Ab initio test study of the N2…H2 and N2…He van der Waalsdimers

Mary C. Salazar* , JoseL. Paz, Antonio J. Herna´ndez

Departmento de Quı´mica, Universidad Simo´n Bolıvar, Apdo. 89000, Caracas 1080A, Venezuela

Abstract

Quantum chemical fully ab initio conformational calculations was performed for the weakly bound van der Waals N2…H2and N2…He dimers in the framework of the supermolecule approach. The counterpoise-corrected interaction energies werecomputed through fourth order MBPT using basis sets constructed to give accurate values for the electric moments, polariz-abilities and dispersion energy contributions. The best size-to-performance ratio basis set found in the present study, predictsthe T-shaped structure to be the most stable configuration for N2…He, with a well depthDeof 2.68 meV at a minimum distanceRe of 3.44A, in close agreement with calculations performed with larger basis sets. The relative stability of the configurationsstudied for N2…H2 were: collinear structure. parallel structure. T-shaped structures. The collinear structure represent themost stable configuration, with aDe value of 8.35 meV at a equilibrium distanceRe of 7.60a0. q 1999 Elsevier Science B.V. Allrights reserved.

Keywords:Van der Waals dimers; Dinitrogen; Dihydrogen

1. Introduction

In much of the recent years, there was intenseresearch work on the study of van der Waals (vdW)molecules, which are accessible both, to detailedexperimental and to reliable quantum mechanicalstudies. The recent advances on our knowledge oflaser techniques for state selective chemistry, and ofhighly sophisticated molecular beam techniques, havelargely contributed to the experimental progress [1]and aim to gather information on the properties ofvdW molecules. Although these techniques haveshown to be a powerful tool for studying electroni-cally excited rare gas vdW dimers, their applicationfor studying the intermolecular interaction betweennon polar diatomic–diatomic vdW molecules asN2…H2 has not yet been reported because of their

small induced dipoles moments, and thus, becausethey exhibit only very weak absorption spectra.More traditional techniques, involving long absorp-tion paths along equilibrium gas samples at lowtemperatures and moderate pressures, was usedrecently to study N2…H2 by McKellar [2] in the infra-red regions corresponding to H2 vibrational and rota-tional frequencies, respectively. Although his reportimproves previous studies [3,4], the true conformationof N2…H2 is still unknown, and further assignmentand analysis requires feedback from more detailedspectra, and from calculations using realistic potentialenergy surfaces. In spite of the relative simplicity ofthe diatomic–diatomic vdW molecules, relatively fewtheoretical studies of N2–H2 were reported so far[5,6]. The N2…rare gas vdW interaction has receiveda similar amount of attention, since this system consti-tutes a good prototype for more complex molecularspecies interacting with rare gases, N2…He being the

Journal of Molecular Structure (Theochem) 464 (1999) 183–189

THEOCH 5999

0166-1280/99/$ - see front matterq 1999 Elsevier Science B.V. All rights reserved.PII: S0166-1280(98)00550-8

* Corresponding author.

simplest complex to study. Thus it has become abenchmark system for dynamic properties of thegaseous state of the mixture [7].

Theoretical studies of vdW interaction energieshave essentially followed two directions. The firstregards the interaction between the subsystems as aperturbation, and partitions the energy into terms suchas electrostatic, repulsion, polarization, induction, anddispersion. The second approach considers the inter-acting subsystems as a supermolecule [8,9]. The inter-molecular perturbation approach is far more suitablefor calculations of weak interactions because theinteraction energy is obtained directly, rather than asthe difference between two numbers usually severalorders of magnitude larger than the interaction energyitself. Nevertheless, since all the highly effective abinitio methods developed for single-molecule calcula-tions are in principle applicable without change, thevast majority of calculations of the interaction energyof vdW complexes are carried out at present using thesupermolecule approach.

The availability of powerful computers and thedevelopment of efficient computational algorithmshas made possible the quantum mechanical study ofvdW interactions of small to medium size molecules[10,11]. Despite many technical difficulties, the abinitio methods offer a sound basis for the calculationof vdW potential energy surfaces valid over the wholerange of molecular distances and orientations.Computational experience shows that large and flex-ible basis sets have to be used in both cases to obtainthe required accuracy [12–26]. The requirementsimposed on the basis sets follow directly from a quali-tative classification of the vdW interaction energy inthe various ranges of intermolecular separations.Long-range interactions, where the electronicexchange is negligible, lead to the requirement thatelectrostatic and the dispersion components of theinteraction energy of the vdW complex must be accu-rately represented. The short-range exchange repul-sion, being a direct consequence of overlap effects,requires an adequate description of the valence elec-tron density, both close and far from the nuclei. Mostdifficulties occur in the region of moderate overlapbetween the charge distributions of the interactingsubsystems, where the use of multicenter basis setsoverestimates the interaction energy of the vdWcomplexes, which is mainly attributable to what is

known as the basis set superposition error (BSSE)[10,11] arising from the tendency of basis orbitalson one system to be variationally mixed with thoseon the others, giving spurious energy improvements.Nevertheless, the BSSE problem is nowadayssurmountable, and the function counterpoise methodof Boys and Bernardi [27] was proved to be thecorrect approach by Gutowski and Chalasinski [28].

In the present paper, we have performed ab initiotest studies on the N2–H2 and N2–He vdW potentialenergy surfaces using, as starting point, the mediumsized POL1 basis set described by Sadlej [29,30], onthe basis of its success describing electrical properties[29–36] and potential energy surfaces[13,14,26,37,38] of weak interacting systems, andthe GLS extended basis set described by Garrison etal. [39] for He. The main purpose of this work is tofind a basis set with the best size-to-performance ratiopossible, which will allow very important computa-tional savings in time and resources for the long-termobjective of the present research study, namely theinvestigation of the bonding and electronic propertiesof the ground and excited electronic states of vdWdimers.

2. Method of calculation

The interaction potential was obtained in the frame-work of the supermolecule approach at the SCF andMBPT levels of approximation [40,41] for the totalenergy

E � ESCF 1 EMBPT; �1�

where the correlation energy is split into contributionsthat are owing to different orders of MBPT

EMBPT � EMBPT2 1 EMBPT3 1 EMBPT4: �2�

Currently, going beyond the full fourth-order\discretionary{}{}{}MBPT treatment does not seemfeasible. It has also been shown that in order to obtainquantitatively meaningful results incomplete higher-order MBPT calculations must be avoided, thus thecomplete fourth-order MBPT approximation wasadopted in the present study. With the linked clustertheorem automatically satisfied on each order ofperturbation, this MBPT scheme is certainly sizeconsistent [40,41], one of the best and more

M.C. Salazar et al. / Journal of Molecular Structure (Theochem) 464 (1999) 183–189184

economical methods available for a reliable calcula-tion of interaction energies within the supermoleculeapproach, and it has several advantages compared toother methods owing to its uniform and systematictreatment of the electron correlation contributions[41,42].

The interaction energy (IE) was defined as:

IE R� � � E N2…H2 He� �; Rÿ �

2 E N2…X; Rÿ �

2 E X…H2 He� �; Rÿ �

; �3�

whereE(N2…X; R) and E(X…H2(He); R) are usedhere to indicate that the monomer energies are derivedin the total dimer centered basis set. This amounts toapplying the counterpoise procedure of Boys andBernardi [27] to correct for the BSSE at both theSCF and the MBPT levels of approximation at eachmolecular configurationR. Through fourth-order inthe correlation perturbation, the interaction energy(3a) can be expressed in terms of the followingcomponents [25] at any particular geometrical config-urationR:

IEMBPT 4� � � IESCF 1 IEMBPT2 1 IEMBPT3 1 IEMBPT4:

�4�

The necessary MBPT correlation energies werecalculated using the BRATISLAVA package[42], interface to the MUNICH molecular code[43] for the Gaussian integrals, SCF eigenvectorsand energies, molecular properties, and four-indexmolecular integral transformation calculations,respectively.

3. Results and discussion

3.1. N2–He

The selection of the starting one-particle basis set ofthe contracted Gaussian-type orbital (CGTO) func-tions is crucial. We have employed the medium-sizepolarized basis set of the CGTO type devised bySadlej [29, 30], comprising [10.6.4/5.3.2] GTO/CGTO for the nitrogen atom, and the basis set ofthe CGTO type described by Garrison et al. [39],comprising [10.2/5.2] GTO/CGTO for helium, werethe notation used on the left side of the braces repre-sents the number of primitives gaussian of character s,p and d, respectively, with the correspondingcontracted gaussian orbitals written on the left side.These basis sets, referred to as the POL1 and GLSbasis in the present study, accounts for the diffusenessof the valence part of the wave function, leads to acorrect calculation of intermolecular electrostaticforces, and to a negligible secondary BSSE [44],which justifies the use of the standard counterpoisecorrection [27]. Tao and Pan [45,46] showed thatbond functions can be very effective in recoveringdispersion energies, provided that an adequate set ofatom-centered basis functions is included. Recentcalculations [47,48] of vdW interactions, includingthe most accurate N2…He ab initio potential energysurface description to date by Hu and Thakkar [49],have used bond functions. Bond functions are the bestchoice for true vdW complexes (bonded mainly bydispersion interactions) like N2…He, thus they arewell suited to complement the POL1 and GLS basissets.

Three different basis sets were explored in thissection. The first corresponds to the POL1 and GLSatom-centered basis set described earllier, which isreferred as POL11 GLS for brevity. The secondand third were obtained by supplementing the basisset described previously with bond function takenfrom Tao and Pan [45–48]: 3s (a � 0.9, 0.3, 0.1),3p (a � 0.9, 0.3, 0.1), 2d (a � 0.6, 0.2), 1f (a � 0.3).The bond functions were placed at the midpoint of theinternuclear vectorR, which joins He with the centerof mass of N2.

The final MBPT(4) counterpoise-corrected inter-action energies were computed according to Eq. (3)for all possible configurations of N2 and He. In the

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Table 1Counterpoise-corrected interaction energy between of N2…He atR � 7ao and u � 90. All energies are inmhartree,No representsthe number of orbital functions

Basis seta No IEMBPT(4)

POL11 GLS 59 -57.52POL11 GLS 1 {3s 3p 2d} 81 -76.08POL11 GLS 1 {3s 3p 2d 1f} 88 -82.71A 1 {3s 3p 2d} 137 -80.79B 1 {3s 3p 2d 1f} 116 -80.58

a Basis sets A and B are described in Ref. [49]. The functions inbraces are bond functions.

supermolecule calculations, N2 was kept rigid at theirexperimental equilibrium bond length of 1.4a0. Thedimer geometry is specified byR, representing thedistance between the center of masses of N2 and Heatom, and the polar angle of orientationu of the vectoralong the N2 bond with respect toR.

Table 1 shows the MBPT(4) counterpoise-corrected interaction energy between He and N2 atu � 908 andR� 7a0, computed with the three basissets obtained by combining the POL1 and GLS atom-centered basis set with the bonds functions of Tao etal. [45–48], where we have included for comparisonthe best two results of Hu and Takkar [49]. Theseresults shows that including bond functions to thePOL11 GLS atom-centered basis increases dramati-cally the interaction energy, mainly owing to increas-ing the dispersion correlation contribution. This canbe understood because these basis sets wereconstructed to give accurate values for the electricmoments and polarizabilities, but they are lackinghigher polarization functions important for the disper-sion forces arising through electron correlation. Thebasis set POL11 GLS 1 {3s 3p 2d 1f} has the bestsize-to-performance ratio of all basis shown inTable 1.

The performance of the POL11 GLS 1 {3s 3p 2d1f} basis set in calculating the interaction energy

of N2…He is depicted in Fig. 1, which shows howthe well depthDe changes withu , compared withthe same results from Hu and Thakkar [49] using theB 1 {3s 3p 2d 1f} basis set. This figure shows theT-shaped structure to be the most stable configuration,with a well depthDe of 2.68 meV at a minimumdistanceRe of 3.44 A for the present calculation, inclose agreement with theDe value of 2.58 meV at aRe

value of 3.43 A, as found by Hu and Thakkar [49] forthe same geometry. These values are also in goodagreement with previous empirical or semiempiricalcalculations recounted by Beneventi et al. [50]. Theother two basis sets explored in this study and shownin Table 1, also predict the T-shaped structure to bethe most stable configuration, with a well depthDe of1.57 meV at a minimum distanceRe of 3.57 A for thePOL11 GLS basis set, and aDe value of 2.36 meV atRe� 3.46 A, for the POL11 GLS1 {3s 3p 2d} basisset.

The POL11 GLS 1 {3s 3p 2d 1f} basis set alsopredicts a linear configuration with aDe value of2.15 meV at a minimum distanceRe of 3.97 A,which also compares well with theDe value of2.00 meV atRe� 3.98 A, as found by Hu and Thakkar[49] for the same geometry. The anisotropy inDe,defined asDDe � De L� �2 De T� �j j; where L and Tdenote the linear and the T-shaped structures,

M.C. Salazar et al. / Journal of Molecular Structure (Theochem) 464 (1999) 183–189186

Fig. 1. Distance optimized interaction energy curve of N2…He as a function ofu , compared with the same results of Ref. [49].

respectively, is found to be of 0.53 in this basis, ascompared with a value of 0.58 found previously [49].This also leads to the conclusion that the bond func-tions in our basis set led to a description of the lineargeometry that is more accurate than that for the T-shaped structure.

3.2. N2–H2

The results of previous section are of main impor-tance for the study of N2–H2 because:

1. they indicate that the use of the Sadlej’s basis setsopen the attractive possibility of reducing the

dimension of the calculation and still retainingthe main qualitative features of the calculatedintermolecular energy, and

2. it allows very important computational savings intime and resources.

In our opinion, the potential surface of N2–H2 has notbeen the subject of any realistic calculation so far, andwe have decided to further test the reliability of thePOL1 basis set for nitrogen and hydrogen atoms[29,30] and the GLS basis set for He, on the calcula-tion of the potential energy surface of N2…H2.

The final MBPT4 counterpoise-corrected interac-tion energy was also computed according to Eq. (3)for all possible configurations of N2 and H2. In thesupermolecule calculations, these two units werekept rigid at their experimental equilibrium bondlengths of 1.4a0 and 2.068a0 for H2 and N2, respec-tively, so that only a rigid rotor potential surfaceresults. The coordinate system used for the calcula-tions is shown in Fig. 2, whereR represents thedistance between the center of masses of N2 and H2,b1 andb2 are the polar angles of orientation of thevectors along the N2 and H2 bonds with respect toR,and corresponds to the angle of torsion. All the config-urations represented by 25 different values of the pairof angles (b1, b2) � (08,08), (08,22.58), (08,458),(08,67.58), (08,908), (22.58,08), (22.58,22.58),

M.C. Salazar et al. / Journal of Molecular Structure (Theochem) 464 (1999) 183–189 187

Fig. 2. Coordinate system for N2…H2.

Fig. 3. Interaction energy surface of N2…H2.

(22.58,458), (22.58,67.58), (22.58,908), (458,08),(458,22.58), (458,458), (458,67.58), (458,908),(67.58,08), (67.58,22.58), (67.58,458), (67.58,67.58),(67.58,908), (908,08), (908,22.58), (908,458),(90.58,67.58) and (908,908), respectively, were studiedin the present contribution. For each value of (b1, b2),geometry optimizations were carried out only withrespect to the intermolecular parameterR, and theH2 and N2 molecules were kept in the same plane.The equilibrium bond distances (Re), the well depths(De), and the dissociation energies (Do), were obtainedby fitting the MBPT(4) counterpoise-corrected inter-action energy points, calculated according to Eq. (3),to a eighth-order polynomial in the stretching coordi-nate R, analytically continued with a seventh-orderpolynomial on 1/R (from 1/R6 to 1/R12) in the asymp-totic R region. The final ‘‘minimum’’ interactionenergy surface (De, b1,b2) of N2…H2 is shown inFig. 3, where the range ofb1 andb2 was extendedbeyond 90 by symmetry considerations.

Fig. 3 shows that the collinear configuration, (b1,b2) � (08,08), corresponds to the most stable confor-mation in our calculations, with a well depth of8.35 meV at a minimum distanceRe of 7.60ao. Zero-point energies, calculated from the fitted potentialcurves using the numerical Numerov–Cooley proce-dure [51], by treating the N2…H2 vdW molecule as adiatomic system with only one degree of freedomR,shows that the calculated dissociation energies of thiscollinear conformer corresponds to aDo of 4.12 meV,with only one vibrational state supported by this linearconfiguration.

Fig. 3 also shows that the parallel configuration,(b1, b2) � (908,908), represents a local minimumcorresponding to the second most stable structure ofthe present IE surface, with a well depth of 5.26 meVat a value ofRe of 6.80ao. Here, the (b1, b2) �(458,458) conformation represent a transition struc-ture, with a well depth of 2.54 meV at a value ofRe

of 7.70ao, which gives rise to a barrier high of5.81 meV along the minimum energy pathb1 � b2,in going from the linear to the parallel structure, andof 2.72 meV in the opposite direction.

The T-shaped structures, (b1, b2) � (908,08) and(b1, b2) � (08,908), represent the less stable config-urations in Fig. 3, with values ofDe of 1.98 and2.90 meV, at values ofRe of 8.20ao and 7.30ao,respectively.

Calculations performed with the POL1 basisaugmented with a set {3s3p2d} of the bond functionsof Tao et al. [45–48], confirm the above relative orderof the conformers. In particular, well depths of 10.06,7.41 and 3.59 meV are obtained for the collinear,parallel, and the (b1, b2) � (458, 458) transition struc-ture, respectively, increasing the barrier high alongthe minimum energy pathb1 � b2, in going fromthe linear to the parallel structure from 5.81 to6.47 meV, and from 2.72 to 3.82 meV in the oppositedirection.

Finally we should add at this point that, contrary tothe findings of Schinke et al. with respect to CO…H2

[52], the present energy surface for N2…H2 isdependent on the torsion angle, showing energydifferences between structures withf � 08 andf �908, for any given conformation, slightly larger than1 meV. In particular, the parallel structure (b1, b2) �(908,908) shows a change inDe from 5.26 to 4.03 meVwhen it is changed from 08 to 908. Similarly, in the(b1, b2) � (45,45) structure, this corresponds to achange inDe from 2.54 to 3.92 meV under the sametransformation.

In summary, although its reliability is welldocumented, the present test calculations usingthe POL1 1 GLS basis set are to be taken onlyas a qualitative guide. In order to increase theirpredictive value, one is forced in practice toimprove this basis sets in order to describe thedispersion energy of the electronic states involvedmore accurately, which in turn, imposes extremedemands on the computational resources. For rela-tive small vdW systems as N2…He, the presentstudy indicates that the use of the POL11GLS 1 {3s 3p 2d 1f} basis, owing to its optimumsize-to-performance ratio, could be used withoutaffecting the accuracy of the calculated intermole-cular energy. For larger vdW systems like N2…H2

even this basis set is somehow large, if one isinterested in describing the interaction energysurface for the ground and electronically excitedstates. Although larger basis sets remain to beinvestigated, the present study seems to indicatethat the use of the POL1 basis opens the attractivepossibility of reducing the size of the polarizationbasis set without affecting the accuracy of thecalculated intermolecular energy for the N2…H2

vdW dimer.

M.C. Salazar et al. / Journal of Molecular Structure (Theochem) 464 (1999) 183–189188

Acknowledgements

The authors thank the Deutsche Forschungs-gemeinschaft (DFG), the Consejo Nacional de Inves-tigaciones Cientı´ficas CONICIT (Grant S1-95000503), the Decanato de Investigaciones de laUniversidad Simo´n Bolıvar (Grant G-13), the Alex-ander von Humboldt-Stiftung, and the Max-PlanckGesellschaft, for the continued support to this researchproject.

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