Model MG-FSGO calculations of short range interactions

Journal of Molecular Structure, 94 (1983) 267-273 THEOCHEM Elsevier Science Publishers B.V., Amsterdam -Printed

MODEL MG-FSGO CALCULATIONS INTERACTIONS

in The Netherlands

OF SHORT RANGE

G. F. MUSSO, S. BISIO, G. FIGARI and V. MAGNASCO Zstituto di Chimica Zndustriale dell’Universitri,

16132 Genova (Italy)

(Received 16 August 1982)

ABSTRACT Ab initio MG-FSGO calculations on Hz-H, with increasingly accurate floating Gaussian bases show that fully optimized wavefunctions of intermediate size do describe quite accurately the intermolecular interactions in short range and their orientation dependence. INTRODUCTION

Few short range atom-molecule and molecule-molecule interactions have been studied so far using the Floating Spherical Gaussian Orbitals (FSGO) model devised by Frost [l].A number of calculations, mostly on the collinear arrangement of 3 and 4 hydrogen atoms, of importance in the study of the potential energy surface of the most elementary exchange reactions, have been reported at different times by the Linnett group w--41.

Since just a single FSGO per bond, even if accurately optimized in its nonlinear variational parameters, can be expected to give a rather poor description of the distribution of the electronic charge inside the molecule (the details of which are important in determining the short range interaction), we present in this paper ab initio calculations of the short range repulsive interaction between two Hz molecules in their ground state within the context of the Multi-Gaussian (MG) FSGO scheme [5,6]. This rather simple system has been used by us in the past as a convenient model for studying the orientation dependence of short range intermolecular interactions [ 7,8] , a problem with interesting implications in the theory of rotational barriers, and is today very important in connection with the increasingly accurate determinations of collisional differential cross sections in experiments with molecular beams [ 9, lo] . Emphasis in our work is not on the determination of the potential energy surface for Hq, but rather in testing the reliability of the simple FSGO approach and its improved MG version in comparison with our previous calculations based on a bond-orbital (BO) scheme [ 73 and with the rather accurate SCF results recently given by Ree and Bender [ll] . 0166-1280/83/$03.00

0 1983 Elsevier Science Publishers B.V.

268 METHOD

A one-determinant wave function of two doubly occupied local orbitals was used, the description of each orbital being systematically improved through a limited extension of the floating Gaussian basis. The number M of FSGOs in each local orbital was allowed to take the values 1,3,5 (M = 1 is the simple Frost approach), giving for the ground state of an isolated Hz molecule a total molecular energy of -0.95593, -1.10139 and -1.12683 a.u.* , respectively [ 121 (the corresponding Hartree-Fock value [13] being -1.13367 a.u.). The geometry of the bimolecular system was described in terms of the two parameters D and 0, specifying the intermolecular separation and the orientation of one molecule with respect to the other, while the bond length R of the single molecule was left as the only geometrical parameter subject to variational optimization. Each Hz molecule was hence left free to modify its equilibrium internuclear distance (R,,) under the repulsive interaction arising from its partner molecule. For each (D, 0)configuration the total molecular energy was optimized by refined numerical methods with respect to (i) all linear and nonlinear orbital parameters, and (ii) the bond length R of the single Hz molecule. The total number of independent variational parameters resulting therefrom was 3,7,11 at 0 = 0” and 5,15,25 at 19fO”, for M = 1,3,5 respectively. Energy minimization resting on the variation theorem was best performed through a derivative based optimization:method [ 141, using compact analytical formulae for the electronic and nuclear gradients of the total MG-FSGO energy recently given by Musso [6,15] . For each given value of M, the intermolecular interaction energy was then obtained as the difference between the total molecular energy of the bimolecular system and twice the appropriate energy of the isolated Hz molecule (E,). All the calculations were performed on the digital Burroughs B-6810 computer of the Computation Centre of the University. RESULTS AND DISCUSSION

The geometry of the bimolecular system is sketched in the top left frame inside Fig. 1. To facilitate comparison with previous work [ 7,8,16] the intermolecular separation was expressed in terms of the ratio D/R,, where % = 1.4166 a.u. is the bond distance for an isolated Hz molecule described by the Weinbaum wavefunction [7] . The numerical results obtained in the present calculations are collected in Tables l-3. The one-configuration interaction energies (a.u.) resulting from the increasingly accurate MG-FSGO bases are given in Table 1 for different separations D/R,, and relative orientations 8 of the two Hz molecules. It can be *Unless otherwise stated, atomic units (a.u.) are used throughout. Energy a.u. (hartree) = 27.21 eV = 2625.4 kJ rnoF ; length a.u. (bohr, q,) = 0.0529177 nm.

269

o.20’!-0.20

0.40

0.60

0.80

z

Fig. 1. Ground state of an isolated H, molecule. Electron density profiles (a.u.) along the internuclear axis according to different MG-FSGO wavefunctions (origin at the centre of the molecule). Dotted curve (a): exact calculation of ref. 20. The geometry of the bimolecular system is sketched in the top left frame inside the figure.

seen that the calculated interaction energies are all reasonably close among themselves, the 3G values being rather similar to the accurate results of Ree and Bender [ll] . The latter values (in parentheses, fourth row in Table 1) have been obtained by fitting a 6-term polynomial through SCF values given in ref. 11 (P geometry, Table V of ref. 11; D/R,, = 4.0 is not included since it falls outside the range of intermolecular distances in ref. 11). Our MG results are also in fair agreement with the SCF-values given by Tapia et al. [16] . Care must be taken, however, in comparing absolute values of interaction energies obtained from MO-SCF calculations, because of the possible occurrence of basis set over extension errors [ 17,181, due to the limited size of the basis set used for the individual molecules. While this error can be reasonably expected to be small for the basis used by Ree and Bender [ll] (their E, differing from the Hartree-Fock value [13] by only -0.00496 a.u.), the same error is expected to be larger for the latter calculation, since the Gaussian basis of Tapia et al. [16] gives an E, (-2.25100 a.u.) which differs from the Hartree-Fock value [ 131 by as much as -0.01634 a.u.

270 TABLE 1 H,--H, ground state. One-configuration interaction energies (a.u.) in short range according to increasingly accurate bIG-FSGO bases, for different separations (D/Et,) and relative orientations (0, degrees) of the two molecules (Ro = 1.4166~ in ail cases)

e

DIR, 2.0

2.5

3.0

3.5

4.0

0.05755* 0.06153b 0.0652Sc (0.05798)d

0.01942 0.01855 0.01988 (0.01859)

0.00602 0.00539 0.00591 (0.00554)

0.00169 0.00159 0.00172 (0.00156)

0.00054 0.00049 0.00051

60

0.04656 0.05220 0.05594

0.01602 0.01585 0.01728

0.00479 0.00456 0.00514

0.00119 0.00129 0.00146

0.00031 0.00036 0.00041

120

0.03463 0.04091 0.04456

0.01174 0.01266 0.01406

0.00334 0.00367 0.00422

0.00087 0.00103 0.00121

0.00019 0.00028 0.00034

180

0.02991 0.03690 0.04056

0.01014 0.01160 0.01295

0.00292 0.00343 0.00394

0.00073 0.00099 0.00115

0.00022 0.00029 0.00034

0

au.) C5G (E. = -2.25366 a.u.) dIntera.u.) b3G (E. = -2.20279 *lG (E. = -1.91187 polated values from accurate SCF calculations from ref. 11 (E, = -2.26238 a.u.).

TABLE 2 H, -H, ground state. One-configuration results (kJ mol” ) according to increasingly accurate MG-FSGO bases for the energy difference AE = E(e = o”, Da) - E(e = 180°, C,) in the relative orientation of the two molecules M

DIR,

1G 3G $-SCF

[7]

2.0

2.5

3.0

3.5

4.0

72.57 64.68

24.36 18.25

8.14 5.17

2.52 1.56

0.81 0.51

68.13 64.82

18.88 18.22

5.09 5.17

1.48 1.39

0.45

Our calculated interaction energies are also consistently lower than our previous BO-SCF results [7] * based on a minimal set of four 1s STOs (E, = -2.25597 a.u.). The main reason for this can be traced back to the lack of any bond length optimization in the molecules under interaction in our previous BO calculations. *The BO interaction energies in Table III of ref. 7, erroneously quoted there as SCF, refer instead to nonpolar BOs. The differences are anyway very small and only apprebeing 0.07033, ciable for D/R, = 2.0 (correct BO-SCF values at 0 = 0,60,120,180° 0.06033,0.04830,0.04438 a.u.) and DIR, = 2.5 (correct values: 0.02166,0.01902, 0.01562,0.01448).

271 TABLE 3 Hz-H, ground state. Geometry (R, bohr) of each individualH, molecule under interaction from optimized MG-FSGO calculationsof increasingaccuracy

e

,DIR, 2.0

2.6

3.0

3.5

4.0

1.4338 (-2.8%) 1.353b (-3.1) 1.344= (-3.3)

1.449 (-1.7%) 1.376 (-1.5) 1.368 (-1.5)

1.460 (-0.9%) 1.389 (-0.6) 1.381 (-0.6)

1.468(-0.4%) 1.394 (-0.2) 1.386 (-0.2)

1.472 (-0.1%) 1.396 (-0.1) 1.388 (-0.1)

60

1.464 (-1.3) 1.372 (-1.8) 1.362 (-2.0)

1.457 (-1.1) 1.383 (-1.0) 1.374 (-1.1)

1.464 (-0.7) 1.391 (-0.4) 1.383 (-0.5)

1.470 (-0.3) 1.395 (-0.2) 1.387 (-0.2)

1.473 (-0.1) 1.396 (<-O.l) 1.388 (e-0.1)

120

1.472 (-0.1) 1.391 (-0.5) 1.380 (-0.6)

1.466 (-0.6) 1.389 (-0.6) 1.381 (-0.6)

1.468 (-0.4) 1.393 (-0.3) 1.385 (-0.3)

1.472 (-0.2) 1.396 (-0.1) 1.388 (-0.1)

1.473 (-0.1) 1.397 (<-O.l) 1.389 (<-O.l)

180

1.481 (t0.5) 1.397 (-C-0.1) 1.386 (-0.2)

1.469 (-0.3) 1.392 (-0.4) 1.383 (-0.4)

1.470 (-0.3) 1.394 (-0.2) 1.386 (-0.2)

1.472 (-0.1) 1.396 (-0.1) 1.388 (-0.1)

1.473 (<-O.l) 1.397 (<-O.l) 1.389 (<-O.l)

0

*lG (f&,

= 1.474 q,). b3G (Roti = 1.397). C5G (Roti = 1.389).

Table 2 collects the oneconfiiation results (in kJ mol-‘) for the energy difference AE = E(B = 0",DZh)- E(8 = 180”, C,,) (“barrier”values). It can be seen that the fully optimized 3G results are essentially of the same quality as the more accurate 5G ones. On the contrary, the simple Frost (M = 1, simple FSGO) wavefunction, which surprisingly enough gave quite reasonable interaction energies despite the roughness of the model, is seen to give “barrier” values which are noticeably larger (up to over 60%) than the MG ones. Similar results have been observed in calculating barriers to internal rotation by the simple FSGO method[l9]. This is not surprising if we look at the isolated molecule electron density profiles plotted in Fig. 1 for different MG-FSGO wavefunctions (data from ref. 12). It can be seen that the simple FSGO wavefunction (curve 1) gives a very crude approximation to the exact density [20] (dotted curve a) only at the midpoint of the bond. The 3G and 5G wavefunctions, on the contrary, try to reproduce the curve better and also the cusps at the protons, which are expected to give important contributions to the repulsive overlap density whose differential behaviour determines the barrier [ 7,211. Curve 13 gives the electron density obtained with a MG-FSGO wavefunction with M = 13, whose molecular energy (-1.13359 au.) [12] is extremely close to the Hartree-Fock value (-1.13367 a.u.) [ 131. The electron density profile for M = 13 closely approximates the accurate electron density for Hz obtained by Kolos and Roothaan [20] using a 40-term James-Coolidge type wavefunction. Table 3 gives the resulting geometry R of each individual Hz molecule under interaction. It can be seen that the Hz molecule contracts under the

272

repulsive short range interaction (the % variations are given in parentheses, to facilitate comparison, a minus sign denoting contraction with respect to R,,p3, a result similar to that found by Ree and Bender [ll].A closer examination of the optimal values of the orbital parameters shows that, correspondingly, the single Hz molecule charge distribution bends slightly outwards. Increasing D/R,, contraction decreases for all MG at 19= 0,60”, while at 13= 120,180” the maximum contraction occurs at D/R,, = 2.5.1G elongates by 0 5% in the coplanar tram confiiuration at D/R,, = 2 .O. Upon increasing 8 at a given D/R,, the contraction always decreases, the effect being smaller at larger D/R, and quite close for 3G and 5G. CONCLUSIONS

The fully optimized MG-FSGO approach, at a moderate level of sophistication of the floating Gaussian basis (M = 3), offers a fairly accurate description of the repulsive part of the interaction potential in the HZ-H2 system as well as of its change with the mutual orientation of the two molecules. The relaxation of the molecular charge distribution of each molecule under interaction seems welI reproduced at the MG level, and may be of considerable importance in studying similar effects occurring in molecules undergoing internal rotation. The efficiency of the numerical methods developed in our laboratory for doing derivative-based optimizations [6,15] ensures the possibility of performing, if necessary, calculations with more extended bases of floating Gaussians. The accurate determination of the repulsive part of the whole potential energy surface would then be possible, with a reasonable compromise between cost of computation and accuracy in the optimization of a large number of variationally relevant orbital and geometrical parameters. The simple Frost model (M = 1, FSGO), although giving unexpectedly fair results for the interaction potential, fails to reproduce its differential change with intermolecular orientation, showing that it cannot describe at a sufficient level of accuracy the small penetration of the charge clouds of the interacting molecules [ 7,21,22] . ACKNOWLEDGMENTS

Financial support by the Italian National Research Council (C.N.R.) and the Ministry of Public Education (M.P.I.) under grants in Theoretical Chemistry is gratefully acknowledged. REFERENCES 1 2 3 4

A. A. Frost, J. Chem. Phys., 47 (1967) 3707,3714. P. H. Blustin and J. W. Linnett, J. Chem. Sot., Faraday Trans. 2,70 (1974) 327. L. P. Tan, Ph. D. Thesis, Cambridge, 1974. H. H. Huang and J. W. Linnett, J. Chem. Sot., Chem. Commun., (1976) 135.

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