Article pubs.acs.org/JPCA

Small and Efficient Basis Sets for the Evaluation of Accurate Interaction Energies: Aromatic Molecule−Argon Ground-State Intermolecular Potentials and Rovibrational States Hubert Cybulski Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland

Angelika Baranowska-Łączkowska Institute of Physics, Kazimierz Wielki University, Plac Weyssenhoffa 11, PL-85072 Bydgoszcz, Poland

Christian Henriksen Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lynbgy, Denmark

Berta Fernández* Center for Research in Biological Chemistry and Molecular Materials (CIQUS), University of Santiago de Compostela, E-15782 Santiago de Compostela, Spain S Supporting Information *

ABSTRACT: By evaluating a representative set of CCSD(T) ground state interaction energies for van der Waals dimers formed by aromatic molecules and the argon atom, we test the performance of the polarized basis sets of Sadlej et al. (J. Comput. Chem. 2005, 26, 145; Collect. Czech. Chem. Commun. 1988, 53, 1995) and the augmented polarization-consistent bases of Jensen (J. Chem. Phys. 2002, 117, 9234) in providing accurate intermolecular potentials for the benzene−, naphthalene−, and anthracene−argon complexes. The basis sets are extended by addition of midbond functions. As reference we consider CCSD(T) results obtained with Dunning’s bases. For the benzene complex a systematic basis set study resulted in the selection of the (Z)Pol-33211 and the aug-pc-1-33321 bases to obtain the intermolecular potential energy surface. The interaction energy values and the shape of the CCSD(T)/(Z)Pol-33211 calculated potential are very close to the best available CCSD(T)/aug-cc-pVTZ-33211 potential with the former basis set being considerably smaller. The corresponding differences for the CCSD(T)/aug-pc-133321 potential are larger. In the case of the naphthalene−argon complex, following a similar study, we selected the (Z)Pol-3322 and aug-pc-1-333221 bases. The potentials show four symmetric absolute minima with energies of −483.2 cm−1 for the (Z)Pol3322 and −486.7 cm−1 for the aug-pc-1-333221 basis set. To further check the performance of the selected basis sets, we evaluate intermolecular bound states of the complexes. The differences between calculated vibrational levels using the CCSD(T)/(Z)Pol33211 and CCSD(T)/aug-cc-pVTZ-33211 benzene−argon potentials are small and for the lowest energy levels do not exceed 0.70 cm−1. Such differences are substantially larger for the CCSD(T)/aug-pc-1-33321 calculated potential. For naphthalene− argon, bound state calculations demonstrate that the (Z)Pol-3322 and aug-pc-1-333221 potentials are of similar quality. The results show that these surfaces differ substantially from the available MP2/aug-cc-pVDZ potential. For the anthracene−argon complex it proved advantageous to calculate interaction energies by using the (Z)Pol and the aug-pc-1 basis sets, and we expect it to be increasingly so for complexes containing larger aromatic molecules.

I. INTRODUCTION van der Waals complexes are characterized by an interaction dominated by dispersion and therefore are frequently chosen as models to study physical processes where this interaction plays a major role. For instance, complexes formed by an aromatic © 2014 American Chemical Society

Received: August 18, 2014 Revised: October 15, 2014 Published: October 15, 2014 10288

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N2−Ne13 dimers, and linear chains of the CO−(HF)n and N2− (HF)n complexes14 employing new bases derived from the LPol-n (n = ds, dl, fs, fl) basis sets of Baranowska and Sadlej15 and the augmented polarization-consistent aug-pc-n (n = 1, 2) (denoted apcn in the following) bases of Jensen16 to obtain accurate IPESs for these complexes. In all the cases the bases were also augmented with midbond functions.17−19 For the CO−Ne complex we additionally considered the evaluation of interaction induced electric dipole moment and polarizability surfaces.12 To further test the potentials we used them to calculate the vibrational spectra of the complexes taking results obtained with Dunning’s series of bases as a reference. These new results were in very good agreement with the experimental data. In the present work we extend our studies to larger van der Waals complexes. To start with we have selected the benzene− argon dimer because of the accurate data, both theoretical and experimental, available for this system. Additionally, because of its size this complex is an obvious starting point to study complexes formed by rare-gas atoms and larger aromatic molecules, like the naphthalene− and anthracene−argon complexes. The aim of the present study is to obtain small and flexible bases that are able to yield accurate potentials and intermolecular states for aromatic molecule−rare gas van der Waals complexes. For this, in particular, basis sets have to be sufficiently polarizable, what is also important in the evaluation of molecular electric properties. That was why we turned our attention to the electric-property oriented basis sets of the Pol family.15,20−22 The idea behind the polarized (Pol) sets, developed over the years by Sadlej and co-workers, was based on a simple physical model of a harmonic oscillator perturbed by an external electric field. The close resemblance of the Schrödinger equation solutions for the harmonic oscillator to the Gaussian-type functions commonly used in basis sets led to the model of generation of appropriately designed polarization functions which were next added to a carefully chosen source basis set. In the original model, the external electric field was static,23 and led to Pol basis sets.20 After generalization to the case of an oscillating electric field perturbation,24 the ZPol22 and LPol-n15 basis sets were developed, for calculations of linear and nonlinear electric properties of molecular systems, respectively. In contrast to our earlier work, here we focus on the smallest basis set of the Pol family, that is the ZPol set. Although it contains only the first-order polarization functions for the valence shell, at the same time it is appropriately diffuse to properly describe linear electric properties and represents a reasonable compromise between accuracy and computational cost. A detailed description of the method of generation of the Pol basis sets can be found in ref 15 and references cited therein. In comparison to correlation-consistent basis sets, Jensen’s polarization-consistent bases have the advantage of the importance of higher angular momentum functions decreasing almost geometrically instead of arithmetically,16,25 therefore, being more efficient from the computational point of view. This has been confirmed by promising results in our previous work on weakly bonded complexes.11−13 For the benzene− and naphthalene−argon complexes we start from the Pol and ZPol series of bases and the aug-pc-1 basis set of Jensen (denoted here as apc1) and extend them with different sets of midbond functions,17,18 in order to obtain efficient bases for evaluating interaction energies. As reference

molecule and rare gas atoms have been used to investigate solvation or adsorption of molecules.1,2 Due to its relatively small size, the benzene−argon van der Waals complex has been widely studied, both from the experimental and the theoretical points of view. Accurate ab initio intermolecular potentials are available for the ground3−5 and the lowest singlet6 and triplet7 excited states. The S0 intermolecular potential energy surface (IPES) has been evaluated with several correlation methods and basis sets (see ref 3 and references cited in ref 5). Two IPESs calculated by employing coupled-cluster singles and doubles including connected triple corrections [CCSD(T)] model are available: the first one evaluated with the augmented correlation-consistent polarized valence double-ζ (aug-ccpVDZ) basis set extended with a set of 3s3p2d1f1g midbond functions (denoted 33211 in the following),4 and the second obtained with the aug-cc-pVTZ basis set where the 33211 midbond set was also added.5 The CCSD(T)/aug-cc-pVTZ33211 surface is characterized by two symmetric global minima of −390.1 cm−1, with the argon atom located on the benzene C6 axis at distances of ±3.536 Å, and six local minima in the benzene molecule plane with energies of 169.5 cm−1 from the bottom of the potential, located at distances of 3.803 Å from the centers of the C−C bonds. To test the potentials, the intermolecular energy level structure of the complex was evaluated. Both IPESs4,5 gave very accurate results and improved those previously available. The best IPES available for the naphthalene−argon complex was obtained in 2011,8 using the second-order Møller−Plesset (MP2) correlation method and the aug-cc-pVDZ basis set. Since the MP2 method tends to overestimate the interaction energy, the potential was further corrected with respect to a few selected CCSD(T) interaction energies calculated mainly near the global minima. The IPES has four equivalent minima characterized by energies of −493 cm−1. The minima appear at distances of 3.435 Å from the naphthalene molecule plane and ±0.43 Å from the center of the molecule along the symmetry axis of naphthalene that passes through three C−C bonds. The intermolecular vibrations were evaluated, and some experimentally observed bands reassigned. In 2013, a simple IPES based on electronic structure data but with different form allowing some natural extension to model multiple argon atoms and the intramolecular dynamics of naphthalene was obtained.9 This potential was derived from CCSD(T)/aug-cc-pVDZ ab initio interaction energies. The potential was applied to collisional energy transfer and matrix isolated IR spectroscopic studies. Calvo kindly sent us the code for the potential; however, we were not able to reproduce the behavior from ref 9. Calvo confirmed that there were problems with the potential, and he is currently working on this. To the best of our knowledge, for the anthracene−argon complex neither interaction energies nor IPESs are available. Taking into account the studies described above and those carried out previously on a considerable number of van der Waals complexes, we could conclude that the use of the CCSD(T) method together with Dunning’s series of bases (aug-cc-pVXZ, X = D, T, Q, ... denoted here as aXZ) yields very accurate intermolecular energies and potentials; however, such calculations quickly become computationally prohibitive when going to larger complexes. Thus, it is desirable to find smaller but still flexible basis sets for the evaluation of accurate interaction energies and properties of weakly bonded complexes. Recently, we have dealt with small complexes, like those formed by two rare-gas atoms,10 the CO−Ne,11,12 and 10289

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counterpoise method.27 The monomer geometries are kept fixed during the calculations. For the benzene molecule we use the most accurate experimental values available (RCC = 1.397 Å and RCH = 1.080 Å).28 For naphthalene (Figure 1) we consider the accurate geometries obtained by Baba et al. by ultra-highresolution laser spectroscopy and ab initio calculations.29 The anthracene geometry is that in ref 30.

we consider aTZ-33211 interaction energies (for the benzene− argon dimer they are taken from ref 5). Each of the selected basis sets are then employed to evaluate a considerable set of interaction energies, selected in order to be able to give accurate intermolecular potentials. For each IPES, parameters of an appropriate analytic function are fitted to the calculated ab initio points, and next used in calculations of vibrational spectra. In the case of the anthracene−argon dimer, due to the size of the system, calculations of the whole IPES are still too computationally demanding and only a restricted basis set study is carried out. For this system it is not possible to use a larger basis set than aDZ-33211 and, thus, these interaction energies are used as reference. The paper is organized as follows: In Section II we describe the basis set selection procedure, in Section III the details of the obtained IPESs are presented, and in Section IV the intermolecular vibrational level structure is discussed. Each section starts with a short description of the computational details, followed by an analysis and discussion of the corresponding results. In Section V we summarize and present our final conclusions.

Figure 1. Numbering of the effective positions for the naphthalene molecule.

To obtain the interaction energies we employ the CCSD(T) method considering its good performance in our previous work.5,6 The calculations are carried out in the frozen core approximation, using the Dalton31 and Molpro32 programs. The (Z)Pol basis set used here is constructed from the Pol set for the Ar atom33 and the ZPol set for the C and H atoms.22 The apc2 basis set is not considered in the present study since in the case of the benzene−argon complex it corresponds to 298 functions and therefore is larger than aDZ with a 33211 midbond function set (257 functions). The latter basis set was proved to be large enough to reproduce the complex interaction energies and rovibrational spectra in a very efficient way.6 It is well-know,3 that the use of midbond functions speeds up interaction energy convergence with respect to basis set limit. Thus, in the present study the basis sets are also extended with additional sets of midbond functions placed in the middle of the van der Waals bond. Although energy convergence toward basis set limit is much faster, the apparent disadvantage of the use of midbond funtions is that this convergence is not as smooth as in the hierarchy of bases without midbonds. To check whether convergence is correct for the benzene−argon complex, we have completed our results with the aXZ (X = T, Q) bases (without midbond functions). The bases considered in this study are listed in Table 1. For all of them we evaluate interaction energies at the given geometries. nmqrtu denotes a set of nsmpqdrf tguh midbond functions. The exponents of the different sets used are given in the Supporting Information. The main results of the basis set analysis are reported in Table 1. For the benzene−argon dimer the aTZ-33211 interaction energies are much closer to the aQZ results than those obtained with the aDZ-33211 basis set. Therefore, we can consider the aTZ-33211 IPES as reference for further analysis. The use of midbond functions is crucial in both cases: the (Z)Pol and apc1 bases. The results obtained with the basis sets without midbond functions are far from convergence. A comparison of the values in Table 1 proves that the (Z)Pol33211 and apc1-33321 basis sets give interaction energies closest to the aTZ-33211 ones in the regions of the global minima. To analyze the performance of the selected basis sets for geometries outside the global minima, we evaluate interaction energies at the other four geometries. For comparison we complete the results with those obtained with

II. BASIS SET SELECTION STUDY The most common procedure to select basis sets is to carry out basis set convergence studies using hierarchies of bases. In ref 3 we used the aXZ (X = D, T ,Q) series of bases of Dunning et al.26 and the corresponding set of bases extended with midbond functions for the benzene−argon complex. In the present work, we have taken into account the results of our previous investigation on smaller van der Waals complexes, for which we showed a good performance of the extended Pol, apc1, and apc2 bases in obtaining accurate interaction energies,5,10 and we have decided to employ them in the study of larger systems. As intermolecular coordinates we use the (x,y,z) Cartesian coordinates of the Ar atom position vector R with the origin in the aromatic molecule center of mass. The aromatic molecule lies in the xy plane. For the benzene−argon dimer as references we take the CCSD(T)/aXZ-33211 (X = D, T) available potentials4,5 and evaluate interaction energies at five selected intermolecular geometries. The selected intermolecular geometries are those that correspond to distances R = 3.1000, 3.2000, 3.5547, 5.0270, and 6.0000 Å with the Ar atom located on the benzene C6 axis. The existing intermolecular potential8 for the naphthalene− argon complex is based on MP2/aDZ ab initio points further corrected by using some CCSD(T)/aDZ-33221 results. Since the MP2 method is well-known for overestimating interaction energies and the aDZ basis set is the smallest in Dunning’s basis set series, we decided to test these results by employing a larger basis set, i.e. the aTZ extended with the 33211 set of midbond funtions. To carry out the basis set analysis, we select a geometry close to the absolute minimum (i.e., R = 3.582 Å). For the anthracene−argon dimer there is no existing CCSD(T) potential. After some test calculations carried out with the aDZ-33211 basis set for five intermolecular distances ranging from R = 3.2 to 3.6 Å with a step equal to 0.1 Å, we chose the geometry (R = 3.4 Å) corresponding to the lowest interaction energy to further perform the basis set selection study. Ground state interaction energies are evaluated within the supermolecular approach, that is as differences between the complex energies and the sum of the energies of the monomers. They are corrected for basis set superposition error through the 10290

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Table 1. Basis Set Selection Study at Selected Geometriesa C6H6−Ar

C10H8−Ar

basis set

N

R = 3.1000

R = 3.2000

R = 3.5547

R = 5.0270

R = 6.0000

aTZ aQZ aDZ-33211 aTZ-33211 (Z)Pol-33211 (Z)Pol-3321 (Z)Pol-33111 (Z)Pol-3322 (Z)Pol-33221 (Z)Pol-333221 apc1-332211 apc1-3331 apc1-33311 apc1-33321 apc1-333211 apc1-33322 apc1-333221 apc1-333222

464 840 257 502 214 205 209

17.34 −59.45 −23.21 −63.77 −22.35 22.51 −4.24

−158.08 −218.68 −198.54 −225.24 −199.31 −161.67 −183.87

−355.31 −380.25 −386.97 −389.58 −389.68 −370.30 −378.89

−87.11 −89.99 −92.38 −91.70 −95.53 −92.06 −92.35

−30.21 −30.96 −31.27 −31.54 −32.09 −31.52 −32.23

188 166 175 182

4.43 34.06 −21.85 −33.27

−175.47 −155.05 −198.92 −207.28

−370.02 −370.29 −383.17 −385.96

−82.89 −80.20 −84.93 −83.27

−28.75 −23.22 −26.97 −26.35

C14H10−Ar

N

R = 3.5820

N

R = 3.4000

367 732 298 289 293 296 305 343

−480.56 −479.60 −484.00 −472.72 −473.21 −478.26 −494.27 −498.94

477

−548.86

380 389

−546.27 −570.44

248 259 257 268 279

−458.93 −471.18 −478.12 −480.03 −480.39

325 323 334

−540.58 −546.52 −553.56

CCSD(T) interaction energies are given in cm−1, and intermolecular distances R in Å. N is the number of basis set functions. See text for basis set notation.

a

calculated energy (−546.52 cm−1) is even closer to the reference result (−548.86 cm−1).

the aXZ-33211 (X = D, T) basis sets. We can conclude that in the regions of the global minima the (Z)Pol-33211 basis set performs similarly to the aTZ-33211 basis set, while the apc133321 values are closer to the aDZ-33211 ones. This suggests that the (Z)Pol-33211 basis set should yield a better IPES and rovibrational spectra than apc1-33321. Nevertheless, the latter basis set has only 182 functions and we think it is worthwhile to test its performance in the evaluation of the potential and intermolecular states. Considering this, for the benzene−argon system we selected the apc1-33321 and (Z)Pol-33211 basis sets to obtain the IPES. In the case of the naphthalene−argon complex we reduce the basis set study to one geometry close to the absolute minima, and following a similar analysis as for the benzene−argon dimer, we selected the (Z)Pol-3322 and apc1-333221 bases. The further extension of these midbond functions by a set of diffuse functions overestimates the interaction energy in the case of the (Z)Pol basis set, but has an insignificant effect when using the apc1 set. Due to the large size of the anthracene−argon complex, basis set analysis for this system is very restricted. The largest basis set that we were able to employ was the aDZ basis set with the 33211 midbond functions. From the results obtained for the benzene−argon complex, one can see that the aDZ-33211 interaction energy is still not converged: the larger aTZ-33211 basis yields an interaction energy ca. 3 cm−1 lower. In the case of the naphthalene−argon complex the results seem more converged: both basis sets yield interaction energy values very close to each other, with the difference being ca. 1 cm−1. Keeping this in mind, we believe that in the case of the anthracene−argon dimer the aDZ-33211 interaction energy is already reasonably close to the basis set limit. The interaction energies calculated for the anthracene−argon complex, presented in Table 1, indicate that the value obtained with the (Z)Pol-3322 basis set (−546.27 cm−1) is close to the reference result (−548.86 cm−1), while the (Z)Pol-33221 set clearly overestimates the interaction energy. The apc1-33322

III. INTERMOLECULAR POTENTIAL ENERGY SURFACES For the benzene− and naphthalene−argon complexes, we fit the interaction energies to many-body expressions with exponential terms. A. IPES for Benzene−Argon. A coordinate system is selected, such that the benzene molecule resides in the xy plane, with the center of mass at the origin and two of the carbon atoms on the y-axis. We denote the position of the jth carbon atom cj = (xj, yj, 0), where the numbering is chosen so that taking a tour from c1 to c2 to ... to c1 corresponds to going once around a circle in the positive direction. The coordinates of the argon atom are denoted by R. As a first adjustable parameter, we introduce the dimensionless scaling s, and set vj = scj. When s > 1 and not too large, vj lies on a C−H bond, making it an effective position of the corresponding C−H unit in the molecule. We set dj = ∥vj − R∥/dil, where ∥vj − R∥ is the Euclidean distance between vj and R, and dil is an adjustable parameter. We also let Dj = 10 exp(−dj). The fit U(R) is a function of these distances and thus a many-body expansion. However, we account for the symmetry before we write up an expression. The group of symmetries of the hexagon (the dihedral group D6) acts on the virtual positions by permuting the index, i.e., a symmetry h ∈ D6 maps vj to vσ(j) where σ is a permutation of the numbers 1, 2, ..., 6. The 12 permutations obtained in this way is the group of permutations S. Since D6 is generated by a rotation by one-sixth and by a reflection, S is generated by the permutation, which written in cycle notation, reads (1 2 3 4 5 6) (corresponding to a rotation by one-sixth of a turn) and (1 6)(2 5)(3 4) (corresponding to a reflection). For a set of exponents K ⊂ 60 , and parameters ak, k ∈ K, we let 10291

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Table 2. Parameters of the Potential Fits C6H6−Ar (Z)Pol-33211

apc1-33321 s = 1.69438

dil = 1.51729 Å −103.32 300.62 33071.84 1235.46 7642.06 10411.30 −53192.93 −4714.06 1679.22 −40649.89

a0,0,0,0,0,1 a0,0,0,0,1,1 a0,0,1,0,0,2 a0,0,0,0,0,4 a0,0,0,0,2,2 a0,0,0,1,3,1 a0,0,1,0,1,3 a0,0,2,0,1,2 a0,0,0,1,2,3 a0,1,0,2,0,3

−25.50 −2144.96 2106.07 19949.04 397524.33 −417443.24 −15834.05 −8910.01 1406.62 −10740.08

a0,0,0,0,0,1 a0,0,0,1,0,1 a0,0,1,0,0,2 a0,0,0,1,0,3 a0,0,1,1,0,2 a0,1,0,1,0,2 a0,0,0,1,1,3 a0,0,1,2,0,2 a0,0,0,0,2,4 a0,0,0,2,2,2 C10H8−Ar

(Z)Pol-3322

a0,0,0,0,0,2 a0,0,1,0,0,1 a0,0,0,0,0,4 a0,0,0,2,0,2 a0,0,2,0,0,2 a0,0,0,0,0,5 a0,0,0,2,1,2 a0,0,0,0,1,5 a0,0,0,2,1,3 a0,0,1,0,0,5

66.52 838.33 −1896.86 −32790.07 25207.65 1438.52 33545.98 −1540.90 7424.74 4542.66

apc1-333221

dil = 1.64083 Å −61.89 −32.62 −88.69 41.53 −5851.87 318.01 −1831.65 −8272.18 2811.21 −2325.56 3912.41 211.70 3788.41 6188.01 −134.55

a0,0,0,0,0,0,0,0,0,1 a0,0,0,0,0,0,0,0,2,0 a0,0,0,0,0,0,0,0,4,0 a0,0,0,0,0,0,0,0,1,2 a0,0,0,2,0,0,0,0,2,0 a0,0,0,0,4,0,0,1,0,0 a0,0,0,0,4,0,2,0,0,0 a0,0,0,1,0,1,0,0,1,0 a0,0,0,0,0,1,0,0,2,2 a0,0,0,0,3,0,1,1,0,0 a0,0,0,2,0,2,0,0,0,1 a0,0,0,0,0,0,1,0,4,1 a0,0,0,0,1,4,0,0,0,1 a0,0,0,2,0,3,0,1,0,0 a0,0,1,0,0,0,2,0,3,0

s = 1.61261

dil = 1.50772 Å

−281.99 −599.04 −35406.51 −10763.62 4667.70 78864.02 −36654.76 −41.87 112432.79 −64071.38

a0,0,0,0,0,2 a0,0,0,1,0,1 a0,0,1,0,1,1 a0,0,0,0,1,3 a0,1,0,1,0,2 a0,0,0,2,0,3 a0,0,1,0,2,2 a0,0,0,0,1,5 a0,0,2,1,0,3 a0,2,0,2,0,2

dil = 1.75123 Å −47.33 −114.45 −626.06 −3480.07 1129.97 −152103.41 9207.14 9329.83 179.41 7451.47 −1928.13 608.43 −7806.07 2660.78

a0,0,0,0,0,0,0,0,1,0 a0,0,0,0,0,0,0,0,0,3 a0,0,0,1,0,0,1,0,0,0 a0,0,0,0,1,0,0,0,0,3 a0,0,0,0,4,0,0,0,1,0 a0,0,0,0,3,0,0,0,0,3 a0,0,0,1,0,1,0,0,0,1 a0,0,0,1,0,2,0,0,0,1 a0,0,0,0,1,2,2,0,0,0 a0,0,0,2,0,0,0,2,1,0 a0,0,0,2,2,0,0,1,0,0 a0,0,0,0,1,0,1,4,0,0 a0,0,0,0,2,0,0,3,0,1 a0,0,0,3,0,0,2,1,0,0

a0,0,0,0,0,0,0,0,1,0 a0,0,0,0,0,0,0,0,3,0 a0,0,0,0,0,0,0,0,2,1 a0,0,0,0,0,0,0,1,0,3 a0,0,0,1,0,0,4,0,0,0 a0,0,0,3,0,0,0,0,3,0 a0,0,0,1,0,1,0,2,0,0 a0,0,0,0,0,0,3,1,0,1 a0,0,0,0,1,0,1,3,0,0 a0,0,0,1,0,0,0,3,0,1 a0,0,0,3,1,0,0,1,0,0 a0,0,0,0,1,0,0,0,3,2 a0,0,0,1,0,0,2,0,3,0 a0,0,0,3,0,0,0,2,0,1 a0,0,1,2,0,0,0,3,0,0

−20.66 −69.03 200.15 −1308.78 1580.98 60930.41 −5900.76 −109.03 3894.08 −19705.39 −2654.92 7141.30 110.64 −14547.55 2701.66

a0,0,0,0,0,0,0,0,0,2 a0,0,0,0,1,0,0,0,0,1 a0,0,0,0,1,0,2,0,0,0 a0,0,0,0,3,0,0,1,0,0 a0,0,0,0,5,0,0,1,0,0 a0,0,0,0,1,0,0,0,2,1 a0,0,1,0,1,0,0,2,0,0 a0,0,0,0,0,1,0,0,1,3 a0,0,0,0,2,1,2,0,0,0 a0,0,0,1,1,0,3,0,0,0 a0,0,0,0,0,0,0,1,2,3 a0,0,0,0,2,1,0,3,0,0 a0,0,0,1,0,3,2,0,0,0 a0,0,0,3,0,0,1,2,0,0

−59.38 −1069.46 −588.53 1377.18 98.84 −1387.75 8050.88 5557.10 −1114.74 −1747.49 −941.18 1374.60 1125.18 −3548.62

Table 3. Minima on the IPESsa global minima C6H6-Ar

C10H8-Ar a

local minima

basis

x

z

E

x

y

ΔE

aDZ-33211 aTZ-33211 (Z)Pol-33211 apc1-33321 (Z)Pol-3322 apc1-333221

0 0 0 0 0.465 0.449

3.555 3.536 3.556 3.540 3.501 3.498

−387.0 −390.1 −389.6 −386.6 −483.2 −486.7

5.029 5.013 5.026 5.055 3.713 3.697

0 0 0 0 4.432 4.461

170.0 169.5 162.0 194.6 271.8 299.1

Energies are in cm−1 and coordinates in Å. The local minima energies are relative to the global minima.

U (R) =

∑ k∈K

=

∑ k∈K

1 ak 12

1 ak 12

only the Euclidean distance goes into computing dj; using C−H effective positions, there is no need to account for anisotropy. We discarded interaction energies over 1000 cm−1, and fitted the function U parameters to the interaction energies. The final parameters are given in Table 2. The fits are based on a total of 22 adjustable parameters. For the (Z)Pol-33211 IPES the rms error is 0.269 cm−1. The found function has minima at the geometries x = y = 0 and z = ±3.556 Å where the energy is −389.6 cm−1, and six local minima on the xy plane. The latter are located at x = 5.026 Å, y = z = 0, and the five geometries one obtains by rotating this point n-sixth of a turn, for n = 1, 2, ..., 5 around the z-axis. At these points the energy is −227.6 cm−1. For the IPES computed with the apc1-33321 basis set the fit has a rms of 0.285 cm−1, and minima at x = y = 0 and z = ±3.540 Å, where the energy is −386.6 cm−1. There are

∑ Dσk(1)Dσk(2)···Dσk(6) 1

6

2

σ∈S

∑ D1k

σ(1)

k

k

D2 σ(2)···D6 σ(6)

σ∈S

(1)

E.g., if k = (2, 1, 0, 0, 0, 0) then the inner sum in the expression for U equals D12D2 + D22D3 + ··· + D62D1 + D1D22 + D2D32 + ··· + D6D12

(2)

Notice that U is a polynomial in the Dj, and the polynomials we can obtain by the expression will automatically respect the hexagonal symmetry of the benzene molecule. Additionally, 10292

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Figure 2. Benzene−argon: Contour plots of the ab initio IPESs fitted by the analytic functions with the parameters specified in Table 2 in the (a) yz , (b) xz , and (c) xy planes. The values of subsequent contours differ by 30 cm−1.

essentially local minima at x = 5.055 Å (and at the other five symmetry equivalent positions) with energies of −192.0 cm−1. The results are summarized in Table 3, where they are also compared to the reference values. We can observe that at the global minima the (Z)Pol-33211 interaction energies are the closest to the aTZ-33211 values, but the intermolecular bond distances are closer to the aDZ-33211 ones, being the apc133321 values in this case are those that agree better with the aTZ-33211 reference. Regarding the local minima, the aTZ33211 and aDZ-33211 energies with respect to the IPES minima are quite similar, and the corresponding bond distances differ by only 0.016 Å, therefore the shape of both potentials

can be expected to be quite similar. In the case of the new potentials the (Z)Pol-33211 IPES is slightly flatter than the aTZ-33211 one at the local minima, which are located closer to the C−C bonds. At the local minima positions the apc1-33321 potential is considerably different from the others. These features of the IPESs are clearly visible in Figure 2, where contour plots of the three IPESs are depicted. Although the overall shape of the three potentials is similar, only the contours of the (Z)Pol-33211 and aTZ-33211 potentials lie close and in some zones nearly cover each other. This is especially well pronounced for the contours in the two planes perpendicular to the benzene molecule plane, namely the xz 10293

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Figure 3. Naphthalene−argon: Contour plots of the ab initio IPESs fitted by the analytic functions with the parameters specified in Table 2 in the (a) yz, (b) xz, and (c) xy planes. The values of subsequent contours differ by 100 cm−1 in parts a and b and by 50 cm−1 in part c.

and yz planes (Figure 2, parts b and a, respectively). The contours of the apc1-33321 potential indicate its slightly different topology with respect to the other two potentials. It is shallower and the positions of the minima are slightly shifted: for the global minima they are shifted closer to the benzene molecule, while the local minima appear slightly further with respect to the aTZ-33211 potential results. B. IPES for Naphthalene−Argon. We select a coordinate system, such that the naphthalene molecule resides in the xy plane, with the center of mass at the origin and such that the two central carbon atoms lie on the y-axis. Let vj j = 1, 2, ..., 10 denote the positions of the hydrogen atoms, except for v3 and

v8 which denote the coordinates of the two central carbon atoms, see Figure 1. The coordinates of the argon atom are denoted by R. As before, we set Dj = 10 exp [−(∥vj−R∥/dil)], where dil is an adjustable parameter. A symmetry of the molecule acts by permuting the indexes of vj; e.g. a reflection in the yz plane sends vj to vσ1(j), where σ1 is the permutation that can be succinctly written as (1 5)(2 4)(6 10)(7 9) in cycle notation. Similarly, reflection in the xz plane corresponds to the permutation σ2 = (1 10)(2 9)(3 8)(4 7)(5 6). The group of permutations S generated by σ1, σ2 has four elements. 10294

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Table 4. Calculated Intermolecular Bound States for the Benzene−Ar Dimer, Using the CCSD(T) Calculated Potentialsa k

energy

0 1 2 3 4 0 1 2 3 4 0

39.18 67.87 93.66 116.32 116.64

65.24 100.98 117.60

0.00 33.94 64.98 93.00 93.10 118.00

40.52 69.93 96.42 119.76 120.05

67.11 104.01 121.14

apc1-33321 78.15 122.28 128.76

39.47 (39.99) 68.09

65.25 (65.39) 101.31

aTZ-33211b 76.17 (77.11) 118.90

93.86 116.38 116.91 (117.17)

117.81

125.28

0.00 (0.00) 33.03 (32.8) 63.22 90.36 90.61

1 2 3

4 a

0.00 33.06 63.23 90.40 90.52 114.63

(Z)Pol-33211 75.65 118.66 125.05

95.26 130.53

109.04

118.99

127.10

98.16 134.36

112.48

122.68

130.93

95.51 (96.03) 130.92

109.72

119.28

127.36 (128.2)

114.68

Experimental values taken from ref 35 are shown in parentheses. The energies are given in cm−1. bTaken from ref 5.

For a set of exponents K ⊂ 10 0 , and parameters ak, k ∈ K, we let U (R) =

∑ k∈K

=

∑ k∈K

1 ak 4

1 ak 4

IV. INTERMOLECULAR STATES The intermolecular bound states are calculated by employing the BOUND program.34 The atomic masses of hydrogen and argon are taken as 1.007 825 032 07 and 39.962 383 1225 amu, respectively. The benzene moments of inertia calculated from this data equal 177.617 086 and 88.808 543 amu·Å2, while for the naphthalene molecule they are 571.832 782, 409.862 155, and 162.014 709 amu·Å2. The values of the intermolecular levels obtained with our potentials are collected in Tables 4 and 5 for the benzene− and naphthalene−Ar complexes, respectively. For comparison we

∑ Dσk(1)Dσk(2)···Dσk(10) 1

10

2

σ∈S

∑ D1k

σ(1)

k

k

D2 σ(2)···D10σ(10)

σ∈S

(3)

E.g., if k = (2, 1, 0, 0, 0, 0, 0, 0, 0, 0) then the inner sum in the expression for U equals 2 D12D2 + D4 D52 + D9D10 + D62D7

(4)

Table 5. Calculated Intermolecular Bound States for the Naphthalene-Ar Dimer, Using the CCSD(T) Calculated Potentialsa

We discarded interaction energies over 1000 cm−1, and fitted the function U parameters to the rest. The final parameters are given in Table 2. The fits are based on a total of 30 adjustable parameters. The stationary points are summarized in Table 3. For the (Z)Pol-3322 IPES, the rms error is 0.299 cm−1. The found function has minima at the geometries of (±0.465, 0, ±3.509) Å where the energy is −483.2 cm−1. The four symmetry equivalent local minima are located in the naphthalene plane at (±3.713, ±4.432, 0.000) Å with energies of 271.8 cm−1 with respect to the global minima. For the apc1-333221 IPES, the rms error is 0.395 cm−1 and the minima are at the geometries of (±0.449, 0, ±3.498) Å with an energy of −486.7 cm−1. The four local minima correspond to a geometry (±3.697, ±4.461, 0.000) Å with energies of 299.1 cm−1 with respect to the global minima. In Figure 3 the new potentials are compared to the MP2 surface of ref 8. The MP2 absolute minima are deeper than the CCSD(T) and the intermolecular distances shorter, due to the well-known MP2 overestimation of the interaction energies. Both in the proximity of the global minima and outside this region the CCSD(T) and the MP2 results differ significantly.

energy

a

10295

(Z)Pol-3322

apc1-333221

MP2/aDZb

0.0 5.8 15.3 24.7 25.6 29.7 36.5 38.4 41.7 46.6 47.9 49.5 50.3 55.1

0.0 6.1 15.8 25.1 26.5 30.8 37.7 40.0 43.0 48.6 50.1 51.1 51.7 57.4

0.0 6.9 16.4 26.5 27.2 35.3 37.8 44.0 45.3 49.2 51.7 54.8 56.1 59.6

The energies are given in cm−1. bTaken from ref 8. dx.doi.org/10.1021/jp508317z | J. Phys. Chem. A 2014, 118, 10288−10297

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also include the results obtained with the aTZ-33211 ab initio potential5 for the benzene−Ar dimer and the MP2/aDZ IPES8 for the naphthalene−Ar complex. For consistency, the bound states are recalculated with the above moments of inertia. In the case of the benzene−Ar dimer we also include the experimental data available.35 The benzene−Ar bound states evaluated with the (Z)Pol33211 potential are close to the aTZ-33211 ones. The energies of the vibrational levels calculated with the former potential lie mainly below those obtained with the latter, with the greatest difference being smaller than 0.70 cm−1. This could be expected, since, as was pointed out in the previous section, these two potentials are very similar. The differences between the apc1-33321 and the aTZ-33211 basis set potentials in the benzene−Ar case are more substantial and this is reflected in the calculated apc1-33321 intermolecular levels which lie higher than the aTZ-33211 basis set reference results. For the lowest states the differences are larger than 1 cm−1 and increase with the energy level, rising up to 3.60 cm−1 for the values presented in Table 4. The experimental assignment of the benzene−Ar bound states partially follows that made in ref 35. Since it was shown that the assignment of the higher states seems to be doubtful, the two highest observed states were related to the calculated states which are closest in energy. In the case of the naphthalene−Ar complex the situation is somewhat different. Since the apc1-333221 potential is deeper (in the global and local minima region) than the (Z)Pol-3322 one, the intermolecular bound state energies calculated with the former potential are larger. This can be clearly seen from the values collected in Table 5: the first transitions are 5.8 and 6.1 cm−1, respectively. This difference increases with increasing energy and for the highest calculated state it is 2.3 cm−1. A comparison of our results with the existing potential for the naphthalene−Ar complex reveals that the fact that the latter potential is substantially deeper is reflected in the values of the calculated rovibrational states. For instance, if one compares the results for the (Z)Pol-3322 and MP2/aDZ IPESs, the differences can reach up to 5.8 cm−1. This can result not only from the differences in the potential depths, but also from those in their overall shapes.

reproduce the intermolecular potential correctly. The differences between the CCSD(T)/apc1-33321 and the CCSD(T)/ aTZ-33211 potentials are more pronounced: the former is by ca. 3 cm−1 shallower in the position of the global minimum but in the local minima this difference increases to 25 cm−1. This can be clearly seen from the depicted contours of the three potentials. For the naphthalene−argon complex the calculated CCSD(T) potentials differ significantly from the MP2 surface in ref 8. To further check the performance of the calculated potentials, we evaluate the intermolecular vibrational states of the complex. For the benzene−argon dimer, the energy levels clearly indicate that the CCSD(T)/(Z)Pol-33211 and CCSD(T)/aTZ-33211 potentials are indeed very similar: the differences for the calculated vibrational energies using these two potentials do not exceed 0.70 cm−1. For the CCSD(T)/ apc1-33321 IPES such differences are much larger and increase up to 3.60 cm−1. In the case of the naphthalene complex the results for the two obtained potentials are similar; however, the differences in the rovibrational states increase with increasing energy. A comparison of our results with the existing MP2 surface from ref 8 proves that the latter potential is significantly different, being much deeper than ours. We also present some results for the anthracene−argon complex. From the basis set selection study we can conclude that both the (Z)Pol-3322 and apc1-33322 bases are good choices for further evaluation of the IPES of the complex. This study proves that the use of the small and flexible (Z)Pol and apc1 basis sets in combination with midbond functions and a high-level correlation ab initio method for calculations of interaction energies for van der Waals systems of small to medium size leads to IPESs of accuracy comparable to those obtained with relatively large correlate-consistent basis sets (e.g., of Dunning’s type).

V. SUMMARY AND CONCLUSIONS We evaluate CCSD(T) IPESs for the benzene− and naphthalene−argon van der Waals complexes, using the (Z)Pol-33211 and apc1-33321 basis sets for the benzene−Ar complex and the (Z)Pol-3322 and apc1-333221 bases for the naphthalene−Ar complex. These bases were selected from systematic basis set studies carried out in order to obtain smaller basis sets than Dunning’s, but at the same time flexible enough to correctly describe the surface topologies. We considered the sets of Sadlej et al. and the augmented polarization-consistent bases of Jensen; these bases were further extended by adding midbond functions. As reference we took the available CCSD(T) aXZ-33211 (X = D, T) potentials for benzene−argon and aXZ-33211 (X = D, T) interaction energies for naphthalene−argon. The new basis sets are considerably smaller than Dunning’s bases of double- and triple-ζ quality. The interaction energy values and the potential shapes are close to those previously available. However, for benzene−argon there is a clear similarity between the CCSD(T)/(Z)Pol-33211 and the CCSD(T)/aTZ-33211 potentials, while the apc1-33321 basis set is not able to

Corresponding Author

■

ASSOCIATED CONTENT

S Supporting Information *

Table of the exponents of the different sets used. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank F. Calvo for providing us the potential they used in their study of the naphthalene−argon complex.. This work was supported by the Spanish Ministerio de Ciencia e Innovación (CTQ2011-29311-C02-01 project), the Polish National Science Centre (project No. 3714/B/H03/2011/40), and the Foundation for Polish Science within the Homing Plus programme (Homing Plus/2010-1/2), cofinanced from the European Regional Development Fund within the Innovative Economy Operational Programme. This research is a part of the program of the National Laboratory FAMO in Toruń, Poland. This research is partially supported by the Foundation for Polish Science TEAM Project cofinanced by the EU European Regional Development Fund. Centro de Supercomputación de Galicia (CESGA) is acknowledged for computational resources. 10296

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