Journal of Molecular Structure (Theochem) ,287 (1993) 139- 147 0166-1280/93/%06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

139

MBPT study of the CO-HZ van der Waals interaction

Mary C. Salazara, Antonio J. Hernhdez*va, Geerd H.F. Diercksenb

‘Department0 de Quimica, Universidad Simdn Bolivar, Apdo. 89000, Caracas 1080A, Venezuela bMax-Planck-Institut fir Astrophysik, karl-Schwarzschild-Str. 1, D-8046, Garching bei Mtinchen, Germany

(Received 12 October 1992; accepted 9 March 1993)

Abstract

The many-body perturbation theory (MBPT) is employed for the calculation of the intermolecular interaction potential for the linear CO. . . Hz dimer in the framework of the supermolecule approach, using the POLl basis described recently by Sadlej. This basis set, designed to reproduce electric properties of molecules accurately, leads to a correct calculation of the counterpoise-corrected interaction energies, in excellent agreement with results obtained in the larger basis set used in this study for comparison, and with the best theoretical values available to date. The minimum of the MBPT(4) interaction energy curves occurs at 8.14 and 8.03 q,, with well depths of 9.16 and 10.66 meV for the POLl and larger basis set calculations, respectively. In agreement with very recent spectro- scopic results, a vibrationally bonded ground state is also predicted for this linear dimer.

Introduction

Knowledge about intermolecular interaction potentials is required for the interpretation of a wide variety of chemical, biological, and physical phenomena. The origin of weak interactions, with interaction energies hardly exceeding several hun- dreds of wavenumbers, is based on the work of van der Waals and his well known state equation for real gases, and through the rise of quantum theory on the fundamental work of Heitler and London [l], and London’s famous description of the weak intermolecular dispersion forces that keep rare gas atoms together [2,3]; these result in what are nowadays termed van der Waals (vdW) complexes.

Quantitative knowledge of vdW interaction potentials has become more important recently because of the development of high resolution spectroscopic techniques used to gather infor- mation on the properties of vdW molecules [4], which requires improved theoretical understand-

* Corresponding author.

ing of the forces that determine their structure. The availability of powerful supercomputers and the development of efficient computational algo- rithms have also made possible the quantum mechanical study of vdW interactions of small to medium sized molecules [5,6]. Despite many tech- nical difficulties, the fully ab initio methods offer a sound basis for the calculation of vdW potential energy surfaces, valid over the whole range of molecular distances and orientations. They typi- cally fall into two categories: those which regard the interacting atoms or molecules as a ‘super- molecule’ [7]; and those which treat the intermolec- ular interaction as a small perturbation [8]. The perturbation approach is at first sight far more suitable for the calculation of weak interactions because the interaction energy is calculated directly, rather than as the difference between two large almost identical numbers, i.e. the difference between the total energy of the supermolecule and the energy of the subsystems. Nevertheless, because all the ab initio methods developed in single-mole- cule calculations are in principle applicable without

140 M.C. Salazar et al.lJ. Mol. Struct. (Theochem) 287 (1993) 139-147

change, the vast majority of calculations of the interaction energy of vdW complexes are carried out at present using the supermolecule approach. Computational experience shows that large and flexible basis sets must be used in both cases to obtain the required spectroscopic accuracy [9- 241. In practice, the basis set size must be kept within reasonable limits, especially for medium sized and large molecules.

The requirements that must be met by the basis sets follow directly from a qualitative classification of the vdW interaction energy in the various ranges of intermolecular separations. Long range inter- actions, where the electronic exchange is negli- gible, lead to the requirement that electric multipole moments and polarizabilities of the indi- vidual subsystems and the dispersion energy inter- action in the vdW complex must be accurately represented. The short range exchange repulsion, being a direct consequence of overlap effects, requires an adequate description of the valence electron density, both close to and far from the nuclei. Most difficulties occur in the region of mod- erate overlap between the charge distributions of the interacting subsystems, where the use of multi- center basis sets often leads to overestimates of the interaction energy of the vdW complexes, which is mainly attributable to what is known as the basis set superposition error (BSSE) [25], arising from the tendency of basis orbitals on one system to be variationally mixed with those on the others, giving spurious energy improvements.

The success of the medium-size polarized basis set developed by Sadlej [26] in calculating dipole and quadrupole moments, and field and field- gradient polarizabilities at SCF and correlated levels of approximation for conventional closed shell molecules is well documented [26-331. This basis set, designed to reproduce the electric proper- ties of molecules accurately, leads to a correct calculation of intermolecular electrostatic forces, to a negligible secondary BSSE [34], accounts for the diffuseness of the valence part of the wave- function, and offers the attractive possibility of reducing the size of the atomic basis without affect-

ing the accuracy of the calculated electrical moments and polarizabilities. These advantages have already been exploited in recent perturbation theory calculations of the hydrogen bonding in co* . . . HFandNzO... HF [24], and in the poten- tial energy surface of the H20-H2 and Ar-CH4 vdW systems [ 10,l I]. The reliability and predictive value of the medium sized extensive polarized basis set derived by Sadlej [26] is further tested in the present work in the calculation of the potential interaction energy of the CO-H2 vdW dimer by means of the many-body perturbation theory (MBPT) [35-371. All contributions to the total correlation energy up to fourth order due to single (S), double (D), triple (T), and quadruple (Q) exci- tations from the reference SCF determinant are included. Calculation of intermolecular interao tion energies by means of MBPT have several advantages compared to other methods [38] owing to its uniform and systematic treatment of the electron correlation contributions, and to its computational simplicity and effectiveness.

Previous studies on the CO. . . H2 potential sur- face have been performed by combining large basis set SCF calculations with semiempirical damped long range dispersion coefficients [39], using inter- molecular perturbation theory up to and including second order [40,41], and by calculations done in the framework of the electron gas model [42]. Of these studies, the complete potential surface of Schinke et al. [39] has given the best overall descrip- tion of the experimental differential cross sections and differential energy loss spectra [43-451. It has also successfully described in detail the vibration energy transfer from Hz to CO [46] and the pres- sure broadening cross section for CO. . . Hz [47].

In the present paper, we have studied the CO.. . Hz system again on the basis of the long term objective of the present research study, namely the investigation of the vibrational- rotational spectra for the ground electronic state of the CO ... Hz vdW dimer from accurate fully ab initio calculations of its potential energy sur- face. We feel that previous semiempirical model potentials may lack general validity insofar as

M.C. Salazar et al./J. Mol. Struct. (Theochem) 287 (1993) 139-147 141

they depend either on the functional forms chosen or on the properties used in fitting the parameters. The main motivation for undertaking the study of the spectroscopic properties of the CO.. . Hz system follows from the observation that, despite the enormous development of high-resolution spectroscopic techniques used to gather infor- mation on the bonding properties of vdW com- plexes [4], it was not until very recently that high resolution spectra of CO. . . H2 (and N2 . . . HZ) had been recorded using long path absorption spectro- scopy of cooled equilibrium gas samples [48]. However, neither CO ...H2 nor NZ - ..H2 have been the subject of calculations realistic enough to reproduce their spectra.

Calculations

The interaction potential has been obtained using the SCF and MBPT approaches [36-381 for the total energy

R = RSCF + EMBPT, (1)

where the correlation energy is split into contri- butions that are due to different orders of MBPT

E MBPT = E MBPTZ + E MBPT3 + E MBPT4 (2)

Currently, going beyond the full fourth order MBPT treatment does not seem feasible. It has also been shown that in order to obtain quanti- tatively meaningful results, incomplete higher order MBPT calculations must be avoided [49,50]. Thus the complete fourth order MBPT approximation has been adopted in the present study. With the linked cluster theorem automati- cally satisfied on each order of perturbation, this MBPT scheme is certainly size-consistent [36,37] and one of the best methods available for a reliable calculation of interaction energies within the super- molecule approach.

The interaction energy (IE) has been defined as

IE(R) = E(CO... H2;R)- E(CO...X;R)

- E(X... H2;R), (3)

where E(C0 . . ' X; R) and E(X . . . H2; R) are used here to indicate that the monomer energies are derived in the basis set of the entire complex, i.e. the energy of the CO (or Hz) molecule is deter- mined using not only its own molecular orbitals, but also those of the H2 (or CO) molecule (by setting its nuclear charges to zero) for any given geometry R of the supermolecule. This amounts to applying the counterpoise procedure of Boys and Bernardi [25] to correct for the BSSE at both the SCF and the MBPT levels of approximation at each molecular configuration.

Here, the quantity

BSSE(R) = BSSE(CO;R) + BSSE(H2;R)

where

(4a)

BSSE(C0; R)= E(CO)- E(CO...X)

and

(4b)

BSSE(H2;R) = E(H2) - E(X . . . H2) (4c)

gives a measure of the importance of the super- position effect for CO and Hz, respectively.

The interaction energy defined by Eq. (3), through fourth order in the correlation pertur- bation, can be expressed in terms of the compo- nents at any particular geometrical configuration R [22] as

IEMs”@) = IEscF + IEMam2 + IEMat”rs

where

+ IEMBPT4 (5a)

IEMBPT2 = IEY, (5b)

IEMBPT3 = IEY, (5c)

IEMBPT4 = IE& + IE& + IEL + IE$ + IE; (5d)

with the superscripts S, D, T, and Q referring to single, double, triple, and quadruple excitations from the reference Hartree-Fock determinant [49,51]. Some approximate forms of the SDTQ- MBPT(4) interaction energy can be obtained by neglecting in Eq. (5d) certain classes of excitations [51]. For instance, the SDQ-MBPT(4) scheme obtained by deleting the IE& term in Eq. (5d)

142 M.C. Salazar et al./J. Mol. Struct. (Theochem) 287 (1993) 139-147

stresses the importance of triple substitutions in the calculated fourth order interaction energy. The necessary MBPT correlation energies were calcu- lated using the BRATISLAVA package [52], interfaced to the MUNICH molecular package [53] for the Gaussian integrals, SCF eigenvectors, molecular properties, and four-index molecular integral transformation calculations.

In all calculations reported in this paper we have employed the medium sized polarized basis set devised by Sadlej [26,27]. It comprises [10.6.4/ 5.3.21 GTO/CGTOs for C and 0 atoms and [6.4/ 3.21 GTO/CGTOs for hydrogen. This basis set, referred to as the S basis, leads to a correct calcu- lation of intermolecular electrostatic forces and to a negligible secondary BSSE [34], which justifies the use of the standard counterpoise correction [25]. The larger basis set used here for comparison was designed to describe accurately the electric moments and polarizabilities of CO and Hz, and to minimize the influence of the BSSE on the scat- tering studies undertaken by Schinke et al. [39]. This basis set, referred to as the SD basis, consists of [11.7.2/9.6.2] GTO/CGTOs in the case of C and 0 atoms and [6.2/5.2] GTO/CGTOs for hydrogen.

Table 1 Electric properties of CO and Hz (all values in atomic units)

Results and discussion

The calculated dipole and quadrupole moments, and dipole polarizabilities of CO and H2 in the S and SD basis sets that follow from the SCF and

MBPT(4) energies used in the framework of the finite field approach [50] are displayed in Table 1. All these properties were calculated relative to their center of mass, using bond lengths of 1.40 and 2.132 a.u. for the H2 and CO molecules, respec- tively, and the z axis coinciding with the molecular axis. The values listed in Table 1 show that the agreement between the electrical properties calcu- lated in both basis sets is fairly good. This agree- ment follows the general trend found in recent publications [29,32,54], where the performance of Sadlej’s basis sets was tested against reference calculations by other authors and against experi- mental results. Of particular importance here is the fact that the relatively small basis set of Sadlej recovers almost the same portion of the MBPT(4) correlation contribution as in large basis set calcu- lations. The rather poor SCF result found for the quadrupole moment of Hz follows certain specific features of Sadlej’s basis set for hydrogen [26] and has been reported as an exception [32]. It is also

Molecule Property Method S basis SD basis

HZ

co

SCF MBPT(4) SCF MBPT(4) SCF MBPT(4) SCF MBPT(4) SCF MBPT(4) SCF MBPT(4) SCF MBPT(4)

0.438a 0.397a 4.3gb 4.35b 6.55b 6.57b

-0.0987b 0.0925b

ll.llb 11.76b 14.47b 15.55b 1.506” 1.515a

0.475 0.434 4.09 4.06 6.53 6.53

-0.0979= 0.0963’

11.03= 11&V 14.46’ 15.42’ 1.524’ 1 .520c

’ SCF/MBPT(4) calculations from Ref. 32. b SCF/MBPT(4) calculations from Ref. 29. ’ SCF/MBPT(4) valence-shell calculations from Ref. 54.

M.C. Salazar et al/J. Mol. Strut. (Theochem) 287 (1993) 139-147 143

The total interaction energy between CO and H2 was computed at the SCF and MBPT level of approximation using the S and SD basis sets. In the supermolecule calculations the units were kept

Fig. 1. Coordinate system for the OC . . . Hz dimer. rigid with bond lengths of 1.40 and 2.132 a.u. for H2 and CO, respectively, in the coordinate system

important to note that the dipole moment of CO shown in Fig. 1. The CO . . . H2 potential surface of

has the wrong sign in the SCF approximation and Schinke et al. [39] shows that the linear configur-

it is only corrected when electron correlation is ation corresponds to the most stable geometry in

included. CO. - . HZ, so we have chosen the linear structure

Table 2 The distance-dependence of various approximations to the counterpoise-corrected interaction energy IE and to the estimates of the basis set superposition error BSSE’

Distance R (a.u.)

SCF MBPT(2) MBPT(3) MBPT(4) SDQ-MBPT(4)

Basis S 7.0 7.25 7.50 7.75 8.00 8.14 8.25 8.50 8.75 9.00 9.50

10.00 12.00 14.00 20.00 40.00

Basis SD 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.50

10.00 12.00 14.00 40.00

36.78 (24.26) 12.36 (56.00) 13.69 (55.57) 10.25 (57.50) 13.39 (55.76) 21.65 (23.01) 2.08 (52.60) 3.19 (52.24) 0.32 (54.03) 2.88 (52.41) 12.06 (21.77) -3.74 (49.29) -2.80 (48.99) -5.21 (50.65) -3.10 (49.15)

6.09 (20.56) -6.76 (46.04) -5.92 (45.81) -7.97 (47.34) -6.22 (45.95) 2.47 (19.36) -8.04 (42.88) -7.29 (42.72) -9.02 (44.13) -7.58 (42.85) 1.14 (18.70) -8.28 (41.15) -7.56 (41.02) -9.16 (42.37) -7.85 (41.15) 0.35 (18.18) -8.30 (39.80) -7.61 (39.71) -9.10 (41.02) -7.89 (39.84)

-0.83 (17.00) -7.99 (36.82) -7.35 (36.81) -8.63 (38.00) -7.61 (36.92) -1.43 (15.81) -7.39 (33.94) -6.80 (34.00) -7.90 (35.10) -7.05 (34.10) -1.68 (14.63) -6.67 (31.17) -6.12 (31.30) -7.08 (32.31) -6.35 (31.41) -1.67 (12.39) -5.23 (26.16) -4.76 (26.40) -5.48 (27.26) -4.96 (26.51) -1.43 (10.48) -4.01 (22.07) -3.61 (22.39) -4.16 (23.12) -3.77 (22.50) -0.56 (5.46) - 1.45 (11.60) -1.25 (11.85) -1.46 (12.20) -1.32 (11.91) -0.22 (1.67) -0.59 (3.62) -0.49 (3.75) -0.59 (3.82) -0.53 (3.75) -0.02 (0.02) -0.09 (0.04) -0.06 (0.04) -0.08 (0.04) -0.07 (0.04)

0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)

35.24 (2.38) 7.71 (7.05) 8.68 (7.03) 5.11 (7.06) 8.77 (6.89) 20.45 (2.04) -1.51 (6.05) -0.67 (6.03) -3.66 (6.06) -0.68 (5.91) 11.16 (1.73) -6.44 (5.15) -5.68 (5.12) -8.21 (5.14) -5.76 (5.02) 5.44 (1.45) -8.75 (4.33) -8.03 (4.31) -10.19 (4.32) -8.17 (4.22) 1.99 (1.21) -9.49 (3.62) -8.81 (3.59) -10.66 (3.59) -8.99 (3.51)

-0.01 (1.02) -9.36 (3.00) -8.71 (2.97) - 10.29 (2.96) -8.90 (2.89) -1.10 (0.87) -8.76 (2.47) -8.14 (2.43) -9.51 (2.42) -8.34 (2.37) -1.64 (0.75) -7.95 (2.02) -7.37 (1.97) -8.54 (1.97) -7.56 (1.92) -1.85 (0.65) -7.07 (1.65) -6.52 (1.61) -7.55 (1.56) -6.72 (1.56) -1.78 (0.50) -5.43 (1.10) -4.95 (1.07) -5.73 (1.05) -5.13 (1.03) -1.49 (0.39) -4.11 (0.77) -3.69 (0.75) -4.29 (0.74) -3.85 (0.73) -0.56 (0.15) - 1.43 (0.26) - 1.22 (0.26) - 1.46 (0.25) - 1.30 (0.25) -0.22 (0.03) -0.60 (0.06) -0.49 (0.06) -0.60 (0.05) -0.53 (0.05)

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

aEnergies are in millielectronvolts. The total BSSE = BSSE(C0) + BSSE(H2) are in parentheses.

144 M.C. Salazar et al/J. Mol. Struct. (Theochem) 287 (1993) 139-147

(7, = 72 = 0 in Fig. 1) as the first test case in the size-consistency [37] of the given MBPT order present study. contributions.

The distance-dependence of different approxi- mations to the counterpoise corrected interaction energy (IE) calculated according to Eq. (3), and to the estimate of the total basis set superposition effect (BSSE) in both basis sets calculated accord- ing to Eqs (4a)-(4c) are presented in Table 2 for the linear 0-C.. a H-H dimer as a function of the distance R between the center of masses of CO and HZ. In all cases the sum of the energies for the two isolated subsystems coincides with the appropriate values calculated for the super- molecule at R = 40 a.u., indicating simply the

The final SCF, MBPT(2), MBPT(3), and MBPT(4) counterpoise-corrected interaction potentials for O-C - a . H-H are plotted in Figs 2a and 2b for the S and SD bases, respectively. The shape of the interaction energy curves calculated by using the incomplete SDQ-MBPT(4) scheme and the position of the minima is almost the same as for the MBPT(2) approximation. It follows from Figs 2(a) and and 2(b) and from Table 2 that the interaction energies determined at MBPT(2), MBPT(3) and MBPT(4) levels of approximation differ only slightly and that the MBPT(4) level of approximation produces the most dominant attractive IE terms. The negligible difference between MBPT(2) and MBPT(3) is due to the small and positive IEY term in Eq. (5), which cor- responds to about 0.7meV at the vdW minimum for both basis sets. Among fourth order IE contri- butions, the attractive IE& term in Eq. (5) is domi- nant. Because of near compensation of the IE& and the IE$, + IEE terms the interaction energies determined at MBPT(2) and MBPT(4) levels of approximation differ only by the triple excitations represented by the IEL term, which corresponds to about 1.3 and 1.7 meV at the vdW minimum for the S and SD bases, respectively.

0.00

-10.00

7.00 9.00 9.00 10.00 11.00 12.00 13.00 14.00

R(a.u.)

IE(meV) 40.00

L O-C---H-H - IE-MBPT(2)

30.00 - R ,...a . ..a* - - IE-MBPT(3) + IE-MBPT(4)

t\ - IE-SCF I 20.00

10.00

0.00

-lo.w

7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00

R(a.u.)

Fig. 2. SCF, MBPT(Z), MBPT(3), and MBPT(4) counter- poise-corrected interaction potentials calculated in a, the S basis set; b, the SD basis set.

All correlation corrections shown in Table 2 suffer from a rather large basis set superposition effect, being one order of magnitude larger for the S than the SD basis set. Its magnitude estimated by the counterpoise method is either comparable to (SD basis) or even larger than (S basis) the corre- sponding values of IE for different methods. Never- theless, the counterpoise corrected IE values shown in Table 2 agree within 0.2 and 1.5 meV around the vdW minimum at the SCF and MBPT(4) levels of approximation, respectively, in both basis sets, and all IE curves exhibit regular behavior for all values of R. This is better illustrated in Fig. 3, where the relative values of the counterpoise-corrected IE potential curves for 0-C. . . H-H at the SCF and the MBPT(4) levels of approximation are plotted for the S and SD bases, respectively.

M.C. Salazar et al./J. Mol. Struct. (Theochem) 287 (1993) 139-147

IE(meV)

Ii +- IE-SCF S BASIS

30.00

R

O-C---H-H . . . . .

R . . . . .

20.00

145

7.00 8.00 9.00 10.00 11 .oo 12.00 13.00 14.00

R(a.u.)

Fig. 3. SCF, and MBPT(4) counterpoise-corrected interaction potentials calculated in the S and SD basis sets.

It clearly follows from Fig. 3 that the SCF level of approximation is practically repulsive and recovers only a small portion of the O-C . . . H-H interaction energy. The minima of the SCF inter- action energy curves occur at about 9.0a.u., with small depths of about 1.68 and 1.85meV for the S and SD basis sets, respectively. This result agrees with the simple picture of a true vdW dimer bonded mainly by dispersion forces. The minima of the MBPT(4) interaction energy curves occur at 8.14 and 8.03a.u., with a well depth of 9.16 and 10.66meV for the S and SD bases, respectively. These fully ab initio values agree very well with the minimum distance of 7.99a.u. and the well depth of 9.2meV predicted for the same O-C - - . H-H linear configuration by Schinke et al. [39], which represent the most reliable comparison results to date.

Fitting of the present fully ab initio MBPT(4) interaction energy points in Table 2 to a sixth

order polynomial in the stretching coordinate R about the predicted equilibrium distances allowed us to obtain the intermolecular zero-point energy of the linear 0-C. . . H-H anharmonic oscillator by treating the complex as a diatomic system with only one degree of freedom [55]. The calculated zero-point energies correspond to 5.2 and 6.2meV for the S and SD basis sets, respectively. In agreement with recent spectroscopic results [48], these values do not exceed the corresponding well depths of 9.16 and 10.66meV predicted in this study, indicating that at least one vibrationally bonded state is possible for this linear dimer. In contrast, recent results of Dykstra and co-workers [56], using the so-called molecular mechanics for clusters (MMC) fitting method predicted no bonded vibrational states for this vdW dimer.

In summary, the counterpoise-corrected inter- action energy data calculated in the POLl basis set for the linear O-C . -. H-H dimer is in excel-

146 M.C. Salazar et al./J. Mol. Strut. (Theo&em) 287 (1993) 139-147

lent agreement with the results obtained in the larger basis set used in this study for comparison. The agreement is 0.11 a0 and 1.5 meV in the mini- mum and well depth, respectively, for the calcu- lated MBPT(4) interaction energy curves in both basis sets. These fully ab initio values of 8.14 and 8.03u,, for the minimum of the MBPT(4) inter- action energy curves with a well depth of 9.16 and 10.66meV for the POLl and larger basis set calculations, respectively, agree very well with the minimum distance of 7.99a.u. and well depth of 9.2meV predicted for the same O-C ... H-H linear configuration by Schinke et al. [39]. In agree- ment with recent spectroscopic results [48], the pre- sent results indicate that at least one vibrationally bonded state is possible for this linear dimer. Although configurations different from the linear OC . . . Hz structure remain to be investigated using the POLl basis set, this result opens the very attractive possibility of reducing the size of the polarization basis set without affecting the accuracy of the calculated intermolecular energy curves of the OC . . . Hz vdW dimer.

Acknowledgments

The authors thank the Deutsche Forschungs- gemeinschaft (DFG), the Consejo National de Investigaciones Cientificas (CONICIT), the Decanato de Investigaciones of the Universidad Simon Bolivar, and the Max-Planck Gesellschaft for the continuous support of this research project.

References

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