Intermolecular interactions in the N2-N2 dimer

M OLECULA R PHY SICS , 1998, V OL. 95, N O. 3, 477± 481

Intermolecular interactions in the N2 ± N2 dimer By SIMONE RA UGEI, GIA NNI CA RDINI and VINCENZO SCHETTINO Laboratorio di Spettroscopia Molecolare, Dipartim ento di Chimica, Via Gino Capponi 9, 50121 Firenze, Italy, and European Laboratory for Nonlinear Spectroscopy (LENS ) , Largo E. Fermi 2, 50125 Firenze, Italy ( Received 20 January 1998; accepted 11 May 1998 ) Intermolecular interactions at the MP2 level f or the nitrogen dimer are reported. The results are compared with density f unctional calculations, showing the limits of the approximation adopted f or the exchange and correlation f unctionals. Preliminary results of ab initio molecular dynamics on solid nitrogen are reported.

1. Introduction A considerable amount of theoretical work has been done on molecular crystals [1]. This has been made possible mainly by the availability of simple, yet su ciently accurate model potentials for the interm olecular interactions in molecular crystals [1]. Generally, the latter are presen ted in the form of non-bonded atom± atom interactions implemented with electrostatic potentials expressed as charge± charge potentials or as multipole expansions. Despite their simplicity, these model potentials have been demonstrated to be capable of reproducing with notable accuracy the structura l and dynamic properties of many com plex system s. However, in several cases, and in particular whenever the structura l and dynamic properties of interest cover a wide range of temperatures and pressures, such that the intermolecular interactions that come into play range from large interatom ic separations to the shortest atom± atom distances, more complex and sophisticated model potentials have been found necessary to account for all the subtleties of the crystal properties. A case of particular interest in this respect is the nitrogen crystal: this has a com plex phase diagram [2± 4] with the high pressure phases not yet fully understood and the object of severa l investigations [5± 10]. In order to reproduce correctly the structura l and dynamic properties of solid nitrogen in its di€ erent forms a detailed knowledge of the interaction potential between nitrogen molecules is of basic importance. The nitrogen± nitrogen interaction potential has been studied using semi-empirical and ab initio methods. Only pair additive potential models have been used, ranging from the simple isotropic Lennard-Jones (LJ ) plus point quadrupole [11] to very complex models [12]. Nevertheless, no potential model thus far has been able to reproduce all the properties of the known phases of the nitrogen crystal with su cient accuracy to be taken as 0026± 8976/98 $12 . 00

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reliable in predicting the behaviour of the less characterized high pressure phases. On the whole the nitrogen crystal can be considered as a signi® cant test case in studying the intermolecular interactions in molecular crystals. A promising approach to the intermolecular interactions in condensed phases tailored to the experimental conditions in consideration has been through the computation of the time evolution of the system by molecular dynamics (MD ) using the forces obtained for each con® guration by a full ab initio calculation of the interaction potential between the molecules in the crystal. This approach, proposed by Car and Parrinello (CP) , is known as ab initio or ® rst-principles MD [13]. This method has been improved over the years starting from its initial formulation, and has been shown to be a valuable tool to investigate (with success ) the structura l and dynamic properties of many system s of chem ical and physical interest, ranging from clusters [14± 18] to condensed phases [19± 22], from ordered to disordered solids [23± 26], and from solutions [27± 36] to chem ical reactions [37± 40]. The CP method is based on density functional theory (DFT ) [41, 42] using an expansion of the Kohn± Sham wavef unctions in term s of plane waves. For practical reasons only the valence electrons are described fully, while nuclei and core electrons are treated with pseudopotentials. DFT is in principle exact, but its practical applications are subject to approximations bound to the adopted form for the exchange and correlation functionals. The simplest functionals correspo nd to the local density approximation (LDA ) [41, 42] and the largest proportion of DFT calculations has been perf ormed in this framework. LDA functionals give good results for strongly bound system s, but overestim ate the bond energy [41, 42, 35]. A slight improvement has been realized with the use of gradient corrected (GC ) methods 1998 T aylor & Francis Ltd.

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[41, 42] that give a better agreem ent for the bond lengths and bond energies, and have been shown to be essential for describing hydrogen bonds [43, 29, 31, 32, 35, 36, 44]. The purpose of the present paper is to report on a comparative study of the intermolecular interactions of the N 2 - N 2 dimer in various con® gurations, using ab initio methods with various level of approximation, including the Mù ller± Plesset correlation energy truncated at second-order level (MP2 ) [45] and DFT approaches (LDA and GC ) . The limitations of the density functional approach to account correctly for the weak and essentially long range interactions that are typical of molecular crystals are discussed. A comparison of the results obtained indicates the improvements that are necessary in the exchange± correlation functionals to provide for a correct description of the N 2 - N 2 interaction. 2. Results and discussion The intermolecular poten tial between two nitrogen molecules has been computed using the standard methods of computational chemistry. The calculations have been perf ormed using the program Gaussian 94 Revision D3 [46], keeping the NÐ N intramolecular disÊ . Three di€ erent relative molecular tance ® xed at 1.106 A orientations in the N 2 - N 2 dimer have been examined (® gure 1) : molecules parallel with the molecular axes perpendicular to the axis joining their centre of mass (P) ; molecules perpendicular with molecular axes perpendicular to the axis joining their centre of mass (X ) and molecules perpendicular with one molecule aligned onto the axis joining the centre of mass of the molecules

Figure 1.

Schematic of the the structures of the dimers N 2 - N 2 considered in the present work.

(T ) . We have computed the energy of the dimer ( E d ) as a function of the intermolecular distance, keeping the molecules rigid. The interaction energy ( E i ) is computed in the standard way using the counterpoise method [47] to eliminate the basis set superposition error (BSSE ) : E i

=Ed

-

E 1

-

E 2

,

where E 1 ( E 2 ) are the energy of the isolated molecule 1 (2 ) computed in presence of the basis set of the other molecule. It is well established that the calculation of the interaction potential between molecules is strongly basis dependent, and it is essential to use a large basis set augm ented with di€ use orbitals [48]. We have chosen to use the largest basis set compatible with our computer resurces, i.e., a Dunning’ s correlation consistent basis set with triple-zeta augmented with one d and one f di€ use function (A UG-cc-pV TZ ) [46, 49, 50]. Some calculations were perf ormed also using the standa rd basis set 6-31G*, obtaining sim ilar behaviour in the repulsive region but large di€ erences (about a factor 2 ) in the well depth. The correlation energy has been taken into account perf orming the calculation both at the MP2 level and by the density functional approach. It was not felt necessary , for the purposes of the present work, to extend the perturbative calculations at the MP4 level, since the MP3 and M P4 corrections generally are of opposite sign [48] and will change the results by only a few per cent. The MP2 results have been compared with the DF calculations in the LDA approximation and with the gradient correctio ns (GC ) of Becke [51] and Perdew [52] (BP) and Becke± Y ang± Lee± Parr [53] (BLY P) . The results of the M P2 calculations are reported in ® gure 2 where they are compared with the semiempirical potentials of Murthy et al. [54] and of Etters et al. [5]. In the ® gure the ab initio results obtained by Berns and van der A voird [55] and by Jonsson et al. [56], using a di€ erent basis set and a perturbative approach, are included for comparison. It can be seen that the MP2 results essentially are in good agreem ent with other approaches or models for the long range part of the interaction. Theref ore, this part of the interaction potential, mainly respo nsible for the crystal properties at low temperature, can be considered to be well established. The M P2 results are comparable with the calculations of Berns and van der A voird [55] both in the well depth and in the repulsive region, particularly in P and X con® gurations. The di€ erences in the case of the T con® guration can be ascribed to the increased importance of the electron overlap that is not accounted for fully by perturb ative treatm ents of the kind used by Berns et al. [55]. In conclusion, the present results and a comparison with previous ab initio calculations show that the MP2

Interm olecular interactions in the N2 ± N2 dim er

479

Figure 2. Interaction energy of the N 2 - N 2 dimers: f ull line, MP2 calculations; short-dashed line, semi-empirical potential 5q of Murthy et al. [54]; long-dashed line, semi-empirical potential of Etters et al. [5]; open square, ab initio data of Bern et al. [55]; and open triangles, ab initio data of Jonsson et al. [56].

Figure 3. Interaction energy of the N 2 - N 2 dimers: f ull line, Car± Parrinello calculations in the LDA approximation; open squares, density f unctional calculation in the LDA approximation; dashed line, density functional calculation using BP functional; and open circles, MP2 calculations.

calculation gives a reasonable representation of the N 2 - N 2 intermolecular potential, and can be taken with con® dence as a ref erence potential for comparison with density functional calculations. The MP2 calculations are compared with the DFT results in ® gure 3. The DF results are reported using the LDA and the BP functionals and using the plane wave expansion or the same basis set as in the MP2 approach. Calculations with the BLY P functionals give results indistinguishable from those obtained with the BP functionals. It can be seen that at the LDA level the dissociation energy of the dimers is overestim ated by a factor of almost 2 and the interaction energy decays too rapidly with the distance, compared with the MP2 calculations. GC methods do not give rise to bounded dimers except

for a very small well depth in the T con® guration at Ê . about 7 A A lthough M P2 and LDA results are in any case in qualitative agreem ent in the repulsive region, GC functionals give a too repulsive an interaction at all shorter distances, as can be seen from the inset in ® gure 3. A s already noted, in this region semi-empirical potentials di€ er strongly from MP2 calculations, explaining their general failure to reproduce correctly the high pressure phases of solid nitrogen. W e have also perf ormed the N 2 - N 2 dimer calculation with the CP method using the QMDCP code [57] and adopting a cuto€ of 70 Ry on the plane wave expansion and a M artins± Trouilier pseudopotential [58]. The results agree essentially with those obtained using a large Gaussian basis set and the sam e functionals. This

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con® rms the equivalence of the two method s, once a very large Gaussian basis set corrected for BSSE using the counterpoise method is used. In conclusion, the present calculations of the N 2 - N 2 dimer are in line with previous results on rare gases [59] and benzene dimer [60] at the limits of the density functional treatm ent of weak long range interactions using available exchange correlation functionals. It has been found that using gradient corrected functionals the interaction potential is repulsive for all the dimer con® gurations considered. On the other hand, in the LDA approximation the potential is attractive but the ® t to the calculated MP2 poten tial is only qualitative. A s a test of the LDA approximation to the N 2 - N 2 interaction we have carried out a preliminary ab initio CP sim ulation on the a phase of solid nitrogen at the experimental volume. It has been found that the orientational order is lost at low temperatures (5 K ) after a few hundred femtoseconds. This should be ascribed to two features of the calculated LDA interaction potential, viz. the location of the potential well minimum for too short a separation and the rapid decay of the van der Waals and quadrupole± quadrupole interactions with increasing distance. The failure of the density functional approach to the problem considered in the present paper depends on the essentially short range nature of the correlation functional used compared with the long range character of the van der Waals forces [59]. Meijer and Sprik [60] have suggested applying empirical correctio ns to gradient corrected DF calculations to accou nt properly for the dispersion interaction. Possible prom ising developments of density functional theory to include a proper treatment of weakly interacting system s have been discussed recen tly [61, 62]. The authors would like to thank Professo r M. Parrinello for making available the QMDCP program. This work was supported by the Italian Ministero dell’ Universita e della Ricerca Scienti® ca e Tecnologica (MU RST ) , by the Consiglio Nazionale delle Richerche (CN R ) and by the European Union (Contract No. E RBFMGECT950017 ) .

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