Electronic Structure and Chemical Bonding in the OTiâN2 Complexes

Article pubs.acs.org/JPCA

Electronic Structure and Chemical Bonding in the OTi−N2 Complexes: A Systematic ab Initio and DFT Study Asma Marzouk,†,‡ Bruno Madebène,†,‡ and M. Esmaïl Alikhani*,†,‡ †

UPMC Univ. Paris 06, UMR 7075, Laboratoire de Dynamique, Interactions et Réactivité (LADIR), F-75005 Paris, France CNRS, UMR 7075, Laboratoire de Dynamique, Interactions et Réactivité (LADIR), F-75005 Paris, France

‡

ABSTRACT: Two OTi−N2 complexes, experimentally observed in the TiO + N2 reaction, have been theoretically studied using several density functionals as well as ab initio approaches and various basis sets. The benchmark results calculated with coupled-cluster singles, doubles, and perturbative triples CCSD(T) and sufficiently large correlationconsistent basis set were used to assess the performance of other theoretical models, especially four density functional families, pure functional, hybrid, double-hybrid, and long-range corrected ones. It has been shown that, out of twenty-three density functionals used in this work, only three functionals, namely TPSS0, LC-TPSS, and B2PLYP, are able to reproduce the CC-reference data quantitatively. Particularly, the B2PLYP double-hybrid (with or without addition of empirical dispersion) is the most promising functional, providing the closest results to the reference ones. The nature of bonding within products has been investigated using two topological techniques and a localized orbital approach.

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INTRODUCTION Activation of the NN triple bond (with its high stability, about 226 kcal/mol) is one of the most challenging subjects in chemistry. Especially, coordination of dinitrogen to transition metal centers is often proposed as the initial step of the chemical activation of dinitrogen.1 Dinitrogen binding to the transition metal center actually weakens the NN bond which could lead to its activation.2,3 Recent experimental investigations provide evidence that early transition metal monoxide (ScO4 and TiO5) reacts with N2 to form two different coordination modes, the side-on and end-on bonded complexes, depending on the experimental conditions.6 The same products, along with others, also result from Ti + N2O reactions. In the case of the titanium works, the conclusions point out the existence of both end-on, 3[OTi(η1-NN)], and side-on, 1 [OTi(η2-NN)], complexes as well as spectacular matrix effects. In ref 6, it was found that both forms coexist in argon matrices, but the end-on, 3[OTi(η1-NN)], structure was calculated to be energetically the most stable structure and identified to the triplet state in DFT-based calculations, thus predicted to be the ground state. A large difference between the predicted NN stretching frequency using their methods and that observed experimentally led the authors to probe the addition of Ar atoms in the coordinating sphere near the metal center and calculations with two Ar atoms attached to the side on complex showed best agreement with the experimental NN frequency. The side-on, 1[OTi(η2-NN)], structure was also observed in solid argon but not in a further study in solid neon, the least perturbing matrix medium.5 It could, at this point, thus be logically concluded that the 1[OTi(η2-NN)]structure is a © XXXX American Chemical Society

metastable state, stabilized only in the more polarizable, solid argon medium. In a recent work in our laboratory,7 for the purpose of studying the reactivity of Ti and Ti2 molecules with N2O, both end-on, 3[OTi(η1-NN)], and side-on, 1[OTi(η2-NN)], complexes were observed in solid neon, in disagreement with ref 5. This disagreement could be related to differences in experimental conditions during the sample preparation, either with laser ablation5,6 or thermal deposition,7 inducing differences in photoexcitation during the initial reaction. Also, photoconversion between the two forms and higher product yields added new information, suggesting that 1[OTi(η2-NN)], in which the N2 unit is very significantly perturbed, is, in fact, the ground state. These studies put several interesting questions to theory. In preliminary calculations, we showed that the energy ordering between the two states is very much basis set-dependent: the use of a basis set larger than 6-311+G(d) could switch the energy ordering.7 Also, the nature of the reaction pathway and conversion process toward the most chemically interesting, 1 [OTi(η2-NN)], complex was not explored. Therefore, in order to better elucidate the characteristics of these systems, we reinvestigated the electronic structure at ab initio level in order to establish reference data and at DFT levels to find an appropriate functional able to limit CPU-time and reproduce the ab initio results. Received: January 30, 2013 Revised: April 26, 2013

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dx.doi.org/10.1021/jp4010778 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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We describe, in the first part, the electronic structure of both OTi-N2 isomers at the CCSD(T) level. A systematic DFT study will be reported in the second part, using several functionals and in comparison to the high-level calculation with the goal of exploring possible reaction pathways. Finally, the third part is devoted to the study of chemical bonding from both orbital and topological points of view.

Figure 1. Two studied geometries of 3[OTi(η1-NN)] (3A″) and 1 [OTi(η2-NN)] (1A′) states.

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COMPUTATIONAL DETAILS All calculations have been performed using the Gaussian 09 package.8 Ab initio investigations have been carried out with the second order Moller−Plesset (MP2) approach and at the CCSD and CCSD(T) levels of theory. DFT calculations have been done using pure, hybrid, double hybrid, and long-range corrected functionals. The pure functionals employed, from GGA and meta-GGA families are BLYP,9,10 PBEPBE,11 BB95,12 M06L,13 TPSSTPSS,14 and B97D.15 The hybrid functionals employed are B3LYP,16 PBE1PBE,17 B1B95,12,16 M06,18 TPSSh,19 mPW1LYP,20 wB97X,21 wB97,21 and wB97XD.22 Furthermore, in order to analyze the effect of the exact exchange on the behavior of density functional, we systematically varied the percentage of the exact exchange (HF) in TPSSTPSS by the keyword IOp(3/76 = 0xx000yy00). In this route command, xx + yy =100 and yy denotes the HF contribution. The double hybrid functionals included a secondorder perturbation correction for nonlocal correlation effects: mPW2PLYP23 and B2PLYP24 and with their dispersion variants: mPW2PLYP-D and B2PLYP-D.25 Finally, the longrange corrected functionals used are LC-BLYP, LC-wPBE and LC-TPSSTPSS as implemented in the Gaussian 09 program. The triple-ζ quality extended basis of Pople 6-311+G(2d)26 labeled as Pop(2d) and augmented correlation-consistent basis sets aug-cc-pVXZ (X = D, T) of Dunning et al.27,28 were used. Hereafter, Dunning’s basis sets are labeled as AVXZ (X = D, T). The nature of chemical bonding has been investigated using the ″Natural Bonding Orbital = NBO″29 analysis and two topological approaches [the Bader’s atoms in molecule (AIM)30 theory and the Silvi-Savin’s electron localization function (ELF)].31 The AIM investigations have been carried out using EXT94b part of the AIMPAC suite of programs32 and “AIMAll (Version 12.09.23)”33 on the wfn files obtained at the MP2/Pop(2d) level with the density = current option. The ELF calculations have been done using TopMod software.34 The ELF basins figures have been obtained using cube files with GaussView program.35 NBO analysis36 has been performed using NBO3.1 program as implemented in the Gaussian 09 package at the SCF function obtained at MP2/Pop(2d) level of theory.

significantly red-shifted (631 cm−1). Despite this difference in frequency and geometry, previous computational results5 showed that the two species are almost isoenergetic giving 3 [OTi(η1-NN)] very slightly more stable than 1[OTi(η2-NN)] (ΔE(1A′/3A″) = E(1[OTi(η2-NN)]) − E(3[OTi(η1-NN)]) = 0.5 and 2.0 kcal/mol at the mPW2PLYP/6-311+G(d) and B3LYP/6-311+G(d) levels, respectively). It is appropriate to emphasize that in our previous work7 we investigated the monoreference character of the electronic structure of 1[OTi(η2-NN)]. Using the CASSCF(4,10) method, we showed that the wave function of the side-on structure at its singlet state is almost of monoreference character in which the contribution of SCF reference accounts for 86% and that of double excited configurations for 13% in the CI wave function. It is therefore clear that all monoreference methods are suitable to the study of this compound provided that the dynamic correlation is covered. Accordingly, all the methods used in this paper and especially CCSD(T) are well adapted to this investigation. I. Ab Initio Energetic and Geometrical Study. In order to provide a reliable reference for density functionals calculations, we performed a systematic study with two series of ab initio methods, namely second order perturbation (MP2) and coupled-cluster (CCSD and CCSD(T)) approaches. In Table 1 are gathered the binding energies (De), inter- and intramolecular distances d(Ti−N), and r2(N−N) for both multiplicities, 1A′ and 3A″. The binding energy, De, is defined as follows: De = Ecomplex − [E(OTi 3Δ) + E(N2 1Σg+)]. Table 1. Binding Energies (De) and Some Calculated Structural Parameters at Various ab Initio Approachesba AVDZ 112 bf

c

Pop(2d) 146 bfc

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RESULTS AND DISCUSSION A. Energetic and Structural Considerations. The performance of some modern density functionals for the prediction of spectroscopic data is investigated for the two studied species, already experimentally characterized by Zhou5,6 and in our group,7 namely end-on 3[OTi(η1-NN)] and side-on 1 [OTi(η2-NN)], as displayed in Figure 1. The observed frequencies in neon matrix, at 2249.8 cm−1,5,7 and 1699.4 cm−1,7 have been assigned to the NN stretching frequencies of end-on and side-on species, respectively.7 It is worth noting that compared to the free N2 molecule, the first NN stretch is only slightly red-shifted (83 cm−1), while the second one is

AVTZ 231 bfc

1

De( A′) De(3A″) 3 A″ d r2 1 A′ d r2 De(1A′) De(3A″) 3 A″ d r2 1 A′ d r2 De(1A′) De(3A″) 3 A″ d r2 1 A′ d r2

MP2

CCSD

CCSD(T)

−5.8 −2.8 2.016 1.247 2.537 1.125 −5.3 −2.3 2.012 1.233 2.465 1.111 −9.5 −3.2 2.007 1.232 2.437 1.11

+7 −2 2.057 1.197 2.489 1.112 +8.7 −1.4 2.051 1.183 2.456 1.096 +5.5 −2.2 2.046 1.182 2.424 1.095

−2.6 −3 2.084 1.209 2.424 1.119 −0.9 −2.5 2.072 1.195 2.391 1.103 -4.7 -3.2 2.066 1.185 2.349 1.103

a

Energies are in kcal/mol and distances in Å. bBond-angles and Ti−O distance (see Figure 1) optimized at the CCSD(T)/AVTZ level are α = 93°, β = 173.2°, and r1= 1.641 Å for 3A″ state, and α = 93° and r1 = 1.650 Å for 1A′. cBasis functions. B



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The most significant trends could be summarized as follows: (1) At the CCSD level, 1[OTi(η2-NN)] is found to be unbound (De > 0) while 3[OTi(η1-NN)] is stable by around 2 kcal/mol with respect to OTi(3Δ) and N2(1Σg+) fragments, whatever the basis set. On the other hand, addition of noniterative triple excitations at the coupled-cluster level, CCSD(T), lowers the binding energy for all three basis sets. In particular, we observe that De(1A′) is lowered by nearly 9 kcal/mol, which results in a bound singlet state (De(1A′) < 0). Accordingly, a reliable calculation requires the inclusion of triple excitations at the coupled-cluster level. (2) As shown in Table 1, one can easily point out that our energetic results are clearly basis set dependent. Nevertheless, it is worth noting that the 1[OTi(η2-NN)] is calculated to be slightly more stable than the 3[OTi(η1-NN)] one by 1.5 kcal/ mol only when we use the CCSD(T) approach with an enough large basis set, such as AVTZ. The less extended basis sets, such as AVDZ and Pop(2d), do not give the correct energetic ordering between the two forms, even with CCSD(T) approach. Energetic results calculated at both MP2/Pop(2d) and MP2/AVDZ level agree well with the CCSD(T)/AVTZ ones, while the use of a larger basis set, AVTZ, with MP2 does not improve energetic results (the 1 [OTi(η 2 -NN)] is particularly overbound). It is, of course, well-known that only the CCSD(T) level with a sufficiently large basis set is able to account for subtle balance between different forces that are responsible for structural equilibrium. Hereafter we consider a particular functional as adequate to study both 3[OTi(η1-NN)] and 1[OTi(η2-NN)] structures only if it is able to reproduce two energetic criteria, i.e., De(1A′) ≈ −4.7 kcal/mol and ΔE(1A′/3A″) ≈ −1.5 kcal/mol. (3) Concerning the geometrical parameters obtained at the CCSD(T)/AVTZ level, when we compare the two states (3[OTi(η1-NN)] and 1[OTi(η2-NN)]) to free subunits (TiO and N2), we observe very small geometrical changes for 3 [OTi(η1-NN)], while the structural changes are far from being negligible in the case of 1[OTi(η2-NN)]. For instance, the calculated Ti−O bond lengths are 1.633, 1.641, and 1.650 Å, in free TiO(3Δ), 3[OTi(η1-NN)], and 1[OTi(η2-NN)] species, respectively. The N−N bond lengths are calculated to be 1.104, 1.103, and 1.195 Å in free N2, 3[OTi(η1-NN)], and 1[OTi(η2NN)] species, respectively. Changes on the Ti−O and N−N distances are obviously larger in 1[OTi(η2-NN)] than in 3 [OTi(η1-NN)] with respect to free TiO and N2 molecules. Furthermore, the Ti−N bond lengths are found to be 2.359 and 2.067 Å in 3[OTi(η1-NN)] and 1[OTi(η2-NN)] structures, respectively. Accordingly, with almost negligible structural changes in 3[OTi(η1-NN)] with respect to free TiO and N2, we are tempted to consider this compound as a van der Waals complex. This suggestion does not hold for the 1[OTi(η2-NN)] species for which the Ti−O and N−N bonds lengthen by 0.017 and 0.092 Å compared to those of free molecules. In addition, the smaller Ti−N distance in 1[OTi(η2-NN)] compared to the Ti−N bond length in 3[OTi(η1-NN)] could be explained as a substantial donation and back-donation between the TiO and N2 partners (vide infra). (4) In spite of a notable geometric difference between the two isomers, the energy gap between two forms is small (ΔE = −1.5 kcal/mol at CCSD(T)/AVTZ). To explain it, we represent the TiO + N2 → OTi−N2 reaction by two thermochemical cycles, as displayed in Figure 2 The first cycle implies the existence of an intermediate step formed by two reagents which are relaxed at the product

Figure 2. Thermochemical cycle for the TiO + N2 → OTi−N2 reaction at the CCSD(T)/AVTZ level. Distances are in Å.

geometry but largely separated (d(Ti−N) = 10 Å). The second cycle consists of an intermediate formed by two reagents at their equilibrium geometry but in the product configuration (d(Ti−N) = 2.067 and 2.359 Å for 1[OTi(η2-NN)] and 3 [OTi(η1-NN)], respectively). As shown in Figure 2, the energetic evolution along each of the two reaction channels could be interpreted as follows. (1) In the first thermochemical cycle, the geometrical relaxation of the two partners does not require any energetic cost for 3[OTi(η1-NN)], whereas it is very destabilizing for 1 [OTi(η2-NN)] (+29.6 kcal/mol). This step is followed by a stabilization step leading to the final products. Nonetheless, we should underline that the stabilization energy for 1[OTi(η2NN)] is substantially higher than that for 3[OTi(η1-NN)] (−34.3 vs −3.2 kcal/mol). (2) In the second channel, the interaction between two nonrelaxed partners destabilize the 1[OTi(η2-NN)] structure by 4.9 kcal/mol and stabilize 3[OTi(η1-NN)] by 3.3 kcal/mol. Afterward, the relaxation energy is negligible for the 3[OTi(η1NN)] and moderately important for 1[OTi(η2-NN)] (0.1 vs −9.6 kcal/mol). Consequently, we can point out that the energetic changes during the formation of 1[OTi(η2-NN)] is significantly more important than that of 3[OTi(η1-NN)]. II. Density Functional Results. The De (kcal/mol), ΔE (kcal/mol), d(Ti−N), and r2(N−N) calculated using pure, hybrid, and long-range corrected functionals with the Pop(2d) basis set are listed in Table 2. For energetic data, the CCSD(T)/AVTZ results are also presented for reference. 1. Pure Functionals. Except for BLYP, all other pure functionals give a good energetic ordering: ΔE < 0. However, the calculated gap is found to be the lowest with PBEPBE and the highest with M06L. The PBEPBE value of ΔE is very close to the CCSD(T)/AVTZ value (−1.5 kcal/mol). Compared to the CC-reference data, d(Ti−N) (3A″) was calculated to be underestimated and the r2(N−N) (3A″) overestimated, whatever the functional used. Contrarily, the 1[OTi(η2-NN)] geometry seems to be better described with all pure functionals. The wave function of the 1[OTi(η2-NN)] has been found to be stable with all pure density functionals without contamination.7 Consequently, the use of a pure density functional is not advisable, because of local character of the exchange functionals.37−39 Since the exact exchange (Hartree−Fock exchange) is a nonlocal function, addition of exact exchange should therefore improve our theoretical results. 2. Hybrid Functionals. As shown in Table 2, in comparison with the pure functional energetic results, we observe a global C



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Table 2. Calculated Parameters for the Two Structures with Various Density Functionals Using the Pop(2d) Basis Seta 3

a

1

A″

A′

De (1A′)

ΔE (1A′/3A″)

d

r2

d

r2

CCSD(T)/ AVTZ

−4.7

−1.5

2.349

1.103

2.066

1.185

BLYP PBEPBE B97D BB95 M06L TPSSTPSS

−12.3 −17.2 −26.5 −17.2 −29.6 −17.8

2.043 2.012 2.029 2.017 2.008 2.013

1.200 1.203 1.194 1.202 1.198 1.201

B3LYP PBE0 mPW1LYP wB97 wB97X wB97XD B1B95 M06 M062X TPSSh TPSS0

−4.8 −6.1 −2.2 −13.6 −10.9 −8.4 −4.9 −21.7 +1.9 −13.0 −6.2

2.030 2.001 2.031 2.005 2.010 2.012 2.004 1.999 2.025 2.010 2.004

1.180 1.181 1.176 1.1807 1.177 1.178 1.177 1.188 1.167 1.191 1.179

LC-BLYP LC-wPBE LC-TPSS

−2.4 −3.1 −5.3

1.989 1.988 1.969

1.165 1.177 1.168

Pure Functional +0.2 2.092 1.127 −1.2 2.063 1.127 −6.7 2.041 1.130 −3.2 2.074 1.126 −9.2 2.059 1.123 −4.3 2.066 1.123 Hybrid Functional +1.4 2.127 1.104 +0.4 2.114 1.101 +3.0 2.161 1.098 −7.6 2.209 1.095 −5.3 2.212 1.092 −4.0 2.180 1.094 −0.3 2.145 1.096 −4.5 2.064 1.111 +7.1 2.318 1.086 −3.3 2.080 1.113 −1.2 2.146 1.095 LC Corrected Functional +2.7 2.247 1.081 +0.3 2.251 1.088 −1.0 2.236 1.080

Figure 3. Relative energy variations (De(1A′), De(3A″), and ΔE) in function of %HF calculated by TPSSTPSS(xx%)/Pop(2d).

qualitatively well reproduced when xx =25% (De(1A′) = −6.2 and ΔE = −1.2 kcal/mol). Hereafter, the TPSSTPSS(25%) is denoted as TPSS0, in line with the PBE0 notation which contains also 25% of exact exchange. For the TPSS0 functional, a very weak basis set effect is observed when using the AVTZ basis set for which the corresponding energetics and geometrical parameters are De(1A′) = −6.2 kcal/mol, ΔE = −0.8 kcal/mol, d(Ti−N)(3A″) = 2.145 Å, r2(N−N)(3A″) = 1.095 Å, d(Ti−N)(1A′) = 2.004 Å, and r2(N−N)(1A′) = 1.177 Å. 3. Long-Range Corrected Functionals. It is well-known that one of the common shortcoming of traditional Kohn−Sham functionals is the incorrect long-range behavior of the exchange-correlation (XC) potential.40 In molecules, the XC potential of semilocal functionals decays exponentially along with the density, while the asymptotic form of the exact XC potential is −1/r.41 Usually, this discrepancy is partially reduced in hybrid functionals, where a small portion of the semilocal density exchange functional is replaced with the nonlocal Hartree−Fock exchange. It has been shown that another way to recover the exact −1/r asymptote consists in introducing a range separation into the exchange functional and replacing the long-range portion of the exchange by the Hartree−Fock counterpart.40,42 It seems that this approach is well adapted to describe the repulsive part of van der Waals potentials.43−45 In this work, we have used three long-range corrected functionals (see Table 2). In comparison with the CC-reference data, we note that only the LC-TPSSTPSS predicts 1[OTi(η2-NN)] more stable than 3 [OTi(η1-NN)], whereas it is reversed with LC-wPBE and LCBLYP techniques. It should be noted that we checked our calculations using LC-TPSSTPSS/AVTZ and we found that the LC-TPSSTPSS results are very slightly basis set dependent (LC-TPSSTPSS/AVTZ: ΔE = −0.7 kcal/mol). 4. Double Hybrid Functionals. In a double hybrid functional, first proposed by Grimme,23,24 both exchange and correlation functionals are corrected by introducing a certain amount of exact exchange in the exchange functional and a portion of the correlation energy is calculated using perturbation theory in the correlation functional (thus double hybrid). Usually the amount of exact exchange is much bigger in these methods compared to the simple hybrids, 53% vs 20% in B3LYP for instance. In this way, dispersion forces can be more accurately computed because of the perturbational calculation that partially corrects the shortcoming of Hartree−Fock and density functional methods.46 Consequently we expect to obtain energetic results between the MP2’s ones

Energies are in kcal/mol and distances in Å.

improvement with hybrid functionals except the B3LYP, PBE0, mPW1LYP, and M062X which reverse the 1A′/3A″ energetic ordering, resulting in an overestimation of the stability of 3 [OTi(η1-NN)]. The worst result was obtained with M062X for which the 1[OTi(η2-NN)] is found to be unbound. The De(1A′) calculated with hybrid wB97 family is notably reduced with respect to the pure B97D result. The one parameter B1B95 meta-GGA functional provides, however, an acceptable De(1A′) value, very close to the CCSD(T)/AVTZ value, but also an overbound 3[OTi(η1-NN)] structure reducing the energetic gap, ΔE. For the nonempirical meta-GGA TPSS family, the energetic ordering, as well as De(1A′), is improved with the addition of 10% exact exchange in the exchange functional (case of TPSSh). In order to explore systematically the effect of the exact exchange, we investigated the variation of the energetic data on 3[OTi(η1-NN)] and 1[OTi(η2-NN)] structures. Figure 3 displays the binding energies, De(1A′) and De(3A″), and the energetic gap variation in function of exact exchange contribution in the TPSS group. The HF addition in the TPSS exchange functional has been labeled as TPSSTPSS(xx %)/Pop(2d), where xx% stands for the exact exchange percent. The singlet state, 1[OTi(η2-NN)] is calculated to be more stable than the triplet one, 3[OTi(η1-NN)], when the exact exchange varies from 0% (ΔE = −4.3 kcal/mol) to 30% (ΔE = 0 kcal/mol). We should underline that 3[OTi(η1-NN)] becomes slightly more stable than 1[OTi(η2-NN)] for an exact exchange contribution higher than 30% reaching +2.9 kcal/mol for 40%. Especially, it is worth to noting that the reference data (De(1A′) = −4.7 and ΔE = −1.5 kcal/mol) are D



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Table 3. Basis Set Effect on the Optimized Results Obtained with Double Hybrid Functionalsba Pop(2d) 3

mPW2PLYP mPW2PLY-D B2PLYP B2PLYP-D CCSD(T)/AVTZ a

AVTZ 1

A″

3

A′

1

A″

A′

De

ΔE

d

r2

D

r2

De

ΔE

d

r2

d

r2

−2.8 −3.0 −4.5 −4.7

+0.2 +0.3 −1.8 −1.6

2.288 2.286 2.265 2.257

1.097 1.097 1.102 1.102

2.022 2.022 2.067 2.067

1.192 1.192 1.195 1.195

−3.8 −4.0 −5.5 −5.8 -4.7

−0.3 −0.2 −2.3 −2.1 -1.5

2.275 2.271 2.254 2.251 2.349

1.097 1.097 1.102 1.102 1.103

2.021 2.020 2.025 2.025 2.066

1.191 1.191 1.199 1.199 1.185

Energies are in kcal/mol and distances in Å. bSee Table 2 for the parameters’ definition.

which are overestimated and those of hybrid DFT (B3LYP and mPW1LYP) which are usually underestimated. We used two double hybrid techniques, B2PLYP and mPW2PLYP, which come from BLYP and mPWLYP functionals, respectively. We used also their empirical dispersion corrected variants: B2PLYP-D and mPW2PLYP-D. In addition to the Pop(2d) basis set, we used also a larger basis set, the AVTZ one, in order to recover a large part of correlation energy with perturbational treatment. A careful look at the results reported in Table 3 reveals the following: (a) the empirical dispersion correction has a negligible effect on the energetic and geometrical results. (b) The B2PLYP method describes better our system than the mPW2PLYP one. Particularly, the calculated energetic gap (ΔE) with B2PLYP keeps its sign with both basis sets, while it changes for mPW2PLYP. The performance of the double hybrid functionals could be explained by the presence of the exact exchange and the perturbational correlation. This combination could offset the defaults of MP2 (overbinding the 1[OTi(η2-NN)] geometry) and hybrid functionals (overbinding the 3[OTi(η1-NN)] geometry). Indeed, the B2PLYP, as a combination of MP2 and hybrid BLYP, provides the closest results to CCSD(T) than the mPW2PLYP, a mixture of MP2 and mPWLYP. This could be related to the result of the mPWLYP which largely overestimates the 3[OTi(η1-NN)] stability. 5. Some Conclusions on the DFT Techniques. The Figure 4a displays a summary of the De(1A′) and ΔE(1A′/3A″) as well as three bond lengths calculated with all the DFT functionals used in this paper, in comparison with the CCSD(T) results. On the ground of our results and discussions presented in this section, we can conclude that over twenty-three density functionals used in this work, only three functionals, namely TPSS0, LC-TPSS, and B2PLYP, are able to fairly reproduce the CC-reference data quantitatively. However, concerning the energetic accuracy the B2PLYP > LC-TPSSTPSS > TPSS0 ordering has been put in evidence comparing to the CCSD(T)/AVTZ results. This trend holds for the geometrical parameters (see Figure 4b,c). Concerning the side-on structure, all the four functional families clearly well reproduce the three bond lengths with respect to the CCSD(T)/AVTZ data (see Figure 4b). In the end-on structure, we note also that the N−N and Ti−O bond distances calculated with DFT approach are actually close to the reference data (CCSD(T)/AVTZ) whatever the functional used, while the third bond length (Ti−N) is very sensitive to the functional. We found that the Mean Absolute Error (MAE) decreases along the functional families as follows: 0.293 (pure), 0.199 (hybrid), 0.114 (LC-corrected), and 0.085 Å (double hybrid functionals). Here, we find again the same accuracy

Figure 4. Schematic comparison of energetic (ΔE and De(1A′) in kcal/ mol) (a) and geometrical parameters (three bond distances in angstrom) (b and c) calculated at the CCSD(T)/AVTZ and with different functional families (using Pop(2d) basis set) for both triplet (end-on structure) and singlet (side-on structure) states.

ordering as for the energetic considerations, i.e., B2PLYP > LCTPSSTPSS > TPSS0. B. Chemical Bonding Nature. While the NBO method provides an orbital description of chemical bonding very close E



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Hartree/Å3), and the ellipticity (ε) at the (X, Y) bond critical point, where X = N, Ti and Y = N, O. The oxygen electronic population reaches about 9 electrons in the three compounds. At the BCP(Ti, O), since Δρ > 0 and H(Ti, O) < 0, it indicates that the Ti−O bond should be considered as a closed-shell interaction (Δρ > 0) with a small covalent character (H(Ti, O) < 0). Consequently, the TiO unit almost retains its ionic structure Ti+O− upon complexation leading to two OTi-N2 structures. In addition, the topological parameters (ρ, Δρ, and H) at the BCP(Ti, O) vary very slightly when going from free TiO to two OTi-N2 states indicating very small perturbations of the TiO subunit upon the formation of complexes. For the N−N bonding, the changes between free N2 and the N−N subunit in 3[OTi(η1-NN)] are almost negligible, while we note significant changes between free N2 and 1[OTi(η2NN)]: ρ decreases by 29% and Δρ and H(N, N) are divided by two. The topological parameters on the BCP(Ti, N) almost correspond to that expected for a closed-shell interaction in both OTi-N2 singlet and triplet states . However, we observe that the interaction between TiO and N2 partners should be considered as a van der Waals interaction within 3[OTi(η1NN)] low values of ρ and positive Δρ, as already pointed out by Popelier.48 In contrast, the interaction between TiO and N2 partners within 1[OTi(η2-NN)] is characterized by two BCP(Ti, N). Values of ρ and Δρ (>0) on these BCPs are significantly larger than those in 3[OTi(η1-NN)]. Consequently, we should consider the interaction between TiO and N2 within 1[OTi(η2-NN)] as a closed-shell interaction. Furthermore, it is interesting to emphasize three points. (1) In addition to the charge density, Laplacien of the charge density, and local energy density at the BCP, there is another parameter in the AIM formalism, called “ellipticity” denoted by ε, which provides a quantitative measurement of the anisotropy of the electron density at the BCP. It is therefore used as a measurement of delocalization.49 From the ellipticity values reported in Table 4, we observe a very high value at the BCP(Ti, N) in the 3[OTi(η1-NN)] compound (1 order of magnitude larger than that at the BCP(Ti, N) of 1[OTi(η2NN)]), while its value at the two other BCPs is close to zero. It is obviously indicative of a strong delocalization at the BCP(Ti, N) within the 3[OTi(η1-NN)] compound. (2) Besides the BCPs, we found also an RCP (ρ = 0.09, Δρ = 0.55, H = −0.01) in the TiNN cycle of 1[OTi(η2-NN)]. If the RCP and BCP came together, it would suggest a structural instability which conferring to the compound the potential to

to the traditional Lewis scheme, the topological analysis of a chemical compound provides a chemical insight from wave functions (Hilbert space) in real 3D space. I. AIM Analysis. In the AIM analysis, the topology of the charge density, ρ(r), leads to a partitioning scheme which defines atoms inside a molecule or a molecular complex via the gradient vector field, ∇ρ. We can find critical points (CP) where ∇ρ vanishes. Bond critical point (BCP) indicates the existence of a chemical bonding between two nuclei of a compound in equilibrium geometry. The Laplacian of ρ, Δρ, at the critical point measures to what extent the electron density is locally concentrated if Δρ < 0, or depleted if Δρ > 0. According to the topological theory of AIM, the positive values of the Laplacian at the bond critical point are associated with closedshell interactions (ionic bonds, hydrogen bonds, and van der Waals molecules), whereas, Δρ < 0, indicates shared interactions (covalent bonds). Another criterion, proposed by Cremer, states that the local energy density, H(r) at the BCP, should be positive for ionic bonds and negative for partly covalent bonds.47 Figure 5 displays the Laplacian contour maps for the two studied OTi-N2 isomers. Topological molecular graphs are

Figure 5. Laplacian contour maps of (a) 3[OTi(η1-NN)] and (b) 1 [OTi(η2-NN)] structures, from MP2/Pop(2d) wfn. Solid lines stand for Δρ < 0 and dashed ones for Δρ > 0. The green circles represent BCPs and black one the RCP.

characterized by three BCPs in the 3[OTi(η1-NN)] structure (Figure 5a), whereas one ring critical point (RCP) in addition to the four BCPs were illustrated in Figure 5b for the 1[OTi(η2NN)] structure. We note that the Poincaré−Hopf relationship holds for both molecular graphs. In Table 4 are reported the AIM quantitative data for the two separate fragments and the two bound states. Atomic charges for Ti and O atoms (Q(Ti) and Q(O) and the net electronic charge for TiO moiety in the studied compounds (Qnet(TiO)) are given in electrons. Three BCPs have been investigated: BCP(N, N), BCP(Ti, O), and BCP(Ti, N). Four topological parameters have been listed in Table 4 on each BCP: ρ (in e/ Å3), Δρ (in e/Å5), energy density denoted as H(X, Y) (in

Table 4. Atomic Charges in TiO Subunit (in |e|), and Quantitative Informations [ρ (e/Å3), Δρ (e/Å5), H (hartree/Å3), and ε[ on All Critical Points Obtained within AIM Analysis, Calculated at the MP2/Pop(2d) Level of Theory N2(1Σg+) BCP

RCP Atomic Charges

ρ, Δρ H(Ti, O), ε ρ, Δρ H(N,N), ε ρ, Δρ H(Ti,N), ε ρ, Δρ H(Ti,N,N), ε Q(Ti)/Q(O) Qnet(TiO)

TiO(3Δ)

[OTi(η1-NN)]

0.26, 0.98 −0.17, 0.00

0.26, 0.91 −0.17, 0.01 0.66, −2.50 −1.19, 0.00 0.02, 0.16 0.00, 4.21

21.00/9.00 0.00

21.02/8.93 0.05

0.66, −2.51 −1.18, 0.00

F

3

1

[OTi(η2-NN)] 0.25, 0.86 −0.16, 0.01 0.48, −1.03 −0.60, 0.01 0.11, 0.27 −0.04, 0.15 0.09, 0.55 −0.01 20.30/8.82 0.81



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evolve under perturbation.50 However, the RCP’s position and its charge density suggest a stable cycle for the 1[OTi(η2-NN)]) compound. (3) Finally, the global charge transfer from TiO to N2 is calculated to be 0.05 and 0.81 |e| in 3[OTi(η1-NN)] and 1 [OTi(η2-NN)], respectively. The topological analysis corroborates the energetic discussion developed above. II. ELF Analysis. The ELF topology provides additional quantitative information which complete and almost corroborate the AIM analysis.51 Particularly, although the AIM approach allows a chemical bonding to be detected on the ground of a unique and physical criterion, it does not yield any information about how many electrons should be assigned to a bond. The ELF analysis is very suitable to give such information. Similar to AIM but based on the topology of the electron localization function of Becke and Edgecombe,52 the ELF approach provides a partition scheme of the molecular space into basins of attractors that have a clear chemical meaning. These basins are either core basins [labeled as C(A)] surrounding nuclei or valence basins, V(A) or V(A,B), belonging to the outermost shell and characterized by their coordination number with core basins, which is called the synaptic order. Accordingly, V(A) stands for monosynaptic basin corresponding to the lone pair of A, and V(A, B) for disynaptic one representing the chemical bond between two nuclei A and B. The basin populations are evaluated by integration of the one-electron density over the volume of the

1

[OTi(η2-NN)] structure, while its presence is confirmed even at high value of ELF function in 3[OTi(η1-NN)]. In order to provide a quantitative analysis, the valence basin populations are reported in Table 5. Table 5. ELF Valence Basin Population (in |e|) for the Studied Compounds, Calculated at the MP2/Pop(2d) Level of Theory N2(1Σg+) V(N, N) V(N) V(Ti, N) V(O) V(Ti) Qnet(TiO)

TiO(3Δ)

3.36 3.24 3.24

3

[OTi(η1-NN)] 3.31 3.32 3.25

7.40 1.05

7.30 1.07 0.04

1

[OTi(η2-NN)] 2.16 2.58 2.58 1.82 1.82 7.29 1.12

First, we remark that the V(O) population changes very slightly in going from free TiO to both OTi-N2 compounds. Second, the most significant modification is observed for monoand disynaptic basins surrounding nitrogens in 1[OTi(η2-NN)], whereas they are almost negligible in 3[OTi(η1-NN)]. Particularly, the V(N, N) population decreases by 1.2 |e| in 1 [OTi(η2-NN)] with respect to free N2. A part of the V(N) populations contribute to the formation of two new disynaptic basins V(Ti, N) sharing the whole of V(Ti) population. This process leads in fact to its vanishing. Finally, the net charge transferred from the TiO unit to the N2 partner is calculated to be 0.04 |e| in 3[OTi(η1-NN)] and 1.12 |e| in 1[OTi(η2-NN)]. These values are very close to those obtained with AIM techniques and we note a striking agreement between the outcomes of both topological analyses. III. NBO Analysis. In order to get an insight into the nature of the interaction in 3[OTi(η1-NN)], we performed an NBO analysis on this compound. The most meaningful result is that there is no natural bonding orbital between Ti and N atoms. The bonding orbital scheme within 3[OTi(η1-NN)] structure is almost the same as that of free molecules, except for a secondary donor−acceptor interaction. Upon coordination on the triplet state, there is a very small increase of the Ti−O bond length (1.633/1.641 Å) while the N−N bond distance is left almost unchanged. From a traditional point of view, this could be interpreted as the result of the superposition of two bonding interactions: σ donation from an occupied orbital of N2 to an empty orbital of the metal, and π back-donation from Ti(dδ) into the vacant π*(N2) orbital. The stability of the 3[OTi(η1NN)] complex depends on the balance between these two interactions (see Figure 7-a). These bonding interactions are reminiscent of the classical Dewar−Chatt−Duncanson model for bonding of olefin to transition metal complexes.53,54 From the perturbative analysis of NBO donor−acceptor interactions (the estimated second-order energy lowering ΔE(2)i→j reported on the Figure 7a), one can easily note that the π back-donation is negligible compared to the σ donation in the end-on complex so that the Ti−N coordinate bond is almost governed by the LPN(σ) → LP*Ti(3dσ) delocalization. In contrast, the NBO analysis of the side-on singlet state shows that the whole of molecule should be considered as an unique unit so that the Ti−N bonding could not be analyzed in terms of the donor−acceptor bonds. Interestingly, as shown in Figure 7b, the canonic HOMO orbital is almost a localized orbital which corresponds to a 2e−3c bond formed between N2 and Ti center. This bond is not well described within the NBO

Figure 6. Isosurfaces (for ELF = 0.80) and contours of localization domains in the two OTi-N2 complexes, from MP2/Pop(2d) wfn.

basin. Figure 6 illustrates the ELF localization domains in both 3 [OTi(η1-NN)] and 1[OTi(η2-NN)] complexes. We have superimposed surfaces (with ELF = 0.80) with contours in the TiNN plane of each species. This figure clearly gives rise to some qualitative topological changes upon interaction between TiO and N2 molecules: (1) While the number of valence basins is the same in the 3 [OTi(η1-NN)] structure than in free TiO and N2, we observe a very weak change in the shape of V(N). In contrast, the number of valence basins, as well as their shape, change considerably upon formation of 1[OTi(η 2-NN)]: three monosynaptic basins instead of four and three disynaptic ones instead of one, compared to TiO + N2. (2) The V(N) monosynaptic basins noticeably deviate from their cylindrical symmetry (as a free N2 molecule) in 1[OTi(η2-NN)], especially under the influence of the V(Ti, N) disynaptic basins located on nitrogen atoms bonded to titanium. There is no such modification within 3[OTi(η1-NN)]. (3) It is interesting to note that there is no longer V(Ti) monosynaptic basin in the G



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(3) It was found that the second order Møller−Plesset approach overestimates more and more De(1A′) when the basis set size augment, going from AVDZ to AVTZ. It was shown that the use of a moderate size basis set with MP2 provide energetic results in good agreement with the CC-reference data. (4) Concerning the density functionals, it has been highlighted that over twenty-three functionals used in this work only three are able to describe qualitatively the CCreference results. Particularly, the B2PLYP functional provides quantitatively accurate description of the spectroscopic properties of the studied 3[OTi(η1-NN)] and 1[OTi(η2-NN)]. Its success is mainly due to mutual error compensation between MP2 and hybrid functional. (5) The topological approaches, AIM and ELF, describe interaction between TiO and N2 in 3[OTi(η1-NN)] as a van der Waals interaction leading to a partial preservation of the topological characteristics of the free partners (TiO and N2) upon the product formation. By contrast, important topological changes are obtained in 1[OTi(η2-NN)]. Especially, we have observed the appearance of two new disynaptic basins corresponding to the partially covalent nature of the interaction between TiO and N2 subunits. (6) Furthermore, the weak interaction between TiO and N2 in 3[OTi(η1-NN)] has been confirmed by the NBO analysis which describes this interaction as a donor−acceptor interaction between an occupied orbital on the nitrogen atom and an antibonding one on the titanium center. Indeed, a weak charge transfer from TiO to N2 in 3[OTi(η1-NN)], 0.04 |e| has been found with topological methods as well as with the NBO techniques. (7) Finally, it is interesting to note that the AIM, ELF, and NBO approaches give complementary descriptions of the reaction between TiO and N2, leading to the same conclusion. While the AIM confirms the presence of a bonding via the existence of a bond critical point, the ELF gives information about the electronic population involving the bonding regions. The donor−acceptor scheme determined by the NBO analysis provides a nice interpretation of molecular orbital interaction between fragments of a molecular compound.

Figure 7. Natural bonding orbitals of both 3[OTi(η1-NN)] and 1 [OTi(η2-NN)] structures calculated from the SCF orbitals at the MP2/Pop(2d) level. Orbitals involved in the σ-donation/π backdonation interactions within end-on structure are depicted in panel a. In panel b are reported the NBO and canonic orbitals involved in the Ti−N2 bonding within side-on structure. Classical Lewis bonds are displayed in panel c.

model because of the 2 electrons shared on the three centers (two N and Ti atoms). The presence of metal center makes it a difficult case for the NBO method, even if this method provides a nice description for other electron-deficient ‘bridged’ species such as the diborane.29 Finally, we note that the dipole moment of 3[OTi(η1-NN)], calculated with the CCSD(T) geometry at the SCF level, presents only a minor change compared to that of free TiO (4.9 vs 4.2 D). This is informative on the interaction between TiO and N2 and does not strongly support the hypothesis of a purely electrostatic interaction, but rather suggests a weak donation/back-donation interaction (delocalization) resulting a very small charge transfer. In contrast, the dipole moment of 1 [OTi(η2-NN)] is very large (7.4 D) suggestive of a charge transfer mechanism. This description is in line with the topological analysis.

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AUTHOR INFORMATION

Corresponding Author

*Tel. 33 (0)1 44 27 30 72. E-mail: [email protected].

SUMMARY AND CONCLUSIONS This work presents a detailed theoretical study of two experimentally observed products of the TiO + N2 reaction: 3 [OTi(η1-NN)] (3A″) and 1[OTi(η2-NN)] (1A′). The benchmark data have been obtained at CCSD(T)/AVTZ level serving to check the reliability and accuracy of the most popular density functionals as well as the MP2 method. The nature of chemical bonding within each complex has been investigated in the topological and natural orbital frameworks. The most significant conclusions could be summarized as follows: (1) The CCSD level is insufficient to study such systems, whatever the basis set size. The introduction of the triple excitations is, however, essential to obtain reliable results. (2) At the CCSD(T) level, it was shown that some popular basis sets, such as AVDZ and Pop(2d) are not large enough for these systems. Only large basis sets, such as AVTZ or higher order, are appropriate, giving 1[OTi(η2-NN)] as the ground state with a stability energy equal to −4.7 kcal/mol and a 1 A′/3A″ energetic gap at about −1.5 kcal/mol.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Dr. Laurent Manceron and Dr. Benoit̂ Tremblay for helpful discussions. REFERENCES

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