# EPR Spectroscopy of Cu(II) Complexes: Prediction of g-Tensors Using Double-Hybrid Density Functional Theory

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Benchmark Set of Copper Complexes

**9**and

**12**, which are square pyramidal, and

**18**, which is octahedral.

**I**is the identity matrix. In this work, we express the principal components of the $g$-tensors as $g$-shifts in parts per thousand (ppt), computed as $\mathsf{\Delta}g=\left(g-{g}_{\mathrm{e}}\right)\xb71000$. The largest component is referred to as $\mathsf{\Delta}{g}_{zz}$ or as the parallel $g$-tensor component, $\mathsf{\Delta}{g}_{\parallel}$. We define the perpendicular $g$-tensor component as the average of the two smaller components: $\mathsf{\Delta}{g}_{\perp}=$ ($\mathsf{\Delta}{g}_{xx}+\mathsf{\Delta}{g}_{yy}$)/2.

**1**–

**18**have been experimentally determined from EPR measurements and are given in Table 1. Since Cu(II) complexes have a d

^{9}configuration, square planar and square pyramidal structures usually exhibit strongly axial EPR signals, i.e., $\mathsf{\Delta}{g}_{xx}$ < $\mathsf{\Delta}{g}_{yy}$ << $\mathsf{\Delta}{g}_{zz}$, consistent with the high ${\mathrm{d}}_{{x}^{2}-{y}^{2}}$ character of the singly occupied molecular orbital (SOMO). The g-shifts arise predominantly due to the interaction of the ligating atoms with the unpaired electron, so it depends on the nature of the ligands, the nature of the Cu–ligand bond, and the corresponding bond lengths. In this benchmark set we include complexes with variable coordination spheres and a wide range of $\mathsf{\Delta}{g}_{zz}$ values, from 83 ppt to 283 ppt. The presence of at least one S atom on the copper coordination sphere decreases the magnitude of $\mathsf{\Delta}{g}_{zz}$. It should be noted that mixed ligand complexes exhibit magnetic parameters that are intermediate between the respective complexes with four of the same ligands of each type [8]. This pattern can be observed in the experimental data shown in Table 1.

#### 2.2. Overview of Double Hybrid Density Functionals

_{6}of the dispersion correction. DSD-BLYP [76] and DSD-PBEP86 [77] have been parametrized to reproduce thermochemistry, kinetics, and dispersion forces, and were found to be more accurate than the previously most successful B2GP-PLYP functional.

## 3. Results and Discussion

#### 3.1. Evaluation Criteria

#### 3.2. Performance of Functionals

**1**–

**18**, visualized in Figure 2 bottom, reflect the error of each method on reproducing the relative Δ${g}_{\parallel}$ values of the various structures. They are also described by the SDD value of each method, given in Table 3. ωB2GP-PLYP, ωPBEPP86 and ωB88PP86 have larger SD values than the B2GP-PLYP and PBE0-DH functionals, which means that there is a larger spread of D values with these functionals.

**1**–

**18**obtained with each functional allows a more in-depth analysis of our results (Figure 2 bottom). This diagram shows clearly that complexes

**11**–

**18**, which have at least one S coordinated ligand on copper, are described differently than complexes

**1**–

**10**, which have only N and O ligating atoms. Specifically, most functionals underestimate the value of parallel $g$-tensor component of structures

**1**–

**10**, while they overestimate it for

**11**–

**18**. This behavior is even more pronounced for the HDFs. Figure 2 bottom shows clearly that even though DHDFs also treat differently the Cu–S bond, they not only have smaller MADs, but also the spread of the $\mathsf{\Delta}{g}_{\parallel}$ values is smaller. This implies that DHDFs achieve a more balanced description of Cu-ligand bond covalency and this behavior is transferable among different systems, at least to a larger extent than HDFs.

_{s}(Cu), for six complexes chosen as representative of different copper coordination spheres, i.e., N4, N2O2, O4, N2S2, N3OS and S4 (see Figure S1 for additional examples). Several conclusions can be extracted from these diagrams. First, it can be clearly observed that the value of D(Δ${g}_{\parallel}$) increases following a near linear trend along with ρ

_{s}(Cu). Second, each system has a different ρ

_{s}(Cu) value for which the experimental Δ${g}_{\parallel}$ is reproduced. Even though no single functional predicts this most favorable ρ

_{s}(Cu) value for all complexes, structures with similar coordination spheres are optimally described by the same density functionals (Figure S1). Among the DHDFs included in this study, the B2PLYP and RSX-QIDH predict the smallest and largest ρ

_{s}(Cu) values, respectively, for all complexes, which directly correlates with the observed systematic negative and positive differences from the experimental values, reflected in the large MD(Δ${g}_{\parallel}$) values shown in Figure 3. In addition, B3LYP and PBE0 in almost all cases predict a small spin population on Cu. Notably, accurate $g$-tensor prediction does not necessarily imply prediction of the “correct” spin density, since other factors, such as higher order relativistic contributions, could also alter the D(Δ${g}_{\parallel}$) values. We note that the possible lack of correspondence between the accuracy of DFs in the prediction of various observable properties, such as relative energies or ionization potentials, and the quality of the computed densities has been extensively debated recently [84,85,86].

**11**–

**18**with S coordinating ligands are not very different from complexes

**1**–

**10**which have N and O ligands only. By contrast, this is not observed for RSX-QIDH, the other DHDF of this set that performs poorly, systematically overestimating g-shifts. Therefore, we can conclude that B2PLYP achieves a more balanced description of the Cu-ligand bond covalency between different ligand donors.

## 4. Conclusions

^{5}) where N is a measure of the system size), the computational cost of DHDFs on large (>100 atoms) systems increases more steeply than HDFs in terms of both time and required memory. Hence, for large systems we propose the use of multilevel approaches, where the metal and first coordination sphere ligands are treated with a DHDFT method and the surrounding protein matrix can be treated with a cheaper DFT method. Based on our results, we recommend the use of B2GP-PLYP and PBE0-DH functionals for g-tensor calculations on Cu(II) complexes bearing N, O and S ligands, which are usually encountered in bioinorganic systems. Evidently, the uncertainty of even the best functionals may exceed the uncertainty of experimental values. This suggests that the successful use of these approaches will depend on the type of chemical problem under investigation and in the combination with complementary data. Nevertheless, the present study clearly defines the current state-of-the-art in quantum chemical calculations of g-tensors for Cu(II) systems and encourages further developments in refining double-hybrid DFT for spin density dependent properties.

## 5. Computational Details

**3**,

**10**and

**17**, which have N4, O4 and S4 coordination spheres, respectively, using the B2PLYP and DSD-PBEP86 functionals. In line with previous studies on Cu(II) $g$-tensor calculations [25], the dependence on the basis set is very weak past the polarized triple-zeta level and the use of the largest available ZORA-def2-QZVPP basis set leads to negligible differences (of the order of 1–2 ppt) in the results compared to the triple-zeta basis sets. The maximum difference in computed g-shifts upon use of the quadruple-zeta basis set (5 ppt) was observed for

**17**with the DSD-PBEP86 functional. Since this difference is of the same order of magnitude as the experimental uncertainty, the basis sets used here are judged to be essentially converged for all practical purposes. For the calculation of g-tensors, the DFT functionals tested in this work are: the hybrid functionals PBE0 [100,101,102], BHandHLYP [91], B3LYP [89,90,91], the double-hybrid functionals: B2PLYP [47], mPW2PLYP [69], B2GP-PLYP [70], B2K-PLYP [71], B2T-PLYP [71], PBE-QIDH [72], PBE0-DH [73], DSD-BLYP [76], DSD-PBEP86 [77], and the range-separated double-hybrid functionals: ωB2PLYP [81], ωB2GP-PLYP [81], RSX-QIDH [74], RSX-0DH [75], ωB88PP86 [82], ωPBEPP86 [82], ωB97X-2 [83]. Spin contamination values for HDFs were smaller than 0.01 for all complexes, which suggests that the ground state of all complexes can be adequately described by a single determinant. For the calculations with double-hybrid functionals the NoFrozenCore option was used. In terms of computational costs, we mention as an example that for complexes

**1**(17 atoms)

**, 18**(43 atoms) and

**12**(63 atoms) the PBE0 calculation needs 1, 16 and 48 min, respectively, using 10 processing cores, while the PBE0-DH calculation needs 3, 184 and 927 min, respectively. The part of the calculation that is responsible for most of the additional computational effort is the calculation of the relaxed MP2 response density, which incorporates orbital relaxation and is consistent with first order properties as analytic derivatives. The spin-orbit coupling was treated using the spin-orbit mean-field (SOMF) operator [102] with the 1X-approximation [103,104] (SOCType 3 in ORCA convention). For the construction of the potential one-electron terms were included, the Coulomb term was computed using the RI approximation, exchange terms were incorporated via one-center exact integrals including the spin-other orbit interaction, and without local DFT correlation terms (SOCFlags 1,3,3,0 in ORCA). Picture change effects were also taken into account.

## Supplementary Materials

**1**and

**3**,

**14**and

**16**, and with similar coordination spheres

**12**and

**13.**Tables S1–S19: Detailed results for each functional.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Structures of the eighteen Cu(II) complexes studied in this work. Ligand abbreviations: py = pyridine; iz = imidazole; en = ethylenediamine; gly = glycine; ox = 3-quinolinolato; sac = salicylaldehyde imine; acac = acetylacetone; bpy = 2,2′-bipyridine; dpa = 2,2′-dipyridylamine; dpc = dipicolinate ligand; mpsme = anionic form of the 6-methyl-2-formylpyridine Schiff base of S-methyldithiocarbazate; sct = N(4)-cyclohexyl thiosemicarbazone; dbpy = 4,4′-dimethyl-2,2′-bipyridine; spt = 5-nitrosalicylaldehyde piperidylthiosemicarbazone; tct = thiophene-2-carbaldehyde thiosemicarbazone; DMF = N,N-Dimethylformamide; eLcys = N,N′-ethylenebis(L-cysteine); tox = 3-quinolinethiolato; dtc = dimethyl-dithiocarbamate, ttcn = 1,4,7-trithiacyclononane.

**Figure 2.**(

**Top**): Mean absolute difference (MAD), mean difference (MD) from the experimental values of the parallel $g$-tensor component calculated with each functional. The respective standard deviations (SD) are plotted in pink. (

**Bottom**): Differences (D) from the experimental values of the parallel $g$-tensor component of each one of the 18 Cu(II) complexes of the set calculated with each functional. Each bar color represents a complex.

**Figure 3.**Correlation of the differences, D(Δ${g}_{\parallel}$), of the calculated $g$-tensor from the experimental value of the parallel $g$-shift with the Cu spin populations computed with the respective functional for complexes

**1**,

**6**,

**10**,

**16**,

**12**and

**17**, which have different copper coordination spheres.

**Table 1.**Experimental $\mathsf{\Delta}g$ values (ppt) of the Cu(II) complexes

**1–18**studied in this work. Molecular structures of the complexes are shown in Figure 1. The complexes are arranged in groups according to the copper-coordinating atoms for each subgroup, which are also shown for convenience.

Complex | Coord. Core | $\mathbf{\Delta}{\mathit{g}}_{\mathit{x}\mathit{x}}$ | $\mathbf{\Delta}{\mathit{g}}_{\mathit{y}\mathit{y}}$ | $\mathbf{\Delta}{\mathit{g}}_{\mathit{z}\mathit{z}}$ | Ref. | |
---|---|---|---|---|---|---|

1 | [Cu(NH_{3})_{4}]^{2+} | N4 | 45 | 45 | 239 | [56] |

2 | [Cu(py)_{4}]^{2+} | 51 | 51 | 261 | [56] | |

3 | [Cu(iz)_{4}]^{2+} | 45 | 45 | 260 | [56] | |

4 | [Cu(en)_{2}]^{2+} | 39 | 39 | 203 | [56] | |

5 | [Cu(gly)_{2}] | N2O2 | 50 | 50 | 265 | [56] |

6 | [Cu(ox)_{2}] | 50 | 50 | 200 | [57] | |

7 | [Cu(sac)_{2}] | 48 | 48 | 238 | [58] | |

8 | [Cu(acac)(bpy)]^{+} | 48 | 55 | 251 | [59] | |

9 | [Cu(dpa)(dpc)] | N3O2 | 60 | 60 | 252 | [60] |

10 | [Cu(acac)_{2}] | O4 | 58 | 58 | 283 | [61] |

11 | [Cu(mpsme)(NCS)] | N3S | 48 | 48 | 203 | [62] |

12 | [Cu(sct)(dbpy)] | N3OS | 54 | 54 | 165 | [63] |

13 | [Cu(spt)(DMF)] | NO2S | 35 | 46 | 201 | [64] |

14 | [Cu(tct)_{2}] | N2S2 | 27 | 88 | 112 | [65] |

15 | [Cu(eLcys)] | 37 | 37 | 124 | [66] | |

16 | [Cu(tox)_{2}] | 42 | 42 | 136 | [57] | |

17 | [Cu(dtc)_{2}] | S4 | 23 | 23 | 83 | [67] |

18 | [Cu(ttcn)_{2}]^{2+} | S8 | 25 | 25 | 112 | [68] |

Functional | ${\mathit{c}}_{\mathbf{X}}$ | $\mathit{\mu}$ | ${\mathit{c}}_{\mathbf{C}}^{\mathbf{DFT}}$ | ${\mathit{c}}_{\mathbf{C}}^{\mathbf{MP}2}$ | ${\mathit{c}}_{\mathbf{O}}$ | ${\mathit{c}}_{\mathbf{S}}$ |
---|---|---|---|---|---|---|

B2PLYP | 0.53 | - | 0.73 | 0.27 | - | - |

mPW2PLYP | 0.55 | - | 0.75 | 0.25 | - | - |

B2GP-PLYP | 0.65 | - | 0.64 | 0.36 | - | - |

B2K-PLYP | 0.72 | - | 0.58 | 0.42 | - | - |

B2T-PLYP | 0.60 | - | 0.69 | 0.31 | - | - |

PBE-QIDH | 0.69 | - | 0.67 | 0.33 | - | - |

PBE0-DH | 0.50 | - | 0.875 | 0.125 | - | - |

DSD-BLYP | 0.75 | - | 0.53 | - | 0.46 | 0.60 |

DSD-PBEP86 | 0.72 | - | 0.44 | - | 0.51 | 0.36 |

ωB2PLYP | 0.53 | 0.30 | 0.73 | 0.27 | - | - |

ωB2GP-PLYP | 0.65 | 0.27 | 0.64 | 0.36 | - | - |

RSX-QIDH | 0.69 | 0.27 | 0.67 | 0.33 | - | - |

RSX-0DH | 0.50 | 0.33 | 0.875 | 0.125 | - | - |

ωB88PP86 | 0.65 | 0.20 | 0.58 | 0.42 | - | - |

ωPBEPP86 | 0.70 | 0.18 | 0.68 | 0.48 | - | - |

ωB97X-2 | 0.63(6) | 0.30 | 1.00 | 1.00 | 0.44(7) | 0.52(9) |

**Table 3.**Mean difference (MD) and standard deviation of the differences (SDD), mean absolute difference (MAD), standard deviation of the absolute differences (SDAD) and mean absolute percent difference (MAPD), from the experimental values of the parallel and perpendicular $g$-shifts components calculated with each functional. Values are expressed in ppt.

Functional | MD $\left(\mathbf{\Delta}{\mathit{g}}_{\parallel}\right)$ | SDD $\left(\mathbf{\Delta}{\mathit{g}}_{\parallel}\right)$ | MAD $\left(\mathbf{\Delta}{\mathit{g}}_{\perp}\right)$ | MAD $\left(\mathbf{\Delta}{\mathit{g}}_{\parallel}\right)$ | SDAD $\left(\mathbf{\Delta}{\mathit{g}}_{\parallel}\right)$ | MAPD $\left(\mathbf{\Delta}{\mathit{g}}_{\perp}\right)$ | MAPD $\left(\mathbf{\Delta}{\mathit{g}}_{\parallel}\right)$ |
---|---|---|---|---|---|---|---|

PBE0 | −49 | 41 | 6 | 54 | 35 | 14 | 24 |

BHandHLYP | 36 | 49 | 19 | 48 | 38 | 49 | 35 |

B3LYP | −66 | 41 | 9 | 67 | 40 | 19 | 30 |

B2PLYP | −57 | 28 | 12 | 57 | 28 | 31 | 30 |

mPW2PLYP | −38 | 30 | 8 | 42 | 24 | 19 | 21 |

B2GP-PLYP | −10 | 34 | 7 | 31 | 16 | 18 | 18 |

B2K-PLYP | 25 | 41 | 8 | 37 | 31 | 21 | 24 |

B2T-PLYP | −27 | 31 | 7 | 36 | 20 | 19 | 19 |

PBE-QIDH | 31 | 41 | 10 | 40 | 32 | 25 | 27 |

PBE0-DH | −9 | 35 | 6 | 31 | 19 | 15 | 17 |

DSD-BLYP | 28 | 43 | 8 | 39 | 34 | 21 | 25 |

DSD-PBEP86 | 21 | 43 | 7 | 37 | 30 | 20 | 24 |

ωB2PLYP | −19 | 33 | 6 | 34 | 18 | 17 | 18 |

ωB2GP-PLYP | 14 | 37 | 7 | 32 | 24 | 19 | 21 |

RSX-QIDH | 53 | 46 | 15 | 55 | 44 | 37 | 39 |

RSX-0DH | 26 | 43 | 14 | 40 | 31 | 35 | 28 |

ωB88PP86 | −7 | 36 | 7 | 33 | 17 | 19 | 19 |

ωPBEPP86 | 10 | 40 | 7 | 34 | 23 | 20 | 21 |

ωB97X-2 | −16 | 39 | 7 | 38 | 20 | 19 | 20 |

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Drosou, M.; Mitsopoulou, C.A.; Orio, M.; Pantazis, D.A.
EPR Spectroscopy of Cu(II) Complexes: Prediction of *g*-Tensors Using Double-Hybrid Density Functional Theory. *Magnetochemistry* **2022**, *8*, 36.
https://doi.org/10.3390/magnetochemistry8040036

**AMA Style**

Drosou M, Mitsopoulou CA, Orio M, Pantazis DA.
EPR Spectroscopy of Cu(II) Complexes: Prediction of *g*-Tensors Using Double-Hybrid Density Functional Theory. *Magnetochemistry*. 2022; 8(4):36.
https://doi.org/10.3390/magnetochemistry8040036

**Chicago/Turabian Style**

Drosou, Maria, Christiana A. Mitsopoulou, Maylis Orio, and Dimitrios A. Pantazis.
2022. "EPR Spectroscopy of Cu(II) Complexes: Prediction of *g*-Tensors Using Double-Hybrid Density Functional Theory" *Magnetochemistry* 8, no. 4: 36.
https://doi.org/10.3390/magnetochemistry8040036