1 Introduction

Organic radical ions play increasingly important roles in modern biochemistry and material science [1, 2]. Semiquinones are typical organic radical anions being the intermediate form in the redox equilibrium between quinones and hydroquinones. These radicals are present in all life forms as they act as electron-transfer agents in the mitochondrial respiratory chain and in the reaction centers of bacterial and plant photosynthesis [3, 4]. Moreover, o-semiquinones are known to possess chelating ability toward metal ions [5–7], which is particularly important for the activation of electron transfer through interaction with cations acting as Lewis acids [5, 8].

Electron paramagnetic resonance (EPR) spectroscopy has established its important position in investigation of semiquinone radicals in laboratory conditions and in their natural surroundings [8–14]. Also, the formation of a complex between diamagnetic metal ions and semiquinone radicals can be efficiently investigated using the EPR techniques since the g and A tensors are sensitive to the radical–metal ions interaction [5, 6, 15–18].

Recent years have witnessed an increasing interest in the application of theoretical methods to chemical and biochemical systems [19–21]. One of the most significant quantum chemical methods employed in this type of studies are the ones based on density functional theory (DFT) since these methods can be applied to (nearly) real chemical systems. Organic radicals (including semiquinones) have been the subject of successful DFT studies covering diverse environmental factors significantly affecting the radical various properties as well as their EPR tensors (g and A) [14, 22–46]. However, far too little attention has been paid to the interaction between the radicals and diamagnetic metal ions. Previously, we reported the results of a detailed DFT study of the influence of Mg2+ on the o-semiquinone ligands in the formed complexes [47]. In the present work, we aimed to characterize theoretically the effects of the Ca2+ ion on the electronic structure of the o-semiquinone radicals and on the molecular geometries of the resulting complexes, in correlation with both the g tensor components and the characteristics of the previously studied Mg2+ complexes. In order to make the comparison meaningful, exactly the same theory levels and software versions were used here as before. Moreover, to provide a broader insight into the differences between Mg2+ and Ca2+ complexes, their relative stability was estimated. This aspect seems to be highly interesting as ubiquinone (Coenzyme Q10) and 2-palmitoylhydroquinone have been shown to transport various metal ions of biological importance (Mg2+and Ca2+) [48, 49] or of high toxicity (Sr2+ and Ba2+) [50] through membranes. Although the reduced forms of the p-quinones correspond directly to the p-semiquinones, the mechanisms proposed for Q10 assume its transformation into polyhydroxy forms and the metal ion coordination to the oxygens of the ortho hydroxy groups [49]. Therefore, the theoretical approach to the relative stability of the model complexes with Mg2+ and Ca2+ ions is expected to reveal which of the ions can be preferred in the transport across membranes.

In this study, semiquinone radicals with different aromaticity, derived from o-quinone (sq), 9,10-phenanthrenequinone (psq), and 1,10-phenanthroline-5,6-dione (ptsq) (see Fig. 1), were chosen as the model ligands coordinating Ca2+ ions.

Fig. 1
figure 1

Schematic structures of anionic semiquinone radical ligands derived from o-quinone (sq), 9,10-phenanthrenequinone (psq), and 1,10-phenanthroline-5,6-dione (ptsq). In addition, the principal directions of g tensor are shown

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2 Computational details

In the calculations, acetonitrile was selected as a solvent because it was mainly used in the experimental studies [5, 8, 51]. Acetonitrile was included in the calculations by using the continuum solvation models (PCM and COSMO), which have been shown to provide an accurate and efficient approximation to the aprotic solvent effects [22, 28, 52, 53].

All the optimizations of molecular structures were carried out using the Gaussian 09 [54] suite of programs employing the popular UB3LYP hybrid functional [55–57] and the TZVP basis set [58]. The initial geometries for the optimizations of the Ca2+ complexes with o-semiquinone ligands were prepared using the structures determined by X-ray crystallography for similar diamagnetic Ca2+ complexes. The coordination number (c.n.) of Ca2+ in these diamagnetic systems was found to be predominantly 7 [59, 60] and 8 [61, 62]. In order to make the investigation more complete, also the complexes with c.n. = 6 (octahedral) and c.n. = 4 (square planar and tetrahedral) were included in the computational analysis. To complete the coordination sphere of Ca2+, the optimized structures contained six, five, four or two acetonitrile molecules in addition to the chelating semiquinone ligand. The effect of solvation on geometry was covered by employing the integral equation formalism variant (IEFPCM) of Tomasi’s PCM method [63–65]. No symmetry constraints were imposed on the optimization procedures. All the open-shell computations were carried out using the spin-unrestricted formalism. The geometries of the investigated species did not reveal imaginary frequencies. The initial square planar structures underwent convergence to tetrahedral. This result was independent of the o-semiquinone ligand and PCM inclusion.

The ORCA electronic structure package [66] was used to calculate the g tensors and to perform the Löwdin population analysis. In these undertakings, the hybrid (UB3LYP [55–57] and UPBE0 [67, 68]) and generalized gradient approximations functionals (UBP86 [69, 70], UPBE [68], and UOLYP [55, 71]), together with the TZVP basis set [58] were employed. The conductor-like screening model (COSMO) [72, 73] was the continuum solvation model used in the computations. The g tensors were computed using Neese’s CPKS method [74] combined with an accurate mean field approximation [RI-SOMF(1X)] [75] to the Breit–Pauli spin–orbit coupling operator [76, 77]. In this work, all the computed components of the g tensors are given as g-shifts (Δg ij ) in parts per million (ppm):

$$\Updelta g_{ij} = (g_{ij} - g_{\text{e}} ) \times 10^{6} \,{\text{ppm}},$$
(1)

where ij = xx, yy, zz, and g e = 2.002319 is the free electron g value.

In order to examine the relative stability of the radical complexes, Gibbs energies at T = 298.15 K were calculated at the (U)B3LYP/TZVP theory level for the process of radical complex ([ML(c.n.−2)R]+·) formation from a cation complex with acetonitrile ([MLc.n.]2+) and a radical ligand (R−·):

$$[{\text{ML}}_{{{\text{c}} . {\text{n}} .}} ]^{2 + } + {\text{R}}^{ - \cdot } \to [{\text{ML}}_{{({\text{c}} . {\text{n}} .- 2)}} {\text{R}}]^{ + \cdot } + 2{\text{L}},$$
(2)

where L = CH3CN, M = Mg or Ca, and R−· = sq, psq or ptsq (all the structures of the Mg2+ complexes were taken from our former work [47]). The Gibbs energies were calculated for the reaction taking place in the gas phase (given in Eq. 2) and in acetonitrile using the thermodynamic cycle shown in Fig. 2. The solvation energies were obtained from single-point PCM computations. The structures of the acetonitrile molecules (L) and of the cations coordinated to the acetonitrile molecules for various c.n. ([MLc.n.]2+) were optimized as described above, but the restricted formalism (RB3LYP/TZVP) was used for the closed-shell species. The DFT methods have been proved useful in the prediction of the Ca2+ and Mg2+ affinity for nonradical ligands [78–80].

Fig. 2
figure 2

Thermodynamic cycle used in the calculations of the Gibbs energies for the complexes formation in acetonitrile

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3 Results and discussion

Figure 3 shows optimized geometries of the Ca2+ complexes with sq. Similar figures for psq and ptsq are given in Supplementary Materials (Figure S1). These figures also illustrate the rules used for structure naming: (1) The short names of o-semiquinones given in Fig. 1 define the radical molecule; (2) an Arabic numeral following a short name of an o-semiquinone indicates the c.n. of Ca2+; and (3) the letter A in the superscript indicates that the continuum solvation model was included in the calculation. The above rules were also applied to the Mg2+ complexes taken from our previous work [47], but an asterisk * was used for the latter to distinguish them from the Ca2+ complexes.

Fig. 3
figure 3

Optimized structures of the Ca2+ complexes with sq; the atoms numbering shown for sq4 is valid for all the structures discussed in the paper

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3.1 Molecular structure and spin density

From the EPR point of view, it is interesting to see how the formation of the Ca2+ complex affects the spin density of the semiquinone ligands. The changes are illustrated in this article with the Löwdin spin populations (see Table 1). Since the spin density of semiquinones depends on the length of the bonds between hydroxyl oxygens and carbon atoms (RC–O) [14, 22, 34, 52], the impact of Ca2+ complexation on RC–O should be first taken under investigation. Regardless of the c.n., the RC–O distances predicted for the Ca2+ complex with the o-semiquinones are significantly larger than the ones for the uncomplexed radicals, e.g., RC–O increased from 1.265 Å for sq A to 1.271 Å for sq8 A. The increase in RC–O was less significant for the complexes with a higher c.n. The RC–O elongation leads to a more profound spin population on the ipso carbons, as compared with the uncomplexed radical. At the same time, the spin population on the hydroxyl oxygen atoms becomes significantly lower. Importantly, the larger RC–O values (and slightly more significant changes in spin populations) were observed in the case of Mg2+ coordination [47].

Table 1 Lengths of the C–O and O–M (M = Ca or Mg) bonds (in Å) as well as Löwdin spin populations (ρ); all computed at the UB3LYP/TZVP theory level
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An interesting problem seems to be the concentration of spin density on the Ca atom. In each of the investigated complexes, the spin population on Ca was found to be barely noticeable. To ensure that this is not due to the basis set effect, the computations with other basis sets (SVP, TZVPPP, QZVP) were conducted. Regardless of the basis set used, the spin population remained insignificant (see Table S2 in Supplementary Materials). Thus, Ca2+ complexation generates a change of spin populations similar to the one observed by us previously for the Mg2+ complexation [47]. The impact of the two cations on spin populations and RC–O is qualitatively similar to the changes induced by the solvent [14, 22, 31, 34, 52], but it is quantitatively far greater.

It is essential to compare the lengths of the metal–oxygen bonds for Ca2+ and Mg2+ complexes. The RO–Ca values are predicted to increase with the c.n. from 2.489 Å for sq4 A to 2.498 Å for sq6 A and to 2.551 Å for sq8 A (Table 1), similarly to the change in RO–Mg from 2.002 Å for sq4 A* (tetrahedral) to 2.080 Å for sq6 A* (octahedral). It is apparent that the RO–Ca values are significantly higher than the RO–Mg values.

3.2 g Tensor

It is known that the DFT methods might misestimate the covalent character of metal–ligand bonds [81–83]. Although this is more frequently the case of transition metal coordination compounds, we decided to calculate the Δg tensors with a vast array of functionals (UBP86, UPBE, UOLYP, UB3LYP, UPBE0). The comparison of the g tensors presented in Table 2 shows that all the methods, both GGA and hybrid approximations, yield similar outcomes. Therefore, one can expect that the variation of the covalency on the functional is small. The reason for this is the fact that the interaction between the Ca2+ ion and the o-semiquinones is mainly electrostatic in nature; therefore, a minor contribution of the covalency to the computed Δg tensors is not to be significantly dependent on the functional.

Table 2 Δg tensors (in ppm) calculated using the UB3LYP, UBP86, UPBE, UPBE0, and UOLYP functionals
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Figure 1 shows principal axes of the g tensor whose directions remained unaffected by Ca2+ interaction. This result was independent of the examined model complex and the used methodology (various functionals and/or continuum solvent model inclusion).

The Δg zz values were found to be far less sensitive to the complexation (see Table 2) than the perpendicular components (Δg xx and Δg yy ). The Δg zz magnitude tends to rise when the radical interacts with the cation. A good example of this is the semiquinone derived from o-quinone. Δg zz increases from −125 ppm for sq A to −22 ppm for sq4 A, 11 ppm for sq6 A, 27 ppm for sq7 A, and 44 ppm for sq7 A (employing UB3LYP). It is also clear from this data that Δg zz becomes more positive for the complexes with the larger c.n.

As mentioned above, the spin populations on the hydroxyl oxygens markedly decrease upon the o-semiquinone interaction with Ca2+. According to Stone’s qualitative model [84, 85], such spin redistribution ought to reduce the Δg xx and Δg yy values. Our DFT predictions of the Δg tensor prove the correctness of this hypothesis. Regardless of the c.n., the lowering of the perpendicular components is substantial. In general, the effects of Ca2+ and Mg2+ complex formation on the Δg tensor are similar, albeit the interaction between Mg2+ and the o-semiquinones, as revealed by the shorter O–Mg bonds and ΔG 298 (to be discussed below), is clearly stronger compared to Ca2+. The stronger interaction in the case of Mg2+ should be expected to induced more significant decrease of Δg xx and Δg yy . This, however, was not observed, suggesting that the g factor is not a sufficient criterion for the strength of the interaction between an o-semiquinone and metal cation.

Interestingly, the COSMO correction decreases Δg-shifts for the solvated semiquinone, as expected [14, 22, 34, 52], but increases them for the complexed semiquinone. The complex can be considered as comprising of two parts: the semiquinone ligand and the cation with acetonitrile molecules. The COSMO correction stabilizes the latter part of the complex, decreasing the strength of the cation–semiquinone interaction and therefore decreasing the cation effect on the Δg tensor. The weakening of Ca2+–semiquinone interaction is clearly seen in the COSMO-induced RCa–O elongation.

Unfortunately, a direct comparison of the calculated Δg tensor diagonal components with their experimental counterparts is limited as very few high-field experiments have been performed so far. Therefore, we focused on the values of the Δg iso parameters. Since the calculated Δg iso values for all the considered models are close to the experimental ones, it is impossible to determine explicitly on the basis of Δg iso which coordination sphere is preferred in real chemical systems. To answer this question, the Gibbs free energies were calculated for the reaction given in Eq. 2; the results are presented below. On the other hand, a general agreement between the calculated Δg tensor components and ones experimentally determined for similar systems was expected to be a good additional way of verifying the quality of the computations. The choice of pyrroloquinoline quinone (2,7,9-tricarboxypyrroloquinoline, PQQ), a quinone cofactor belonging to a class of dehydrogenases known as quinoproteins, seems to be the most appropriate. PQQ is bonded to the Ca2+ ion and exhibits an EPR spectrum with the Δg xx  = 3,531, Δg yy  = 2,861, and Δg zz  = −199 components [10, 11]. The magnitude of the Δg-shifts predicted by us for the model o-semiquinone complexes is similar, despite the chemical differences. This fact (in combination with the good agreement between experimental and theoretical Δg iso) suggests the high accuracy of the computations.

The distribution of spin density in the o-semiquinones coordinating Ca2+ (and Mg2+) gives rise to an interesting question about the direct contribution of the Ca2+ (and Mg2+) ion to the perpendicular Δg tensor components. Minor spin populations on the metal cations to a certain degree suggest that such a direct impact should be insignificant and only strong indirect effects may be expected. To investigate this problem, we performed a theoretical analysis of the atomic contributions to Δg xx and Δg yy . In order to make this analysis complete, the calculations were also done (for the first time) for the representative Mg2+ complex investigated by us previously [47].

In one-component DFT calculations, the total Δg tensor is given as a sum of three contributions [74, 86, 87]:

$$\Updelta g_{st} = \delta_{st} \Updelta g^{\text{RMC}} + \Updelta g_{st}^{\text{DC}} + \Updelta g_{st}^{\text{PSO}} ,$$
(3)

where Δg RMC is the relativistic mass correction to kinetic energy, δ st is the Kronecker delta function ensuring that Δg RMC contributes only to the diagonal components of the Δg tensor (s = t), \(\Updelta g_{st}^{\text{DC}}\) is the diamagnetic correction and \(\Updelta g_{st}^{\text{PSO}},\) is the paramagnetic spin–orbit term. The values of the three terms are given in Table 3.

Table 3 Individual contributions to the Δg tensor components and the direct contributions of selected atoms obtained employing the one-center approximation
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Irrespective of the interactions with the metal ion, the predicted Δg RMC and \(\Updelta g_{st}^{\text{DC}}\) values were found to be of minor magnitude. Moreover, their opposite signs lead to the mutual cancelation of the two terms whereby the Δg xx and Δg yy components are dominated by \(\Updelta g_{st}^{\text{PSO}}\). In this case, an accurate approximation of the atomic contributions to the Δg tensor can be obtained via the breakdown of \(\Updelta g_{st}^{\text{PSO}}\) into the contributions from the particular atoms. Since the mean field approximation to the molecular spin–orbit coupling operator employed in this work [RI-SOMF(1X)] [75] takes into account the mulicenter terms (except for the exchange part), these terms have to be neglected to obtain the atomic contributions. Considering that such an omission may cause significant errors [75], in Table 3, the \(\Updelta g_{st}^{\text{PSO}}\)(1c) values calculated in the one-center approximation are compared with the ones calculated including the multicenter terms (\(\Updelta g_{st}^{\text{PSO}}\)), revealing only a limited deviation.

The contributions from all the atoms for sq were calculated at the UB3LYP/TZVP theory level and are shown in Fig. 4. Both perpendicular components are dominated by the contributions from the oxygens. This is in agreement with the previous reports for p-semiquinone [34] and the phenoxyl radical [35]. The contributions from carbon atoms, even from these in the ipso positions, are considerably small. Moreover, the contributions from the different carbons to Δg xx have opposite signs, which leads to their mutual cancelation.

Fig. 4
figure 4

Contributions of the particular atoms to Δg xx (green) and Δg yy (maroon) for sq; calculated at the UB3LYP/TZVP theory level

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The inclusion of the COSMO model (sq A) results in a significant decrease in the contributions from the oxygens to the Δg xx and Δg yy components (see Table 3). After the attachment of Mg2+ or Ca2+ to sq, barely noticeable contributions of the metal atoms were predicted. Thus, the observed diminution of Δg xx and Δg yy upon the complex formation is exclusively the result of the reduced contributions from the oxygens. Consequently, the impact of a diamagnetic metal ion on the Δg tensor of the semiquinone radical can be described as indirect; a metal ion does not bring any significant direct contribution, but causes a decrease in the contributions from hydroxyl oxygen atoms.

Table 3 is quite informative in another way. As it was mentioned above, Δg zz increases on complex formation, and this table shows that this increase is related primarily to the rise of \(\Updelta g_{\text{zz}}^{\text{DC}}\) and secondarily to the growth of \(\Updelta g_{\text{zz}}^{\text{PSO}}\) term.

As it was demonstrated in our former systematic study of Mg2+ complexes with o-semiquinone ligands [47], breaking down of the dominant \(\Updelta g_{st}^{\text{PSO}}\) term into the contributions originating from the particular excited states can be fruitful for the understanding of the Δg tensor changes on metal ions complexation. To provide similar insight in this work, the alternative one-component method proposed by Schreckenbach and Ziegler [86], as implemented in the ADF package [88], was used. In this method [86]:

$$\Updelta g_{st}^{\text{PSO}} = \sigma_{st} + \Updelta g_{st}^{{{\text{PSO}},\text{occ}{-}\text{occ}}} + \Updelta g_{st}^{{{\text{PSO}},\text{occ}{-}\text{vir}}} ,$$
(4a)
$$\sigma_{st} = \frac{1}{2c}\sum\limits_{\gamma = \alpha ,\beta } {2m_{\gamma } } \sum\limits_{i = 1}^{{n_{\gamma } }} {n_{i}^{\gamma } } \sum\limits_{\lambda ,\nu }^{{2{\text{M}}}} {d_{\lambda i} d_{\nu i} \left\langle {\chi_{\lambda } \left| {\left( {\overrightarrow {R_{ \lambda}} \overrightarrow {R_{\nu }}} \right)_{s} h_{t}^{01} } \right|\chi_{\nu } } \right\rangle } ,$$
(4b)
$$\Updelta g_{st}^{{{\text{PSO}},\text{occ}{-}\text{occ}}} = \sum\limits_{\gamma = \alpha ,\beta } {2m_{\gamma } } \sum\limits_{i,j = 1}^{{n_{\gamma } }} {n_{i}^{\gamma } S_{ij}^{1,s} \left\langle {\varPsi_{i} \left| {h_{t}^{01} } \right|\varPsi_{j} } \right\rangle ,}$$
(4c)
$$\Updelta g_{st}^{{{\text{PSO}},\text{occ}{-}\text{vir}}} = 2\sum\limits_{\gamma = \alpha ,\beta } {2m_{\gamma } } \sum\limits_{i = 1}^{{n_{\gamma } }} {n_{i}^{\gamma } } \sum\limits_{a}^{\text{vir}} {u_{aj}^{1,s} \left\langle {\varPsi_{i} \left| {h_{t}^{01} } \right|\varPsi_{a} } \right\rangle } .$$
(4d)

The term σ st has been shown to be numerically irrelevant [86, 89], and it is therefore neglected in the further discussion; \(\Updelta g_{st}^{{\text{PSO,occ}{-}\text{occ}}}\) are the couplings between occupied orbitals and \(\Updelta g_{st}^{{\text{PSO,occ}{-}\text{vir}}}\) between occupied and virtual ones; \(h_{t}^{01}\) is the paramagnetic spin–orbit operator defined in [86]; Ψ i and Ψ a are occupied and virtual Kohn–Sham orbitals, respectively; the orbitals are expanded into the set of 2M basis functions {χ λ }; the expansion coefficients are the d λi ; \(n_{i}^{\gamma }\) is the occupation number of Ψ i ; \(S_{ij}^{1,s}\) and \(u_{aj}^{1,s}\) are the first-order occupied–occupied and occupied–virtual coefficients, respectively; and the coefficients 2m γ afford the correct signs for α and β spins (m α  = ½ and m β  = −½). All the Δg tensor calculations performed with the ADF package were spin-unrestricted, based on the scalar Pauli Hamiltonian and employing the UBP86 functional in concert with the standard all-electron Slater-type TZP basis set. The ADF program was used because the implementation included in it allows to analyze \(\Updelta g_{st}^{{\text{PSO,occ}{-}\text{vir}}}\) in terms of single excitations.

The \(\Updelta g_{st}^{{\text{PSO,occ}{-}\text{vir}}}\) term is usually the most important one [89], and it is shown here to dominate the perpendicular components of the uncomplexed sq radical and the studied complexes (see Table 4). Δg zz is very small in magnitude because of the insignificant coupling between occupied and virtual orbitals. Nonetheless, \(\Updelta g_{\text{zz}}^{{\text{PSO,occ}{-}\text{vir}}}\) is slightly increased after the Ca2+ or Mg2+ complex formation. To meaningfully discuss the contribution of excited states to the Δg xx and Δg yy components, the possible excitations were classified into three groups: (1) from the doubly occupied orbitals to the SOMO (D → S); (2) from the SOMO to the virtual ones (S → V); and (3) from the doubly occupied orbitals to the virtual (D → V). The last group is expected to bring small contributions to the Δg tensor of organic radicals as contributions of these excited states arise from the spin polarization solely. The contributions of the three groups of excited states to \(\Updelta g_{st}^{{\text{PSO,occ}{-}\text{vir}}}\) are listed in Table 4 and visualized in Fig. 5.

Table 4 Contributions of excited states to the g tensor; all calculated at the UBP86/TZP theory level with the ADF package employing the method proposed by Schreckenbach and Ziegler [86]
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Fig. 5
figure 5

Graphical illustration of various excited states contributions to Δg xx (a) and to Δg yy (b). In addition, molecular orbitals connected to the excited states giving significant contributions to the Δg tensor are shown (c). Labels to the orbitals were given according to the results for sq

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The perpendicular components for sq are dominated by the contributions from D → S excited states, in accordance with the report for p-semiquinone [52]. This group of excited states is the preeminent one also after the Ca2+ and Mg2+ complexation; however, its contribution is significantly decreased. For the o-semiquinone radical anion sq, Δg xx and Δg yy components are mainly prevailed by the contributions from HOMO-2 → SOMO and HOMO → SOMO excited states, respectively (the both excited states are of the D → S type). The formation of a complex between Ca2+ and sq results in the significantly reduced contributions of these two excited states, and consequently, Δg xx and Δg yy decrease on the complex formation. Interestingly, after the complex formation, the HOMO → SOMO contribution becomes nearly negligible. The isosurfaces of SOMO, HOMO, and HOMO-2 are shown in Fig. 5c.

3.3 Relative stability

It would be interesting to find which metal cation, Mg2+ or Ca2+, forms more stable complexes with o-semiquinone ligands. The enthalpies \(\left( {\Updelta H_{\text{gas}}^{298} } \right)\), the entropies \(\left( {\Updelta S_{\text{gas}}^{298} } \right)\), the Gibbs energies in the gas phase and in acetonitrile (\(\Updelta G_{\text{gas}}^{298}\) and ΔG 298, respectively) and the changes in the solvation energies \(\left( {\sum {\Updelta G_{\text{sol}}^{298} } } \right)\) calculated for the radical complexes formation (Eq. 2) are reported in Table 5.

Table 5 Selected thermodynamic properties calculated at the (U)B3LYP/TZVP theory level according to the thermodynamic cycle shown in Fig. 2
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First, it is sensible to compare the Gibbs energies calculated for the complex formation taking place in acetonitrile (ΔG 298) and in the gas phase (\(\Updelta G_{\text{gas}}^{298}\)). The \(\Updelta G_{\text{gas}}^{298}\) values are significantly more negative, whereby the formation of the semiquinone complex with Mg2+ or Ca2+ from the free anionic radical and cation complexed by acetonitrile molecules tends to be energetically more beneficial in the gas phase. This can be explained by the fact that the stability of ions is increased by solvation more significantly than the stability of uncharged molecules. Considering that the complex formation in Eq. 2 is inseparable from the reduction of the charge (one 2+ cation and one 1− radical anion are the substrates and one 1+ radical complex is the product), the solvation of the substrates is expected to be energetically more beneficial than the solvation of the products, which should lead to the lowering of the radical binding affinity in the solution. This presumption is clearly corroborated by the positive change of solvation energy \(\sum {\Updelta G_{\text{sol}}^{298} }\) (see Table 5). Despite the positive \(\sum {\Updelta G_{\text{sol}}^{298} }\) values, the resulting ΔG 298 is predicted to stay negative.

For the Ca2+ complexes, the most negative ΔG 298 values were obtained for c.n. = 6, strongly suggesting the formation of the complexes with this c.n. in real chemical systems. On the other hand, for c.n. = 4, the ΔG 298 values are the least negative, indicting a low probability of such Ca2+ complex formation. In contrast, for the Mg2+ ion, c.n. = 4 is shown to be energetically more profitable as the predicted ΔG 298 values are considerably lower than for the complexes with c.n. = 6. To understand why the higher c.n. are energetically beneficial in the case of Ca2+ ions, one has to compare the ion radius of the both cations. Ca2+ has significantly greater radius (1.00 Å) [90] than the Mg2+ (0.72 Å) [90]; in consequence, Ca2+ may be surrounded by a larger number of ligands without significant steric repulsion between them.

Another point of concern is the relative stability of the complexes containing various o-semiquinone ligands. In general, the relative stabilities of Ca2+ coordination compounds with sq and psq radical ligands are comparable; however, in the case of Mg2+, the stability of the psq complexes, clearly indicated by the more negative ΔG 298 value, is noticeably increased as compared with sq. The coordination of both cations to the ptsq ligand results in lower stability than the coordination to sq and psq. This can be explained by the fact that the interaction between o-semiquinone radicals and Mg2+ and Ca2+ cations is mainly electrostatic in nature, whereby the stability of the formed complexes decreases as the negative charge located on the hydroxyl oxygens atoms is reduced. In the case of ptsq, the negative charge on the O atoms should be moderately diminished as compared with sq and psq, since the ptsq molecule contains two N atoms that are additional attractors of the negative charge. This can be illustrated using the Löwdin atomic charges. For the hydroxyl oxygens of sq and psq, they are predicted to be −0.32 and −0.31, respectively, while for ptsq, just −0.27.

Perhaps the most interesting is the relative stability of the Ca2+ and Mg2+ complexes with o-semiquinones since the two cations usually coexist in natural systems and so competition between them is expected to occur. As it can be seen in Table 5, the most negative ΔG 298 values are predicted for Mg2+ complexes with c.n. = 4. This fact strongly suggests that the formation of the o-semiquinone complexes with this cation is more favorable, and therefore, Mg2+ can be expected as preferred over Ca2+ in the mechanism of cation transport through membranes [49].

In our opinion, it is always sensible to confront theoretical results with more general ideas, here with the Pearson hard and soft acids and bases concept (HSAB) [91]. According to HSAB, certain metal ions (hard Lewis acids) exhibit high affinity for oxygen donor ligands. Thus, the harder the Lewis acid, the stronger the preference for O donors. For the complexes of Mg2+ and Ca2+ with o-semiquinones, the Mg2+ > Ca2+ stability order is expected as Mg2+ is considered to be a moderately harder acid than Ca2+ due to the same +2 charge but noticeably smaller size. To summarize, in spite of its limitations, HSAB gives a qualitative answer being in agreement with the results yielded by DFT methods.

4 Conclusions

This paper has provided a detailed insight into the interaction between o-semiquinone radicals and Ca2+ ions. Good agreement between the calculated and experimental g iso parameters, supported by accordance of the calculated g tensors with the experimental data for PQQ, suggests that DFT methods are suitable not only for theoretical examination of this parameter but also may provide insight into the molecular and electronic structure of the radical species interacting with diamagnetic metal ions. In other words, this good agreement between the theoretical and experimental Δg iso values might be treated as an indication that the other predicted properties (spin distribution, structural parameters as RC–O, RCa–O) properly characterize the real systems.

The conducted computations revealed that, in general, the effects of Ca2+ and Mg2+ complex formation on the Δg tensor are similar, although the interaction between Mg2+ and the o-semiquinones, as revealed by the shorter O–Mg bonds and more negative ΔG 298, is clearly stronger compared to Ca2+. The stronger interaction in case of Mg2+ should be expected to induce more significant decrease of Δg xx and Δg yy , but this was not observed. Therefore, this study have shown that the g factor is not a reliable criterion for the strength of the interaction between an o-semiquinone and diamagnetic metal cation.

The calculated atomic contributions to the Δg tensor indicate that the impact of the metal ion (Ca2+ or Mg2+) on the Δg tensor of o-semiquinone radicals is mainly indirect. Although the metal ion brings only a barely noticeable direct contribution, it causes a significant decrease in the contributions of hydroxyl oxygens to the Δg xx and Δg yy components. In addition, the contributions of various excited states to the Δg tensor were analyzed. It was shown that the decrease of Δg xx and Δg yy on the complex formation is the consequence of reduced contributions of HOMO-2 → SOMO and HOMO → SOMO excited states.

Another important observation is that the general stability of the Mg2+ complexes is higher than that of the complexes with Ca2+. Therefore, in the transport mechanism through membranes with Q10 playing the role of the transfer agent [49], Mg2+ ions can be expected to be favored over Ca2+.