Introduction

Since the discovery of the Haber-Bosch process in 1909, considerable efforts are focused on the basic understanding of nitrogen fixation. This operation is achieved by chemisorption of N2 on a metal catalyst’s surface, where the unsaturated valence shell of the metal establishes charge transfer interactions with the relatively inert N2 molecule. An important step in this research was the discovery of the first ruthenium dinitrogen complexes in the 1960s and since then strenuous efforts were made on the synthesis of transition metal dinitrogen complexes [1, 2]. Particular attention was paid on coordinatively unsaturated complexes involved in the catalytic reduction of N2 [3,4,5,6,7].

In general, coordinatively unsaturated metal dinitrogen complexes of metals have low stability and thus can be synthesised, isolated and studied only under extreme conditions. The most efficient tool in this research proved to be the matrix-isolation technique, combined initially with vibrational and electronic spectroscopy. Examples on transition metal complexes include the matrix-isolation IR study of FeN2 [8], Ni(N2)x (x = 1–4) [9,10,11,12,13,14], Pd(N2)x (x = 1–3) [10] and Pt(N2)x (x = 1–3) [11]. Support for the proposed structures and bonding properties were provided by, over the years increasingly sophisticated, quantum chemical calculations, e.g. FeN2 [15], Ni(N2)x (x = 1–4) [12, 14, 16,17,18] and WNx (x = 1–9) [19]. Related studies with rare earth elements include the co-deposition of the laser-ablated metals and N2 gas in cryogenic matrices with a posteriori investigations by matrix-isolation IR spectroscopy [20,21,22,23]. Relevant compounds include various organometallic lanthanide dinitrogen complexes with fully characterised examples for the entire series of lanthanides [24,25,26,27].

Dinitrogen complexes of metals can exhibit different binding geometries of N2: end-on terminal, end-on bridging between two metals, side-on terminal, side-on bridging and side-on end-on bridging. They were theoretically investigated by DFT calculations on transition metal complexes [28].

In contrast to the extended theoretical studies on transition metal complexes [12, 14,15,16,17,18,19, 28], for the rare earth dinitrogen Ln(N2)x species, only three theoretical reports were found in the literature.

Very recently, Zhao et al. studied the whole lanthanide series of Ln(N2) molecules using the mPW3PBE hybrid exchange-correlation functional in conjunction with quasi-relativistic 4f-in-valence pseudopotentials for the lanthanides and 6-31G* basis set for nitrogen [29]. The covered molecular properties were the geometries, HOMO-LUMO energy gaps, and magnetic and electronic properties. These calculations predicted the end-on 4La(N2) structure to be favoured by 47 kJ/mol over 2La(N2) with a side-on structure. From the bonding properties, the π-type HOMO orbital and the atomic charges were presented.

Teng and Xu performed DFT calculations using the BP86 and B3LYP functionals in order to assist the interpretation of the matrix-isolation IR spectra of the reaction products formed by laser-ablated La co-deposited with N2 [23]. In the spectra, several species were identified and their electronic structures, dissociation energies and vibrational data were reported. On the basis of isotope experiments and computed data, the authors concluded on the dominant presence of end-on 4La(N2) in the argon matrix.

The only theoretical study on complexes of a lanthanide atom with more than one N2 ligands is an early SCF/3-21G computational study of holmium complexes by Ermilov et al. [30] using a 4f-in-valence pseudopotential [31] for Ho. Beyond the low-quality computational level, this study has also other limitations: only quartet spin multiplicity and complexes up to six N2 ligands were considered. However, the small energy difference between the quartet and sextet Ho [32] may readily be overcompensated by N2 bonding, particularly in the larger complexes. Therefore, the sextet Ho(N2)x species should not be neglected. (Note that Zhao et al., in contrast, studied only the sextet HoN2 [29].) Most striking is, however, the instability of HoN2 at the SCF/3-21G level: in contrast to chemical intuition, the computations predicted dissociation to HoN + N [30].

The goal of our ongoing research is the elucidation of structural, energetic and spectroscopic properties of Ln(N2)x complexes by state of the art quantum chemical calculations. In the present work, the results on lanthanum complexes are reported, which metal is the prototype of the rare earth elements.

The neutral lanthanum atom has an open-shell [Xe]4f05d16s2 electronic structure with three valence electrons. The ground electronic state is a doublet (2D3/2) [32], in which the single 5d electron (in contrast to the filled 6s2 subshell) is suited for electron-sharing interactions. The first quartet state (4F3/2) lies higher in energy by 2668 cm−1 and has an electron configuration of [Xe]4f05d26s1 [32]. Here, the two singly occupied 5d orbitals can facilitate two orbital overlapping interactions in contrast to the single interaction by the doublet ground state.

Description of the results will start with the analysis of the end-on complexes, because literature data on complexes of transition and rare earth metals with N2 show generally the preference of this structure [12, 14,15,16,17,18, 23,24,25,26,27,28]. Subsequently, the related side-on isomers will be discussed and compared with the end-on isomers.

Computational details

The DFT computations have been performed with the Gaussian09 suit of programs [33] using the B3LYP [34, 35] exchange-correlation functional in conjunction with the quasi-relativistic small-core 4f-in-valence pseudopotential [36]. The valence basis set treating the 4s4p4d4f5s5p5d6s orbitals had the contraction scheme of [14s13p10d8f6g]/[10s8p5d4f3g] [37]. For nitrogen, the correlation-consistent cc-pVTZ basis set [38] was applied. In order to account for dispersion interaction in the model (particularly the large) structures, the B3LYP functional has been extended with the D3 version of Grimme’s dispersion correction using the original D3 damping function (D3) [39].

The computed model structures are neutral. Doublet and quartet spin multiplicities have been probed according to the number of expected unpaired electrons. In most cases, these different electronic structures resulted in different spatial structures. The minimum character of the obtained stationary points on the potential energy surface has been confirmed in all cases by frequency analysis. The unpaired electrons in the model structures required spin unrestricted calculations. The spin contamination was checked in all cases and for the quartets, always the theoretical value of 3.75 was found. For the doublets, mostly the theoretical value of 0.75 was obtained, in a few cases deviating up to 0.78.

The atomic charges and orbital populations were assessed using the natural bond orbital (NBO) model [40] by means of the NBO 6.0 code [41, 42].

In order to assess the reliability of the electronic structures predicted by B3LYP, single-point relativistic multireference calculations were performed on selected La(N2)x species on their B3LYP optimised geometries. The properties in focus included the character of the main electron configuration (whether it agrees with the electronic properties from the B3LYP calculations), its contribution (%) in the multi-determinant wavefunction and the relative energies of the doublet and quartet states.

The complete active space self-consistent field (CASSCF) method [43] has been used to generate molecular orbitals and reference functions for subsequent multiconfigurational second-order perturbation theory calculations of the dynamic correlation energy (CASPT2) [44, 45] with frozen 1 s for N, and up to 3d for La.

In the CASSCF calculations, the scalar relativistic effects were taken into account using the second-order Douglas-Kroll-Hess Hamiltonian [46, 47]. The all electron basis sets of atomic natural orbital type, developed for relativistic calculations (ANO-RCC) with the Douglas-Kroll-Hess Hamiltonian [46, 47], had contraction schemes of 24s21p15d5f3g2h.8s7p5d2f1g [48] and 14s9p4d3f2g.4s3p2d1f [49] for La and N, respectively, corresponding to triple-zeta valence plus polarisation (TZP) quality. In the treatment of two-electron integrals, the Cholesky decomposition method [50] was applied.

The active space has been constructed on the basis of state-averaged test calculations using 5 roots. Based on the observed occupations [51] and cost considerations in case of the larger model structures, the following active spaces (electron/orbital) were applied in the calculations of the various structures: 7/10 for La(N2) while 11/12 for La(N2)2 and La(N2)6. For these CASPT2 calculations, the MOLCAS 8.2 code [52, 53] was applied.

In general, the electronic structures from the B3LYP calculations were confirmed by CASPT2. The dominant electron configurations in the CAS were > 85% and > 70% for the quartets and doublets, respectively. The minor contributions in the multi-determinant wavefunction appeared only by a few percent, suggesting that the title molecular systems can be reasonably modelled by DFT. This refers particularly to the structure and bonding interactions. On the other hand, the mostly very small energy differences between the doublet and quartet forms should be treated with caution.

Results and discussion

Geometry and bonding of end-on structures

A compilation of the end-on structures is shown in Fig. 1 while the bond distances and selected bond angles are given in Table 1.

Fig. 1
figure 1

Structures of end-on La(N2)x complexes optimised by B3LYP calculations

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Table 1 Relative energies and selected bond distances of the end-on type La(N2)x complexes from B3LYP calculations
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In case of the small La(N2)x complexes (x = 1–4), the doublet and quartet states have very similar structures with somewhat different bond distances and angles. As the complexes are formed from neutral La atom and N2 molecules, the arrangements of the N2 ligands around La are governed by orbital interactions. These are donor-acceptor interactions facilitated by the low-lying empty valence atomic orbitals (primarily 5d) of La and anti-bonding orbitals of N2. Therefore, both N2 → La (ligand-to-metal donation) and La → N2 (metal-to-ligand back donation) charge transfers might be possible in these complexes. The charge transfer processes are governed by the spatial arrangements of the interacting donor and acceptor orbitals. They differ in the various complexes due to the different number of ligands and different spatial structures. In addition, they depend also on the (doublet or quartet) spin state of La, as it defines the population of the valence La orbitals.

Both the doublet and quartet La(N2) complexes have a linear structure. The La-N2 bond distances of 2.4 and 2.3 Å, respectively, are somewhat smaller than the sum of single-bond covalent radii of La (1.80 Å) and N (0.71 Å) [54]. The NN bond distances of around 1.13 Å in the two complexes are between the NN triple and double-bond lengths of 1.08 Å and 1.20 Å, respectively, derived from the known triple- and double-bond radii of N (0.54 and 0.60 Å, respectively [55]). Hence, the geometrical parameters suggest a near single-bond character of the La…N2 interaction, while a considerably weakened triple NN bond. It is noteworthy that the present 4La-N2 bond distance deviates by 0.04 Å from that obtained with the mPW3PBE functional by Zhao et al. [29]. The deviation from the BP86 and B3LYP results of Teng and Xu [23] is of similar magnitude. The shorter B3LYP 4La-N2 bond distance from the latter study is probably due to their smaller basis set.

Regarding the relative stabilities of doublet and quartet La(N2), the present B3LYP (similarly to previous DFT [23]) calculations predicted the quartet state to be more stable than the doublet one by ca. 6 kJ/mol, while the CASPT2 computations favoured the doublet state by a similar magnitude (cf. Table 1). This small energy difference, however, is within the uncertainties of the applied computational levels. Therefore, they alone are not suitable to draw unambiguous conclusion on the ground state of La(N2). The known doublet ground state of the La atom [32] is not determinative in this case either, because orbital interactions can readily stabilise low-lying excited states.

The favoured stability of quartet La(N2) as predicted by DFT is supported by matrix-isolation IR data [23]. In the spectrum of the reaction products of laser-ablated La and N2 gas (deposited in Ar matrix), an absorption band at 1749.8 cm−1 has been reported and assigned to NN stretching of end-on 4La(N2). Note that the B3LYP functional tends to overestimate this vibrational frequency (see ref. [23] and the present 1889-cm−1 value in the Supplementary Information) while BP86 provided a better performance [23]. In another related studies, the absorption bands in this range of the matrix-isolation IR spectra seem to be erroneously attributed to side-on Ln(N2) species [20,21,22]. The more extensive donor-acceptor interactions in the η2-coordinated side-on Ln(N2) result in a larger elongation of the NN bond (vide supra); therefore, lower NN vibrational frequencies are expected.

The donor-acceptor interactions in end-on La(N2)x are analysed on the basis of characteristic molecular orbitals and natural atomic charges shown in Figs. 2 and 3 and Table 2, respectively. Table 2 includes also the populations of the La valence orbitals.

Fig. 2
figure 2

Characteristic bonding orbitals of end-on La(N2)x (x = 1, 2) complexes

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Fig. 3
figure 3

Selected bonding orbitals of end-on La(N2)x (x = 3–8) complexes

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Table 2 Selected NBO atomic charges and population of La valence orbitals in the end-on La(N2)x complexes from B3LYP calculations
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The La-N2 bond is established jointly by donor-acceptor and electrostatic interactions. In the view that the complexes are formed from neutral components, the former interaction is likely the major one. Analysis of the molecular orbitals revealed a π-type bonding interaction between the 5dπ orbitals of La and the 2p orbitals of the adjacent N atoms (cf. Fig. 2). The overlap of these π-type orbitals is most efficient in a linear La-NN arrangement, which explains the linear structure. The shapes of the orbitals over the N2 moiety correspond to anti-bonding orbitals. Hence, the found bonding molecular orbitals describe La → N2 charge transfer. Molecular orbitals corresponding to N2 → La charge transfer could not be found either among the B3LYP or the CASSCF orbitals. Accordingly, a ligand-to-metal donation does not seem to be characteristic for the end-on La(N2) complexes. The B3LYP and multireference CASPT2 calculations agree in the above described bonding scenarios, supporting the reliability of B3LYP molecular properties. Note that similar bonding situation was reported in the Ni(N2)2 complex with 3dδ donor orbitals of Ni [14]. The different spatial forms of the La and Ni donor orbitals can be responsible for some structural differences found in their larger complexes (vide infra).

The main difference in the bonding between the doublet and quartet La(N2) is that in the doublet species, there is only one 5dπ-2p orbital overlap formed by the single unpaired 5dδ electron of La, while in the quartet, two orthogonal 5dπ-2p orbital overlaps appear, formed by both unpaired 5dδ electrons of La. The remaining valence electrons of the two species form non-bonding (lone pair and lone electron, respectively) molecular orbitals.

The consequence of the La → N2 charge transfer is the partial positive charge of La and a negative one of the N2 ligand (cf. Table 2). This facilitates an electrostatic attraction contribution to the bonding.

The different number of one-electron bonding orbitals in the two electronic states results in characteristic differences in the molecular parameters. The atomic charge of La is larger in quartet La(N2) due to the transferred larger charge to the N2 anti-bonding orbital (cf. Table 2). The geometrical consequence of the latter interaction is the NN bond distance lengthened more in the quartet than in the doublet species (Table 1) with respect to the NN bond of the free N2 molecule (1.091 Å, calculated at the applied B3LYP level). Accordingly, the stronger quartet La-N2 bond is shorter by 0.1 Å than that in doublet La(N2).

Similar charge transfer interactions appear in La(N2)2, where both the doublet and quartet La atoms establish two orthogonal orbital overlaps with the N2 moieties (Fig. 2). For that, one of the two 6s electrons of the doublet La state needs to be promoted (with preserved spin) to a 5dδ orbital; thus, the electron configuration of La in La(N2)2 is 5d26s1. Accordingly, the calculations indicate two orthogonal one-electron three-centre donor-acceptor bonds and a lone non-bonding electron on La. Due to the identical bonding scenario, the bond distances of the doublet and quartet La(N2)2 are nearly the same and slightly longer than in quartet La(N2).

The non-bonding La 6s electrons present in La(N2) and La(N2)2 are not retained in the larger complexes. The larger number of ligands (x = 3–8) require additional orbitals getting involved in the bonding; therefore, the electron configuration of La in these complexes is 5d36s0. In all of these complexes, three (in most cases) delocalised one-electron π-type bonding orbitals were found. Examples are shown in Fig. 3.

While the linear structures of La(N2) and La(N2)2 agree with those of the Ni analogues [12, 14, 17, 18], the presently obtained structures of La(N2)3 and La(N2)4 differ from the planar D3h and tetrahedral Td structures of Ni(N2)3 and Ni(N2)4, respectively [18]. The pyramidal La(N2)3 structure of both spin multiplicities (Fig. 1) is stabilised by more-or-less balanced delocalised bonding orbitals. This balanced delocalisation results in close La-N2 bond distances in both the doublet and quartet La(N2)3 (cf. Table 1).

In the C2v La(N2)4 structures, the considerable 0.1-Å difference between the axial and equatorial La-N2 bond distances (Table 1) can be explained by the characteristically different orbital interactions. The shorter axial bonds are formed by two orthogonal, over the vertical N-La-N moiety delocalised bonding orbitals, while the equatorial N2 ligands are connected to La by a single delocalised bonding orbital only (cf. Fig. 3).

The bonding in the tetragonal pyramid (C4v) 4La(N2)5 has some resemblance to that in La(N2)4: the shorter axial La-N2 bond is formed by two orthogonal three-centre orbitals, while the longer equatorial bonds are established by a single five-centre bonding orbital (cf. Fig. 3). The consequence is again a 0.1-Å difference between the axial and equatorial La-N2 bond distances. In contrast, the La-N2 bond distances of 2La(N2)5 vary in a small range (cf. Table 1) as a result of considerably balanced delocalised bonding orbitals. The different bonding scenario results in a quite high energy difference (20 kJ/mol, cf. Table 1) between the characteristically different structures of the doublet and quartet La(N2)5 species.

For the quartet La(N2)6, the B3LYP computations predicted a highly symmetric octahedral (Oh) structure. The bonding is established by three equivalent (over LaN4 moieties delocalised) orthogonal molecular orbitals; an example is shown in Fig. 3. The doublet La(N2)6 is slightly distorted from Oh to D4h symmetry, but the bonding scenario is analogous.

Interestingly, a stable La(N2)7 structure was found for the doublet state only; it has Cs symmetry (Fig. 1). Quartet La(N2)7 seems to be unstable: the geometry optimisations starting from several reasonable initial structures converged to an La(N2)6•N2 adduct in which one N2 moiety is too far from La for any bonding interaction.

Highly symmetric structures were obtained for the doublet (D4h) and quartet (D4d) La(N2)8. The equivalent La-N2 bond distances imply highly delocalised symmetric bonding orbitals in both structures. The one, established by the 5dσ La orbital positioned in the C4 rotational axis, is delocalised over all the eight La-N2 bonds (cf. Fig. 3). This orbital is common in both the doublet and quartet states. The other two bonding orbitals, however, are characteristically different in the two states. In 2La(N2)8, both orbitals are delocalised over all the eight La-N2 bonds, while in 4La(N2)8, the delocalisation occurs only either in the upper or in the bottom part of the molecule, hence over four La-N2 bonds only (cf. Fig. 3).

The formation of larger La(N2)x (x > 8) complexes is unlikely due to the steric crowding around La. Indeed, probing La(N2)9, the ninth N2 ligand escaped from the first coordination sphere of La during geometry optimisation.

The above introduced La(N2)x structures can be compared with the theoretical structures of Ho(N2)1–6 complexes obtained in simple SCF/3-21G calculations by Ermilov et al. [30]. The unreasonable HoN…N result was already noted in the “Introduction” section (vide supra). Other significant differences appear in the structure of quartet Ho(N2)3 and La(N2)3: the former was predicted to be planar (D3h) in ref. [30] while pyramidal (C3v) by the present calculations. Similarly, the C4v Ho(N2)4 structure is in disagreement with the strongly bent La(N2)4 possessing C2v symmetry. On the other hand, the D4h Ho(N2)6 is quite close to the Oh La(N2)6 structure. On the basis of the expected marginal effect of the partly filled Ho 4f subshell on the bonding interactions, the described differences are likely due to the different theoretical levels and/or to the smaller size of the Ho atom.

An overview of the data in Tables 1 and 2 does reveal some characteristic trends in the molecular parameters of the La(N2)x species:

  1. 1.

    The La-N2 bond distances are increased gradually with the increase of x (cf. Table 1). The difference between the largest and smallest La-N2 bonds is 0.35 Å. This large change can be understood by both the distribution of the La bonding electron densities among more N2 ligands and the increasing steric interactions with increasing x.

  2. 2.

    The NN bond distance shows an opposite (decreasing) trend with increasing x, though up to only 0.03 Å due to the rigidity of the triple bond. The shortening of the NN bonds correlates well with increasing NN stretching frequencies (cf. Supplementary Information).

  3. 3.

    The correlation of the two bond distances reflects the geometrical consequences of the charge transfer interactions in La(N2)x: a stronger La-N2 bond means larger charge transfer to the N2 anti-bonding orbital, weakening in this way the NN bond. Altogether, the computed bond distances support a considerably weakened NN triple bond in the small complexes, while only a slightly weakened one in the larger (x ≥ 6) species.

In Table 2, some interesting features can be observed in the atomic charge data. First of all, the atomic charge of La is positive in all the complexes. This is the consequence of the La → N2 charge transfer decreasing the electron density around La. The variation of the La charge shows a convex curve with a maximum around x = 3. In La(N2)8, the charge drops nearly to the value of La(N2). In contrast, the magnitude of the negative charge of the attached N decreases gradually from x = 1 to 8. The charge of N′ is considerably smaller and has values between − 0.08 and + 0.04 e. These data refer to notable ionic interactions between La and N2 in the smaller complexes which, however, weaken in the larger species.

The shown orbital (charge transfer) interactions are established by the valence electrons of La. The La valence orbital populations in the two spin states of La(N2) resemble the different populations in doublet and quartet La atom, respectively. In contrast, in the larger complexes, the La populations in the doublet and quartet La(N2)x pairs are quite similar. Characteristic is the above-mentioned redistribution of the 6s electron density to the 5d orbitals with increasing size of the complexes. From La(N2)3 on the population of the 6s orbital is between 0.1 and 0.2 e, while that of the 5d orbital is around 2 e. This is undoubtedly due to the spatial suitability of the 5d orbitals for simultaneous overlaps with the orbitals of more than one N2 ligands. The 4f orbitals are marginally populated—except for the quartet La(N2)8, where the spatial arrangement of the N2 ligands may require a pronounced hybridisation.

Energetics of end-on structures

Table 3 shows the energies of formation (in terms of ΔH0, ΔH298 and ΔG298) corresponding to the reactions La (N2)x−1 + N2 = La(N2)x. They were evaluated from the computed absolute energies corrected for unscaled zero-point vibration (ΔH0), as well as from enthalpies and Gibbs free energies at 298.15 K and 1 atm (ΔH298 and ΔG298, respectively) obtained on the basis of the ideal gas model. The latter data do not include thermal corrections for the electronic energy. Because of the change of species number in the reaction, the molar work term Δ(pV) = ΔnRT (Δn = − 1) was also considered.

Table 3 Energies (kJ/mol) of the reactions La(N2)x−1 + N2 = La(N2)x from B3LYP calculations
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At 0 K, all reactions are exothermic indicating the favoured formation of the complexes up to La(N2)8. Note that even the largest La(N2)8 complex has a substantial stability with respect to the smaller complexes. The stability trend is approximately linear up to x = 6. The trend is broken at La(N2)7, which complex has a considerably smaller formation energy as another indication of the less favoured sevenfold coordination of N2 ligands around La. This can be due both to the unfavourable directionality of the valence La orbitals for the required orbital interactions and to the increased steric repulsion with respect to the smaller complexes. The latter steric interactions may be responsible for the somewhat smaller formation energy of La(N2)8 too.

The thermal contributions at 298 K increase somewhat the magnitudes of the reaction energies (see ΔH298 values in Table 3). In contrast, the entropy effect introduced in the ΔG data decreases them considerably, particularly those of the larger complexes. Upon this latter effect, the formation of La(N2)7 becomes endotherm.

Properties of side-on structures

In the side-on La(N2)x complexes, the La atom interacts with both N atoms of the N2 ligands. The optimised structures of these complexes are depicted in Fig. 4 while selected molecular data are given in Table 4.

Fig. 4
figure 4

Structures of side-on La(N2)x complexes optimised by B3LYP calculations

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Table 4 Relative energies and selected bond distances of the side-on La(N2)x complexes from B3LYP calculations
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According to the present B3LYP calculations (similarly to the experience on related Ni complexes [14, 18]), the side-on La (N2)x structures are generally less favourable than the end-on ones. Exception is the doublet La(N2)3 being at 0 K by 23 kJ/mol lower in energy than the most stable quartet end-on La(N2)3. The thermal and entropy effects (298 K) can change somewhat (up to 13 kJ/mol) the relative energies. In terms of Gibbs free energies, also the side-on doublet La(N2)2 becomes more stable than its end-on quartet isomer. The relative stability of the side-on isomers with respect to the end-on ones decreases with the size due to the increasing steric repulsion between the ligands. The steric interactions limit the size of such structures to x = 6. Geometry optimisations from various initial structures of La(N2)7 resulted in the escape of one N2 from the first coordination sphere of La.

The steric interactions introduce considerable limitations also in the relative arrangement of the N2 ligands, particularly in the larger complexes. As a result of these effects, the side-on La(N2)x structures are in general less symmetric than the end-on ones. High symmetry was found up to x = 3 and for 4La(N2)6 (cf. Table 4).

Both the doublet and quartet side-on La(N2) have C2v symmetry due to the in-plane π orbital interaction between the La 5dπ and N2 π* anti-bonding orbitals (Fig. 5). This is a two-electron bond in the doublet La(N2), while a one-electron bond in the quartet. The difference is characteristically demonstrated by the considerably (by 0.13 Å) shorter La-N2 and by the somewhat longer (0.04 Å) NN bond distances of the doublet. The other donor-acceptor interaction corresponding to N2 → La charge transfer could not be found either among the B3LYP or the CASSCF orbitals.

Fig. 5
figure 5

Selected bonding orbitals of side-on La(N2)x complexes

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Though the CASPT2 and B3LYP calculations agree in the main characteristics of the electronic structure and bonding scenario in the two states, they predict opposite stability trends as found for the end-on structures too (vide supra): at the CASPT2 level, the doublet is more stable by 8 kJ/mol, while B3LYP favours the quartet species by 2 kJ/mol (cf. Table 4). The CASPT2 calculations resulted in 89% contribution of the main configuration, which means a very small multiconfigurational character of doublet La(N2). Hence, the different trend may not be attributed solely to the deficiency of the B3LYP level for static correlation energy, but rather to the superimposed differences between the two theoretical levels.

It should be noted that the side-on 2La(N2) structure obtained with the mPW3PBE functional by Zhao et al. has an La-N2 bond distance shorter by 0.1 Å with respect to the present B3LYP value and is less stable than their end-on 4La(N2) ground state by 47 kJ/mol [29]. Similar deviation in the geometry is observed by comparing the present B3LYP 4La(N2) structure with the BP86 one of Teng and Xu [23], while the relative stabilities are comparable.

In the matrix-isolation IR study by Willson and Andrews [21], IR bands in the range of 1550–1580 cm−1 were attributed to end-on Ce(N2), Gd(N2) and Sm(N2) species. However, they would better fit to the side-on Ln(N2) isomers for which lower vibrational frequencies are expected (see Supplementary Information). This is the consequence of the η2-Ln-N2 bond, lengthening the NN triple bond with respect to the η1-coordinated end-on isomers (cf. Tables 1 and 4).

In contrast to La(N2), the orbital interactions seem to be somewhat stronger in the side-on quartet La(N2)2 than in the doublet one, as can be seen in the shorter La-N2 distances (by 0.03 Å) and longer (by 0.01 Å) NN bonds with respect to those of the doublet ones. The difference can be explained by the different symmetries of the two structures. In the planar C2v doublet, each La 5dπ electron interacts only with one N2 (Fig. 5) establishing two one-electron bonds. In the quartet similarly, but the two one-electron bonds are orthogonal due to the D2d symmetry. The third La valence electron remains non-bonding in both species.

In spite of the above noted shorter La-N2 distances, the quartet La(N2)2 is less stable than the doublet one (cf. Table 4). Moreover, the B3LYP calculations underestimate the relative stability of these side-on species with respect to the end-on isomers as compared to CASPT2 theory. The latter calculations predicted relative energies of − 52.9 and − 18.9 kJ/mol for doublet and quartet La(N2)2, respectively, with respect to the most stable end-on form.

Comparing the side-on La(N2)2 structures with those of transition metal complexes, the D2d structure of the quartet La(N2)2 agrees with that of side-on Ni(N2)2 obtained by DFT calculations by Guan et al. [18]. Apparently, the structure depends on the number and character of valence d electrons: The 5d2 subshell of quartet La has two unpaired electrons (ready for orbital interactions) similarly to the partly filled 3d8 subshell of Ni. In contrast, the 5d1 subshell of doublet La can provide only one electron for the bonding. Consequently, the computed structures of quartet side-on La(N2)3 and La(N2)4 agree too with those of the respective Ni(N2)x (x = 3–4) complexes [18].

The optimised equilibrium structure of quartet La(N2)3 has one N2 with perpendicular orientation to the other two N2-s. This is requested by the orthogonality requirement of the bonding 5d orbitals belonging to the three alpha spin electrons in quartet La. The requirement for full orthogonality is released in doublet La(N2)3, where the La valence 5d electrons consist of two with alpha and one with beta spin. The consequence is a different structure: here, all the three N2 ligands are arranged parallel with respect to each other. The cylinder-type C2v structure is formed by one three-centred and two five-centred orbital interactions (cf. Fig. 5). This structure has a considerable stability (vide supra and Table 4).

The larger side-on La(N2)x (x = 4–6) complexes, due to the less favourable asymmetric bonding scenarios and steric interactions, have ground states considerably higher in energy than the respective end-on isomers (cf. Table 4). The B3LYP and CASPT2 calculations agree in around 100 kJ/mol less stabilities of the side-on La(N2)6 species.

The larger doublet structures can be derived from the cylinder-type structure of doublet La(N2)3 by adding an N2 to each of the terminal positions (x = 4 and 5, respectively), while in La(N2)6 the cylinder is formed by four N2 moieties (cf. Fig. 4). Similarly, the larger quartet structures can be derived from quartet La(N2)3: the characteristic feature of these structures is that they contain parallel-positioned N2 pairs (two in x = 4 and 5, while three in x = 6) with N-La-N angles of ca. 95° between the N2 moieties of a pair. The N2 pairs take orthogonal relative positions with respect to the other pairs. Such a ligand arrangement is quite unusual and reflects well the orthogonality of the three bonding 5d orbitals of quartet La.

The computed η2 La-N2 bond distances agree in magnitude with the sum of single-bond covalent radii of La (1.80 Å) and N (0.71 Å) [54] and are slightly larger than the La-N2 bonds in the respective end-on isomers (cf. Tables 1 and 4). The computed NN bond distances agree with a double-bond character in the small side-on complexes (x = 1–3), while with a somewhat weakened triple bond in the large ones (x = 4–6).

Table 5 lists the natural atomic charges in the side-on La(N2)x complexes. Because the charge transfer interactions are more efficient with two nitrogens than with a single nitrogen in the end-on complexes, the La → N2 charge transfers to the N2 anti-bonding orbitals are more pronounced in the side-on isomers. Consequently, these complexes are more ionic than the end-on ones. The positive charge of La is by ca. 60% larger than in the respective end-on isomers. The variation with x shows again a convex curve with the maximum at x = 3.

Table 5 Selected NBO atomic charges and populations of La valence orbitals in the side-on La(N2)x complexes from B3LYP calculations
Full size table

The La valence orbital populations indicate considerable non-bonding 6s populations in La(N2), which is decreased in the larger complexes. Parallel, the 5d populations increase somewhat in the larger complexes, though they remain below the end-on ones mostly by ca. 0.4 e. On the other hand, the 4f orbitals are more populated (up to 0.2 e) in the side-on complexes than in the end-on ones. This implies a slight hybridisation between the La 5d and 4f orbitals in the side-on structures.

Conclusions

In the presented work, the structures and bonding properties of end-on and side-on La(N2)x (x = 1–8 and 1–6, respectively) complexes were investigated by B3LYP calculations. For the x = 1, 2 and 6 species, single-point relativistic multireference CASPT2 calculations were carried out on the optimised B3LYP geometries. These calculations confirmed the electronic structures obtained by the B3LYP exchange-correlation functional. The dominant (mostly > 80%) electron configurations indicate a reasonable reliability of the DFT structures and bonding data. The mostly very small energy differences between the doublet and quartet isomers, however, should be treated with care.

From the two coordination modes, in general, the structures with end-on coordination were found to be more stable. The preference is particularly pronounced in the larger complexes due to the considerable steric repulsion in these side-on structures. The first coordination sphere is filled by eight and six N2 ligands in the end-on and side-on type species, respectively. The computed bond distances correspond to weak single La-N2 bonds, to double or considerably weakened NN triple bonds in the small complexes, while to weakened triple bonds in the larger (x ≥ 6) ones.

The main bonding interaction is the donation of La 5d valence electrons to anti-bonding orbitals of N2. These directional interactions determine the equilibrium molecular structures which are in several cases less logic from steric point of view. The La → N2 charge transfers resulted in partial charges up to + 1.5 e of La and − 0.5 e of N. These generate notable electrostatic attraction between La and the N2 ligands, which bonding interaction contributes to the stability of the complexes.