hybrid functionals, only pure GGA functionals have been used in this particular comparison.

3.3

Accuracy of the spin-rotation approach

As the present approach relies on relatively small energy differences, the numerical accuracy of the noncollinear two-component energy calculations is obviously of importance. To estimate it, we have performed additional computations for the linear complex CrH. Due to symmetry, the transversal parameter (E) of the ZFS should be exactly zero, which is only the case if calculations with spin oriented along x and y directions give exactly the same energy. With the present settings, the computed transversal parameter is about 10−3 cm−1 (where the axial ZFS parameter is about 1 cm−1 ), providing a reasonable estimate of the achievable accuracy for the other parameters evaluated in this work. Notably, this should also be sufficient after inclusion of two-electron SO terms, which are known to diminish the anisotropic interactions.

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4

Results

Before discussing the results for the model complexes, we note that the models have been chosen to evaluate the consistency of the present approach rather than for best agreement with experimental data. To reduce computational effort, the complexes have been simplified by replacing the full OOCR – ligands by OOCH – . Additionally, the structures were optimized for the high-spin state and within C3 symmetry (which requires us to only evaluate one rather than three separate exchange-coupling tensors). We will nevertheless make some comparisons with experimental data to provide an idea on how realistic the present approach is.

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J 1c-DKH hS 2 i corr.b 34.07 22.62 11.93 1.15 2c-DKH 34.16 22.60 12.02 1.41

D 0.17 -0.17 -0.18 0.12

aniso, asymm. E E/D 0.04 0.24 -0.05 0.29 -0.05 0.28 0.04 0.33 d -0.12 -0.36 -0.52 -0.24

z

D 0.29 0.34 0.33 0.22

siZFS E 0.09 0.02 0.02 0.03

siZFS E/D D 0.31 0.41 0.06 0.32 0.06 0.32 0.14 0.27

(Ga replacement) E E/D 0.02 0.05 0.01 0.03 0.01 0.03 0.06 0.22


17

J 1c-DKH hS 2 icorr.b 51.64 42.57 32.99 17.93 2c-DKH 51.94 42.80 33.18 17.96

D 0.33 0.34 0.24 0.17

aniso, E 0.02 0.06 0.01 0.03 asymm. E/D dz 0.06 -1.12 0.18 -1.13 0.04 -0.94 0.18 -0.56 D -6.32 -8.10 -7.43 -9.15

siZFS E -1.27 -1.52 -1.40 -1.64

siZFS E/D D 0.20 -6.75 0.19 -8.53 0.19 -7.80 0.18 -9.49

(Ga replacement) E E/D -0.86 0.13 -1.09 0.13 -1.02 0.13 -1.17 0.12

a

J aniso, asymm. 2 b 1c-DKH 1c-DKH hS icorr. 2c-DKH D E E/D dz TPSSh 87.73 87.40 87.10 0.58 0.02 0.03 -2.67 B3LYP 67.75 67.57 67.17 0.46 0.06 0.13 -3.06 PBE0 55.59 55.48 55.25 0.29 0.03 0.10 -2.32 BHLYP 26.32 26.30 26.17 0.12 0.01 0.08 -1.36 a Results obtained from scalar relativistic HS and BS states using equation 14. and BS states using equation 15.

siZFS D E E/D 0.86 0.05 0.06 1.14 0.28 0.25 0.79 0.21 0.27 0.36 0.11 0.31 b Results obtained

siZFS (Ga replacement) D E E/D -0.54 -0.10 0.19 -0.75 -0.06 0.08 -0.54 -0.05 0.09 -0.30 -0.01 0.03 from scalar relativistic HS

Table 3: Computed parameters of the full exchange interaction tensor for [Fe3 O(OOCH)6 (H2 O)3 ]+ in C3 symmetry using different functionals (all data in cm−1 ).

TPSSh B3LYP PBE0 BHLYP

1c-DKH 51.76 42.64 33.03 17.94

a

Table 2: Computed parameters of the full exchange interaction tensor for [Mn3 O(OOCH)6 (H2 O)3 ]+ in C3 symmetry using different functionals (all data in cm−1 ).

TPSSh B3LYP PBE0 BHLYP

1c-DKH 34.16 22.66 11.95 1.15

a

Table 1: Computed parameters of the full exchange interaction tensor for [Cr3 O(OOCH)6 (H2 O)3 ]+ in C3 symmetry using different functionals (all data in cm−1 ).

Page 17 of 36 Journal of Chemical Theory and Computation


4.1

Isotropic exchange couplings

The results for [Cr3 O(OOCH)6 (H2 O)3 ]+ , [Mn3 O(OOCH)6 (H2 O)3 ]+ , [Fe3 O(OOCH)6 (H2 O)3 ]+ (cf. Figure 1) are provided in Tables 1, 2, and 3, respectively. In all cases the isotropic exchange couplings calculated at two-component level differ slightly from one-component scalar-relativistic DKH BS-DFT results. This may reflect spinor relaxation at the twocomponent level or the necessary averaging over several gBS and gHS states (the parameter is found as trace of the full tensor). Nevertheless, all three techniques compared to estimate isotropic exchange (one-component DKH with Noodleman’s equation 14 or with Yamaguchi’s hS 2 i-corrected equation 15, or Noodleman-type two-component DKH) give very similar results within less than 1 cm−1 . This suggests that a Noodleman-type spin-projection procedure should give reasonable results also for the other parameters. As expected, the isotropic exchange coupling depends appreciably on the exchangecorrelation functional used, decreasing with increased exact-exchange admixture in hybrid functionals, as has been observed previously. 61–63 For example, the large exact-exchange contribution incorporated in the BHLYP functional (50%) leads to a rather low isotropic exchange coupling. Experimental isotropic exchange couplings for the present complexes are not available, but data for related systems (differing mostly in the substituents R of OOCR – ) exist (see, e.g., ref. 8 for a review). The series of [Cr3 O(OOCR)6 (H2 O)3 ]+ complexes has generally quasi-trigonal structures with only small deviations from C3 symmetry. They feature only moderate isotropic exchange couplings between 18 cm−1 and 26 cm−1 , specifically 22 cm−1 for [Cr3 O(OOCMe)6 (H2 O)3 ]+ . Differences between isotropic exchange couplings due to deviations from trigonal symmetry are about 1-3 cm−1 . Our computations at B3LYP level are in almost perfect agreement with the experimental data, while other functionals underestimate (BHLYP, PBE0) or overestimate (TPSSh) the experimental values (Table 1). Isotropic exchange coupling constants of [Mn3 O(OOCR)6 (L)3 ]+ (L=H2 O) do not seem to be available experimentally. Many manganese(III) compounds containing [Mn3 (µ3 -O)]7+ cores form isosceles triangles and are fitted by two J values that differ significantly. Equilat18


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eral triangles are formed by complexes containing pyridine, 3-methylpyridine and imidazole ligands at the terminal positions. 64 All of these compounds can be fitted by one isotropic exchange interaction of about 12 cm−1 . The calculated values (except for the BHLYP results) are too high in comparison here (Table 2), which may not be surprising given the structural differences. Indeed, vibrational frequency analysis confirms that the C3 -type structure is not minimum in this case but has two low-lying imaginary frequencies involving motion of the manganese ions). Lowering of the symmetry to quasi-C2 gives a local minimum. Two of the three Mn-Mn bonds are now longer by about 0.12 ˚ A, compared to the C3 structure. Lower isotropic exchange couplings of about 6-14 cm−1 (scalar relativistic BS DKH results, cf. Table S4 in Supporting Information) are then obtained, in better agreement with experiment. In case of the iron system, experimental data are available for the [Fe3 O(OOCMe)6 (H2 O)3 ]+ cluster. 65,66 Here the fitted isotropic exchange coupling is about 60 cm−1 if J12 = J23 = J31 is assumed, or 76 cm−1 and 58 cm−1 when J12 6= J23 = J31 . The B3LYP and PBE0 results provide again the best agreement with experiment.

4.2

The anisotropic exchange coupling tensor

The calculated axial and transversal anisotropic exchange parameters tend to be very small. In most cases the axial parameter does not exceed 0.5 cm−1 . This is less than 1.5% of the isotropic exchange coupling, showing that the strong-exchange case, where isotropic exchange is much larger than the remaining contributions, is not affected by the inclusion of the anisotropic exchange couplings. The transversal parameter for [Cr3 O(OOCH)6 (H2 O)3 ]+ is close to the E/D=1/3 limit, where small changes of the tensor, e.g. with different exchange-correlation functionals, may switch the sign of D. For the two other systems the E/D ratio tends to be below 0.1, and no sign changes are found. For the manganese and iron complexes, the anisotropic exchange exhibits a rather similar dependence on exact-exchange admixture in the functional as discussed above for the isotropic exchange. Only [Cr3 O(OOCH)6 (H2 O)3 ]+ features appreciably 19



smaller changes of the anisotropic contribution. Experimental data for anisotropic exchange do not seem to be available, so that we cannot compare the rather small computed values with experiment. We may, however, obtain error estimates of the current procedure by evaluating the anisotropic tensor also from fitting parameters for an overdetermined system of equations (details and results are given in Supporting Information, section S2). The fitted parameters obtained from the simpler, exactly determined equations agree well with the results from the overdetermined fitting procedure. This confirms the stability of the simplified procedure used in throughout most of this work, and of the underlying model.

4.3

Antisymmetric (Dzyaloshinskii-Moriya) exchange

It can be shown that in the case of the D3h symmetry only the z-component of the DM vector has a non-zero value. 33 This is generally the case, when the two centers in question are related by a mirror plane and lie together in another plane (for more details see 67). While this does not hold strictly for the present C3 case, the transverse components are clearly expected to be small (see below). The largest DM exchange parameter (dz ) is observed for the iron complex (up to ca. 3 cm−1 ), the smallest for the chromium complex (less than 0.5 cm−1 ). A dependence on exact-exchange admixture in the functional is observed here too. For example, the larger admixture in BHLYP compared to B3LYP gives a smaller d value. Antisymmetric exchange parameters for related trichromium complexes have been investigated in detail experimentally. 8,68 The effect was found to be small but non-negligible (0.1-0.35 cm−1 68–72 ). Agreement with our computed values (0.12-0.52 cm−1 ) is good. While antisymmetric exchange does not seem to be available for manganese complexes related to our model system, data are available for the Fe3 O core in iron complexes. The experimental value for [Fe3 O(OOCEt)6 (H2 O)3 ]+ 73 is 1.4 cm−1 . Estimates for the related hexanuclear complex [Fe6 Na2 O2 (O2 CPh)10 (pic)4 (EtOH)4 (H2 O)2 ](ClO4 )2 are between 2 cm−1 and 4 cm−1 , 66 bracketing the present computed values. The perpendicular vector components have been estimated only at B3LYP level (see 20


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Table S2 in Supporting Information). The results confirm that dx and dy are small (less than 5% of the dz component).

4.4

The single-ion zero-field splitting tensor (siZFS)

In keeping with the assumption of a strong-exchange case, the computed axial and transversal siZFS parameters for [Cr3 O(OOCH)6 (H2 O)3 ]+ and [Fe3 O(OOCH)6 (H2 O)3 ]+ are much smaller than the isotropic exchange coupling, but they are twice larger than the anisotropic exchange coupling parameters (Tables 1-3). The axial parameter is below 0.4 cm−1 and below 1.2 cm−1 for the chromium and iron systems, respectively, with E/D ratios below 0.3. In contrast, the manganese complex is found to exhibit axial siZFS values above 6 cm−1 , which is a sizeable fraction (ca. 20% for all functionals except BHLYP, where the ratio is 50%) of the isotropic exchange. The relatively large siZFS reflects the Jahn-Teller distortion of the high-spin d4 MnIII ions in the crystal field, resulting in low local symmetry. This is confirmed by analysis of the optimized structure: the distorted octahedral structure is characterized by differences between Mn-O bond lengths of about 0.25˚ A, which is much larger than for the chromium (0.01˚ A) and iron (0.04˚ A) complexes. In case of the siZFS, the dependence on the functional is less clear-cut (Tables 1-3), with almost no dependence exhibited by [Cr3 O(OOCH)6 (H2 O)3 ]+ , an irregular but modest dependence for [Mn3 O(OOCH)6 (H2 O)3 ]+ , and a sharp decline of the axial parameter with the largest exact-exchange admixtures for [Fe3 O(OOCH)6 (H2 O)3 ]+ . This reflects the local character of the siZFS, which essentially signals the way in which the electronic levels at the given ion are split by SO coupling. To evaluate the soundness of obtaining the siZFS parameters from noncollinear spin states in the way described above, additional tests involving diamagnetic substitution have been performed. That is, two of three paramagnetic MIII ions in each complex have been replaced by diamagnetic GaIII . Subsequently, similar routines involving two-component spinor rotation in six directions (cf. eqs. S24-S29 in Supporting Information) have been ap21



plied to evaluate the siZFS parameters of the remaining paramagnetic ion in the substituted complexes. Diamagnetic replacement gives almost identical results as the full procedure for the trichromium complex. This suggests that the assumption of locality of the site-spin magnetization, needed for the mapping onto the effective spin Hamiltonian, is well justified in this case. That is, the local electronic structure at the remaining CrIII ion in the substituted complex is essentially unaltered by the Ga substitution. Reasonable agreement between siZFS parameters obtained by the full noncollinear procedure and by diamagnetic replacement is found also for [Mn3 O(OOCH)6 (H2 O)3 ]+ . The differences are clearly larger than for the chromium complex, but they are still below 2% of the siZFS values obtained with the full gBS-DFT method. Notably, the influence of the restricted C3 symmetry (compared to the fully optimized C2 structure) for the manganese complex is much smaller for the siZFS parameters than seen above for the isotropic exchange coupling. More detailed comparisons of the siZFS for diamagnetically replaced complexes for both symmetries are given in Table S4 (Supporting Information). Table 4: Comparison of the siZFS (in cm−1 ) for [MnGa2 O(OOCH)6 (H2 O)3 ]+ obtained at two-component level (DKH4) against second-order perturbation results (Pederson-Khanna method with van W¨ ullen prefactors) using either a bare one-electron or the effective AMFI and SOMF spin-orbit operators (including spin-other-orbit terms)

2c-DKH4-1e PK-1e PK-SOMF PK-AMFI

BLYP DsiZF S EsiZF S -8.56 -0.95 -6.97 -0.97 -1.99 -0.28 -2.01 -0.26

BP86 DsiZF S EsiZF S -7.52 -0.95 -6.31 -0.94 -1.80 -0.27 -1.81 -0.25

PBE DsiZF S EsiZF S -7.61 -0.95 -6.36 -0.95 -1.81 -0.27 -1.83 -0.25

Much poorer agreement between the full gBS-DFT treatment and diamagnetic-replacement results is found for [Fe3 O(OOCH)6 (H2 O)3 ]+ . In all cases the sign of the interaction is altered by diamagnetic substitution. Replacement leads also to lower absolute values of the axial siZFS parameter. Population analysis using Becke’s partitioning 74 (as implemented in TURBOMOLE) shows that for this system, the assumption of localized spin centers ceases 22


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to hold. The spin density is clearly delocalized onto the ligand atoms and even over the central oxo core.

4.4.1

Evaluation of spin-orbit operators

As the SO operators provided by the current two-component DKH4 implementation in TURBOMOLE determine to some extent the accuracy of the anisotropic contributions, we have additionally computed the ZFS for the diamagnetically replaced complexes using secondorder perturbation theory (PK approach with van W¨ ullen prefactors) with the ORCA code, comparing different SO operators. As the coupled-perturbed KS equations for hybrid functionals did not converge in some cases, we provide this comparison for pure GGA functionals. Table 4 summarizes results for [MnGa2 O(OOCH)6 (H2 O)3 ]+ , where the ZFS contributions are largest. Results for the other two complexes are provided in Supporting Information (Table S3). The two-component ZFS values with TURBOMOLE are even larger than the perturbational results with only the bare one-electron operator used in ORCA, consistent with the neglect of two-electron SO contributions in the current TURBOMOLE implementation (two-electron contributions are known to reduce the SO effects). Using the spin-orbit mean field (SOMF) approximation 75 of ORCA to incorporate accurately the two-electron terms gives significantly lower ZFS values. This has to be kept in mind when considering the quantitative accuracy of the results, and future evaluations will have to include the two-electron SO terms adequately.

4.5

The magnetic dipole-dipole interaction

So far we have exclusively focused on the exchange interaction tensor between two centers. It is worth evaluating also the importance of magnetic dipole-dipole (DD) interactions. This goes outside our noncollinear energy determinations but can be done straightforwardly, provided we have local g-tensors. Details on the evaluation of DD tensor elements are described elsewhere. 76 Since we do not yet obtain g-tensors wihin the gBS methodology (this is planned 23



in subsequent work), we obtained the local g-tensors from second-order perturbation theory computations in ORCA, using diamagnetic replacement. The B3LYP functional, an uncontracted def2-TZVP basis set with the DKH2 Hamiltonian and an SOMF(1X) approximation to the spin-orbit matrix elements has been used. These preliminary computations for all components of the DD tensor show that contributions to the isotropic and antisymmetric interactions are negligible (less than 10−5 [cm−1 ]). This results from essentially isotropic local g-tensors for the present systems. Only contributions to the anisotropic tensor are sizable. DD Table 5 collects the D and E parameters obtained during diagonalization of (Dex aniso + Daniso )

or of Dex aniso only. The DD contribution affects the anisotropic interaction by up to 0.07 [cm−1 ] (less than 35% of the exchange contribution). As the exchange contribution is likely overestimated due to the missing SO contributions, DD contributions to the anisotropic part of the exchange tensor are thus expected to be nonnegligible. Table 5: Effects of magnetic dipole-dipole interactions on the anisotropic exchange tensor at B3LYP level

[Cr3 O] [Mn3 O] [Fe3 O]

5

exchange contribution ex.+dip.-dip D E D -0.17 -0.05 0.23 0.34 0.06 0.28 0.46 0.06 0.53

contributions E 0.08 0.04 0.10

Conclusions

The approach presented in this work provides access to the full exchange interaction tensor between the centers of trinuclear spin-coupled systems in a consistent way. The equations for the mapping procedure have been derived for the case of C3 symmetry but can be straightforwardly generalized to any system size, site spin, and point group symmetry. All parts of the interaction tensor, including single-ion zero-field splitting, are obtained on the same footing from the computed energies obtained in non-collinear two-component quasirelativistic DFT calculations, using a generalization of the concept of broken-symmetry DFT. That 24


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is, the energies of the non-collinear gBS-DFT states are mapped onto a suitable uncoupled (“many-spin”) effective spin Hamiltonian to find its parameters. While a comparison with experimental data was indirect due to simplifications in the chosen model structures, the agreement obtained indicates the essential soundness of the presented approach. The methodology offers an alternative to previous approaches to single-ion zero-field splittings, and it provides access to the rarely examined antisymmetric Dzyaloshinskii-Moriya exchange interaction. In the latter case, the energies of two (or more) helical states are compared, generalizing a method proposed by Takeda et. al. to larger and more complex molecular systems. One advantage of the present approach, in spite of the computational burden involved, is the variational inclusion of SO coupling. This will be of particular importance for complexes involving 4d or 5d transition-metal centers or possibly also f-block elements, where a perturbational treatment of SO coupling may not be valid anymore. Notably, while we have used quasirelativistic two-component methods, the present approach can be extended straightforwardly to four-component calculations. We should also keep in mind the limitations of this methodology: as it corresponds to a generalization of broken-symmetry DFT, it also inherits some of the limitations of BSDFT. For example, the invariance of the E0 energy in the effective spin Hamiltonian (see equation 7) has to be ensured. Here we have ensured invariance by choosing pairs of the most similar states (in particular gHS-gBS pairs or pairs of two helical states with effective spin in the same direction). This introduces possible additional errors requiring further study. Obviously, the need of a mapping to an effective spin Hamiltonian is another limitation. Such a mapping may not always be straightforward, and the identification of localized spin centers is a requirement that needs to be examined. This is not a specific problem of the gBS-DFT approach but of any methodology involving an effective spin Hamiltonian in spin-coupled systems. An aspect relevant for systems containing heavier atoms with larger SO effects is to retain

25



the desired orientations of the local spinors during the SCF process. Based on experience with other noncollinear DFT computations, one option is to include local external magnetic fields into the Fock matrix that favor certain spinor orientations. The field strength is decreased during the SCF process, providing the desired unperturbed energies.

6

Supporting information

Detailed description and derivation of the mapping procedure used, as well as some additional results, are part of Supporting information available free of charge via the Internet.

7

Acknowledgments

This project has been funded by DFG project KA1187/13-2 and by the Polish Ministry of Science and Higher Education under Grant 1317/1/MOB/IV/2015/0 (”Mobility Plus” program).

26


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a development of University of Karlsruhe and

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Spin-Frustrated

System,

34

and


[Cr3 (µ3-O)-(µ2-PhCOO)2 (µ2-

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Noncollinear Two-Component Quasirelativistic Description of Spin

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