1. Introduction
The coordination chemistry of copper has been extensively studied for its biological role [
1,
2,
3,
4], synthetic and catalytic applications [
5,
6,
7], and recently, for its pharmacological action and the potential use of its compounds in medicine [
8,
9,
10]. The most stable oxidation states of copper are +I (3d
10, diamagnetic) and +II (3d
9, paramagnetic) [
11], which often—and particularly in the living systems and in catalysis—give interconversion between each other. In normal conditions, the state +II is more stable than +I, which disproportionate in aqueous solution, and specific ligands are necessary to stabilize the lowest oxidation state [
12].
Due to the non-spherical symmetry and Jahn–Teller effect expected for a d
9 ion, the stereochemistry of Cu(II) complexes is characterized by non-rigid structures and covers several distorted geometries (the phenomenon often named “plasticity”) [
13,
14]. Species with coordination numbers 4, 5, and 6, and geometries square planar, square pyramidal, bipyramidal trigonal, compressed and elongated octahedral with all the possible distortions, have been observed [
15]. Several instrumental techniques are available for the study of Cu(II) compounds, among which UV-Vis and circular dichroism (CD) spectroscopy, but the most used is electron paramagnetic resonance (EPR). Electron paramagnetic resonance technique can be applied, in principle, to any paramagnetic systems with one or more unpaired electrons; the background of EPR is similar to nuclear magnetic resonance (NMR), but the magnetic field splits the energy of the α and β spin states of the electron instead of those of the nuclei [
16]. For a mononuclear paramagnetic compound with only one unpaired electron, two spin Hamiltonian parameters characterize its EPR spectrum: the
g factor and the hyperfine coupling constant (HFCC), indicated with
A, between the unpaired electron and nuclei (
giso and
Aiso in an isotropic spectrum, and
gx,
gy,
gz and
Ax,
Ay,
Az in an anisotropic spectrum) [
16]. For Cu(II) complexes the coupling of the electron is with the nuclei of the isotopes
63Cu and
65Cu, whose natural abundance is 69.1% and 30.9%, respectively. The
g and
A tensors are strongly perturbed by the chemical environment around Cu(II) ion and their variation can give information on the structure of a specific species. In particular, the values of
gz and
Az are more sensitive to the change in the coordination sphere than the
x and
y components. In general, with increasing the electron donor capability of the ligands, a decrease of the experimental value of
gz and an increase of
Az are observed. For example, Peisach and Blumberg [
17] found a linear relationship between the experimental
gz and
Az values for CuO
4, CuNO
3, CuN
2O
2, CuN
3O, and CuN
4 equatorial coordination with different electric charge of the species. The presence of ligands in the axial positions causes an increase of
gz and a decrease of
Az. Moreover, because of the plasticity of copper and the possibility to have several distortion degrees (for example, in an elongated octahedron the distance between copper and the axial ligands can be more or less long) a variation—not always easily predictable—in the values of
gz and
Az is expected. This results in more complicated situations compared to other metal ions such as vanadium(IV) (3d
1), for which the “additivity relationship” allows one to predict the value of
Az within 3 × 10
−4 cm
−1 [
18].
Therefore, methods to help in the interpretation of the experimental spectra and to correlate the values of
gz and
Az with the structure of a specific Cu(II) compound are desirable. Among the possible tools, density functional theory (DFT) [
19] has reached an enormous popularity and a number of reviews have been published over the last years [
20,
21,
22,
23,
24,
25,
26]. Density functional theory is a theory of the electronic structure based on the electron density ρ(
r) rather than the many-electron wavefunction and allows to determine the properties of a system using a functional dependent only on ρ(
r) with relatively short computational times. With an appropriate choice of the functional and basis set it is possible to perform calculations on large molecules with high efficiency. Up to now, unfortunately, for most of systems there are no rules to choose the correct DFT protocol to obtain the best agreement with the experimental data: an efficient functional is fundamental but it depends on the molecular/spectroscopic properties and on the metal under examination, the basis set must be large enough and include—if necessary—polarization and diffuse functions and relativistic effects, and in some cases the role of the solvent must be considered through continuum solvation models such as polarizable continuum model (PCM) [
27] or solvation model based on density (SMD) [
28]. Therefore, on the basis of the actual knowledge in the literature, the best protocol for a DFT calculation must be obtained case by case, and a criterion to establish which level of theory will give the best agreement with the experiment under consideration is lacking. Density functional theory methods allows to calculate the structure, energy, and molecular properties with great accuracy. Concerning the spectroscopic behavior of the transition metal ions, IR, UV-vis, CD, and magnetic circular dichroism (MCD) spectra were simulated, but only for nuclear magnetic resonance (NMR) and EPR—and UV-Vis in some rare cases [
29]—the agreement with the experimental data can be quantitative [
30,
31,
32,
33,
34,
35,
36,
37].
For transition metal species, the prediction of
A is strongly influenced by the choice of the functional, and it has been shown that for each metal ion it is necessary to validate that which gives the best performance [
38]. At the moment, a unique functional which works efficiently for the entire series of transition metals is not available. For example, for some iron(III) complexes (e.g., with nitrosyl ligands) the OLYP functional performs better than meta-generalized gradient approximation (meta-GGA) and hybrid functionals such as B3LPY and TPSSh [
39]; for Mo(V) species hybrid functionals with 30–40% of exchange admixture give a good agreement with the experimental data [
40], and for V(IV)O compounds, the half-and-half hybrid BHandHLYP functional is able to predict
Az for neutral species formed by (N,O) ligands with a mean absolute percent deviation (MAPD) from the experimental value lower than 5% [
36]. Generally, meta-gradient functionals underestimate the value of
A and an elevated fraction of the Hartree–Fock (HF) exchange, which increases the spin polarization, is necessary to predict values in line with the experiments. Over the last decade, new meta-GGA, hybrid, and double hybrid functionals have been tested in the prediction of EPR spin Hamiltonian parameters; for instance, the hybrid version of meta-GGA TPSS, i.e. TPSSh, represents a good alternative to the “usual” B3LYP. In the double hybrid functional B2PLYP [
41], which consists of B88 exchange combined with the LYP correlation functional through two empirical parameters controlling the amount of HF exchange and PT2 correlation, the HF exchange is quite large with respect to the most used hybrid functionals (53%, compared with 20% of B3LYP and 25% of PBE0, for example); it has been successfully tested on “bare” V(IV) complexes reaching a MAPD value of 3.3% from the experimental values of
A [
37].
For the calculations of HFCC
A, three contributions must be taken into account: the Fermi contact (
AFC), the anisotropic or dipolar hyperfine interaction (
AD), and the second-order spin-orbit (SO) coupling (
ASO) [
34]. Generally, the most critical parameter is the Fermi contact, which is directly proportional to the spin density at the nucleus (
), and, for this reason, a good model must be able to predict the spin polarization in the electron distribution of the core and valence shells due to the unpaired electron in the
d orbitals [
42,
43]. The SO contribution increases going toward the right of the first transition series and becomes important for late 3
d elements, such as Co [
44], Ni [
45], and mainly Cu [
34,
44].
is in the approximate range 4–8% with a mean of ca. 6% compared to the sum (
) for V(IV)O [
36], while it increases for Cu(II) species; for example, a value of ca. 25% has been calculated with BP86 functional for [Cu(acac)
2] [
34]. Therefore, as it was pointed out in the literature, the SO contribution should be included in the
A calculation [
34,
44]. Van Lenthe et al. [
46] calculated the zero-order regular approximation (ZORA) to the Dirac equation to determine the SO coupling; this approach, that has the limitation that spin polarization and SO effects cannot be introduced contemporaneously, has been overcame by Neese in the coupled-perturbed Kohn–Sham implementation of SO contribution to HFCC [
34]. The validation of HFCC on a series of 3
d transition metal complexes agreed well with the experimental data; the main error was related to the underestimation of the Fermi contact, whose prediction can be improved by varying the functional [
34].
For the first row of transition metal complexes, the scalar relativistic (SR) contributions are of little importance or negligible [
35], and they must be considered only for the elements of the second and third series, as demonstrated in a systematic study for 4
s, 5
s, and 6
s radicals [
47].
Over the last years, several papers have been published on the calculation of
g and
A values for Cu(II) complexes, but to the best of our knowledge the only systematic study is from Neese [
34], who found a MAPD below 5% using B3LYP functional for 10 different species. However, he tested only B3LYP and BP86, and therefore, any comparison with other or more recent functionals is lacking. Larsen and co-workers [
48,
49,
50] dedicated several papers to Cu(II) compounds but the number of cases considered was always below five structures.
In this paper a systematic study on the DFT prediction of
g and
A values for fourteen Cu(II) complexes with different total charge, coordination geometry, and type of donors is presented. The experimental values of
Ax,
Ay,
Az, and
gz for these complexes are listed in
Table S1 of the Supplementary Materials. Eleven functionals were examined and their performance compared with a statistical analysis based mainly on the MAPD value. The most problematic cases were also evaluated in detail and some methods to improve the prediction are also proposed. Finally, the best computational conditions were applied to the calculations of
gz and
Az values to the species formed upon the binding of Cu(II) ion to the N-terminal region of human serum albumin (HSA) and to fragments 106–126 and 180–193 of prion protein (PrP).
4. Conclusions
Many metal species play a fundamental role in bioinorganic chemistry, biochemistry, and medicine, including copper containing complexes. Several instrumental techniques are available for their characterization, but in the case of paramagnetic species, EPR is the most used tool because the values of spin Hamiltonian parameters, gz and Az, depend mainly on the equatorial donors bound to the metal. However, in some situations the interpretation of the spectra is not trivial because the variation of gz and Az is not always easily predictable and can be influenced by various factor, such as the distortion of the geometry and the presence of ligands in the axial positions. Often Cu(II) complexes fall in this range of situations. The computational methods, in particular based on DFT, have been demonstrated very useful to confirm the experimental evidence or to provide hypotheses to explain the spectroscopic measurements.
In this study, we provided a simple method to predict the spin Hamiltonian of copper(II) species and discussed how to overcome the most complicated cases. The results suggested that, in the framework of DFT calculations, it is not possible to propose a general method to obtain a satisfactory agreement with the experimental data. For example, for V(IV)O and non-oxido V(IV) species the functionals BHandHLYP and B2PLYP, respectively, perform better than the others, and unlike Cu(II) species, the prediction of the second-order spin-orbit contribution is often not necessary. Therefore, the best combination of functional and basis set must be reached case by case and, up to today, there are no rules to know in advance which level of theory could give the best agreement with the experimental spectroscopic data.
For Cu(II) complexes examined in this paper, the functionals B3LYP and PBE0 show the best performance in the calculation of Az and gz, respectively. They can be coupled with a triple-ζ basis set such as 6-311g(d,p) to obtain a satisfactory agreement with the experimental data. These computational conditions could be applied to clarify also the coordination environment of Cu proteins or in other biological systems. Finally, we would like to highlight that, when the quality of the optimization is not good, the prediction of Az worsens significantly, and a wider basis set must be used or the solvent effect must be considered.