An ab initio and DFT study of (N2)2 dimers

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An ab initio and DFT study of (N2)2 dimers

COURONNE, Olivier, ELLINGER, Yves

Abstract

The structure of van der Waals dimers (N2)2 is studied using ab initio and density functional calculations. The potential energy surfaces corrected a priori for basis set superposition errors are necessary for determining both the geometries and the vibrational frequencies. With both ab initio MP2, MP4 and DFT PW91–PW91 levels of theory, the T-shaped and canted conformations appear to be the most stable, within 1–5 cm−1 of each other. The DFT PW91–PW91 dissociation energy is 67 cm−1 and an upper limit to the barrier to internal motion is 30 cm−1, both in excellent agreement with the values deduced from IR measurements.

COURONNE, Olivier, ELLINGER, Yves. An ab initio and DFT study of (N2)2 dimers. Chemical Physics Letters , 1999, vol. 306, no. 1-2, p. 71-77

DOI : 10.1016/S0009-2614(99)00431-5

Available at:

http://archive-ouverte.unige.ch/unige:2734

Disclaimer: layout of this document may differ from the published version.

1 / 1

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Ž . Chemical Physics Letters 306 1999 71–77

ž /

An ab initio and DFT study of N

_{2 2}

dimers

O. Couronne

⁾

, Y. Ellinger

Laboratoire d’Etude Theorique des Milieux Extremes, Ecole Normale Superieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France´ ˆ ´ Received 7 December 1998; in final form 6 April 1999

Abstract

Ž .

The structure of van der Waals dimers N_{2 2} is studied using ab initio and density functional calculations. The potential energy surfaces corrected a priori for basis set superposition errors are necessary for determining both the geometries and the vibrational frequencies. With both ab initio MP2, MP4 and DFT PW91–PW91 levels of theory, the T-shaped and canted conformations appear to be the most stable, within 1–5 cm^y1of each other. The DFT PW91–PW91 dissociation energy is 67 cm^y¹ and an upper limit to the barrier to internal motion is 30 cm^y¹, both in excellent agreement with the values deduced from IR measurements.q1999 Elsevier Science B.V. All rights reserved.

1. Introduction

A large part of the mass of the universe is concen- trated in dust particles. Due to the low temperature Ž10–50 K of a number of celestial objects cold. Ž molecular clouds, comets, rings of giant planets, etc.. these particles behave as cold surfaces on which light molecules condense, forming what is known as ice mantles. This generic term, besides water, covers all types of organic or non-organic materials such as CH , NH , CH OH, CO, CO , or N whose accre-₄ ₃ ₃ ₂ ₂ tion at surfaces is at the origin of the growth of interstellar grains. Before dealing with the structure of molecular solids and the possible chemistry that takes place at their surface under the extreme conditions prevailing in space, we found it necessary to consider the primary process of accretion, namely the formation of isolated dimers.

)Corresponding author. Present address: Department of Physi- cal Chemistry, University of Geneva, 30 quai Ansermet, CH-1211 Geneva 4, Switzerland. E-mail: [email protected]

The first dimer, the study of which is reported in

Ž .

this Letter, is N_{2 2}; it may play an important role in the atmospheric chemistry of Neptune and Titan.

Experimental evidence is limited but suggests that the most stable dimer has either a T-shaped or X-shaped geometry, with the centers of mass of the monomers separated by approximately 3.7–4.2 A˚ w1,2 . From a theoretical point of view, the Nx Ž _{2 2}.

w x dimers have been studied with ab initio 3–8 and

w x

model potential 9–12 calculations. Most of these studies have predicted up to four local minima T-Ž shaped, X-shaped, parallel and canted but the ener-. getic ordering varies with the method.

The Møller–Plesset level of theory is usually accepted as a good starting point to account for the correlation effects required to describe weak interac-

w x

tions in van der Waals systems 13–15 . We will use the results obtained with this methodology for comparison with DFT calculations on this type of compounds. Although DFT gives remarkable results for solid structures as well as for covalent and ionic systems, it is not completely clear yet if it can

Ž .

PII: S 0 0 0 9 - 2 6 1 4 9 9 0 0 4 3 1 - 5

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( ) O. Couronne, Y. EllingerrChemical Physics Letters 306 1999 71–77 72

systematically deal with van der Waals super-

w x

molecules 16,17 even if clear-cut results have been

w x

obtained for rare gas dimers 18–20 , complexes w x between molecular nitrogen and benzene 21 and

w x Ž .

Cl –C H₂ ₂ ₄ 20 . Beyond the test case of the N_{2 2} dimer, the present work is intended to probe the quality of DFT calculations in view of future studies on N , CO and CO ices.₂ ₂

2. Computational background

The ab initio calculations were performed at the MP2 and MP4 levels of theory as implemented in

w x

the GAUSSIAN94 package 22 . The standard 6-31 qG⁾ basis set was used. It has already proven reliable for these types of problems as well as for the description of molecular systems with a diffuse

w x

wavefunction such as negative ions 23 . Density

Ž .

functional theory DFT calculations were performed

Ž . w x

applying the Kohn–Sham KS formalism 24 . Two parametrizations of the exchange-correlation func- tionals which include gradient corrections within the

Ž .

generalized gradient approximation GGA were

w x w x

used: B3LYP 25,26 and PW91–PW91 27 . DFT calculations were performed with GAUSSIAN98 w28 .x

In the DFT calculations, the atomic orbitals were constructed from the following contraction patterns:

Ž7111r411r1⁾.w29 . This basis set is the closest tox the basis used in ab initio calculations as shown by the comparison made with the results previously

w x obtained on weakly interacting species 21 . The grids considered for the computations comprised 128 radial shells. At the end of the SCF procedure, each radial shell had 302 angular points.

One of the main characteristics of the present study is that the considered potential energy surface ŽPES is corrected for the basis set superposition.

Ž .

error BSSE according to the counterpoise method w30 . The corrected energy is obtained as:x

EcorrsEDŽBBD,rD.y

Ý Ž

EMŽBBD,rD.

Ms1 ,2

yEMŽBBM,rD.

.

, Ž .1 where the indices M and D refer to the monomer and dimer, respectively. The first term stands for the

energy of the dimer in the molecular basis BB_D at geometry r . The second term is a summation run-_D ning over the energies of the monomers, each being calculated in the basis BB_D at their geometries in the dimer. The last term is a summation running over the energies of the monomers, calculated in the basis BB_M of each monomer at their geometries in the dimer. The BSSE-corrected PES is used to determine the geometry as well as the vibrational frequencies.

The binding energy is given by:

DEsE^corry

Ý

E^MŽBB^M,r^M.. Ž .2

Ms1 ,2

It is now well established that this correction is essential for the determination of molecular struc-

w x tures bound by van der Waals interactions 31 . But unlike many studies where the BSSE correction is applied to the equilibrium structure, i.e. after the minimization has been completed, it is applied here to the entire optimization process. Therefore, we obtain a different surface, the minimum of which is the correct equilibrium geometry. We will show that, in these compounds where cohesion is dependent on weak intermolecular forces, neglecting BSSE or applying an inadequate correction can lead to dramati- cally erroneous results.

As the BSSE-corrected energy is a sum of ener- gies, it is possible to define a corrected Hessian H_corr as the matrix of the second derivatives of the BSSE- corrected energy with respect to the nuclear coordi-

w x nates, namely 32 :

HcorrsHDŽBBD,rD.y

Ý Ž

HMŽBBD,rD.

Ms1 ,2

yHMŽ BBM,rD.

.

, Ž .3

where the terminology is the same as for the cor- rected energy and H indicates a Hessian matrix.

Each second derivative matrix can be computed either analytically or numerically. Frequencies and normal modes corrected for BSSE are then obtained by diagonalization of the force constant matrix F_corr which is derived from H_corr. In such a way, the

Ž .

zero-point energy ZPE obtained from these frequencies is also corrected for BSSE.

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3. Results

The geometry of the dimer in any conformation investigated can be defined by four parameters, namely the distance between the centers of mass R, two angles u₁ and u₂, and one dihedral angle w ŽFig. 1 . Six types of conformations of N. Ž _{2 2}. have been studied: linear, parallel, T-shaped, X-shaped,

Ž .

canted and crystal Fig. 2 , constrained to DD`h,CC2Õ, DD_{2 h}, DD_{2 d} and CC₁ symmetries, respectively.

3.1. Symmetry constrained optimization

The first step in our study consists in an optimiza- tion along the R coordinate. In these calculations the internuclear distance in the N molecule is the same₂ for the dimer as for the monomer. The position of the minimum R₀ and the binding energy E_min are

Ž .

obtained by fitting n-order ns2–10 polynomials to 3–11 points.

The BSSE correction appears to be essential for a reliable evaluation of both the equilibrium geometry

Ž .

and ZPE Fig. 4a .

More precisely, Table 1 shows that R_min is systematically shifted toward longer distances, with a correction varying from 0.2 to 0.4 A according to the˚ type of dimer. The smaller corrections appear in the DFT treatment. It can also be seen that the binding energy is systematically larger with the PW91–PW91 functional than in ab initio MP2 and MP4 calculations. In all cases it is greatly reduced when the BSSE correction is applied about 80% for all con-Ž formations . The BSSE-corrected potential energy. curves obtained using MP2, MP4, DFT B3LYP and DFT PW91–PW91 are presented in Fig. 3 for the T-shaped conformation which will be shown below

Fig. 1. General representation for a dimer. M and M1 2 are the middle points of monomers 1 and 2. R is the distance between

Ž .

these two points.u1 andu2 are the angles between M M1 2 and the molecular axis. w is the dihedral angle between the two monomers.

Ž .

Fig. 2. Different types of N2 2 structures studied, as defined by

Ž .

the triplet u1,u2,w.

to be one of the two stable structures of equal energies.

At the MP2 level, all dimers are found to be local minima, the T-shaped and canted conformations being the most stable ones with equivalent energies.

Parallel and X-shaped conformations are found to be very close to each other, and the linear conformation

Table 1

Equilibrium distances and binding energies along the R coordi-

y1 ˚

Ž .

nate DE in cm and Rmi nin A

BSSE BSSE

uncorrected corrected

Ž .

R_{mi n} DE R_min DEqBSSE MP2r6-31qG)

T-shaped 4.09 273 4.33 69

Canted 4.05 225 4.29 70

X-shaped 3.55 248 3.97 39

Parallel 3.70 203 3.99 36

Linear 4.87 154 5.19 15

MP4r6-31qG)

T-shaped 4.11 266 4.40 58

Canted 4.11 217 4.35 59

X-shaped 3.64 221 4.11 28

Parallel 3.73 191 4.10 28

Linear 4.84 181 5.17 16

PW91–PW91r Ž7111r411r1).

T-shaped 3.99 305 4.25 132

Canted 3.90 296 4.17 138

X-shaped 3.50 264 3.81 103

Parallel 3.57 234 3.86 88

Linear 4.80 195 4.94 16

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Fig. 3. BSSE-corrected potential energy curves along the inter-

Ž .

molecular coordinate R see Fig. 1 for the T-shaped structure:

Ž . Ž . Ž . Ž .

MP2 ^, MP4 q, B3LYP e and PW91–PW91 =.

is the least stable one. Binding energies at the MP4 level slightly differ from MP2, but do not inter- change the relative stabilities of the dimers.

For DFT calculations, all dimers were found to be non-bonded at the B3LYP level, implying that this functional is unable to reproduce dispersion forces for this type of molecular dimer. On the other hand, the PW91–PW91 results compare much better with the MP2 ones. The positions of the minima are close, although systematically shifted by 0.10–0.15 A to-˚

Ž ˚

wards shorter distances except the 0.25 A shift for the linear dimer , the binding energy being a little. larger. Very similar results have previously been obtained for the N -benzene van der Waals complex₂ w21 .x

The crystal conformation, as it appears in the a

Ž .

phase Fig. 2 is the third most stable arrangement after the T-shaped and canted conformationsŽDEs

y1 .

56 cm at the MP2qBSSE level . The distance between the two centers of mass optimized at the MP2qBSSE level of theory is R_mins4.2 A. The˚ experimental distance in thea phase of the crystal is

˚ Ž

4.0 A corresponding to a lattice parameter as5.7

˚. A .

The a phase crystal conformationŽu₁s90,u₂s 35.3, ws54.7 is not far from either the T-shaped. Žu₁s0,u₂s90,ws0 or canted. Žu₁s45,u₂s45, ws0 conformation. It can then be inferred that. molecules tend to adopt T-shaped andror canted conformations but twist due to crystal constraints.

This situation, however, has no physical meaning for

the dimer and will be discussed later when compar- ing larger N clusters to the crystal structure.₂

3.2. Minimum energy structures andÕibrational fre- quencies

In order to assess the stability of the different dimers, frequencies have been computed on the corresponding MP2, MP4 and PW91–PW91 BSSE-corrected PES. As seen above, the force constant matrix corrected for BSSE is derived from a sum of Hessian matrices. Each Hessian can be computed either analytically or numerically on the MP2qBSSE and PW91–PW91qBSSE surfaces, but only numerically for the MP4qBSSE one. For the numerical calculations, we used a two-point differentiation. In order to obtain reliable results, especially for low

Ž .

frequencies, we had to use small displacements dx together with high accuracy in the energy value ŽdE_accuracy.. We found, from comparison between analytical and numerical computations at the MP2 level, that the parameters dxs0.005 A and˚ dE_accuracys10^y⁷ a.u. give the best match between the two methods. MP4qBSSE frequencies were then computed with these parameters.

Frequencies have been obtained for the linear, parallel, X-shaped, canted and T-shaped conformations. All the frequencies of the T-shaped and the canted conformations are real, which confirms that

Ž .

these structures are two real minima Table 2 . By

Table 2

Ž ^y1.

Internal modes cm of the potential surface for the T-shaped and the canted structures. Geometries are equilibrium geometries for each surface and force constant matrices were computed on the PESs corrected a priori for BSSE. The MP2 and PW91–PW91 force constant matrices were computed analytically, and the MP4 force constant matrix were computed numerically see text forŽ details.

n_{MP 2} n_MP4 nPW91 – PW91

T-shaped

Bending b₂ ;5 ;5 26

Stretching a₁ ;5 ;5 27

Bending b₂ 26 16 37

Out of plane b₁ 26 21 39

Canted

Bending b2 10 8 27

Out of plane b1 40 32 61

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Table 3

Comparison between theoretical binding energies for the T-shaped

Ž ^y1.

and the canted conformations energies in cm

MP2 MP4 PW91–PW91 T-shaped

Ž .

DEqBSSE 69 58 132

Ž .

DEqBSSEqZPE_corr 38 34 67 Canted

Ž .

DEqBSSE 70 59 138

Ž .

DEqBSSEqZPE_corr 40 34 71 Ž .w x

Experiment D_e 1 66

contrast, all the angular frequencies for the parallel, X-shaped, linear conformations are imaginary, which implies that these conformations are not stable.

Fig. 4. Stretching of the T-shaped conformation at the MP2r6-31 qG⁾ level. The dashed line represents the uncorrected energy and the plain line the energy corrected for BSSE. a DifferenceŽ . between a posteriori correction of the BSSE 2 on the minimumŽ . of the non-corrected surface 1 and a minimization process on theŽ .

Ž . Ž .

BSSE-corrected surface =. b ZPE corrections on the BSSE- uncorrected and BSSE-corrected PES.

Table 2 shows the vibrational frequencies for the internal modes of the most stable conformations:

T-shaped and canted. ZPEs can be evaluated from these frequencies. The ZPE is a non-negligible quan- tity when compared to the binding energy: it represents up to 50%. The final binding energies are given in Table 3. Particularly good agreement is obtained between the DFT PW91–PW91 value and that of 66 cm^y¹ deduced from IR measurements by Long et al.

w x1 .

3.3. A prioriÕersus a posteriori BSSE correction of binding energy

One of the main points in this study is the a priori inclusion of the BSSE correction in the minimization process. This approach is seldom used, which is entirely justified when large energy differences are implied. In the present study, the energy differences involved are too small for this technique to be used.

Ž .

Using the T-shaped dimer as an example Fig. 4a shows the result of an a posteriori BSSE correction on a minimum corresponding to an uncorrected PES.

At the MP2r6-31qG⁾ level reported here, the error in the dissociation energy is 13 cm^y¹, which represents 20% of the binding energy. The equilibrium distance between the centers of mass is under- estimated by 0.24 A. In other cases, such as the˚

Table 4

Ž ^y1.

Comparison between internal modes cm before and after BSSE correction of the potential surface for the T-shaped struc-

Ž .

ture D2 h. Geometries are equilibrium geometries for each surface. The MP2 and PW91–PW91 force constant matrices are computed analytically, and the MP4 force constant matrix are

Ž .

computed numerically see text for details

n_{MP 2} n_MP4 n_PW91yPW91 Before BSSE correction

of the PES

Stretching a₁ 44 44 34

Out of plane b₁ 63 64 64

After BSSE correction of the PES

Bending b2 26 16 37

Out of plane b1 26 21 39

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X-shaped conformation, the error in R increases up to 0.4 A.˚

Using non-coherent ZPE is even worse. Fig. 4b shows two types of vibrational corrections, one computed on the corrected potential surface and hereafter

Ž .

referred to as ZPE_corr top , the other obtained from

Ž .

the uncorrected surface bottom .

Significant differences appear between the frequencies depending on whether they are corrected

Ž .

for BSSE or not Table 4 . After correction, the low frequencies corresponding to the motion of one N₂ molecule with respect to the other are lower than before correction. It was found that the stretching of the N bond is not affected by the formation of the₂ dimer as anticipated from the conservation of theŽ N–N bond length when going from the isolated monomer to the dimerized species . It can be in-. ferred from the curvature of the potential surfaces on Fig. 4a, that the second derivatives, and thus, the force constants, decrease when the BSSE correction is taken into account.

By definition, BSSE is always negative. Since its magnitude decreases monotonically with intermolecular separation, it is quite obvious that correcting for BSSE results in a decrease of the PES curvature. A shallower surface implies lower frequencies for intermolecular modes and consequently a lower ZPE correction. For instance, at the MP2r6-31qG⁾ level, ZPE_corr amounts to 31 cm^y¹ in the T-shaped dimer whereas the value directly obtained from the uncorrected PES is equal to 88 cm^y¹ which is 57 cm^y¹ higher. The present study clearly shows that erroneous results are obtained in such cases due to a critical overestimation of the vibrational frequencies.

Table 5

Binding energies for the T-shaped conformation: comparison between a priori and a posteriori correction of the BSSE energies inŽ

y1 .

cm ;=: instable

MP2 MP4 PW91–

PW91

DE 273 266 305

a posteriori

DEqBSSE 56 41 103

DEqBSSEqZPE = = 9

a priori

Ž .

DEqBSSE 69 59 132

Ž .

DEqBSSEqZPEcorr 38 34 67

It can be seen in Table 5 that the T-shaped dimer is no longer bound at the MP2 and MP4 levels in these conditions the same situation is found for the cantedŽ conformation ..

4. Concluding remarks

In this contribution, we report a study of the

Ž .

structure and stability of the N_{2 2} dimer at canted, T-shaped, X-shaped, parallel and linear structures plus that taken from the crystal. We have compared ab initio MP2 and MP4 results to the DFT ones. The present work provides further confirmation that the B3LYP functional is unable to properly describe the bonding forces in weakly interacting systems. By contrast, the exchange-correlation functional PW91–

PW91 gives results which are not only close to the MP values but, even more satisfactorily, are in excellent agreement with IR measurements.

The PW91–PW91 exchange-correlation functional, although still being semi-local and not cor- rectly describing the R^y⁶ behaviour of the interac- tion energy at large distances, provides a very good approximation in the range of medium distances characteristic of the equilibrium structures of weakly interacting molecules. Moreover, this rather good performance of the PW91–PW91 functional, compared to that of B3LYP, can be attributed to its

w x exchange part 21 .

We have shown that PESs and ZPEs should be determined with explicit consideration of BSSE in the optimization procedure and calculations of the vibrational frequencies, a posteriori BSSE corrections possibly lead to erroneous equilibrium geometries, ZPE and binding energies. Under these conditions, the T-shaped and canted dimers are the only stable structures at the MP2, MP4 and DFT PW91–

PW91 levels of theory, contrary to previous studies where equilibrium geometries were determined using PESs which were not BSSE corrected. It should be mentioned at this point that the same two conformations were found separated by 5 cm^y¹ in a recent study at the CCSD T level with a posteriori BSSEŽ .

w x

corrections 33 . With proper a priori correction, the

Ž . ^y¹

binding energy of the N_{2 2} dimer is 38 cm at the MP2 and MP4 levels of theory and 67 cm^y¹ for the DFT PW91–PW91 functional, in remarkable agreement with experiment.

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The energy difference between the two stable

Ž ^y¹ .

conformations 1–6 cm all corrections considered is beyond the accuracy which can reasonably be expected when internal motions are treated at the

Ž .

harmonic level no coupling between vibrations . No significant barrier could be found between the T- shaped and canted conformations, in agreement with

Ž . w x

the CCSD T study 33 , which suggests that both conformations belong to the same potential well of the internal motions. The next saddle points, X- shaped and parallel can be seen as transition points

Ž .

for out-of-plane and in-plane non-concerted motions. We have verified that there is a direct path of decreasing energy connecting the parallel and X- shaped structures to the minima. These structures are

y1 Ž .

about 30 cm 43 K higher than the equilibrium minima at all levels of theory. This value is in

y1Ž .

agreement with the 15–30 cm 22–43 K deduced from IR experiments 1 and is consistent with thew x permanent flip of the molecules in the crystal.

Acknowledgements

The authors would like to thank T.A. Wesołowski and Prof. N.C. Handy for stimulating remarks and suggestions in the course of this study. The financial support of CNRS Programme National de Planetolo-´

Ž .

gie PNP is greatly acknowledged. Part of the calculations were performed at IDRIS and CNUSC com- puting facilities.

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