Introduction

The Cu(I), Ag(I) and Au(I) as transition metals d10 cations adopt various coordination modes with different ligands corresponding to outstanding linear, trigonal planar and tetrahedral molecular structures [1,2,3,4] with probable applications as phosphors in organic light emitting diodes (OLEDs) for Cu(I) complexes [5, 6].

The linear two-coordinated Cu(I) complexes targeted in this investigation have been experimentally explored and theoretically analyzed [7, 8], particularly, those of monodendate nitrogen ligands which are represented. These types of complexes are rarely characterized due to their relative instability due to electronic factors [9,10,11,12,13], but the sterically hindered ligands enhance their stability [9].

However, the linear Cu(I) carbene complexes have been extensively investigated owing of their interesting photophysical characteristics resulting from ligand-to-ligand charge transfer [14,15,16], where several of them have been used as emitters [17, 18] exhibiting interesting luminescence properties [19,20,21]. Indeed, linear di-coordinate copper complexes of cyclic(alkyl)(amino)-carbenes (CAAC)CuX (X = Cl, Br, I) display photoluminescence as the first species highly luminescent [22] and showing remarkable photophysical properties when compared to their analogous N-heterocyclic carbene complexes [23].

The linear Cu(I) and Ag(I) alkene or alkyne compounds are scarcely characterized by XR-diffraction [24, 25], despite that this class of compounds is the first reported among organometallic ones, contrary to their homologous of Au(I) which are more isolated and characterized [26,27,28,29,30,31].

The XCuL’ and (LCuL’)+ (L = CO, PH3, NH3, L’ = NHC, C2H2, C2H4, C4H6, C6H6, and X−  = Cl, CH3, and CN) complexes of 14-MVE (metal valence electrons) count are investigated throughout this theoretical study and are compared to their homologous LML’ compounds known to adopt a linear structure with L-M-L’ angle of about 180° [32, 33].

Within this work, the molecular and electronic structures were investigated for each L’ ligand in relationship with its bonding towards the XCu or (LCu)+ metallic moiety based on the identification of different interactions, where X− and L as terminal ligands are isoelectronic units, but with different properties; it is what we ought to elucidate.

Furthermore, in order to assess the σ-donation and π-backdonation of various used ligands, the interactions have been evaluated between XCu and L’ fragments on one hand and between LCu+ and L’ on other hand using the energy decomposition analysis (EDA) [34,35,36] combined with natural orbital for chemical valence (EDA-NOCV) [37,38,39,40,41] analysis taking into account the impact of X or L ligand composing the XCu or LCu+ fragments of 12-MVE count. The EDA provides a connection between the physical rules of quantum mechanics and a conceptually simple explanation of nature chemical bonding. Besides, the EDA-NOCV method connects the frontier orbital theory of Fukui [42] and the orbital symmetry rules of Wood-Ward and Hoffmann [43] resulting from DFT calculations.

Thus, we present a detailed analysis of the coordination between acetylene as alkyne, ethylene, butadiene and benzene as alkene or NHC and the XCu’ or (LCu)+ metallic fragment, where a complete rationalization of bonding is provided of this kind of complexes with respect to X, L, and L’ ligands in order to establish the similarities and discrepancies of the occurred interactions.

Theoretical methods

The density functional theory (DFT) calculations have been performed to optimize all molecular structures at the BP86 [44,45,46,47] and B3LYP [48, 49] computational levels. The electron correlation was treated within the local density approximation (LDA) in the Vosko–Wilk–Nusair parametrization [50]. The triple-ξ polarization basis set for all atoms and the frozen core approximation for BP86 functional using the ADF2022.01 program [51] developed by Baerends and co-workers [52,53,54,55,56]. Frequency calculations have been performed on the optimized structures to authenticate that the obtained structures correspond to energetic minima [57, 58]. The natural population-based (NPA) and Wiberg bond indices [54, 59] were obtained from calculations implemented in the NBO 7.0 program [60, 61]. The EDA-NOCV [37,38,39,40,41] method results from the combination between natural orbitals for chemical valence (NOCV) [37,38,39,40,41] and energy decomposition analysis (EDA) [34,35,36] was applied to decompose the deformation density correlated to the bond formation into various components of the chemical bonding. The EDA-NOCV scheme divides the orbital interactions between the interacting fragments into pairwise contributions of the most relevant molecular orbitals. Furthermore, The Mayer bond orders (MBO) [62,63,64] have been calculated which can be seen as an expansion of the Wiberg bond indices implemented in the ADF2022.01 program [51].

Results and discussions

Free ligands

The C2H2, C2H4, C4H6 and C6H6, and NHC as free ligands were optimized to get information about their structures (Fig. 1) and their π-electrons localization. Indeed, the C2H2 (acetylene) as alkyne molecule displays a C–C bond distance of 1.231 Å and a linear C–C-H angle of 180° matching well with a triple bond as known in the literature [64]. However, C–C bond distances in the plane C2H4 (ethylene) molecule and C4H6 (butadiene) as alkene are of 1.332 and 1.343 Å corresponding to a double bond describing a localized scheme comparable to the experimental values and reproducing theoretical findings [66,67,68], but in C6H6 (benzene) aromatic ring, the average C–C bond distance is of 1.398 Å matching well with a delocalized scheme and corresponds to 1.5 bond order as obtained in previous works [69,70,71,72,73,74,75,76,77].

Fig. 1
figure 1

BP86-optimized structures of free ligands with C–C and C-N bond distances in Å

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XCUL’ and (LCuL’)+ optimized structures

The optimized structures of the XCuL’ and (LCuL’)+ compounds are shown in Figs. 2 and 3, and their relevant geometrical parameters obtained by means of BP86 and B3LYP are gathered in Tables S1 and S2, and compared to those experimentally observed when they are available. Firstly, we discuss the structures of XCu(C2H2), XCu(C2H4), XCu(C4H6), XCu(C6H6), and XCu(NHC) displaying large HOMO–LUMO gaps to get information about the change of the Cu-C(1) and Cu-C(2) bond distances related to the coordination mode of the XCu metallic fragment induced by the introduction of L’ ligand. Besides, the C(1)-C(2) bond distances calculated within the free L’ ligands are compared to their corresponding ones in XCuL’ compounds. In recent work, it has been shown that the X-Cu-L’ angle encountered in XCu(CAAC)] (X = Cl, Br, I, and CAAC = cyclic(alkyl)(amino)carbene) compound [19, 77] showed a linearity of L-Cu-L bond angle comparable to those computed for CH3Cu(NHC) and CNCu(NHC) varying from 161° to 180°. One can remark that for each XCu fragment the Cu-C(L’) bond distance varies in function of the variation of the X ligand. The shortest Cu-C(L’) bond distance of 1.891 Å corresponds to the Cu-C in CH3Cu(NHC) comparable to those found in previous works [24, 78], while the longest one corresponds to the Cu-C in CNCu(C6H6) of 2.175 Å as given in Table S1, comparable to those experimentally reported in the literature [79] relative to Cl-Cu bond distance of 2.099 Å, which is similar to those computed for ClCuL’ compounds ranging from 2.089 to 2.097 Å. It is interesting to mention that for each L’, the shortest Cu-C(L’) is obtained in ClCuL’ compounds followed respectively by those in CH3CuL’ and CNCuL’ ones, giving rise to η2-coordination mode in the presence of C2H2, C2H4, C4H6, and C6H6. Considering the results gathered in Table S1, there is a perfect correlation between the Wiberg bond indices (WBI), Mayer bond orders (MBO), and the bond distances shortening or lengthening of the Cu-L’ bond distances. Indeed, the WBI and MBO values tend to become larger when the bond distances become short as well illustrated in Table S1. So, the shortest Cu-C(L’) bond distance is obtained in the case of the NHC compounds corresponding to WBI of 0.51, 0.46, 0.43, and 0.45 and MBO of 0.89, 0.85, 0.83 and 0.84 when it is attached to ClCu, CH3Cu, CF3, and CNCu metallic fragments, respectively. However, the WBI values for Cu-C(C6H6) bond fall to 0.29 (0.14 + 0.15), 0.28 (0.14 + 0.14), 0.32 (0.16 + 0.16), and 0.36 (0.18 + 0.18) in the presence of CN−, \({\text{CF}}_{3}^{-}\), \({\text{CH}}_{3}^{-}\), and Cl− anions, respectively, corresponding to the longest Cu-C bond lengths. Besides, Mayer bond orders listed in Tables S1 and S2 show large values compared to those of WBI ones, putting emphasis on a σ-dative L → Cu bond, i.e., the interaction are interpreted as a dative or donor–acceptor bond [80].

Fig. 2
figure 2

The lowest BP86-optimized XCuL’ structures of singlet state and their atom numbering adopted throughout the paper. The C–C and C-N bond distances are given in Å

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Fig. 3
figure 3

The lowest BP86-optimized (LCuL’)+ structures of singlet state and their atom numbering adopted throughout the paper. The C–C and C-N bond distances are given in Å

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A higher positive charge at the Cu(I) cation signifying a considerable charge donation and vice versa in all XCu(NHC) complexes. The findings relative to the electronic configuration show obviously that there is a correlation between the charge of the Cu(I) cation and the 4s and 3d orbitals’ populations (Table S1). The most populated 4s orbital is encountered in the XCu(NHC) compounds regardless the X ligand indicating considerable σ-donation from the HOMO orbital of the NHC ligand towards the antibonding vacant XCu orbital mostly of 4s character. However, the 3d orbital is weakly depopulated in agreement with a very weak π-backdonation into the vacant π*C-N antibonding orbital of the NHC ligand as clearly given in Table S1.

It is interesting to mention that the C = C bond length modifications when it is coordinated to different XCu fragments undertaking somewhat elongations in accordance with the L’ ligand. Indeed, the major C = C elongations are obtained in the cases of C6H6 and C4H6 of 1.424 and 1.382 Å compared to those of free ligands of 1.398 Å and 1.343 Å, respectively. Besides, the C–C-H linear angle of the free C2H2 ligand (Fig. 1) undertakes deformation in various XCu(C2H2) complexes and becomes bent one by 16° as elucidated in Fig. 2.

For each XCuL’ compound, the C = C elongations obey the following tendency in accordance with the L’ ligand: C6H6 > C2H2 ≈ C2H4 > C4H6. Similarly, the (LCuL’)+ compounds exhibit comparable tendencies than those observed for XCuL’, but with somewhat differences concerning the Cu-C(L’) and C(1)-C(2) bond lengths.

In (LCuL’)+, the Cu-C(L’) bond distances are slightly longer by about 0.05 Å than those obtained in their analogous of XCuL’ as gathered in Table S2. Moreover, the Cu-C bond distances in (COCuL’)+ are weakly longer than those encountered in (PH3CuL’)+. The C = C bond distances within these compounds undergo elongation as sketched in Fig. 3. All (LCuL’)+ structures display very large HOMO–LUMO gaps ranging from 2.96 to 5.18 eV (BP86) or 4.36 to 6.15 eV (B3LYP) predicting stable compounds.

Furthermore, L-Cu-L’ angles in all (LCuL’)+ structures are linear or deviate slightly from the linearity with regard to their values ranging from 161 to 180° as described for related complexes [81].

Energy decomposition analysis

The Morokuma–Ziegler energy decomposition analysis (EDA) [34,35,36] is a powerful method for a quantitative explanation of chemical bonding and largely used recently [82,83,84,85,86,87,88,89,90,91,92,93]. Within this work, the EDA is applied to describe the nature of the interactions between XCu and L’ on one hand (Table 1) and between (LCu)+ and L’ on other hand (Table 2). The total bonding energy ΔEBond is considered as the combination between the preparation energy ΔEprep (or strain energy, deformation energy) and the interaction energy ΔEint as summarized in Tables 1 and 2. The preparation energy is the quantity of energy that is necessitated to deform XCu and L’ on one hand and LCu+ and L’ on other hand from their equilibrium structures to the geometries they have in the XCuL’ and [LCuL’]+ complexes (it is identified as the difference between the isolated fragments and the states found in the final complex). Thus, the preparation energy (ΔEprep) is the difference between ΔEint and ΔEBond: ΔEprep = ΔEBond − ΔEint. Furthermore, for a molecule AB composed of A and B molecular fragments, the preparation energy (ΔEprep) can be expressed as follows: ΔEprep = [E(A)AB − E(A)] + [E(B)AB − E(B)], where E(A)AB and E(B)AB are respectively the energies of A and B fragments calculated in the optimized AB molecule, whereas E(A) and E(B) are respectively the energies of A and B in their optimized states corresponding to the isolated structures. The most important conclusion that arises from the calculations of the preparation energy ΔEprep is that its contribution is weak in the ranges 0.5–4.4 and 0.1–2.8 kcal/mol for XCuL’ and [LCuL’]+ complexes, respectively, as gathered in Tables 1 and 2. The most important value of ΔEprep of 4.4 kcal/mol corresponds to the deformation of the C2H2 ligand which losses its linearity through the bending of C–C-H angle shifting from 180° (free ligand) to 163° in ClCu(C2H2. However, it has been observed that C2H4, C4H6, C6H6, and NHC ligands are less distorted, particularly the NHC fragment composing the LCu(NHC) complex which it does not undergo any deformation behaving as rigid molecule as evidenced in Table 1. The LCu+ and L’ in (LCuL’)+ are relatively less sensitive to the deformation, in relationship with their ΔEprep values gathered in Table 2 compared to those obtained for XCuL’ compounds.

Table 1 The ∆Eint, ∆Eprep, and ∆ECP (counterpoise) energies of BP86 functional arising from the interaction between XCu (X = CH3, CF3, CN, Cl) and L’ (L’ = C2H2, C2H4, C4H6, C6H6, NHC) fragments in kcal/mol. Contribution (%) of each ∆Eelstat and ∆Eorb component into the total attractive energy (∆Eelstat + ∆Eorb) is given in parentheses and the electronic populations of σ and π orbitals derived from the natural population analysis are given between square brackets
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Table 2 The ∆Eint, ∆Eprep, and ∆ECP (counterpoise) energies of BP86 functional arising from the interaction between (LCu)+ (L = PH3, NH3, CO) and L’ (L’ = C2H2, C2H4, C4H6, C6H6, NHC) fragments are given in kcal/mol. Contribution (%) of each ∆Eelstat and ∆Eorb component into the total attractive energy (∆Eelstat + ∆Eorb) is given in parentheses and the electronic populations of σ and π orbitals derived from the natural population analysis are given between square brackets
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Table 1 gathers the results of the energy decomposition analysis (EDA) arising from the interactions between XCu and L’. The EDA scheme decomposes the bonding interaction energy ΔEint into three terms of energies: the classical electrostatic interactions ΔEelstat and the orbital term ΔEorb as attractive terms and the ΔEPauli repulsion exchange term are comparable. Thus, the Cu-L’ interaction energies ΔEint collected in Table 1 have different negative amounts, but display the same trend recorded for all XCuL’ compounds. The ΔEint shows that for each L’ ligand, the strongest interactions are calculated for ClCu fragment corresponding to the highest absolute value and obeying the decreasing order ClCu > CNCu > CF3Cu > CH3Cu. Really, Table 1 and the plots of Fig. 4 demonstrate that the interaction energies ΔEint of the CICuL’ complexes are about 5, 10, and 15 kcal/mol lower than the values of their corresponding obtained for CNCuL’, CF3CuL’, and CH3CuL’ complexes, respectively, except for the NHC ligand whose values are comparable for the ClCu and CNCu fragments, as clearly illustrated. For each XCu fragment, the most significant interactions are found for NHC ligand and the weakest ones are encountered for C6H6, while the C2H2, C2H4, and C2H6 ligands exhibit comparable energies as well summarized in Table 1.

Fig. 4
figure 4

a, b ΔEint energy (in kcal/mol) evolution for XCuL’ (X = Cl, CN, CH3, CF3, and L’ = C2H2, C2H4, C4H6, C6H6, NHC) compounds obtained by interactions between each XCu and the five L’ ligands

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In each case of the XCuL’ compounds, the largest contribution into the ΔEint value stems from the ΔEPauli repulsive term, but it is overbalanced by the total attractive contribution (ΔEelstat + ΔEorb) giving rise to stabilizing interactions as highlighted by the negative values of ΔEint. The proportion of the ΔEelstat electrostatic attractive energy is higher than that of the ΔEorb one which comes from the orbital relaxation and orbital mixing between the interacting fragments. The lowest ΔEelstat contribution is observed for all C6H6 compounds which ranges from 44% (CNCu(C6H6) to 60% (CH3Cu(C6H6); however, it substantially enhances to 77% for the NHC ligand in all XCuL’ compounds. The main part of the repulsive term ΔEPauli as an exchange repulsion stems from the interaction between occupied orbitals of the XCu fragment and the occupied π orbitals of the L’ ligand in the cases of C2H2, C2H4, C2H6, and C6H6 and the occupied σ-type orbital of NHC. The results given in Table 1 show clearly that NHC is the strongest σ-donor, but the weakest π-acceptor as reported in previous works [77]. It is interesting to mention the comparable behavior of C2H2, C2H4, and C4H6 ligands regarding ΔEint and its components, while C6H6 behaves differently due probably to the π-electrons delocalization within its aromatic ring. Indeed, the benzene ring loses its aromaticity as witnessed by the C–C bond distances elongation to an average value of 1.42 Å compared to that of the free ligand of 1.398 Å and corroborated by its HOMA (Harmonic Oscillator Model of Aromaticity) [94, 95] value of 0.82 conducting to a decreased aromaticity.

It is worth to mention that the interaction energies have been corrected from basis set superposition error (BSSE) with the counterpoise (CP) method developed by Boys and Bernardi [96]. The ΔEint (without BSSE correction) and ΔECP (with BSSE correction) are given in Tables 1 and 2. The BSSE introduces a nonphysical attraction between the two interacting fragments. Thus, the counterpoise correction usually leads to intermolecular complexes less stable [97]. For all complexes treated in this study, the ΔEint is slightly lower in energies than ΔECP, where the difference between them do not exceed 1.0 and 2.3 kcal/mol for XCuL’ and [LCuL’]+, respectively, indicating a negligible influence on interactions, so, in this case, the CP should weakly or even negligibly correct the optimized geometries as well as the interaction energies. It becomes visible from Tables 1 and 2 that the ΔECP energies arising from total interaction between fragments with BSSE correction obey the same order than that found for ΔEint ones without BSSE correction (Fig. 5).

Fig. 5
figure 5

a, b ΔEint energy (in kcal/mol) evolution for (LCuL’)+ (L = CO, NH3, PH3, and L’ = C2H2, C2H4, C4H6, C6H6, NHC) compounds obtained by interactions between (LCu)+ and the five L’ ligands

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One of the best characteristics of EDA-NOCV is to decomposing the total orbital interactions (ΔEorb) into pairwise interactions providing the strength and each type of orbital, where the total σ and π orbital contributions into the covalent bonding are gathered in Tables 1 and 2.

For CH3Cu, the σ-bonding energy towards the L’ ligands change drastically from C2H2, C2H4, and C4H6 to C6H6 and more to NHC, i.e., they are comparable for C2H2, C2H4, and C4H6, but they enhance for C6H6 and more for NHC. We can specify that the proportion of σ and π orbital energies give a higher σ contribution into the ΔEorb component than that of π one, particularly for C6H6 and more for NHC ligands.

The ΔEorb component is comparable in both XCuL’ and (LCuL’)+ compound types, but the absolute values of ΔEPauli and ΔEelstat are larger in the former species than in the latter, despite the fact that the ΔEint are less significant in XCuL’ than in (LCuL’)+ due to the difference of natural charges of the interacting fragments in each case. Table 2 shows that the positive charge (NPA) on Cu(I) in the PH3 complexes has lower values in presence of NHC traducing the strong σ-donation into the vacant metallic orbital. This agrees with the energy decomposition of the orbital interaction term (Table 2), demonstrating that the former species have a higher ΔE(π) contribution than the latter molecules. The EDA analysis of the Cu-C(L’) bonding in the presence of CF3 and CH3 demonstrates disproportional contributions from electrostatic and covalent interactions and contributions to the latter are less than those to the former. The Cu-L’ bonds of the PH3 and CH3 ligands have less covalent character and a significant lower degree of π bonding in CF3 that in CH3, where a same tendency has been observed for organometallic complexes coordinated in η2-fashion to L’ ligand (L’ = C2H2, C2H4) [98] and related alkyne copper(I) complexes [99].

For ample details regarding the σ-donation and π-backdonation, Tables 1 and 2 gather their amounts obtained for the interacting fragments in XCuL’ and (LCuL’)+ complexes.

The values given in Fig. 6 highlight electron transfer between the frontier orbitals of the interacting fragments namely CNCu (left side) and NHC (right side), which give information about σ-donation and π-backdonation of the involved orbitals demonstrating a strong σ interaction between 18a orbital of NHC and 12a orbital of CNCu metallic fragment emphasizing a considerable donation of 0.49e of the former and the population by 0.48e of the latter, while the π-backdonation is weakly stressed between the 19a orbital and 21a one by an electron transfer of 0.07e as described in previous works for related NHC ligands [77].

Fig. 6
figure 6

MO diagram obtained by interactions between CNCu and (NHC) fragments with their frontier orbital populations

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One can see, that for each XCu or LCu+ metallic fragment, only one high-lying orbital is involved in the interactions with the L’ ligand composed of\({\text{d}}_{{\text{z}}^{2}}, {\text{d}}_{{\text{x}}^{2}-{\text{y}}^{2 },}\text{and }{\text{d}}_{\text{xy}}\). As example, for the CNCu(NHC) compound, only the 19a orbital of the CNCu metallic fragment composed of \({d}_{{x}^{2}-{y}^{2}}\)(55%) and \({p}_{y}\)(19%) of Cu(I) cation that interacted due to its orientation towards the 21a orbital of the NHC ligand, which acquires 0.07e. Similarly, one low-lying vacant σ* antibonding orbital composed mainly of the s orbital of Cu(I) received electron through a σ-donation. At the right side, the NHC ligand presents one high-lying 18a occupied orbital of σ-type composed of s (33%) Px (32%) and py (19%) carbon atomic orbitals which interacted strongly with the 22a metallic fragment orbital composed of a mixing of s (45%), px (23%), and py (14%) Cu(I) orbitals, beside one antibonding low lying π* orbital. The NHC ligand can be considered Fisher carbene [100] coordinated to a singlet transition metal via a dative carbene to metal σ-donation and a dative metal to carbene π-backdonation.

The breakdown of ΔEorb into pairwise interactions for CNCuL’ (L’ = C2H2, C2H4, C4H6, C4H6, and NHC) complexes is taken as example, which gives rise principally to two orbital contributions ΔE(σ) and ΔE(π) (Table 1) that can be pinpointed by combining deformation densities Δρ(σ) and Δρ(π), respectively, and their associated MOs (Fig. 7). For CNCu(NHC), the NOCV contributions to the deformation density clearly show that the σ-component corresponds to the donation from the lone electron pair of carbon as carbene center towards the sp-hydrid orbital of Cu(I) cation as illustrated in Fig. 7e (Top), corresponding to a strong orbital stabilization energy of − 20.5 kcal/mol. While the π-backdonation is due to the electron transfer from the metallic fragment into the empty π* orbital of NHC exhibiting carbon 2p character, it was revealed to be relatively weak with an orbital stabilization energy of − 8.7 kcal/mol. This π-backdonation is weakened by the presence of π*CN receiving a part of the π-backdonation as described in previous work for related complexes [8]. Accordingly, the orbital interactions are governed by the σ-bonding representing 60% as a contribution into the total orbital interaction. These results are nicely comparable to those obtained by Frenking [83] for the complexes (CO)5Cr = CHR (R = H, CH3, CH = CH2, Ph, C≡CH).

Fig. 7
figure 7

The contour of the deformation densities Δρ(σ) and Δρ(π) that correspond respectively to ΔE(σ) and ΔE(π) of the complexes CNCu(C2H2) (a), CNCu(C2H4) (b), CNCu(C4H6) (c), CNCu(C6H6) (d), and CNCu(NHC) (e) and their corresponding NOCV orbitals in their singlet state. Isosurface values are 0.0015 au for Δρ(σ) and Δρ(π). The eigenvalues ǀνnǀ give the size of the charge migration in electron. The direction of the charge flow of the deformation densities is green → yellow

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The interaction diagram for CNCu(C2H2) displayed in the left side (a) of Fig. S1(a) shows the involvement of the same orbitals of the CNCu fragment encountered in the case of the CNCu(NHC) compound, but with different contributions. Indeed, the C2H2 ligand acts as σ-donor through only one orbital of π-type, which is considerably depopulated from 2e to 1.67e in favor of the sp-hybrid orbital of CNCu unit receiving 0.38e. This NOCV pairwise contributes to the deformation density Δρ(σ) through an orbital stabilization energy ΔΕ(σ) of − 23.7 kcal/mol (the top of Fig. 7a). Nevertheless, the π-backdonation is assumed by one high-lying filled d-type orbital of CNCu fragment (20a) towards the empty 8a π* orbital reaching an occupation of 0.17e corresponding to one NOCV couple giving rise to an orbital stabilization energy of − 17.3 kcal/mol (the bottom of Fig. 7a).

The MOs diagram given in the right side (b) of Fig. S1(b) matches up to the CNCu(C2H4) one showing the interactions between the CNCu and C2H4 fragments, stressing clearly the involvement of one orbital of the C2H4 fragment namely 8a that undertakes significant occupation decreasing from 2e to 1.69e as illustrated in the left side of Fig. S1(b), through σ-donation towards the vacant sp-hybrid 22a orbital of the CNCu metallic fragment receiving 0.41e as illustrated in the (right side of Fig. S1(b)); thus, this NOCV pairwise contribute into the total ΔΕorb by − 21.1 kcal/mol as a stabilization energy ΔΕ(σ) (the top of Fig. 7b). Inversely, the low-lying π* antibonding 9a orbital of C2H4 is involved as an acceptor orbital interacting with the donor 20a orbital of CNCu. This NOCV couple is responsible of the π-backdonation via a stabilization energy of − 18.8 kcal/mol as shown in Fig. 7c (bottom). As can be seen for Table 1, these values show comparable molecular orbital populations for both C2H2 and C2H4.

Finally, for the CNCu(C6H6) compound, the interaction diagram (b) of Fig. S2(b) records the weakest σ-donation from the 21a orbital of the C6H6 fragment acting as σ-donor towards the 22a orbital of CNCu fragment which acquires an occupation of 0.43e. Also, the π-backdonation from the 20a orbital of the CNCu moiety into the 22a vacant π* orbital of C6H6 is the weakest one compared to those obtained for C2H2, C2H4, and C4H6 ligands, in accordance with energy of − 5.3 kcal/mol, which contributes weakly in the deformation density Δρ(π) as sketched in Fig. 7d (bottom).

However, the NHC ligand turned out to be strong σ-donor, but weak π-acceptor as given in Tables 1 and 2. For the NHC ligand, the NOCV contributions to the deformation density clearly (Fig. 7e) show that the σ-component corresponds to the donation from the lone electron pair of carbon as carbene center, enhanced further by an electron transfer from ancillary halogen atoms. The π-bonding is due to the backdonation from the metal into the empty orbital of X, mostly exhibiting chlorine 3p character in accordance with Cl-Cu-L’ bent angle.

It is habitually assumed that the σ bonding contribution for the d.10 metal ethylene and acetylene complexes prevails over the π-backdonation contribution as observed in previous work [101]

Consequently, the L’ → metal σ-donation is twice larger than the metal → L’ π-backdonation for L’ = C2H2, C2H, and C4H6, and it is five or six times larger when L’ = C6H6 and NHC.

Natural bond orbital analysis

The charge transfer contribution of an NBO pair into the total interaction energy is estimated by the second-order perturbative energy correction [102, 103]. The results collected in Table S3 give emphasis to LP → σ*, σ → LP*, LP → LP*, and σ → σ* donor–acceptor charge transfers occurred between different entities composing the XCuL’ and LCuL’+ (X = CH3, Cl, CN, L = NH3, L’ = C2H2, C2H4, C4H6, C4H6, and NHC) complexes (σ and σ * correspond to bonding and antibonding orbitals, respectively, and LP and LP* designate lone pair and vacant nonbonding NBOs, respectively). The occupancies of LP and σ designate their population after electron transfers towards the LP* and σ* empty NBOs, while the occupancies of LP* and σ* denote the fractional number of electron transferred from donor to acceptor.

Generally, we observe that LP → σ *, σ → σ*, σ → LP*, and LP → LP* donor–acceptor charge transfers vary from one complex to another and do not obey a specific rule. Indeed, it is noted that the maximum energies corresponding to LP → σ* donor–acceptor giving rise to stabilizing energies between the Cu(I) and the interacting ligands, namely L’, X, or L. The donor–acceptor charge transfers are numerous and important in relationship with the magnitudes of the second-order perturbative energies (E2); for this reason, the values under 10 kcal/mol are not considered. Firstly, the number, the types and the magnitude of the charge transfer of NBO pairs are comparable for CH3CuL’ (L’ = C2H2, C2H4, and C4H6), where the second-order perturbative energies (E2) are in the range 10.6–28.7 kcal/mol matching well with weak or moderate depopulation of one Cu lone pair (LP) from 2e to 1.83e towards C(1)-C(2) σ * NBO which acquires a population of 0.16e. On the other hand, the C(1)-C(2) σ NBO undergoes a depopulation from 2e to 1.84e transferring 0.12e to the vacant Cu (LP*) NBO. It is worth noticing comparable results for CH3Cu(C2H4) and CH3Cu(C4H6) as gathered in Table S3. However, the number of these types of charge transfers is higher in CH3Cu(C6H6), in accordance with the presence of six C–C delocalized bonds, but with comparable values of energies. Secondly, as clearly observed from Table S3, the second-order perturbative energies (E2) enhance in the presence of NH3 or NHC as illustrated by the charge transfers namely N(1) → N(1)-C(1) (70.8 kcal/mol) in NH3Cu(NHC), N → Cu (74.8 kcal/mol) and C(1)-C(2) → Cu (76.8 kcal/mol) in NH3Cu(C6H6), N → Cu (87.3 kcal/mol) and C(1)-C(2) → Cu (95.6 kcal/mol) in NH3Cu(C2H2), N → Cu (78.3 kcal/mol) and C(1)-C(2) → Cu (81.0 kcal/mol) in NH3Cu(C2H4), N → Cu (82.8 kcal/mol) and C(1)-C(2) → Cu (89.7 kcal/mol) in NH3Cu(C4H6), N(1) → C(2)-C(3) (99.0 kcal/mol) in ClCu(NHC), N(1) → C(2)-C(3) (67.9 kcal/mol) in CNCu(NHC), N(1) → N(1)-C(1) (66.7) in CH3Cu(NHC), and C(7) → Cu (215.7 kcal/mol) in CNCu(NHC). The second-order perturbative energies (E2) values show their dependence on the number of the nitrogen atoms involved in the interactions. The strength and the weakness of charge transfers are related to the gain or the loss of electron between the donor and the acceptor moieties. Consequently, based on the stabilization second-order perturbative energies (E2) are large for NHC and substantially decrease for those of alkenes and alkynes as L’ ligands in the presence of CH3 as X ligand and also increase in the presence of NH3 as L ligand and in the presence of CN and Cl as X ligands. The second-order perturbative energies (E2) and the charge densities Increase the charge transfers regardless the L’ ligand amounts the decreasing order of the charge transfers for different complexes is given as follows: Consequently, based on the second-order perturbative energies (E2) amounts, the decreasing order of the charge transfers for different complexes is given as follows: ClCu(C6H6) ≈ CNCu(C6H6) > [NH3Cu(C2H2)]+ ≈ [NH3Cu(C2H4)]+ ≈ [NH3Cu(C4H6)]+ ≈ [NH3Cu(C6H6)]+  > [NH3Cu(NHC)]+ ≈ CH3Cu(NHC) > ClCu(NHC) > CNCu(NHC) > ClCu(C2H2) ≈ ClCu(C2H4) ≈ ClCu(C4H6) > CH3Cu(C6H6) > CNCu(C2H2) ≈ CNCu(C2H4) ≈ CNCu(C4H6) > CH3Cu(C2H2) ≈ CH3Cu(C2H4) ≈ CH3Cu(C4H6). Therefore, the highest values correspond to the ClCu(C6H6), CNCu(C6H6), [NH3Cu(C2H2)]+, and [NH3Cu(C2H4)]+ complexes, while the weakest ones are for CH3Cu(C2H2), CH3Cu(C2H4), and CH3Cu(C4H6) complexes.

Conclusion

DFT calculations of the optimized structures and energy decomposition analysis of XCuL’ and (LCuL’)+ complexes were employed to provide a relationship between the bond lengths, the X-Cu-L’ and L-Cu-L’ bond angles, the Wiberg indices, Mayer bond orders, bonding energies, and the Cu-L’ bonding character. The Mayer bond order offers an appropriate and computationally competent tool of summing all the contributions to the bond, giving rise to a bond order around 1, except for the (C6H6) ligand, in agreement with the Cu-C bond distances. The energy decomposition analysis indicates that the interactions that occurred for various L’ ligands are more electrostatically than covalently bonded to the Cu(I) center, particularly in the presence of C6H6 and NHC ligands. The ΔEprep energies highlight weak or negligible deformation of the interacting fragments. Also, ΔECP corrected energies show a weak impact of the BSSE on the ΔEint energies. The ΔEelstst electrostatic contribution for all η2-coordination complexes is found slightly around 55 to 60% of the total attractive interaction energy ΔEelstst + ΔEorb is more important than that of the orbital one, while this contribution enhances to an average percentage of 75% for all η1-C(NHC) complexes. The Cu-C bonds of the XCuL’ and (LCuL’)+complexes are nearly two-thirds ionic and one-third covalent character, except for C6H6 and NHC ligands exhibiting more ionic character. The σ-bonding contributes more than 50% into the total orbital interaction overtaking those of π type resulting from σ and π electron transfers. The EDA method gives rise to a coherent scheme of the nature of the chemical bonding, while its extension to the EDA-NOCV offers a link between molecular orbital diagrams and pairwise orbital interactions contributing.

The σ-donation from the filled πC-C bonding orbital to the vacant sp(Cu) acceptor orbital is larger than back-donation from filled metal d-orbitals to the π*C–C or π*C-N.

It is worth noting that the enhancement of π-backbonding reinforces the L-M-L’ bending leading to the increased of steric repulsion, which is reduced in the presence of CO and CN− ligands. Besides, the Cu-NHC π-backbonding is enhanced in presence of chlorine substituent.

The more linear X-Cu-L’ and (L-Cu-L’)+ angles are obtained in the presence of NHC ligand independently on the L or X ligand. Consequently, the ionic character is more pronounced for the XCuL’ compounds than that described for the (LCuL’)+ ones. Based on the NPA, the Cu oxidation state is comprised between 0 and + I (average d10s0/+1 configuration), highlighting the strength of the σ- and π-type interactions, particularly in the NHC cases.