Dinitrogen complexes N₂L₂ (L = N₂, CO, CS, NO⁺, CN⁻)

Abstract

Quantum chemical calculations using ab initio methods and density functional theory have been carried out on the equilibrium structures and the vibrational spectra of the (valence) isoelectronic compounds N₂L₂ (L = N₂, CO, CS, NO⁺, CN⁻). The molecules have a trans-periplanar arrangement of the L₂ ligands at the N₂ unit. The complexes with L = N₂, CO, NO⁺, CN⁻ are predicted as thermodynamically unstable for dissociation into N₂ + 2L with ΔG²⁹⁸ value lying in between −257 kcal mol⁻¹ (L = NO⁺) and −73 kcal mol⁻¹ (L = CO), but the adduct N₂(CS)₂ is calculated as slightly stable with ΔG²⁹⁸ = 4 kcal mol⁻¹. The homolytic dissociation reaction into two fragments N₂L₂ → 2 NL is energetically less favorable than the heterolytic fragmentation N₂L₂ → N₂ + 2 L, which proceeds synchronously but asymmetrically. The activation barriers for the fragmentation reaction N₂L₂ → N₂ + 2L have values between ΔG^≠(298 K) = 17 kcal mol⁻¹ for L = N₂ and ΔG^≠(298 K) = 84 kcal mol⁻¹ for L = CS. The calculated vibrational frequencies suggest that the molecules N₂L₂ can be identified by the IR active antisymmetric stretching mode ν_as of the ligands L, which is blue shifted for L = CO (Δ = 55 cm⁻¹) and L = NO⁺ (Δ = 118 cm⁻¹) but it is red shifted for L = CS (Δ = −242 cm⁻¹) and L = CN⁻ (Δ = −133 cm⁻¹) relative to the ν_as mode of L = N₂. The analysis of the bonding situation reveals that there is a total charge donation L→(¹Γ-N₂)←L in all complexes, ranging between 1.38 e (L = CN⁻) and 0.56 e (L = N₂), except in the dication with L = NO⁺, where a small backdonation in reverse direction L←(¹Γ-N₂)→L with 0.10 e is calculated. EDA-NOCV calculations of N₆ show that the best description of the bonding situation is given in terms of dative interactions N₂→(¹Γ-N₂)←N₂ between central N₂ in the excited (1)¹Γ_g singlet state and the terminal N₂ fragments in the ¹Σ_g⁺ electronic ground state. In contrast, the best description of the complexes with L = CO, CS, NO⁺ is calculated for the interactions between the central N₂ in the ⁵Σ_u⁺ quintet state and the terminal ligands in the symmetry-adapted (L)₂ quintet state. For N₂L₂ with L = CN⁻, it is found that the bonding is best described for the interaction between N₂⁻ in the electronic quartet (⁴Σ_u⁺) state and the terminal (L)₂⁻ ligand as symmetry-adapted quartet. In contrast to the common bonding model for N₆ using Lewis structures N⁻=N⁺=N–N=N⁺=N⁻, the donor–acceptor model N₂→(N₂)←N₂ explains that the lowest activation barrier is found for the concerted cleavage of the two formal double bonds, leading to the experimentally observed dissociation into 3 N₂.

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Introduction

In 1964, Appel and Schöllhorn reported that triphenylphosphinazine, which was sketched with the formula Ph₃P=N–N=PPh_3, is a thermally stable diamagnetic species that has a melting point of 184°.¹ However, neither precise structural information about the compound was given in the work, nor in their further study.² Later, quantum chemical calculations showed that N₂(PPh₃)₂ is thermodynamically unstable for the release of N₂ by ∼90 kcal mol⁻¹,³ which raised doubts about the structure of the isolated compound. A subsequent X-ray structure analysis confirmed that the species is indeed triphenylphosphinazine, which has an antiperiplanar arrangement of the phosphine groups in Ph₃P-(N₂)-PPh₃ with bending angles P–N–N of 107^o, and a long N–N bond of 1.497 Å.⁴ The theoretical analysis of the bonding situation suggested that the N₂ moiety binds through its highly excited (1)¹Γ_g state, where the out-of-plane π and π* orbitals are doubly occupied and the in-plane π MO is vacant, which leads to strong dative interactions Ph₃P→(N₂)←PPh₃ with N₂ as a double Lewis acid. The same type of dative interactions was suggested for the related compound with NHC (N-Heterocyclic Carbene) ligand, NHC→(N₂)←NHC, which was reported to have an antiperiplanar arrangement of the NHC groups and a long N–N bond of 1.415 Å.⁵

Very recently, Qian, Mardyukov and Schreiner (QMS) reported the synthesis, via gas-phase reaction, of the new nitrogen allotrope N₆, which was trapped in low-temperature argon matrices at 10 K and as a film at liquid nitrogen temperature of 77 K.⁶ The molecule was identified by IR and UV-vis spectroscopy, and by ab initio calculations, which predict a structure where two N₃ fragments are bonded in a trans-arrangement through a long (1.460 Å) N–N bond. The viewpoint of N₆ as the dimer of N₃ is reasonable, because hexanitrogen was synthesized by treating AgN₃ with Cl₂, which yields ClN₃ that reacts with AgN₃ and leads to the formation of the new nitrogen allotrope N₆. Hexanitrogen was heralded as a molecule of the year 2025 wth a long N–N bond connecting two N₃ fragments.⁷ But the structural similarity to the N₂(PPh₃)₂ and N₂(NHC)₂ species led us suspect that the N₆ species is another example of the compound class N₂L₂ where the ligands L are bonded through dative interactions L→(¹Γ-N₂)←L. Dinitrogen N₂ is generally known as weakly bonded ligand, but the highly excited (1)¹Γ_g state of the central N₂ is a strong σ acceptor and strong π donor, which is capable to bind two N₂ ligands. The (1)¹Γ_g state of N₂ is 294.3 kcal mol⁻¹ above the X¹Σ_g⁺ ground state,⁸ which, however, does not occur in N₂L₂ as a free species, but as a reference state that is strongly stabilised by orbital interactions.

Another acceptor, which binds even eight N₂ ligands in the octa-coordinated complexes M(N₂)₈ (M = Ca, Sr, Ba), is the alkaline-earth atom M in an excited triplet state with (n−1)d² electron configuration via strong M→(N₂)₈ π backdonation.⁹

The new findings prompted us to investigate the electronic structure of N₆ in terms of dative bonding N₂→(¹Γ-N₂)←N₂ and to compare the homolytic and heterolytic bond dissociation with quantum chemical methods. We also calculated the (valence) isoelectronic compounds N₂L₂ (L = N₂, CO, CS, NO⁺, CN⁻). Here, we report about the equilibrium geometries, bond dissociation energies (BDEs), and the vibrational spectra of the molecules. We also present a thorough analysis of the nature of the chemical bonds using a variety of methods. The results may be useful as a guideline for future experimental studies.

Methods

The geometrical optimizations, followed by the frequency calculations of N₂L₂ (L = N₂, CO, CS, NO⁺, CN⁻) compounds, were first carried out at the M06-2X/cc-pVTZ level¹⁰ in their respective singlet and triplet spin states. Taking the most stable spin state (singlet), further reoptimizations and frequency calculations were performed at the CCSD(T)-Full/cc-pVTZ level.¹¹ These calculations were done using the Gaussian 16 program.¹² Natural atomic charges and Mayer bond order¹³ were calculated using NBO7 (ref. 14) and Multiwfn programs,¹⁵ respectively.

Energy decomposition analysis (EDA)¹⁶ in conjunction with the natural orbital for chemical valence theory (NOCV)¹⁷ was carried out at the M06-2X/TZ2P-ZORA¹⁸//CCSD(T)/cc-pVTZ level using the ADF 2020 package.¹⁹ The ZORA method considers relativistic effects, which are unimportant for this work, but further work by us on heavier analogues requires a uniform level of theory. In the EDA-NOCV²⁰ analysis, the intrinsic interaction energy (ΔE_int) between two fragments is dissected into three distinct energy components, as follows:

ΔE_int = ΔE_elstat + ΔE_Pauli + ΔE_orb (1)

The electrostatic ΔE_elstat term represents the quasiclassical electrostatic interaction between the unperturbed charge distributions of the prepared fragments. The Pauli repulsion, ΔE_Pauli accounts for the energy change during the transformation from the superposition of unperturbed electron densities of the individual fragments to a wavefunction that explicitly adheres to the Pauli principle, achieved through the necessary antisymmetrization and wavefunction renormalization. The orbital term ΔE_orb comes from the mixing of orbitals, charge transfer, and polarization between the isolated fragments.

The EDA-NOCV enables the partition of the total ΔE_orb into pairwise contributions of the orbital interactions that are very important to get a complete picture of the bonding. The charge deformation Δρ_k(r), resulting from the mixing of the orbital pairs ψ_k(r) and ψ_−k(r) of the interacting fragments presents the amount and the shape of the charge flow due to the orbital interactions (eqn (2)), and the associated energy term ΔE_orb provides with the size of stabilizing orbital energy originated from such interaction (eqn (3)).

Several papers extensively discussed details of the EDA-NOCV method and its application, offering different perspectives and viewpoints.²¹

Results and discussion

Fig. 1 shows the calculated geometries, atomic partial charges (q), and bond orders (P) of the compounds N₂L₂ (L = N₂, CO, CS, NO⁺, CN⁻) at the CCSD(T)/cc-pVTZ level. All molecules exhibit a trans-periplanar arrangement of the L₂ ligands at the N₂ unit with a C_2h point group and ¹A_g electronic state. The singlet-triplet gap computed at the M06-2X/cc-pVTZ level is very high (39.7–65.7 kcal mol; see Table S1 in SI). The central N–N bond lengths in N₂(N₂)₂ (1.450 Å) and [N₂(CN)₂]²⁻ (1.488 Å) compare favorably with the bond length in H₂N–NH₂ (1.445 Å),⁶ whereas the N–N bonds in N₂(CO)₂ (1.386 Å), N₂(CS)₂ (1.355 Å) and [N₂(NO)₂]²⁺ (1.382 Å) are significantly shorter. The small differences between our values for N₆ and the data reported by QMS are due to the fact that the CCSD(T)/cc-pVTZ calculations were carried out by us with the full core option, whereas the previous study used the frozen core approximation. Geometry optimisations of a syn isomer did not result in another conformational isomer, but rather in the trans form as the only acyclic energy minimum structure. An extensive search of the potential energy surface using various starting geometries gave planar four-membered cyclic structures N₂(CX)₂ (X = O, S) as energy minima, which are > 25 kcal mol⁻¹ less stable than the acyclic N₂L₂ forms. We also located nonplanar four-membered cyclic isomers for all species except for N₆ where the central N–N bond is broken, which are > 40 kcal mol⁻¹ higher in energy than the acyclic N₂L₂ forms. They are shown in Fig. S1 of the SI. Since the present work focuses on the dinitrogen complexes N₂L₂, we do not discuss the cyclic isomers in this work.

Fig. 1 Minimum energy geometries, partial charges (q) and bond orders (P) of the N₂L₂ (L = N₂, CO, CS, CN⁻, NO⁺) compounds and the ligands L at the CCSD(T)/cc-pVTZ level. The bond lengths are given in Å, bond angles in degree. All molecules have a C_2h point group and ¹A_g electronic state.

Table 1 gives the calculated BDEs for the fragmentation N₂L₂ → N₂ + 2L at the CCSD(T)/cc-pVTZ level, which show a remarkably high variation. The purely electronic values suggest that [N₂(NO)₂]²⁺ is even less stable (D_e = −233.3 kcal mol⁻¹) than N₂(N)₂ (D_e = −179.1 kcal mol⁻¹). The exoenergetic values are much lower for [N₂(CN)₂]²⁻ (D_e = −116.7 kcal mol⁻¹) and N₂(CO)₂ (D_e = −49.5 kcal mol⁻¹), and N₂(CS)₂ is even predicted to be energetically stable (D_e = 27.7 kcal mol⁻¹). The corrections by vibrational frequencies and entropic and thermal contributions lead to ΔG²⁹⁸ values where the N₂L₂ compounds are seen as thermodynamically unstable in the order L = NO⁺ > N₂ > CN⁻ > CO, but the molecule N₂(CS)₂ is calculated as thermodynamically stable at room temperature with ΔG²⁹⁸ = 3.6 kcal mol⁻¹.

Table 1 Computed bond dissociation energies D_e, zero-point energies corrected bond dissociation energies D₀, enthalpy change ΔH and free energy change ΔG of the processes (a) N₂L₂ → N₂ + 2L and (b) N₂L₂ → 2NL at the CCSD(T)/cc-pVTZ level. All values are given in kcal mol⁻¹

Complex	De	D0	ΔH	ΔG
(a) N2L2→ N2+ 2L
N2(N2)2	−179.1	−184.1	−182.0	−200.7
N2(CO)2	−49.5	−55.8	−53.7	−73.4
N2(CS)2	27.7	21.8	23.6	3.6
[N2(CN)2]^2−	−116.7	−122.2	−120.1	−139.7
[N2(NO)2]^2+	−233.3	−238.7	−236.5	−256.8
(b) N2L2→ 2NL
N2(N2)2	34.4	29.6	30.5	22.2
N2(CO)2	62.1	58.5	59.1	48.9
N2(CS)2	27.8	24.1	25.0	13.4
[N2(CN)2]^2−	−51.8	−56.8	−55.8	−60.8
[N2(NO)2]^2+	−70.2	−74.5	−73.7	−84.6

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Table 1 also gives the BDEs for breaking the central N–N bond in the fragmentation N₂L₂ → 2 NL at the CCSD(T)/cc-pVTZ level. It becomes obvious that the homolytic bond rupture of the N–N bond is thermodynamically strongly disfavoured compared with the heterolytic cleavage of the L–N₂–L bonds, except for L = CS. The energies of the two fragmentation reactions of the latter species are very similar, but the free energy of the heterolytic process (ΔG = 3.6 kcal mol⁻¹) makes it clearly more favourable than the homolytic fragmentation (ΔG = 13.4 kcal mol⁻¹). The calculated BDEs at the M06-2X/cc-pVTZ level are very similar (Table S2 of SI) to the CCSD(T)/cc-pVTZ values, which indicates that the DFT values are quite reliable.

Fig. 2 shows the reaction profile for the heterolytic dissociation reaction N₂L₂ → N₂ + 2 L along with the calculated activation free energy barriers ΔG^≠(298 K), which vary between 16.9 kcal mol⁻¹ (L = N₂) and 84.2 kcal mol⁻¹ (L = CS). The trend of the ΔG^≠ follows, in general, the reaction energies, except for the very exergonic reactions where L = N₂, NO⁺. The calculated barrier for the fragmentation of N₂ is in good agreement with the value reported by QMS (14.8 kcal mol⁻¹).⁶ The computed value of the T₁ diagnostics suggests that the multi-reference character of the CCSD(T) calculations of the N₂L₂ complexes and the transition states is very low (Table S3, SI), which indicates that the single-reference approach is sufficient. The activation barrier for the homolytic reaction N₆ → 2 N₃ was reported by QMS to be significantly higher (ΔG^≠(298 K) = 26.1 kcal mol⁻¹) than the heterolytic process.⁶ We carried out energy calculations with stretched N–N distances of the other L₂L₂ species, which suggest that the barriers are also higher than for the heterolytic fragmentation. Since the NL fragments are also less stable than N₂ + 2 L (Table 1), it is unlikely that the homolytic reaction course plays a role in the fragmentation reaction of the N₂L₂ species.

Fig. 2 Energy profile for the reaction N₂L₂ → N₂ + 2L at the CCSD(T)/cc-pVTZ//M06-2X/cc-pVTZ level.

The higher barrier for breaking the bond in the central N₂ moiety of N₆ compared to breaking the (N₂)–(N₂)₂ bonds is surprising, given the calculated bond orders, which are much higher and have shorter distances in the latter bonds than the former one. It turns out that the energy required to break a bond depends not only on the strength of the bond, but also on the reorganisation of the electronic structure of the fragments during the cleavage reaction. The N₃–N₃ bond in N₆ is weaker than the (N₂)–(N₂)₂ bonds, but the electronic charge migration during rupture of the latter heterolytic process is energetically more favourable than the homolytic reaction. The lower barrier for breaking the (N₂)–(N₂)₂ bonds than the N₃–N₃ bond supports the use of our bonding model for N₆ in terms of dative interactions N₂→(N₂)←N₂ rather than the more conventional model using Lewis structures N⁻=N⁺=N–N=N⁺=N⁻, since it accounts for the experimental finding that N₆ directly dissociated into 3 N₂. The same applies for N₂(CO)₂, which photolytically dissociates into N₂ + 2 CO whereas cleavage into NCO radical was observed only to a small extent.^22a

Examination of the transition state structures for the dissociation reaction reveals surprising features (Fig. 3). The geometries of [N₂L₂]^≠ possess a non-planar staggered geometry with a syn conformation of the ligands except for [N₂(CS)₂]^≠, which exhibits a nearly linear NNCS moiety with one CS ligand, where the bond to the second CS ligand is significantly stretched. This implies that the fragmentation reaction may possibly be a two-step process in which an intermediate product, NN-L is formed along the dissociation reaction. We calculated the intrinsic reaction coordinate starting from the transition states and found that it smoothly connects to the reaction products N₂ + 2L. The fragmentation reaction is thus predicted as a concerted but highly asynchronous process where the two N₂–L bonds break one after the other. Note that the central N–N bond of the transition state [N₂L₂]^≠ is clearly shorter than in the equilibrium structures.

Fig. 3 Geometries of the transition states of N₂L₂ (L = N₂, CO, CS, CN⁻, NO⁺) compounds for the dissociation into 2L and N₂ at the M06-2X/cc-pVTZ level. The top view gives the bond lengths and bond angles, and the side view gives the dihedral angles L-N₂-L. The bond lengths are given in Å and the angles in degree.

Hexanitrogen was clearly identified by spectroscopic signals, and the authors presented a careful analysis of the vibrational spectrum of the molecule.⁶ The IR spectrum of N₆ exhibits an intense vibrational band at 2076.6 cm⁻¹, which comes from an asymmetric stretching mode of the terminal N₂ ligands. Fig. 4 shows the calculated IR spectra of the five molecules N₂L₂. They show a similar pattern where the asymmetric stretching mode ν_as of the terminal L₂ ligands, which has the second largest wavenumber of all 12 fundamentals, exhibits the strongest signal. The symmetric stretching mode ν_s with a slightly higher wavenumber is IR inactive. The calculated wavenumber of 2263.7 cm⁻¹ for the asymmetric mode of N₆ is blue shifted for L = CO (Δ = 55 cm⁻¹) and L = NO⁺ (Δ = 118 cm⁻¹), but it is red shifted for L = CS (Δ = −242 cm⁻¹) and L = CN⁻ (Δ = −133 cm⁻¹). The calculated frequency shift of the intense asymmetric stretching mode ν_s is very helpful information for identifying the molecules.

Fig. 4 Calculated IR spectra of N₂L₂ complexes (L = N₂, CO, CS, CN⁻, NO⁺) at the M06-2X/cc-pVTZ level.

In addition to N₆ [N₂(N₂)₂], the complexes N₂(CO)₂ and N₂(CS)₂ have been synthesised and identified spectroscopically and they were also the subject of theoretical works. The dicarbonyl adduct was introduced as diisocynate O=C=N–N=C=O following the conventional descriptions using Lewis structures.²² Photolysis of the matrix-isolated N₂(CO)₂ gives N₂ + 2 CO as products, which agrees with our calculated reaction profile (Fig. 2). The IR spectrum shows one very intensive band at 2200.6 cm⁻¹ for the asymmetric NCO stretching mode besides several signals with lower intensity.^22a The comparison with the IR spectrum of N₆ gives a red shift of 124 cm⁻¹, which agrees with the direction but is higher than our computed value of 54 cm⁻¹. The diisothiocyanate complex N₂(CS)₂ was identified as product of the photolysis of the energetically lower lying isomer S₂(CN)₂ with a characteristic vibrational mode of the IR spectrum at 1910 cm⁻¹.²³ The experimental red-shift compared to the asymmetric stretching mode of N₆ is also given by our calculations, but the observed value of −166 cm⁻¹ is smaller than our computed value of −242 cm⁻¹. The computed wavenumber for [N₂(NO)₂]²⁺ (2382 cm⁻¹) agrees quite well with the experimental value of NO⁺ (2340 cm⁻¹) whereas the calculated value for [N₂(CN)₂]²⁻ (2131 cm⁻¹) is red-shifted compared with an approximately interaction-free anion CN⁻ (2244 cm⁻¹).²⁴ The complete set of the calculated vibrational spectra of the five molecules N₂L₂ is given in Table S4 of the SI.

We analyzed the bonding situation in N₂L₂ with a variety of methods. Fig. 1 shows that the Mayer bond order (MBO) of the central N–N bond is between 0.83 (L = CN⁻) and 0.96 (L = CS), which is a rather small variation considering the differences in the N–N bond length between 1.488 Å (L = CN⁻) and 1.355 Å (L = CS). The atomic partial charges suggest that there is a total charge donation in the neutral complexes L→(N₂)←L with 0.98 e (L = CO), 0.78 e (L = CS), and 0.56 e (L = N₂). There is an even stronger charge donation of 1.38 e in the dianion (L = CN⁻) and a small backdonation in reverse direction L←(N₂)→L with 0.10 e in the dication (L = NO⁺).

We proposed in our earlier study that the diatomic N₂ species in N₂(PPh₃)₂ binds through its highly excited (1)¹Γ_g state where the out-of-plane π and π* orbitals are doubly occupied.⁴ Fig. 5 shows schematically the valence MOs of N₂ in the (1)¹Γ_g state. The donation of the L₂ ligands L→(N₂)←L may take place into the vacant 1π_u′ (bonding) and 1π_g′ (antibonding) orbitals of N₂. Note that the assignment of π symmetry refers to free N₂, which has two mirror planes that contain the atoms. The complexes L-N₂-L have only one mirror plane and the donation is correctly assigned as in-plane σ(+,−) and σ(+,+) orbital interaction.

Fig. 5 (a) Schematic representation of the (1)¹Γ_g state of N₂ and the associated orbitals. (b) Donation of the plus and minus combination of the lone-pair donor orbitals of L into the vacant in-plane π and π* orbitals of N₂. The orbital numbering refers to the valence orbitals of N₂.

There are two occupied out-of-plane π MOs, 1π_u (bonding) and 1π_g (antibonding), in the (1)¹Γ_g state of N₂. If this holds true also for the N₂L₂ complexes in the present study, the number of occupied valence π MOs should be four (π and π* from N₂ and one π from each ligand L) and not three. Examination of the shape of the Kohn–Sham MOs shows that this is indeed the case. Fig. 6 displays the four occupied valence π MOs of N₆. The number of nodes follows the common symmetry rules with zero (HOMO-8), one (HOMO-7), two (HOMO-2), and three (HOMO). The same number of occupied valence π is found for the other N₂L₂ complexes. The complete set of occupied valence MOs of all N₂L₂ complexes is presented in Fig. S2–S6. Fig. 6 shows also the HOMO of the N₂L₂ complexes, which can be identified with the out-of-plane π* orbital of N₂ mixing with the antibonding π* orbitals of the ligands L.

Fig. 6 Top row: plot of the four occupied valence π MOs of N₆. Bottom row: plot of the HOMO of N₂L₂ with L = CO, CS, NO⁺, CN⁻.

A more detailed insight into the nature of the N₂–L₂ bonds is provided by the EDA-NOCV analysis of the interactions between the fragments. In our previous study of N₂(PPh₃)₂, we considered only neutral N₂ in the excited (1)¹Γ_g state and 2 PPh₃ ligands in the electronic ground state.⁴ We analyzed N₂L₂ using various electronic states and charges of the fragments N₂ and 2L. Numerous studies have shown that the strength of the orbital term ΔE_orb, which considers the change in the wave function during bond formation, is a reliable indicator of the optimal fragments for describing the bond between them.^7,25 The fragments with the lowest absolute values of ΔE_orb are the best choice for explaining the bonding interactions. Table 2 shows the numerical results for L = N₂.

Table 2 EDA-NOCV results of the complex N₂L₂ (L = N₂) considering L₂ as one fragment and central N₂ as another fragment at the M06-2X/TZ2P-ZORA//CCSD(T)/cc-pVTZ level. Energy values are given in kcal mol⁻¹

Energies	Orbital interaction	N2 (singlet) + 2L (singlet)	N2 (triplet) + 2L (triplet)	N2 (quintet) + 2L (quintet)	N2^−(doublet) + 2L^+(doublet)	N2^−(quartet) + 2L^+(quartet)
a The values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb.b The values in parentheses give the percentage contribution to the total orbital interactions ΔEorb.
ΔEint		−165.0	−368.7	−401.3	−387.3	−519.0
ΔEPauli		1036.0	1022.8	1028.3	1042.8	1090.1
ΔEelstat^a		−364.9 (30.4%)	−388.8 (27.9%)	−438.1 (30.6%)	−538.1 (37.6%)	−615.1 (38.6%)
ΔEorb^a		−836.1 (69.6%)	−1002.7 (72.1%)	−991.5 (69.4%)	−894.5 (62.4%)	−980.0 (61.4%)
ΔEorb(1)^b	L–NN–L σ-bond (+,−)	−328.7 (39.3%)	−317.3 (31.6%)	−328.7 (33.2%)	−329.8 (36.9%)	−350.4 (35.8%)
ΔEorb(2)^b	L–NN–L σ-bond (+,+)	−296.4 (35.5%)	−296.9 (29.6%)	−314.1 (31.7%)	−261.4 (29.2%)	−337.7 (34.5%)
ΔEorb(3)^b	L–NN–L π-bond (+,−)	−98.6 (11.8%)	−174.9 (17.4%)	−151.4 (15.3%)	−123.1 (13.8%)	−90.9 (9.3%)
ΔEorb(4)^b	L–NN–L π-bond (+,+)	−44.7 (5.3%)	−86.5 (8.6%)	−94.8 (9.6%)	−54.0 (6.0%)	−56.2 (5.7%)
ΔEorb(rest)^b		−67.7 (8.1%)	−127.1 (12.7%)	−102.2 (10.3%)	−126.2 (14.1%)	−144.8 (14.8%)

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It becomes obvious that the chemical bonds in N₆ between the central N₂ moiety and the terminal N₂ species are indeed best described in terms of dative interactions N₂→(N₂)←N₂ between central N₂ in the excited (1)¹Γ_g singlet state and two terminal N₂ fragments in the ¹Σ_g⁺ electronic ground state. The interactions between central N₂ in the ³Σ_u⁺ triplet state and the terminal ligands in the symmetry-adapted (N₂)₂ triplet state, which exhibit a mixture of electron-sharing and dative bonds, give a bigger ΔE_orb value. The same holds for the formation of electron-sharing double bonds between central N₂ in the ⁵Σ_u⁺ quintet state and the terminal ligands in the symmetry-adapted (N₂)₂ quintet state. EDA-NOCV calculations using central N₂⁻ as an anion in the electronic doublet (²Σ_g⁺) or quartet (⁴Σ_u⁺) state and the terminal (N₂)₂⁺ ligand as a cation in the symmetry-adapted doublet or quartet state also result in bigger ΔE_orb values.

Further examination of the dative interactions between central N₂ and the terminal N₂ ligands reveals that they come mainly from N₂→(N₂) N₂ σ donation through out-of-phase (ΔE_orb(1)) and in-phase (ΔE_orb(2)) orbital pairs, which provide 75% of the total orbital (covalent) bonding. The π backdonation N₂←(N₂)→N₂ via out-of-phase (ΔE_orb(3)) and in-phase (ΔE_orb(4)) orbital interactions deliver only 17% of ΔE_orb. The nature of the individual orbital interactions ΔE_orb(1)–ΔE_orb(4) becomes clear when considering the corresponding deformation densities and the associated orbitals, which are shown in Fig. 7. The dominant orbital interactions through (+,−) and (+,+) σ donation N₂→(N₂)←N₂ nicely explain the bond shortening of the central N₂ moiety compared to free N₂ in the ¹Γ state, which has a calculated bond length of 1.608 Å and is 294.3 kcal mol⁻¹ above the X¹Σ_g⁺ ground state.⁸ The donation takes place into the vacant in-plane and out-of-plane π orbitals of N₂, which are bonding orbitals (see Fig. 5). Note that the (1)¹Γ_g state of N₂ is a reference state which is strongly stabilized by the orbital interaction. It is not formed as a free species during the reaction process.

Fig. 7 Plot of the deformation densities, Δρ_(1–4) shown as the electronic charge corresponding to ΔE_orb(1–4) and the related interacting orbitals of N₂L₂ (L = N₂) at the M06-2X/TZ2P-ZORA//CCSD(T)/cc-pVTZ level using NN (¹Γ_g) + L₂ (singlet) as interacting fragments. The eigenvalues ν indicate the size of the charge flow. The direction of charge flow is red → blue. The isovalue for Δρ_(1–4) is 0.003 au.

The numerical EDA-NOCV results of the other N₂L₂ compounds are given in Tables S5–S8 of SI. In contrast to the results for N₆, the lowest ΔE_orb values with L = CO, CS, NO⁺ are calculated for the interactions between the central N₂ in the ⁵Σ_u⁺ quintet state and the terminal ligands in the symmetry-adapted (L)₂ quintet state, which indicate electron-sharing double bonds between the fragments. The central N–N bond in the latter species is much shorter (between 1.355–1.386 Å) than in N₆ (1.450 Å). The lowest ΔE_orb value of N₂L₂ with L = CN⁻ is found for the interactions between N₂⁻ in the electronic quartet (⁴Σ_u⁺) state and the terminal (CN)₂⁻ ligands as symmetry-adapted quartets. The individual orbital interactions are identified by the corresponding deformation densities and the associated orbitals, which are shown in Fig. S7–S14. The latter electron-sharing interactions also lead to four occupied π valence MOs in all five N₂L₂ molecules.

In order to compare the N₂L₂ compounds with each other, we have chosen the same fragments with the central N₂ in the excited (1)¹Γ_g singlet state and two terminal N₂ fragments in the ¹Σ_g⁺ electronic ground state as model for the bonding interactions. Table 3 gives the numerical results. The calculated interaction energies ΔE_int show the same order CS > CO > CN⁻ > NO⁺ > N₂ as the calculated activation barriers ΔE^≠ (Fig. 2), which indicates that the choice of the singlet fragments is a valid model for the trend of the chemical bonds. The orbital (covalent) interactions always make the largest percentage contribution to the chemical bonds, which is particularly high for L = NO⁺ (75%). The breakdown of ΔE_orb into the pairwise orbital interactions reveals that the σ donations through out-of-phase (ΔE_orb(1)) and in-phase (ΔE_orb(2)) orbital pairs are always the largest components of the total orbital bonding. The percentage contribution of ΔE_orb(1) and ΔE_orb(2) is higher for L = CN⁻ and lower for L = NO⁺, which is due to the charges of the ligands. A surprising result concerns the low contribution of the π backdonation ΔE_orb(3) and ΔE_orb(4) for L = CO, which is smaller than for L = N₂. CO is known to be a better π acceptor than N₂ in transition metal complexes.²⁶ The peculiar results for the orbital interactions in the neutral N₂L₂ compounds can be explained with the dominance of the σ donation L→N₂←L over π backdonation L←N₂→L, which comes to the fore by the calculated partial charges (Fig. 1). The energy of the σ lone-pair HOMO suggests that the donor strength of the ligands has the order CS (ε = −10.1 eV) > CO (ε = −12.2 eV) > N₂ (ε = −13.9 eV). But the overall strength of the dative interactions cannot simply be derived from the orbital interactions. The orbital interaction ΔE_orb of the negatively charged CN⁻ is significantly weaker (−972.1 kcal mol⁻¹) than that of NO⁺ (−1088.0 kcal mol⁻¹), but the total interaction energy ΔE_int of the former ligand is clearly higher (−301.4 kcal mol⁻¹) than the latter (−202.5 kcal mol⁻¹). The electrostatic interaction ΔE_elstat but also the Pauli repulsion ΔE_Pauli, which makes the largest contribution to ΔE_int, are equally relevant for the trend of the interatomic interactions. It has been shown that the Pauli repulsion is the crucial factor for the equilibrium geometry of molecules.²⁷

Table 3 EDA-NOCV results of complexes N₂-L₂ (L = N₂, CO, CS, NO⁺, CN⁻) considering L₂ as one fragment and central N₂ as another fragment in the singlet states (¹Γ_g state of N₂ and symmetry adapted singlet state of L₂) at the M06-2X/TZ2P-ZORA//CCSD(T)/cc-pVTZ level. Energy values are given in kcal mol⁻¹

Energies	Orbital interaction	L = N2	L = CO	L = CS	L = NO^+	L = CN^−
a The values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb.b The values in parentheses give the percentage contribution to the total orbital interactions ΔEorb.
ΔEint		−165.0	−306.0	−387.0	−202.5	−301.4
ΔEPauli		1036.0	1400.4	1542.2	1242.3	1155.6
ΔEelstat^a		−364.9 (30.4%)	−530.8 (31.1%)	−595.4 (30.9%)	−356.9 (24.7%)	−484.9 (33.3%)
ΔEorb^a		−836.1 (69.6%)	−1175.7 (68.9%)	−1333.8 (69.1%)	−1088.0 (75.3%)	−972.1 (66.7%)
ΔEorb(1)^b	L–NN–L in-plane(σ)-bond (+,−)	−328.7 (39.3%)	−483.0 (41.1%)	−536.4 (40.2%)	−373.2 (34.3%)	−405.8 (41.7%)
ΔEorb(2)^b	L–NN–L in-plane(σ)-bond (+,+)	−296.4 (35.5%)	−434.3 (37.4%)	−493.4 (37.0%)	−322.3 (29.6%)	−400.0 (41.1%)
ΔEorb(3)^b	L–NN–L out-of-plane(π)-bond (+,−)	−98.6 (11.8%)	−76.1 (6.5%)	−134.6 (10.1%)	−182.1 (16.7%)	−70.4 (7.2%)
ΔEorb(4)^b	L–NN–L out-of-plane(π)-bond (+,+)	−44.7 (5.3%)	−45.2 (3.8%)	−48.0 (3.6%)	−75.5 (6.9%)	−34.6 (3.6%)
ΔEorb(rest)^b		−67.7 (8.1%)	−137.1 (11.7%)	−121.4 (9.1%)	−134.9 (12.4%)	−61.3 (6.3%)

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We also want to comment on the ΔE_orb(rest) term, which is comparatively large in the systems with L = CO, CS, NO⁺. It stems from the relaxation of the fragment orbitals with respect to the isolated species. The central N₂ moieties in the latter complexes have significantly shorter N–N distances than in the ¹Γ_g singlet state (1.606 Å at the M06-2X/cc-pVTZ level), and the electronic relaxation is, therefore, larger than in the complexes with L = N₂, CN⁻ which have longer central N–N bonds (Fig. 1). The impact of the geometry relaxation on the BDE and the stability of molecules has been pointed out by Bickelhaupt in his activation strain model.²⁸ The strong influence of the fragment relaxation and the geometry of the interacting species becomes obvious when the trend of the interaction energy ΔE_int between the frozen fragments (L = CS > CO > CN⁻ > NO⁺ > N₂) is compared with the BDEs that are calculated using the fragments at their equilibrium geometries and electronic ground state (L = CS > CO > CN⁻ > N₂ > NO⁺). Finally, we want to mention that the chemistry of N₂(PPh₃)₂ was recently studied in joint experimental and theoretical works by the groups of Stephan and Grimme, which showed a surprising reactivity of the member of the N₂L₂ compound class.²⁹

Summary and conclusion

Geometry optimizations of the compounds N₂L₂ (L = N₂, CO, CS, NO⁺, CN⁻) at the CCSD(T)/cc-pVTZ level give the energy minimum structures with a trans-periplanar arrangement of the L₂ ligands at the N₂ unit. Attempts to localize a syn isomer failed, and the calculations gave the trans isomer as the only conformational minimum. Energy calculations suggest that all compounds N₂L₂ (L = N₂, CO, CS, NO⁺, CN⁻) may be synthesized and should be observable under appropriate conditions. The complexes with L = N₂, CO, NO⁺, CN⁻ are predicted as thermodynamically unstable for dissociation into N₂ + 2 L with ΔG²⁹⁸ value lying in between −257 kcal mol⁻¹ (L = NO⁺) and −73 kcal mol⁻¹ (L = CO), but the adduct N₂(CS)₂ is calculated as slightly stable with ΔG²⁹⁸ = 4 kcal mol⁻¹. The homolytic dissociation reaction into two fragments N₂L₂ → 2 NL is energetically less favorable than the heterolytic fragmentation N₂L₂ → N₂ + 2 L, which proceeds synchronously but asymmetrically. The activation barriers for the fragmentation reaction N₂L₂ → N₂ + 2 L have values between ΔG^≠(298 K) = 17 kcal mol⁻¹ for L = N₂ and ΔG^≠(298 K) = 84 kcal mol⁻¹ for L = CS. The calculated vibrational frequencies show that all molecules N₂L₂ have a characteristic vibrational mode with the second highest wavenumber ν_as that comes from the antisymmetric stretch of the ligands L, which is blue shifted for L = CO (Δ = 55 cm⁻¹) and L = NO⁺ (Δ = 118 cm⁻¹) but it is red shifted for L = CS (Δ = −54 cm⁻¹) and L = CN⁻ (Δ = −133 cm⁻¹) relative to the ν_as mode of L = N₂.

The analysis of the bonding situation using the charge distribution reveals that there is a total charge donation L→(N₂)←L in all complexes ranging between 1.38 e (L = CN⁻) and 0.56 e (L = N₂), except in the dication with L = NO⁺, where a small backdonation in reverse direction L→(N₂)←L with 0.10 e is calculated. EDA-NOCV calculations of N₆ using the central N₂ moiety and the terminal N₂ ligands as interacting fragments in various electronic states and with different partial charges show that the best description of the bonding situation is given in terms of dative interactions N₂→(N₂)←N₂ between central N₂ in the excited (1)¹Γ_g singlet state and the terminal N₂ fragments in the ¹Σ_g⁺ electronic ground state. In contrast to the results for N₆, the best description of the complexes with L = CO, CS, NO⁺ is calculated for the interactions between the central N₂ in the ⁵Σ_u⁺ quintet state and the terminal ligands in the symmetry-adapted (L)₂ quintet state, which indicates electron-sharing double bonds between the fragments. For N₂L₂ with L = CN⁻, it is found that the bonding is best described for the interaction between N₂⁻ in the electronic quartet (⁴Σ_u⁺) state and the terminal (L)₂⁻ ligand as symmetry-adapted quartet. A comparative analysis of the five N₂L₂ compounds using the same fragments with central N₂ in the excited (1)¹Γ_g singlet state and the terminal L₂ fragments in the ¹Σ_g⁺ electronic ground state reveals that the σ donation L→(N₂)←L makes the largest contribution to the stabilizing interactions and that the π backdonation L←(N₂)←L is much weaker. In contrast to the common bonding model for N₆ using Lewis structures N⁻=N⁺=N–N=N⁺=N⁻, the donor–acceptor model N₂→(N₂)←N₂ explains that the lowest activation barrier is found for the concerted cleavage of the two formal double bonds, leading to the experimentally observed dissociation into 3 N₂.

Author contributions

G. F. and S. P. conceived the project, wrote the draft and finalized it, Y. L., C. D. and L. X. performed the calculations, Y. L. analyzed the data. All authors took part in the discussions and approved the final version.

Conflicts of interest

The authors declare no conflict of interest.

Data availability

The data supporting this article have been included as part of the supplementary information (SI). Supplementary information is available. See DOI: https://doi.org/10.1039/d5sc08399k.

Acknowledgements

GF acknowledges financial support by Nanjing Tech University. SP is grateful for financial assistance by Jilin University.

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