Faraday Symp. Chem. SOC.,1984 19 125-135 Ab Initio Calculations on Weakly Bonded Systems TEUS B. VAN DUIJNEVELDT BY JOOP H. VAN LENTHE,* VAN DAMAND FRANS Theoretical Chemistry Group University of Utrecht Padualaan 8 De Uithof Utrecht The Netherlands AND LOESM. J. KROON-BATENBURG Laboratorium voor Kristal- en Structuur-chemie University of Utrecht Padualaan 8 De Uithof Utrecht The Netherlands Received 16th August 1984 Large-basis CI calculations are performed on the van der Waals complexes Ar-HC1 and (H,O),. It is shown that a reasonable estimate of the CI basis-set superposition error is obtained from a ghost calculation involving the orbitals of the monomer and only the virtual orbitals of the ghost. Both basis-set superposition error corrections and size-consistency corrections are of vital importance to obtain a reliable potential-energy surface.For Ar-HC1 the minima of the potential are predicted within 50 phartree of the experimental surface viz. -804 phartree for the Ar-HCl orientation and -565 phartree for the HC1-Ar geometry. The water dimer van der Waals minimum is estimated to be -4.9 kcal mol-l which is less deep then the experimentally derived minimum of -5.4 kcal mol-l but just within the experimental error limit. The use of supermolecule ab initio CI techniques for the calculation of potential- energy surfaces for weakly bonded (van der Waals) systems has been frought with difficulties.lP6 In 1981 Maitland et a/.’ wrote ‘In general except for the case of He, where significant progress is being made towards the calculation of the complete potential function there is no immediate prospect of calculating an accurate potential.’ Accurate CI calculations require large basis sets for a proper representation of both intra- and inter-molecular correlations so huge numbers of configuration state functions (CSF) are involved. Only in recent years with the advent of the new generation of supercomputers (Cray I Cyber 205) and CI have CI calculations involving lo5CSF and extensive basis sets become routinely possible. It is therefore timely to establish reliable procedures to perform CI calculations on moderately sized van der Waals systems of chemical interest. As one prototype system the Ar-HCl system was chosen. This complex has been extensively studied experimentallyl99 l1 and the uncertainty in the empirical potential- energy surface is thought to be 50 phartree.? The complex is weakly bound the well depth being only 800 phartree (0.5 kcal mol-l) and only a minor part of this is reproduced at the SCF leve1,12 so the system is well suited to test CI schemes.No satisfactory CI surface has yet been published however.’ In order to be useful a scheme has to be applicable to a wide range of van der Waals t 1 hartree = 4.359814 x 10-l8 J. 125 CI CALCULATIONS * /’ / / / CIL (a) (b) Fig. 1. (a)Ar-HC1 coordinates. The HCl centre of mass is the origin for the (R 0) coordinates. (b) (H,O) coordinates. The origin is at the acceptor oxygen atom. systems. Therefore our second test system is the H20 dimer where the interaction energy is an order of magnitude larger and the induction energy constitutes a significant part of the interaction potential.The experimentally derived well depth of 5.4 kcal mol-1 has a large uncertainty of 0.7 kcal m01-l.~ A number of potential- energy surfaces have been calculated using CI the most recent one 37 59 being that calculated by Clementi and Habitz.6 An important systematic defect of supermolecule finite-basis calculations is the basis-set superposition error (BSSE).5yl3 Methods to deal with this problem at the CI level include avoiding it by the choice of CSF,2 ignoring it4 and calculating it using the standard Boys-Bernardi scheme and either correcting for it5 or In this paper we will explore an alternative CI scheme which yields reasonable estimates of the BSSE thus producing reliable results for both the van der Waals systems mentioned above.The importance of either using a size-consistent method or applying size-consistency corrections is demonstrated. For a long time SCF calculations augmented by an estimate of the dispersion energy from second-order perturbation theory14 have been an important ab initio source of potential-energy surfaces. Since these calculations are considerably cheaper than CI calculations it is worthwhile to compare the results of the two approaches. COMPUTATIONAL PROCEDURE The interaction energy for a complex with a geometry defined by a distance R and angle 0 (see fig. 1) is given by where E(R,O) and E(R = 100) are the single- and double-excitation CI (SDCI) energies resulting from dimer calculations at the specified geometry and at large distance and EBssEand E, are the basis-set superposition error and the size-consistency correction respectively.The dimer calculation at a large distance is used instead of the separate monomer calculations to avoid the bulk of the size-consistency error which for our Ar-HCl calculation (basis I) is 0.0107 hartree. E, corrects for the change in the size-consistency error with the geometry in the dimer calculation. EBssE is calculated from size-consistent monomer calculations making the Stone and Price correction15(ca. 50 phartree for Ar-HC1) superfluous. For Ar-HCl we calculated E, using a correction formula due to Pople,16 since it proved to reduce the size-consistency error at infinity to only 250,uhartree an order of magnitude better than the performance of the Davidson f0rmu1a.l~ For the H,O dimer the size-consistency effects were included by employing the CEPA( 1) method as implemented in the ATMOL~direct CI program.VAN LENTHE VAN DAM VAN DUIJNEVELDTAND KROON-BATENBURG 127 Table 1. Basis sets used in the Ar-HC1 and (H,O) calculations ~~~ ~ denotation atom(s) +reference composition basis I" Ar Cl HZ9 basis 11" Ar C1 H29 DZP' 0,HZ1 DZPP 0,HZ1 EZPM 030 H21 7 EZPP 030 H21 7 CH 0,H6 a Bases I and I1 correspond to bases I1 and I11 of ref. (1 2). In all CI calculations the inner shells were kept frozen and the corresponding virtual orbitals were deleted.For Ar-HCl this concerns the Is 2s and 2p orbitals of Ar and C1; for H20only the oxygen 1s orbital was frozen. All CI calculations were performed using the ATMOL program system as implemented on the Cray 1 S in Daresbury/London and Rijswijk the Cyber 205 in Amsterdam and the Cyber 175 of the University of Utrecht. A typical calculation on one geometry of Ar-HC1 took ca. 20 min of Cray 1s C.P.U. time. BASIS SETS The GTO basis sets are characterized in table 1. The Ar and C1 sp basis sets are derived from 12s,9p bases of Huzinagalg and Veillard,20 respectively by replacing the four most diffuse s functions by 6 gaussians and the three most diffuse p atomic orbitals by 4 gaussians in order to obtain a better description of the valence regions and the exchange repulsion between the monomers.The extended-double-zeta basis (EZ) for oxygen is obtained similarly from the double-zeta (DZ) basis.21 The Ar-HC1 polarization functions were spherical harmonic gaussians. The high-exponent C1 d and H p functions were chosen to minimize the HCl Hartree-Fock energy while the diffuse Cl d and H p as well as the Ar d and fand C1 fgaussians were chosen to maximize the dispersion energy at 10 bohrt distance and 0". In basis I1 one d gaussian on Ar is optimized for the intra-atomic correlation while the other as well as the d gaussian on hydrogen is meant to maximize the dispersion energy The H20 dimer calculations employed Cartesian gaussian orbitals. The single set of polarization functions used in the DZP basis is optimized to reproduce good Hartree-Fock multipole moments and polarizabilities.21 The EZPM basis is meant to mimic the basis set used by Clementi and Habitz6 and uses their p exponent of 0.4 for hydrogen.GEOMETRIES The geometry parameters of the dimer are shown in fig. 1. The monomer geometries were kept fixed in all calculations. The HCl distance used is 2.426 bohr,22 the water monomer geometry is taken from e~periment.~~ 1 bohr = 5.2917706 (44)x lo-" m. CI CALCULATIONS BASIS-SET SUPERPOSITION ERROR SCHEMES For the estimation of the BSSE in Hartree-Fock calculations the Boys-Bernardi13 counterpoise (ghost-orbital) method is widely used although there is still an argument as to whether the complete orbital space of the ghost molecule should be included or only the virtual orbitals.For moderately sized van der Waals molecules like the ones studied here it may be possible to circumvent this problem by extending the basis so that the BSSE is negligible. For the large basis sets used in accurate CI calculations the SCF BSSE is indeed very small. However even for our largest basis set the CI BSSE for Ar-HCl is of the same order of magnitude as the well depth of the van der Waals minimum so it is vitally important to obtain a proper estimate. DlMER C1 GHOST C1 (1) I I I I 1 I I occupied A I occupied B r l laTypdl] ;uj+; Fig. 2. Schematic representation of the three CI procedures scheme (1) SCF molecular orbitals; scheme (2) combined monomer molecular orbitals ;scheme (3) combined monomer molecular orbitals.We compare three schemes to perform CI calculations and obtain estimates of the BSSE. They are represented pictorially in fig. 2. (1) The standard Boys-Bernardi recipe the dimer CI calculation is performed in the basis of the dimer SCF orbitals. The ghost-orbital CI calculations employing the complete orbital space are preceded by ghost SCF calculations. (2) In order to be able to separate orbital spaces no dimer SCF calculation is carried out. The dimer occupied space consists of the occupied orbitals of the monomers suitably orthonormalized. The dimer virtual space is obtained by orthogonalizing the combined monomer virtual orbitals onto the occupied space. The BSSE CI calculations employ the occupied space of the monomer augmented by the same virtual orbitals as used in the dimer calculation.In this scheme exactly the same intramolecular excitations that are possible for the monomer in the dimer calculation are also supplied in the ghost calculations. This comes closest to the original Boys-Bernardi concept of employing identical function spaces in monomer and dimer calculations. (3) The dimer CI calculation is identical to that of scheme (2). For the ghost calculations the combined but not yet orthogonalized monomer +ghost virtual spaces are added to the monomer occupied orbitals and then suitably orthogonalized. In this scheme which was originally proposed by Daudey VAN LENTHE VAN DAM VAN DUIJNEVELDT AND KROON-BATENBURG 129 Table 2.Ar-HCl SDCI interaction energies at R = 7.53bohr 8 =0” in various schemes (in phartree)a BSSE scheme pncorrected Ar HCl int 0 -1266 0 0 -261 -1527 -1804 -1 -1266 -906 -237 -261 -383 2 -1107 -655 +lo2 -442 -996 -1063 3 -1107 -761 -142 -442 -646 -785 exptllO -824 a Components are given in basis I. et al.,24the complete unperturbed monomer space is employed to which only the virtual orbitals of the ghost are added. By contrast in scheme (2) the ‘occupied’ orbitals of the ghost are projected out of both virtual spaces. Finally we define a scheme (0) which involves a CI calculation as in scheme (1) and no BSSE correction. The reasoning behind the exclusion of the occupied ghost orbitals in schemes (2) and (3) is as follows.The ghost calculations are meant to simulate the extra freedom (correlation or excitation possibilities) that the monomer obtains in the dimer calculation and so to estimate its artificial energy lowering. Since the occupied orbitals of the other monomer are not available for this purpose in the dimer calculation they should also be excluded from the ghost calculation. The situation is quite different for SCF calculations. It has been 26 that the use of monomer orbitals calculated in the full dimer basis set (large-basis monomers) yields a more consistent first-order interaction energy than when using small-basis monomers implying that large-basis orbitals are the most natural start for a dimer calculation. Furthermore it may be argued that in the dimer SCF calculation at least part of the occupied space of the ghost is available to the monomer since the ghost’s Hartree-Fock function being in its variational minimum is easily ‘pushed away’ during the dimer SCF process.Finally it has been that using the full orbital space in the ghost calculation does yield SCF results that are most independent of the basis set. Considering the argument given above there are two possible choices for the occupied orbitals in the CI calculations in schemes (2) and (3) uiz. either small-basis or large-basis SCF monomer molecular orbitals. In the original Daudey scheme only the small-basis case was considered. The procedure for the large-basis case is exactly as in scheme (3) except that the occupied molecular orbitals are taken from monomer SCF calculations in the full dimer basis.Since the SCF BSSE is small in our Ar-HCl calculations (<50 phartree) there is almost no difference between the two approaches and we will use small-basis monomers. For the H,O dimer there is a substantial difference between the two approaches in some basis sets so the problem will be addressed there. Ar-HCl COMPARISON OF BSSE SCHEMES The results of schemes (O)-(3) for a point close to the experimental minimum are given in table 2. It is obvious that if no correction is applied [scheme (O)] no useful 5 FAR CI CALCULATIONS Table 3. Properties of Ar-HCl obtained with method (3) and size-consistency correctionsa property SCF +dispersion basis I basis 11 e~periment~~*~~,~~ RdO") 7.49 7.73 (0.06) 7.69 (0.06) 7.58 Emin(0") -999 -671 (6) -804 (6) -824 Re(180") 6.64 6.93 (0.00) 7.05 (0.00) 7.04 Emin( 180") -1087 -598 (1) -565 (1) -616 (Pl) 0.38 0.62 (0.02) 0.76 (0.01) 0.67 -32) 0.26 0.35 (0.02) 0.46 (0.04) 0.34 M3OO mb -75.3 -20.0 -25.5 -35 a R in bohr E in phartree and B12in cm3 rnol-'.Fit-error estimates in parentheses. The long-range parts (R > 9) of the fitted CI potentials were replaced by the experimental potential. lo result is obtained. As expected scheme (1) overestimates the BSSE and much too small interaction energies are obtained. Only the schemes where the occupied orbitals of the ghost are omitted yield results close to the experimental value. A fundamental problem with scheme (2) can be observed in table 2.The BSSE for the HCl molecule turns out to be positive i.e. the orbital basis in the BSSE calculation is less suited to describing the correlated monomer than the original monomer basis. This is due to the fact that the occupied orbitals of the ghost are projected out of the monomer virtuals. It is easy to see that in an infinite basis (in which the BSSE ought to be zero) scheme (2) always yields a positive BSSE (the BSSE is defined as the difference in energy between the ghost calculation and the isolated monomer). The problem might be avoided by using as a reference instead of the isolated monomer a monomer whose virtual orbitals are maltreated in the same way as in the ghost calculation. This however would make the calculations even more cumbersome.We therefore deem scheme (3) to be the best procedure to perform CI calculations on van der Waals complexes. At first sight a drawback of schemes (2) and (3) seems to be that the SDCI dimer calculation cannot fully reproduce the mutual polarization effects that are present in the omitted SCF dimer calculation. Indeed in schemes (2) and (3) the uncorrected CI interaction energy is 160 phartree smaller than in scheme (1). However the size-consistency correction (Esc) when added to @z!orrected has the remarkable property that the difference between the schemes is reduced to only 22phartree (table 2). Thus the size-consistency correction seems to be able to replace the SCF relaxation step to a large extent. The same effect is observed in (H,O) calculations.The size-consistency correction is an important contribution to the interaction energy in all schemes. In scheme (3) it is a decisive factor in obtaining a good anisotropy of the potential-energy surface.12 RESULTS We calculated the Ar-HC1 interaction energy using BSSE scheme (3) for three distances (6.6 7.1 and 8.0 bohr) and three angles (0 90 and 180") in both basis I and basis 11. In order to obtain an optimal representation of the potential-energy surface the origin of this grid was laid 0.5 bohr from the Cl atom. In basis I we also calculated three points at 39.2 and at 140.8" the zero points of P3(cos6). The points were fitted with a three-term exponential function for the R coordinate and up to P,(cos 6) and VAN LENTHE VAN DAM VAN DUIJNEVELDT AND KROON-BATENBURG 131 I I I 1 I I 0 45 90 135 180 el" Fig.3. Ar-HC1 potential-energy variation with 6 at R = 7.6 bohr (a)CI (basis 11) (b)HHMS. P2(cos8) in the angle for the basis I and I1 results respectively. Using these fits the lowest vibrational-rotational wavefunction was calculated yielding spectroscopic observables. In table 3 we present a comparison between our CI results and the experimental values. In parentheses we give estimates of the errors caused by the limitations of the fit and the small number of points.12 All errors are surprisingly small indicating that even only 9 points of the potential-energy surface (basis 11) yield a reasonable description of the surface around the van der Waals minimum.Also provided in table 3 are results from SCF +dispersion12 calculations using second-order perturba- tion theory and basis I which being an order of magnitude cheaper were calculated for many points. The 0" well depth and equilibrium distance determined in basis 11 agree well with experiment; the well depth is within the experimental error limits. The remaining difference may be due to the fit error and the lack of polarization functions higher thanf. From data on Ne227 we estimate the contribution to the dispersion energy due to these functions to be ca. 60 phartree at Re yielding an estimated well depth that remains within the experimental error bound of 50 phartree. The basis I1 180" well depth shows a larger difference with the HHM5lO surface; the minimum is even shallower than in basis I which results from a larger exchange repulsion due to the better description of intra-atomic correlation of the Ar atom in basis 11.On the other hand the inclusion of g functions may lead to some lowering. The experimentally determined surface is uncertain in the 180" region so our minimum may well be the more accurate one. The main defect of our calculated potential-energy surface is that the angular barrier is probably too high as may be seen from fig. 3. This would account for the high values obtained for (PI) and (P2) as well as for the fact that the mixed second virial coefficient B, is too small in spite of the essential correctness of the minima. We suppose that the inbalance in the angular behaviour is caused by the fact that our diffuse polarization functions were optimized at O" thus biasing the basis set 5-2 CI CALCULATIONS towards a good description of the 0" region.In both basis sets the perpendicular dipole polarizability of HCl is considerably further from the experimental values than the parallel one. From these polarizabilities we estimate that Ckis 7 % too low with respect to Cl. With a dispersion energy of ca. 1500 phartree near the barrier this accounts for a barrier which is 100 phartree too high. Finally we would like to assess the SCF +dispersion calculations. While they correctly predict two minima one at 0" and one at 180° the balance between 0 and 180" is wrong. The perturbation estimate of the dispersion energy is not sufficiently accurate and yields only part of the total correlation energy.For weakly interacting systems like Ar-HC1 the SCF +dispersion approach therefore yields at best qualitative results. H20 DIMER COMPARISON OF BSSE SCHEMES Since for Ar-HCl scheme (3) was shown to produce the most reliable results all calculations on (H20) were performed according to this procedure. The only variation is in the choice of the occupied monomer orbitals. Both large-basis (L) and small-basis (S) results are shown. Only one CEPA dimer calculation (viz.starting from small-basis molecular orbitals) is used in calculating interaction energies. This is justified because although a dimer SDCI using large-basis monomers is 0.3 kcal mol-1 lower than the small-basis variant the difference between the CEPA calculations is only 0.03 kcal mol-1 in the DZP' basis at the geometry studied.Moreover a CEPA calculation employing dimer SCF orbitals yields an energy only 0.14 kcal mol-l lower than the combined large-basis monomer results. Thus in accordance with the Ar-HCl result the (size-consistent) CEPA calculations are remarkably invariant with respect to the choice of occupied orbitals. In table 4 we present results for a geometry close to the experimentally derived van der Waals minimum. While the uncorrected CEPA interaction energies differ significantly from one another the BSSE correction especially the large-basis variant succeeds well in making the interaction energies more nearly basis-set independent.The BSSE is far from negligible ranging from 2.3 kcal mol-1 in the DZP' basis to 0.55 kcal mol-1 in the EZPP basis. RESULTS The main error sources in our EZPP calculation are the lack off and higher polarization functions and possibly the missing dimer SCF step. As mentioned above the latter step gives an additional 0.14 kcal mol-1 in the DZP' basis. For the Ne dimer it has been demonstrated2' that using a basis up to d yields ca. 80% of the total dispersion energy at the van der Waals minimum implying that addingfand higher polarization functions may deepen the calculated well depth by 0.4 kcal mo1-1.2s We may therefore expect the limiting value for the well depth of the van der Waals minimum of the water dimer to be 4.9 kcal mol-I just within the experimental error limits.Clementi and Habitz6 found an interaction energy of -5.5 kcal mol-l only corrected for the SCF BSSE. The second virial coefficient calculated for their surface is a factor of 2 too high indicating too much attraction. When they decreased the correlation contribution to the interaction energy by an arbitrary factor of 2 the virial coefficient was in good agreement with experiment and the calculated van der Waals minimum was -4.7 kcal mol-l a result supported by our above estimate. Indeed a calculation using their basis set and the present procedure (column headed CH in table 4) gives VAN LENTHE VAN DAM VAN DUIJNEVELDT AND KROON-BATENBURG 133 Table 4. (H,O) CEPA interaction energies at R = 2.96 A 8 = 40” (in hartree)a basis DZP’ DZPP EZPM EZPP CH pncorrected -0.011 504 -0.008 579 -0.008 105 -0.007 824 -0.008 600 int EBSSE (S) -0.002 727 -0.001 212 -0.000 829 EBSSE (L) -0.003 748 -0.001 485 -0.000 969 -0.000 874 -0.001 232 int Eorrected (S) -0.008 777 -0.007 367 -0.006 995 Forrected (L) int -0.007 756 -0.007 094 -0.007 136 -0.006 950 -0.007 369 porrected (L) 74-87 -4.45 -4.48 -4.36 -4.62 int /kcal mol-1 a L and S refer to large- and small-basis monomers respectively.Table 5. (H,O) equilibrium energies and geometries from SCF+dispersion calculations28a SCF +dispersion multipole corrected basis R e E R e E DZP‘ 2.87 52 -5.83 2.9 1 56 -4.97 DZPP 2.86 46 -5.71 2.91 52 -4.93 expt15 --2.98 58 -5.4 a R in A 8 in O E in kcal mol-l. a dimerization energy of -4.6 kcal mol-l much less than their -5.5 kcal mol-l result.The other basis sets in table 4 are seen to reproduce the EZPP result rather closely the largest discrepancy arising for the DZP’ basis. This discrepancy was not expected since calculations at the SCF +dispersion level at a variety of (H,O) geometries had shown excellent agreement between the DZP’ and DZPP results.2s A summary of these results is shown in table 5. Because the SCF multipole moments differ considerably from the far more accurate CEPA ones there is a large error in the electrostatic part of the interaction energy. This error has been compensated in the entries under ‘multipole corrected’. The multipole-corrected DZP’ energies and equilibrium geometries for (H,O) agree well with the corresponding DZPP results.However the resulting Re is much shorter than the experimental value (table 5). Also the resulting DZPP interaction energy is significantly larger than in the present CEPA-CI approach (table 4). These differences seem to be due to the neglect ofintramolecular correlation effects in the SCF +dispersion approach. When we varied R in CEPA calculations in the DZP’ basis at 9 = 30° the well depth came out to be 4.75 kcal rnol-l the same value as in the corresponding multipole-corrected SCF +dispersion calculations (4.74 kcal mol-l) but the equilib- rium distances differed considerably (3.00 as against 2.93 A). This shows that the intramolecular correlation is important in obtaining a correct equilibrium distance.The DZP’ basis through its lack of high-exponent polarization functions under- estimates these intramolecular correlation effects thus leading to a larger CEPA interaction energy than in the EZPP basis. 134 CI CALCULATIONS CONCLUSIONS (i) In dimer CI calculations on van der Waals complexes neither the size-consistency corrections nor the BSSE may be neglected. (ii) The best way to estimate the BSSE in CI calculations is to perform the ghost-orbital calculation in the monomer basis augmented by only the virtual space of the ghost as suggested by Daudey et al.24(iii) If the occupied orbitals in the CI BSSE calculation are obtained in an SCF ghost calculation (large-basis monomers) the resulting interaction energies are more basis-set independent than when using the occupied orbitals of the isolated monomers.(iv) The calculated potential-energy surface for Ar-HCl is well within the experimental error limit of 50phartree at the van der Waals minimum but its anisotropy shows an inbalance. (v) The estimated van der Waals well depth for the H20 dimer is 4.9 kcal molt1 which is considerably less attractive than the experimentally derived result of 5.4 kcal mol-l but just within the bounds of experimental error. (vi) For weak van der Waals complexes like Ar-HCl the SCF +dispersion approach only yields approximate results. 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