Modeling of EPR parameters of copper(II) aqua complexes

Chemical Physics 332 (2007) 176–187 www.elsevier.com/locate/chemphys

Modeling of EPR parameters of copper(II) aqua complexes Katia Julia de Almeida a, Zilvinas Rinkevicius a,*, Ha˚kan Wilhelm Hugosson a, ˚ gren a Amary Cesar Ferreira b, Hans A b

a Department of Theoretical Chemistry, Royal Institute of Technology, SE-10691 Stockholm, Sweden Departamento de Quı´mica, Universidade Federal de Minas Gerais, Av. Antonio Carlos, 6627, CEP-31270-901 Belo Horizonte, Minas Gerais, Brazil

Received 30 September 2006; accepted 15 November 2006 Available online 19 November 2006

Abstract In this paper we report density functional theory calculations of the electronic g-tensor and hyperfine coupling constants of the copper dication in sixfold- and fivefold-coordination models of the first aqueous solvation sphere. The obtained results indicate that the electronic g-tensor of these copper complexes in combination with hyperfine coupling constants of copper in principle can be used to elucidate the coordination environment of the hydrated copper dication. In addition to these results, we have designed a methodology for accurate evaluation of electronic g-tensors and hyperfine coupling tensors in copper complexes, and demonstrate the applicability of this approach to copper dication aqua complexes. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Copper dication; EPR; DFT; g-Tensor; Hyperfine coupling tensor

1. Introduction Aqua complexes of transition metal ions are often regarded as basic model systems for theoretical and experimental investigations of more complex transition metal compounds as they mimic essential features of the larger compounds, while still remaining relatively simple, allowing detailed investigations [1,2]. In particular, these complexes provide valuable insight into the electronic structure of the active sites of enzymes, and have therefore been considered to be suitable prototypes for biological systems [3]. Consequently, aqua complexes of transition metal ions have been well attended to by theoreticians and experimentalists alike, with numerous investigations carried out for the understanding of their geometric and electronic structures and related properties [4]. However, many questions related to the hydration of these ions still remain to be answered and research in this field is continuously and actively pursued. *

Corresponding author. Tel.: +46 8 5537 8418; fax: +46 8 5537 8590. E-mail address: [email protected] (Z. Rinkevicius).

0301-0104/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2006.11.015

The above statements hold well for the copper dication, Cu(II). This transition metal ion is one of the most abundant transition metal ions in biomolecules, and understanding its hydration, i.e. the structure of its aqua complexes, is of major importance. Cu(II) aqua complexes are also frequently employed as prototypes of the larger copper complexes encountered in biomolecules, such as in the electron transfer blue copper proteins and prion proteins. Nevertheless, the electronic structure and the behavior of the Cu(II) aqua complex in different phases remain relatively poorly understood. The conventional picture of the Cu(II) hydration postulates that the first solvation sphere forms a sixfold-coordinated complex. This aqua complex is assumed to have an octahedral ligand configuration which is Jahn– Teller distorted, leading to an elongation of the bonds of the two water ligands in axial positions [6]. However, recently this view was challenged by Pasquarello et al. [7]. In their study, combining neutron diffraction experiments and first-principles molecular dynamics simulations, the hydrated Cu(II) ion was found to have a pyramidal fivefold-coordination. Furthermore, it was shown that previous visible near-infrared absorption [5], X-ray absorption near-

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edge structure [8] (XANES) and nuclear magnetic resonance [9] (NMR) experimental measurement results, which have been used to support the sixfold-coordinated hydrated Cu(II) hypothesis, well can be reinterpreted in a way that is consistent with the picture of fivefold-coordination. More recent investigations on the geometrical structure of the copper dication first solvation sphere, performed using full multiple-scattering theoretical analysis of the copper K-edge near-edge X-ray absorption [10] (NEXAFS) and copper K-edge extended X-ray absorption fine structure [11] (EXAFS) experiments, also provided evidence of fivefoldcoordination. Nevertheless, some experimentalists as well as theoreticians still support the conventional sixfold-coordinated Cu(II) aqua complex formation upon hydration, indicating that the question of the number of water molecules in the first solvation sphere remains open. In order to gain more insight into the electronic and molecular structures and properties of Cu(II) in water solutions, and as a step towards possibly resolving the disparate views on the problem, we present in this article results from a density functional theory (DFT) based investigation of the electronic paramagnetic resonance (EPR) parameters, namely electronic g-tensors and hyperfine coupling constants, in the fivefold- and sixfold-coordinated Cu2+ ion aqua complexes. As the electronic g-tensor and the hyperfine coupling constants are extremely sensitive to the coordination environment of the transition metal ion, depending intrinsically on the surrounding ligand field, we hope that results of the present investigation can provide guidance to further experimental efforts to elucidate the geometrical arrangement and number of water molecules in the first solvation sphere of Cu(II). The collected data should also provide a desirable element of agreement for the determination of the electronic and molecular structure of hydrated Cu2+ ions.

restricted density functional response theory method developed by Rinkevicius et al. [16]. In this formalism, the electronic g-tensor of a molecule is computed as

2. EPR parameters in density functional theory

where E0 and Em are energies of the ground |0i and excited states |mi, respectively, and the summation runs over all singly excited states of the molecule. The spin–orbit contribution to the electronic g-tensor shift involves the evaluation of matrix elements of the one- and two-electron SO operators. However, as is well established, the computation of two-electron operators in DFT is non-obvious (a detailed discussion of this topic can be found in Ref. [20]). In order to avoid this problem, and still obtain the accurate SO matrix elements required for reliable evaluations of g-tensors, we employ the so-called atomic mean field approximation (AMFI) for spin–orbit operators, i.e. the one-electron SO operator matrix elements are evaluated in the ordinary way and the two-electron SO operator matrix elements are computed according to the procedure developed by Schimmelpfennig [21]. Previous works devoted to electronic g-tensors [15,16] as well as other properties have clearly given support to the use of the AMFI SO operators, something that motivates us to use this approximation for the present calculations of electronic g-tensors of Cu(II) aqua complexes.

Methods for computation of electronic g-tensors and hyperfine coupling constants based on density functional theory are well established by now [12–19]. Most of these are capable of predicting EPR parameters of organic compounds quite accurately. However, the situation is less fortunate in the case of transition metal compounds for which the accuracy of electronic g-tensors [14–16] as well as hyperfine coupling constants [17–19] show significant dependence on the approach selected for the particular molecular system. Therefore, a careful selection of methodology is crucial for obtaining reliable results. Among a variety of DFT methods available for the evaluation of electronic g-tensors and hyperfine coupling constants, the approaches based on the restricted Kohn–Sham formalism clearly stand out as the most suitable ones for investigations of transition metal compounds, as they avoid the spin contamination problem common in methods based on the unrestricted Kohn–Sham formalism [16,19]. In this paper we have selected one such method, namely the spin-

g ¼ ge 1 þ DgRMC þ DgGCð1eÞ þ DgOZ=SOðAMFIÞ ;

ð1Þ

where ge is the free electron g-factor. The remaining terms refer to the various parts of the so-called electronic g-tensor shift Dg, which describes the influence of the local electronic environment in the molecule on the unpaired electron(s) compared to the free electron. The g-tensor shift contains two minor contributions, namely the mass–velocity (DgRMC) and the one-electron gauge (DgGC(1e)) corrections to the electronic Zeeman effect, along with the dominating spin–orbit (SO) contribution, DgOZ/SO(AMFI). For most of the molecules, the SO contribution almost solely determines Dg, and an accurate evaluation of this contribution is therefore of paramount importance for reliable calculations. Being the most important contribution, the spin–orbit contribution is unfortunately at the same time the most difficult one to evaluate as it is defined via the linear response function: 1 b SOðAMFIÞ ii ; LO; H DgOZ=SOðAMFIÞ ¼ hh c 0 S

ð2Þ

involving the electron angular momentum operator, c LO, and a sum of one- and two-electron SO operators, b SOðAMFIÞ . Here, we assumed that the linear response funcH tion is computed for the ground state of the molecule with maximum spin projection, S = MS. The linear response function used in the above definition of DgOZ/SO(AMFI) can be written in the spectral representation for two arbib 1 and H b 2 as trary operators H X h0j H b 1 jmihmj H b 2 j0i þ h0j H b 2 jmihmj H b 1 j0i b 1; H b 2 ii ¼ hh H ; 0 E 0  Em m>0 ð3Þ

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The second of the EPR spin Hamiltonian parameters, investigated in this article, is the hyperfine coupling tensor. For this parameter we chose the DFT restricted–unrestricted (RU) approach developed by Rinkevicius et al. [19], and which was earlier derived for Hartree–Fock and for multi-configuration self-consistent field wave functions by Fernandez et al. [22]. The RU approach allows the computation of the lowest order contributions (second order in fine structure constant) to the hyperfine coupling tensor A, namely the isotropic Fermi contact AFC and classical spindipolar Adip contributions. Both are divided into two terms in the RU approach, where the first term is evaluated as an b FC or spin-dipolar expectation value of the Fermi contact H b H dip operator, while the second term is computed as a linb FC or H b dip operator coupled ear response function of H with the triplet gradient of the restricted Kohn–Sham b 0 of the unperturbed molecular system. Hamiltonian H The second term allows for an account of spin-polarization effects in these calculations. This leads to the following AFC and Adip evaluation formulas in the RU approach (detailed description can be found in Ref. [19]): b FC j 0i þ hh H b FC ; H b 0 ii0 ; AFC ¼ h0j H dip dip dip b j 0i þ hh H b ;H b 0 ii0 : A ¼ h0j H

ð4Þ ð5Þ

The RU approach described above for hyperfine coupling constants performs very well for organic compounds, but unfortunately in the case of transition metal compounds it has limited applicability as it neglects higher order spin–orbit contributions, which, as pointed out in the earlier work of Neese [18], become non-negligible in the case of transition metal compounds. Therefore, in order to facilitate an accurate computation of copper hyperfine coupling constants, apart from ordinary AFC and Adip contributions, we also computed the above mentioned spin–orbit contribution ASO, which (according to Neese [18]) is evaluated as a linear response function between the one- and twob SOðAMFIÞ and the nucleus-orelectron spin–orbit operators H c bit interaction operator L N , i.e. 1 c b ð6Þ ASO ¼  hh L N ; H SOðAMFIÞ ii0 : S

metal compounds are accounted for and that a correct physical picture of the hyperfine interaction is reproduced. We are therefore confident that this approach allows us to reliably predict copper hyperfine coupling constants for the sixfold- and fivefold-coordinated water complexes investigated in this paper. 3. Computational details In the present investigation of EPR parameters of copper dication aqua complexes, we employ two models of the first solvation sphere. In the first model, the conventional view of the Cu(II) hydration process is adopted and two different sixfold-coordinated Cu(II) aqua complexes with distinct D2h and Ci symmetry point groups (see Fig. 1) are built. The former corresponds to a Jahn– Teller distorted octahedral water ligand configuration, where the two axial bonds between the copper ion and water ligands are longer than the corresponding equatorial bonds, leading to D2h symmetry. In the second sixfoldcoordinated model, we relaxed the symmetry restriction on the molecular geometry even more in order to estimate the influence of ligand rotations around the axes in their octahedral configuration. The second main model, now assuming pyramidal fivefold-coordinated first solvation spheres of hydrated Cu(II) complexes, is based on the experimental and theoretical findings of Pasquarello et al. [7]. Five different geometrical arrangements of C2v symmetry were investigated, corresponding to square-pyramid (I– III structures in Fig. 2) and trigonal bipyramid (IV and V structures in Fig. 2) configurations of the water ligands. The orientation of water molecules around the Cu(II) center is the main difference between these structures. Geometry optimization of the hexa- and penta-coordinated Cu(II) aqua complexes were carried out at the B3LYP level [23], employing standard Gaussian basis sets: 6-31G(d) for copper [24] and 6-311G** for oxygen and hydrogen [25]. In order to perform this task we used the grid based DFT implementation in the GAMESS-US package, [26] which

Detailed formula and their derivation for the ASO contribution can be found in the work of Neese [18]. The major difference between his and our method is the use of restricted linear response theory in the present calculations, rather than the unrestricted couple-perturbed Kohn–Sham method. Furthermore we, similarly to the electronic g-tensor calculations in the computation of ASO, employ the AMFI approximation for the spin–orbit interaction operator, which is more accurate than the semi-empirical effective charge approximation used by Neese [18]. Therefore, summarizing Eqs. (4)–(6), the hyperfine coupling tensor of copper was here computed in the following way: A ¼ 1AFC þ Adip þ ASO :

ð7Þ

The outlined hyperfine coupling constant approach ensures that all important contributions encountered for transition

Fig. 1. DFT/B3LYP optimized molecular structures of sixfold-coordinated Cu(II) aqua complexes.

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179

Fig. 2. DFT/B3LYP optimized molecular structures of fivefold coordinated Cu(II) aqua complexes.

provides efficient geometry optimization procedures for locating equilibrium geometries. Calculations of the EPR spin Hamiltonian parameters, namely the electronic g-tensors and hyperfine coupling constants of copper, for all investigated Cu(II) aqua complexes were carried out for optimized geometries using the DALTON 2.0 quantum chemistry package [27]. Electronic g-tensors were computed using restricted density functional theory according to the formulas outlined in the previous section of this paper. When evaluating the Fermi contact and spin-dipolar contributions to the hyperfine coupling tensors (see Eqs. (4) and (5)) the restricted– unrestricted approach was used. Higher order spin–orbit contributions to the hyperfine coupling tensor were computed using spin restricted density functional linear response theory (see Eq. (6)). In the evaluation of this contribution to the hyperfine coupling tensor, the spin-polarization effects, which are expected to be of minor importance based on our experience with similar calculations of the spin–orbit contribution to the electronic g-tensors [16], were neglected. All the calculations of EPR parameters described above have been performed using the modified B3LYP exchange–correlation functional reparametrized by Solomon et al. [28,29] to reproduce the correct balance between ionic and covalent bonding in Cu d9 complexes, like the Cu(II) compounds here. This choice of exchange-correlation functional gives reliable d– d excitation energies in the Cu(II) aqua complexes and consequently enables an accurate evaluation of both the electronic g-tensors and hyperfine coupling constants [28,29]. In the electronic g-tensor calculations we employed the IGLO-II basis set for the oxygen and hydrogen atoms, while in the computation of the hyperfine coupling tensors for these atoms we further extended the basis set by uncontracting the s-type functions in the IGLO-II basis [30,31] and adding two tight s-type functions. For copper we used

the CP(PPP) basis set designed by Neese [32] for accurate calculations of hyperfine coupling in transition metal compounds. Our choice of basis set is not compatible with the standard routines for AMFI spin–orbit operator matrix elements implemented in DALTON 2.0, and in order to overcome this limitation we employed the code provided by Schimmelpfennig [21], which is capable to evaluate AMFI SO matrix elements for arbitrary basis sets. 4. Results and discussion 4.1. Geometries: hexa- and penta-coordinated Cu(II) aqua complexes EPR parameters, especially hyperfine coupling constants, are notorious for their strong dependence on the geometrical structure of compounds. This feature is used to determine molecular structure from EPR spectra, but at the same time it necessitates that geometries used in theoretical investigations of EPR parameters closely correspond to the ‘‘real’’ structure of the compounds for which EPR spectra have been measured. Therefore, before discussing the electronic g-tensor and hyperfine coupling constants for Cu(II) aqua complexes, let us assess the quality of the geometries obtained in our B3LYP calculations. The shorter Cu–O bond length in Cu(II) aqueous solutions is well determined using various experimental methods [7,8,10,11,33,4], including EXAFS, XANES and neutron diffraction. Depending on the experimental ˚ to 2.01 A ˚ . However, the sitapproach it varies from 1.95 A uation for the longer Cu–O bond is less fortunate and the experimental methods disagree, estimations of this bond ˚ to 2.39 A ˚ . The geometrical struclength vary from 2.29 A ture of hexa-coordinated Cu(II) aqua complexes is well investigated by theoreticians [6,34–38], since it gives a nice representation of the Jahn–Teller effect, corresponding to

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the well-known E  e problem [39] (copper dication electronic configuration in this complex is (t2g)6 (eg)3, assuming an octahedral ligand field). The solution to this problem predicts an adiabatic potential energy surface of the warped ‘‘Mexican hat’’ shape with three equivalent minima (see Fig. 6 in Ref. [39] for details), where the geometrical structure in the minima corresponds to an octahedron tetragonally distorted along one of the three fourfold axes. Therefore, taking into account the above outlined Jahn– Teller effect, one can expect two elongated Cu–O bonds in sixfold-coordinated Cu(II) aqua complexes. Our geometry optimizations, along with previous calculation results and available experimental data, are tabulated in Table 1. Overall, Cu–O bonds obtained in our B3LYP calculations are in good agreement with results from previous calculations as well as experimental X-ray diffraction and EXAFS data. It is worth noting that our D2h symmetry model, as well as previous B3LYP calculation results, predict a geometrical structure of sixfold-coordinated Cu(II) aqua complexes which is in agreement with experimental data. It can be noted our results are among the best results obtained with computational methods. For example, the Hartree– Fock and sophisticated QM/MM methods encounter difficulties in predicting the axial Cu–O bond distance accurately, and underestimate this bond length compared to experimental data, as well as compared to previous B3LYP calculation results [36,38]. The only exception from this trend is QM/MM calculations in which the QM region was treated at the MP2 level [38], which obtained axial and equatorial bond lengths in very good agreement with experimental results [33,4]. Based on analysis of the geometrical structure of the hexa-coordinated Cu(II) aqua complex, we selected the geometry of the Cu(II) complex obtained with the D2h symmetry as the most suitable model for representing the first solvation shell of the copper dication (assuming six-

fold-coordination of this transition metal ion). In order to gain more support for this choice, we also examined Cu–O bond lengths of hexa-coordinated Cu(II) aqua complexes with octahedral symmetry existing in various crystals and compared them with our B3LYP calculation results for the D2h symmetry model. As is evident from data presented in Table 2, the B3LYP calculations reproduce the pattern of bond length alternations observed in the crystals, and, in agreement with experimental data, predict the two equatorial bonds to be slightly shorter than the other two bonds. This feature is more pronounced in the crystals than in the calculations, due the influence of the crystal environment on the Jahn–Teller distortion. The good agreement between calculated and observed Cu–O bond lengths in crystals further reassured us that the Cu(II) aqua complex geometry obtained using the D2h symmetry model is the one best suited for realistic modeling of the first solvation sphere of hexa-coordinated Cu(II). We are therefore confident that the geometry obtained with the D2h symmetry model will serve well as reliable input data for evaluation of the EPR parameters of hydrated Cu(II). In our B3LYP calculations, the geometries obtained with D2h symmetry apparently resemble experimentally observed geometries more closely than geometries obtained with the alternative Ci model. This finding contradicts the available experimental data [33,4], which indicate a strict Ci micro-symmetry for sixfold-coordinated Cu(II). The main reason for this discrepancy is the presence of additional degrees of freedom for the water molecules, which are introduced in our Ci model by reducing symmetry from D2h to Ci, as they evidently are not present in the crystal environment. One way to improve our Ci model would be to explicitly include a second solvation sphere in order to restrict the axial ligand movement and consequently reduce additional degrees of freedom in the first shell. In conclusion, the Ci model can in principle

Table 1 a,b Hexa-coordinated Cu(II) aqua complex, CuðH2 OÞ2þ 6 , geometrical parameters obtained in B3LYP simulations Method B3LYP (D2h) B3LYP (Ci) B3LYPd B3LYPe HFe HF/MMf a

Req(Cu–O)c 1.97/2.05 2.00/2.00 2.00/2.03 2.02 2.03 2.07

Rax(Cu–O) 2.30 2.26 2.28 2.29 2.15 2.20

Method g

MP2/MM QM/MMh QM/MMi Experimentalj Experimentalk Experimentalk

Req(Cu–O)

Rax(Cu–O)

2.07 2.07 2.07 1.95 1.98 2.01

2.35 2.24 2.20 2.29 2.39 2.33

Calculations performed using 6-31G (d) basis set for copper and 6-311G** basis set for oxygen and hydrogen. All Cu–O bond lengths are in angstroms. c Equatorial Cu–O bond length, Req(Cu–O), values are give for x and y molecular-axis (see Fig. 1). In cases where only averaged Cu–O bond length available, only one value presented. d B3LYP/6-31+G(d) calculation results from Ref. [36]. e B3LYP and HF calculation results form Ref. [38]. Model include first and second solvation spheres of copper dication. f Combined QM/MM simulation results from Ref. [38]. Quantum region treated at the HF level. g Combined QM/MM simulation results from Ref. [38]. Quantum region treated at the MP2 level. h Combined QM/MM simulation results from Ref. [35]. i Combined QM/MM simulation results from Ref. [37]. Quantum region treated at the HF level. j Experimental EXAFS results from Ref. [33]. k Experimental X-ray diffraction data from Ref. [4]. b

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181

Table 2 a Characteristic Cu–O bond lengths in the CuðH2 OÞ2þ 6 complex Compound CuðH2 OÞ2þ 6 CuðH2 OÞ2þ 6

[Ci]

b

[D2h]b (NH4)2Cu(SO4)2 Æ 6H2O K2Cu(SO4)2 Æ 6H2O Rb2Cu(SO4)2 Æ 6H2O Tl2Cu(SO4)2 Æ 6H2O K2Cu(ZrF6)6 Æ 6H2O

Req(Cu–O)

Req(Cu–O)

Rax(Cu–O)

Reference

1.998

2.001

2.255

This work

1.968 1.966 1.943 1.957 1.957 1.967

2.047 2.073 2.069 2.031 2.017 2.024

2.296 2.230 2.278 2.307 2.317 2.327

This work Experimentalc Experimentalc Experimentalc Experimentalc Experimentalc

a

All Cu–O bond lengths are in angstroms. B3LYP calculation results. The following basis sets were used in the simulations: 6-31G(d) basis set for copper and 6-311G** basis set for oxygen and hydrogen. c Experimental X-ray diffraction data from Ref. [41]. b

not be used for EPR parameter calculations, but for the sake of consistency and comparison we will include it in the calculations of the hyperfine coupling constant and electronic g-tensor. Furthermore, the use of two different geometries will allow us to estimate the influence of geometrical structure on EPR parameters in the sixfold-coordinated Cu(II) aqua complexes. The fivefold-coordinated first solvation sphere of the copper dication most likely consists, according to Pasquarello et al. [7], of five water ligands in a trigonal bipyramid arrangement. Therefore this Cu(II) aqua complex, in contrast to the hexa-coordinated one, does not feature ordinary Jahn–Teller distortion, since it has a non-degenerate ground state 2 A01 (the copper ion d9 configuration in a 4 4 1 D3h symmetry ligand field is (e00 Þ ðe0 Þ ða01 Þ ). However, 2 0 one can expect low-lying E excited states (as in other [CuX5]2+ complexes X = NH3, Cl, Br, etc.), which allow efficient vibronic mixing of the ground state 2 A01 with the excited state 2E 0 by e 0 symmetry vibrations, leading to the so-called pseudo Jahn–Teller effect (see discussion in Ref. [40]). In penta-coordinated Cu(II) aqua complexes this effect manifests itself as so-called Berry rotations, i.e. rapid conversions from a trigonal bipyramid configuration of the water ligands to a square-pyramidal configuration. This was also observed in the molecular dynamic simulations

of Pasquarello et al. [7]. Based on the arguments outlined above five models of penta-coordinated Cu(II) aqua complexes were constructed; three corresponding to squarepyramidal configurations of water ligands (the structures denoted I–III in Fig. 2), and two corresponding to trigonal bipyramid configurations (structures denoted IV and V in Fig. 2). The results of the geometry optimization for all these models, along with results from previous calculations and available experimental data, are summarized in Table 3. As expected, water ligands after geometry optimization in models I–III, remain in square-pyramid configurations, with approximately equal equatorial Cu–O bond lengths, i.e. Req ðCu–OÞ  R0eq ðCu–OÞ. In models IV and V, after geometry optimization, one set of equatorial Cu–O bonds, namely Req(Cu–O), becomes approximately equal to the axial Cu–O bond, Rax(Cu–O), indicating a trigonal bipyramid configuration. Independent of the configuration of water ligands (trigonal bipyramid or square pyramid), B3LYP calculations are overall in good agreement with experimental data, but the axial Cu–O bond length is significantly underestimated due to DFT failure to describe the pure electrostatic nature of this bond. Here, we note that also experimental methods based on the assumption of the pseudo Jahn–Teller effect predict a square-pyramidal configuration of water ligands in penta-

Table 3 a,b Cu–O bond lengths in CuðH2 OÞ2þ 5 complex obtained with the B3LYP calculations Bond

Model h,i

Req(Cu–O) R0eq ðCu–OÞi,j Rax(Cu–O) a b c d e f g h i j

Other works c

Experiment d

e

I

II

III

IV

V

B3LYP

B(38HF)P86

CPMD

MXANf

EXAFSg

1.975 1.995 2.162

1.967 2.016 2.169

1.988 1.988 2.161

2.078 1.971 2.052

2.088 1.979 2.006

1.974 2.013 2.170

1.96 2.00 2.13

2.00 2.00 2.45

1.95 1.98 2.35

1.96 – 2.36

Calculations performed using 6-31G(d) basis set for copper and 6-311G** basis set for oxygen and hydrogen. All Cu–O bond lengths are in angstroms. B3LYP/6-31+G(d) calculation results from Ref. [36]. B(38HF)P86 calculation results from Ref. [10]. Car–Parrinello molecular dynamics simulations from Ref. [42]. Cu–O bond lengths obtained from the MXAN data analysis with quality of fit (Rsq = 1.32). See Ref. [10]. Cu–O bond lengths derived from the EXAFS data analysis assuming fivefold coordination model. See Ref. [11]. Req(Cu–O) bond is oriented along x-axis (see Fig. 2). For model III both equatorial bonds, Req(Cu–O) and R0eq ðCu–OÞ, are positioned in xy plane as shown in Fig. 2. R0eq ðCu–OÞ bond is oriented along y-axis (see Fig. 2).

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coordinated Cu(II) aqua complexes [7,10,11]. A similar situation is observed in our calculations, where the energetically most favorable geometrical structure is obtained using model I (see Fig. 2), which corresponds to a square-pyramid arrangement. At the same time, this provides the best agreement between calculated and experimental Cu–O bond lengths of all the models for fivefoldcoordinated Cu(II) aqua complexes investigated here. However, the geometrical structures obtained with the other models (II–V), including the ones featuring a trigonal bipyramid configuration of the water ligands, are energetically relatively close to the structures obtained using model I. Conversions between all these structures are consequently possible, as also expected based on the findings in the molecular dynamics simulations of Pasquarello et al. [7]. Therefore, we employ all calculated B3LYP structures in the evaluation of the EPR parameters of Cu(II) aqua complexes of this kind. Furthermore, as we will include geometries corresponding to both square-pyramid and trigonal bipyramid configurations of the water ligands in the calculations of electronic g-tensors as well as hyperfine coupling constants, this will allow us to assess the impact of Berry rotations on the EPR parameters of penta-coordinated Cu(II) aqua complexes. 4.2. Electronic g-tensors of hexa- and penta-coordinated Cu(II) aqua complexes The computation of electronic g-tensors in transition metal complexes is a challenging task for density functional theory. Therefore, before discussing the electronic g-tensors of solvated Cu(II), it is crucial to assess the performance of our methodology based on restricted density functional response theory for evaluating EPR spin Hamiltonian parameters in compounds containing Cu(II). For this purpose Table 4 summarizes the electronic g-tensors of octahedral Cu(II) aqua complexes, measured in various crystals, along with the electronic g-tensor calculated in this work for a D2h symmetry model.

Table 4 Electronic g-tensor of CuðH2 OÞ2þ 6 complex in various crystals Compound

gxx

gyy

gzz

Reference

a CuðH2 OÞ2þ 6 [D2h] (NH4)2Cu(SO4)2 Æ 6H2O (NH4)2Cu(SO4)2 Æ 6H2O (NH4)2Cu(SO4)2 Æ 6H2O K2Cu(SO4)2 Æ 6H2O Rb2Cu(SO4)2 Æ 6H2O Tl2Cu(SO4)2 Æ 6H2O K2Cu(ZrF6)6 Æ 6H2O

2.066 2.071 2.071 2.059 2.055 2.074 2.070 2.061

2.164 2.209 2.218 2.219 2.154 2.109 2.116 2.139

2.374 2.363 2.360 2.355 2.403 2.420 2.418 2.443

This work Experimentalb Experimentalb Experimentalc Experimentalb Experimentalb Experimentalb Experimentalb

a

Calculations carried out with modified B3LYP exchange-correlation functional. IGLO-II basis with two tight s-type functions used for oxygen and hydrogen atoms and CP(PPP) basis set used for copper atom. b Experimental EPR data from Ref. [41] . c Experimental EPR data from Ref. [43].

As is evident from the presented results, calculated values reproduce the proper ordering of the principle values of the g-tensors observed in crystals, and are overall in qualitative agreement with experimental data. The best agreement of calculated principal values is obtained with the corresponding quantities observed in the (NH4)2Cu(SO4)2 Æ 6H2O crystal. This is expected, since the geometrical structure employed in the calculations closely matches the geometry of Cu(II) aqua complexes determined for this crystal (see Table 2). The only discrepancy in this case is an overestimation of the gyy values, which is most likely caused by the neglect of crystal environment effects in our B3LYP calculations (which were performed in vacuum). In the cases of the other crystals, the agreement between the calculated and observed g-tensors is poorer, but still remains relatively good. The deterioration in agreement is caused by differences between the structures used in the calculations and the ones in the crystals for which the g-tensor was measured. This reemphasizes the importance of reliable geometrical structures for accurate electronic g-tensor calculations. Based on the discussion above, we are confident that the approach selected in this article is capable of accurately predicting the electronic gtensors of Cu(II) complexes, and is consequently well suited for an investigation of the spin Hamiltonian parameters of hydrated copper dications. Results from electronic g-tensor calculations for all the sixfold- and fivefold-coordinated Cu(II) aqua complex structures, discussed in previous sections of this article, are summarized in Table 5. Let us first separately discuss the electronic g-tensors of the hexa- and penta-coordinated complexes. From the sixfold-coordinated complexes we selected the structure obtained with the D2h symmetry model as the most suitable candidate for representing the first solvation sphere (assuming hexa-coordination of Cu(II)). The electronic g-tensor of this structure has a distinct ordering of principal values, namely gzz > gyy > gxx. Furthermore, electronic g-tensor shifts for the gzz and gyy components are of considerable size: Dgzz = 371.7 ppt and Dgyy = 161.7 ppt, indicating a large spin–orbit contribution from the copper ion d orbitals as expected. Therefore, the hydrated copper dication, assuming sixfold-coordination, can be characterized by an electronic g-tensor with large anisotropy, in which the largest component (gzz) points along the axial Cu–O bond direction. The other two components, gyy and gxx, lie in the equatorial plane and are oriented along the corresponding Cu–O bonds axes (see Fig. 1). As well as calculating the electronic g-tensor for hexacoordinated Cu(II) aqua complexes in D2h symmetry, calculations were also performed for the Ci symmetry model. Although this symmetry model is not suitable for describing the first solvation sphere of Cu(II) (see discussion in the previous section of this article), it serves as an instructive example of the dependence of calculated electronic gtensors on geometrical structure. Going from the D2h symmetry model to Ci symmetry, two major changes occur: (1)

K.J. de Almeida et al. / Chemical Physics 332 (2007) 176–187

183

Table 5 2þ a Electronic g-tensor calculation results for CuðH2 OÞ2þ 6 and CuðH2 OÞ5 complexes Model

gxx gyy gzz giso

CuðH2 OÞ2þ 6

CuðH2 OÞ2þ 5

Experimentalb

D2h

Ci

I

II

III

IV

V

2.065 2.164 2.374 2.201

2.136 2.118 2.353 2.202

2.123 2.115 2.343 2.194

2.097 2.152 2.345 2.198

2.105 2.109 2.353 2.189

2.330 2.016 2.227 2.191

2.342 2.041 2.212 2.198

2.200 ± 0.001

a

Calculations carried out with modified B3LYP exchange-correlation functional. IGLO-II basis with two tight s-type functions used for oxygen and hydrogen atoms and CP(PPP) basis set used for copper atom. b Experimental data from Ref. [44].

the axial Cu–O bond lengths decrease slightly, by roughly ˚ ; (2) the equatorial Cu–O bonds become almost 0.05 A equal, where the shorter bonds become longer and the longer bonds become shorter. These changes in geometries lead to a major change in the electronic g-tensor, namely the gxx component becomes almost equal to the gyy component, i.e. gyy  gxx. Taking into account the reorganization of the equatorial water ligands into the square-planar configuration, one can interpret the changes in the electronic gtensor patterns from gzz > gyy > gxx to gzz > gyy  gxx as consequences of the transformation of the Cu(II) complex from octahedral sixfold-coordinated to square-planar fourfold-coordinated. This underscores that reliable geometrical structures of Cu(II) aqua complexes, accurately mimicking the first solvation sphere of this ion, are of paramount importance for a reliable evaluation of the electronic g-tensors. The electronic g-tensors for square-pyramid (models I– III), as well as trigonal bipyramid (models IV and V) fivefold-coordinated Cu(II) aqua complexes, are summarized in Table 5. As expected for the same configurations of ligands (square-pyramid or trigonal bipyramid), all employed structures give closely matching electronic g-tensors. On the other hand, there are big differences in Cu–O bonds lengths and electronic g-tensors when going from one geometry to another (see Table 3 for details). For models representing a square-pyramid configuration of the water ligands, the calculated electronic g-tensors follow the patterns observed in the square-planar complexes, i.e. gzz > gyy  gxx. The only notable discrepancy from this trend is for the structure obtained with model II, where the deviations of the equatorial Cu–O bond lengths are more pronounced, leading to a difference between the gyy and gxx component values as large as 55 ppt. Once again, this shows the sensitivity of the electronic g-tensors of Cu(II) aqua complexes to geometrical structure, and especially to the Cu–O bond lengths. For models representing a trigonal bipyramid configuration of the water ligands in penta-coordinated Cu(II) aqua complexes, electronic g-tensors follow similar patterns as those observed in the hexa-coordinated Cu(II) aqua complexes, i.e. gxx > gzz > gyy, but with components aligned along different axes. The similarity between the hexa- and penta-coordinated complexes is maintained independent of the models used for the copper dication first solvation

spheres (penta- or hexa-coordinated), or the actual configuration of the water ligands (octahedral, square-pyramid or trigonal bipyramid). The largest component, gzz or gxx, is always predicted to be around 2.3, with the largest difference among all calculation results being of about 44 ppt. This component is therefore dominated by a large local spin orbit contribution coming from molecular orbitals dominated by the copper d atomic orbitals, and shows only a small dependence on the water ligand arrangement and positions. In contrast, the two remaining components (gyy and gxx) show a strong dependence on the water ligand configuration, and especially on equatorial Cu–O bonds lengths, indicating that molecular orbitals contributing to g-tensor components include ligand orbitals. Under Berry rotation the electronic g-tensor of penta-coordinated Cu(II) aqua complexes therefore transforms from gzz > gyy  gxx (square-pyramid), to gxx > gzz > gyy (trigonal bipyramid) and back. The EPR spectra corresponding to both conformations – square-pyramid and trigonal bipyramid – should be observed in solid state EPR experiments. Concluding this section, let us discuss the differences between the electronic g-tensors in hexa- and penta-coordinated Cu(II) aqua complexes and the potential applicability of the electronic g-tensor for structure determination of the first solvation sphere of hydrated Cu(II). Overall, the electronic g-tensor of sixfold-coordinated Cu(II) (D2h symmetry model) closely resembles the g-tensor of penta-coordinated Cu(II) with a trigonal bipyramid arrangement of the water ligands. The only noticeable difference is the change in the ordering of the components going from the hexa-coordinated complex to the penta-coordinated one. Therefore, these two aqua complexes have rather similar electronic g-tensors, where the isotropic g-factor of the penta-coordinated complex is slightly smaller than the giso of the hexa-coordinated complex. On the other hand, as described in the previous paragraphs of this section, penta-coordinated Cu(II) with a square-pyramid configuration of the water ligands has a very different electronic g-tensor from the same complex with the trigonal bipyramid configuration of the water ligands, which consequently also is different from hexa-coordinated Cu(II). Furthermore, isotropic g-factors of these complexes are systematically shifted to lower values compared to the hexacoordinated Cu(II) complex (see Table 5). Summarizing,

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in hexa- and penta-coordinated Cu(II) complexes we observed two distinct patterns of the electronic g-tensor components: gzz > gyy  gxx (square-pyramid, penta-coordinated) and gzz > gyy > gzz (octahedral, hexa-coordinated) or gxx > gzz > gyy (trigonal bipyramid, penta-coordinated), where the g-tensor of the octahedral complex differs from trigonal bipyramid by the ordering of the g-tensor components. We also note, that hexa-coordinated complexes feature a slightly larger isotropic g-factor compared to the penta-coordinated complexes (see Table 5). Unfortunately, these substantial differences between the electronic g-tensors of the square-pyramid and of the octahedral Cu(II) complexes can not be exploited in determination of the arrangement of the water molecules in the first solvation sphere of Cu(II) as EPR spectroscopy methods in solution are capable to measure only the isotropic g-factor. Consequently, the electronic g-tensors of Cu(II) aqua complexes have limited applicability for elucidation of the hydrated Cu(II) structure and can be used only in connection with other experimental data, as our calculations indicate that the isotropic g-factors of hexa- and penta-coordinated Cu(II) complexes are rather similar (see Table 5). Taking into account this limitation, a direct comparison of the computed giso values with available experimental data (see Table 5), favors, in our opinion, an octahedral hexacoordination of Cu(II). However, we stress that current calculations of electronic g-tensors in Cu(II) aqua complexes neglect environmental effects as well as the dynamical reorganization of the water ligands. In a following paper, we intend to refine our model of Cu(II) solvation by a dynamic treatment of the water molecules by combining Car–Parrinello molecular dynamic simulations with spin-restricted density functional theory calculations. 4.3. Copper hyperfine coupling constants in hexa- and pentacoordinated Cu(II) aqua complexes Copper hyperfine coupling constants are tabulated in Table 6 for both hexa- and penta-coordinated Cu(II) aqua complexes. As expected, three unique patterns of the principal values of the copper hyperfine coupling tensor were

Table 6 Hyperfine coupling constants of

observed, which correspond to different configurations of the water ligands. For the hexa-coordinated Cu(II) aqua complex with an octahedral configuration of the ligands, the principal values show similar trends as for the electronic g-tensor of this complex, namely |Azz| > |Axx| > |Ayy|, i.e. only the ordering of the equatorial components is interchanged. A similar situation is also encountered for the penta-coordinated copper dication complex with the square-pyramid configuration of the water ligands, where the principal values (|Azz| > |Ayy|  |Axx|) have the same ordering as for the electronic g-tensor (gzz > gyy  gxx). The only exception from this agreement between the ordering of the principal values of the copper hyperfine coupling tensor and the electronic g-tensor is found for the pentacoordinated Cu(II) aqua complex with trigonal bipyramid configuration of the water ligands. In this complex the principal values are ordered as |Axx|  | Ayy| > |Azz| and gxx > gzz > gyy, respectively. Although the hyperfine coupling tensor of the copper (A(Cu)) dications in the Cu(II) aqua complexes in some cases share common features with their electronic g-tensors, this EPR spin Hamiltonian parameter exhibits overall a unique pattern for each of the three configurations of the water ligands in the investigated complexes. Furthermore, the hyperfine coupling constants of hydrated Cu(II) show a more pronounced dependence on the geometrical arrangement of the ligands than does the electronic g-tensor of this ion. Consequently, the copper hyperfine coupling constants are more suitable for determination of the number of water molecules in the first solvation sphere of Cu(II). This is, however, only the case if the components of the hyperfine coupling tensor can be resolved in the EPR measurements, and, similarly to the g-tensor case, only the isotropic hyperfine coupling constants (Aiso) are obtained in the solvent environment of most EPR experiments. Consequently, we can judge the coordination of Cu(II) in water solution only by comparing experimental data with computed Aiso values, while we can not exploit individual components of the hyperfine coupling tensor for this task. A quick inspection of Table 6 reveals that the penta-

63

Cu dication in penta- and hexa-coordinated aqua complexesa,b

Compound

Model

Axx

Ayy

Azz

Aiso

AFC

Adip xx

Adip yy

Adip zz

ASO xx

ASO yy

ASO zz

CuðH2 OÞ2þ 6

D2h Ci I II III IV V

241.26 192.28 123.74 171.38 127.29 304.28 371.92

70.55 146.73 188.63 167.29 131.54 344.5 336.07

406.39 469.52 495.91 485.91 487.89 139.51 80.69

31.53 43.50 61.18 49.08 76.35 33.10 38.85 110.92 ± 1.5

251.33 298.46 307.98 300.68 317.79 273.83 288.73

421.17 317.29 275.67 349.86 312.60 449.51 515.13

146.08 293.28 348.30 271.78 311.11 600.04 572.85

567.25 610.57 623.96 621.64 623.71 150.53 57.72

71.48 173.45 156.05 122.20 132.48 419.06 431.94

175.80 151.91 148.31 196.19 138.22 18.29 51.95

412.19 439.51 436.03 436.41 453.61 284.85 265.76

CuðH2 OÞ2þ 5

Experimentalc

a Calculations carried out with modified B3LYP exchange-correlation functional. IGLO-II basis with two tight s-type functions used for oxygen and hydrogen atoms and CP(PPP) basis set used for copper atom. b All values are given in MHz. c Experimental data from Ref. [44].

K.J. de Almeida et al. / Chemical Physics 332 (2007) 176–187

coordinated square-planar complex (denoted as III) features an Aiso value closest to experimental data. However, this result contradicts our analysis of isotropic g-factors of the Cu(II) complexes, in which we concluded that the hexacoordinated Cu(II) aqua complex is the most suitable model of the first solvation sphere of hydrated Cu(II). However, here we emphasize that our calculations of electronic g-tensors are more reliable compared to the hyperfine coupling constant calculations, as the latter show a very severe dependence on molecular geometry. Therefore more sophisticated modeling (inclusion of vibrational, environmental effects, etc.) of the hyperfine coupling constants are clearly needed in order to improve the reliability of the computational results. Due this reason, we favor hexa-coordination of Cu(II) in water solution as predicted by the electronic g-tensor analysis. In the following paper, we plan to address the shortcoming of our calculations of the hyperfine coupling constants by combining Car–Parinello molecular dynamics simulations with spin-restricted density functional theory calculations, something that we hope will allow us to resolve the apparent contradiction between the present results for the hyperfine coupling constants and electronic g-tensors. In general, the hyperfine coupling constants are well suited for determination of coordination of the Cu(II) compounds and it is therefore important to understand the impact of various contributions on the components of the hyperfine coupling tensor of copper, in this example of Cu(II) aqua complexes. In conventional hyperfine coupling calculations only the Fermi contact and spin-dipolar contributions (first two terms in Eq. (7)) are evaluated. The Fermi contact term (AFC) varies from 251 MHz to 318 MHz in the investigated hydrated Cu(II) complexes, depending on coordination (penta or hexa) and configuration of water ligands (octahedral, square-pyramid and trigonal bipyramid). For the spin dipolar contribution (Adip ii , i = x, y, z) on the other hand, the Adip components of hexa-coordinated Cu(II) aqua complexes follow a |Azz | > |Axx| > |Ayy| pattern (where Azz is negative). In the case of the penta-coordinated complexes, two different patterns of the Adip components are witnessed: |Azz| > |Ayy|  |Axx| (Azz is negative), and |Axx|  |Ayy| > | Azz| (Ayy is positive) for the square-pyramid and trigonal bipyramid type complexes, respectively. For all Cu(II) aqua complexes the ordering and sign patterns of the spin-dipolar contributions are in agreement with the patterns observed in the total hyperfine coupling tensor, indicating the importance of this contribution. However, for hydrated Cu(II), as we mentioned above, due to the orientational averaging in solution, only the isotropic hyperfine coupling constants can, most likely, be measured in ordinary EPR experiments. These are solely determined by the Fermi contact contribution in conventional calculations as the spin-dipolar contribution is a symmetric traceless tensor. The procedure described above for evaluating hyperfine coupling constants, in particular Aiso, is well suited for organic radicals, but fails severely for transition metal com-

185

pounds, as was shown by Neese [18], where the spin–orbit contribution to the hyperfine coupling becomes important (third term in Eq. (7)). For the present Cu(II) aqua complexes, the spin–orbit contribution (denoted as ASO ii in Table 6, i = x, y, z) indeed plays a crucial role in both the principal values of A(Cu), as well as in Aiso(Cu). For the principal values, the spin–orbit contribution (being opposite in sign to the Fermi contact contribution and of similar magnitude), effectively counteracts the Fermi contact contribution in the total hyperfine coupling tensor. The principal values of this tensor therefore retains the ordering observed for the spin-dipolar contribution. For the isotropic hyperfine coupling constants even more dramatic effects of the inclusion of spin–orbit contributions are witnessed. Here, Aiso values obtained taking into account only the Fermi contact contribution are reduced (in terms of absolute values) by roughly 220–250 MHz (depending on the type of Cu(II) aqua complex) due to spin–orbit contributions. Therefore, from the results outlined above, it is evident that for a reliable description of the hyperfine coupling constants of Cu(II) aqua complexes, as well as other Cu(II) complexes, one must take into account the spin–orbit interaction in one way or another. 5. Conclusions We have presented an investigation of the two most important spin Hamiltonian parameters, namely the electronic g-tensor and the hyperfine coupling tensor, in sixfold- and fivefold-coordinated Cu(II) aqua complexes, which model the first solvation sphere of hydrated copper dications. The main aim of this work was to find out if these EPR parameters in principle can be used to determine the Cu(II) coordination in the first solvation sphere. In this way we hope to contribute to the ongoing debate on hydrated Cu(II) coordination, as one experimental finding supports hexa-coordination, while other experimental results advocate penta-coordination of this transition metal ion. Our calculations show that the electronic g-tensor of the sixfold-coordinated Cu(II) aqua complex is similar to the gtensor of the fivefold-coordinated complex with a trigonal bipyramid configuration of the water ligands, but on the other hand that it severely differs from the g-tensor of the fivefold-coordinated complex with a square pyramid configuration. However, this difference between the electronic gtensors of the penta-coordinated complexes can not be exploited for determination of the coordination number, as in solution environment only the isotropic g-factors of the Cu(II) complexes can be determined experimentally. On the other hand, computed giso values are overall rather similar for both hexa- and penta-coordinated Cu(II) complexes, where the octahedral hexa-coordinated Cu(II) complex in our calculations gets the giso value closest to the experimentally observed one. Consequently, we favor hexa-coordination of the hydrated Cu(II). However, here is also appropriate to note that penta-coordination of

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Cu(II) can not be excluded as some penta-coordinated complexes in our calculation feature giso values comparable to the giso value of the octahedral hexa-coordinated complex. Therefore, the electronic g-tensor have limited applicability in determination of hydrated Cu(II) and must be used in connection with other experimentally observed molecular parameters in order to more definitely settle the issue about the coordination number of hydrated Cu(II). Another spin Hamiltonian parameter which has been investigated in this work, is the (copper) hyperfine coupling tensor. In agreement with our expectations, this coupling tensor show pronounced dependence on the Cu(II) coordination environment, and three completely different patterns of principal values have been observed, corresponding to the different encountered configurations of the water ligands. However, comparison of isotropic hyperfine coupling constants of copper obtained in our model systems with experimental data indicated that the penta-coordinated Cu(II) aqua complex with square-planar configuration of the water ligands is a most suitable candidate as a model of the first solvation sphere of Cu(II). This finding contradicts our assignment of the coordination number based on the analysis of the electronic g-tensor of Cu(II) aqua complexes and can evidently be attributed to shortcomings of the methodology (neglect of vibrational, environmental effects, etc.) used in this paper for hyperfine coupling constants. Finally, we would like to discuss two issues concerning calculations using density functional theory of EPR spin Hamiltonian parameters of Cu(II) complexes. First, in this article we demonstrate, and to our best knowledge for the first time, the ability of spin-restricted density functional theory to predict the electronic g-tensor of Cu(II) compounds with accuracy, allowing satisfactory agreement with available experimental results. This result was achieved, overcoming difficulties usually encountered by DFT methods – which previously have underestimated the g-tensors of transition metal compounds with about 40–60% – by employing a modified B3LYP exchange-correlation functional due to Solomon et al. [28,29], designed to reproduce correctly the balance between ionic and covalent bonding in Cu(II) complexes. Second, we reconfirmed the observation of Neese that inclusion of spin–orbit contributions to the hyperfine coupling constants of transition metal compounds is necessary in order to obtain physically meaningful results. This is especially true for the isotropic hyperfine coupling constants, in which spin–orbit contributions counteract the Fermi contact contributions. In copper this leads to several times smaller |Aiso| values compared to values obtained only by taking into account the Fermi contact contribution. Therefore, we wish to emphasize that any calculation of hyperfine coupling constants of transition metals should include the spin–orbit contributions to the hyperfine coupling tensors in order to be reliable and physically meaningful. In future studies, we plan to extend this work on the EPR parameters of hydrated copper dications by taking

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