Potential energy surfaces for dissociative recombination reactions of HCO+ and HCS+

Chemical Physics 126 (1988) 291-300 North-Holland, Amsterdam

POTENTIAL ENERGY SURFACES FOR DISSOCIATIVE RECOMBINATION OFHCO+ANDHCS+

REACTIONS

D. TALBI, F. PAUZAT and Y. ELLINGER Equipe d’Astrochimie Quantique, Laboratoire de Radioastronomie, ENS, 24 rue Lhomond, F- 75005 Paris, France and DEMIRM, Observatoire de Paris, F-92190 Meudon, France

Received 24 March 1988

Adiabatic potential energy surfaces for dissociative recombination reactions of HCO+ and HCS+ are determined in a series of CI calculations which use a localized orbital representation. The design of equivalent n-particle spaces for the two neutral molecules and for the two positive ions permits a comparison of the two systems at the same level of correlation. In both species there is a strong mixing between an excited Rydberg state of s character and a dissociative state leading to the breaking of the CH bonds; this energy profile is repulsive at all CH distances for HCO, it has a minimum for HCS which is sufftciently deep to contain one vibrational level; thus an indirect dissociative recombination should be much easier for HCO+ than for HCS+. Moreover, the crossing point between the ion curve and the dissociative state is such differently positioned in the two species that the direct process should also favor HCO+ dissociation. It is then expected that dissociative recombination should be globally more efficient for HCO+ than for HCS+.

1. Introduction

Because it is one of the most abundant positive ions in interstellar molecular clouds, the formyl cation, HCO+, has been a continuous subject of attention. The studies really started with the proposal by Herbst and Klemperer [ 1,2 ] concerning the assignment of the X-ogen band at 89.189 GHz to this particular species. Due to the lack of experimental information at that time, quantum chemical calculations were undertaken on the problem [ 3-5 1, which corroborated the astrophysical hypothesis. The definitive proof of HCO+ being an interstellar molecule came more recently from the comparison of the radioastronomy measurements with a laboratory microwave experiment [ 6 1. Confirmation of the existence of HCO+ in a number of molecular clouds is important since it is a key molecule in astrochemical models [ 7-9 1. By analogy, HCS+ is supposed to play the same role in sulphur chemistry. Detection in several dense clouds [lo] coupled with the determination of the structure by both quantum mechanical methods [ 11,12 ] and laboratory experiments [ 13 ] confirmed the importance given previously to HCS+ in molecular processes.

The astrophysical measurements, however, pointed to the puzzling result that the abundance ratio HCS+/ CS is two orders of magnitude larger than that of the oxygenated compounds HCO+/CO. This fact, which is most intriguing in view of the very similar chemistry in the two series, has not received any coherent explanation. Most of the reactions implying HCO+ and HCS+ have been studied in the laboratory and the corresponding rate constants have been determined. These ions are mainly formed in ion-molecule reactions with rate constants all found in the range (l5 ) x 1O-9 cm3 s-i; they are essentially destroyed by dissociative recombination with electrons in exothermic reactions that yield neutral products, HCO++e-+CO+H, HCS+ +e--CS+H

(1) .

(11)

Reaction (I) has a rate constant of 10m6cm3 s- ’ at the temperature of the interstellar medium [ 141. This value which has been confirmed in recent experiments [ 15 ] leads to a theoretical abundance ratio of 10m4 for HCO’/CO in agreement with the spatial observations. The rate constant of reaction (II) is not

D. Talbi et al. /Dissociative recombination of HCO+ and HCS+

292

known but generally approximated by that of typical dissociative recombination reaction, namely 10m6 cm3 s- ’ [ 16 1. Using such a value, an abundance ratio of the order of 10m4 is obtained for HCS’/CS which is far from the 10m2 ratio observed in interstellar clouds [ lo]. Since further refinements of the astrochemical models were unsuccessful, Millar [ 17 ] suggested that the dissociative recombination of HCS+ should be slower than expected. The purpose of this first paper is to present the calculations of the potential energy surfaces for the dissociative recombination reactions of HCO+ and HCS+, which amounts to setting comparable and accurate correlated wavefunctions for the corresponding neutral species. It will be shown that the fundamental differences between the two systems are significant enough to justify Millar’s hypothesis.

additional electron occupies a virtual orbital of the positive ions. The orbitals available are the antibonding A*orbitals and the CH* antibonding orbital. Addition of an electron to the ICsystem leads to the 211states of HCO ...(oCO)*(oCH)*(

1nC0)4(2x*CO)’

(2a)

and HCS ...(oCS)2(oCH)2(2xCS)4(3rr*CS)‘.

(2b)

These radicals, however, are not linear and the n system is split into a’ and a” orbitals, the former being stabilized by Renner-Teller splitting. The ground states are bent with the electronic configurations *A’ ...(oCO)2(oCH)2(~“C0)2(pO)2(o)1.

(3a)

...(oCS)*(oCH)*(2n”CS)*(pS)*(o)’

.

(3b)

The second components of *Il (‘A” ) give the lowest excited states of linear geometries 2. Method of calculation

...(oCO)2(oCH)2(7c’CO)2(7c”CO)2(2Tc”*CO)’.

At the qualitative level, the problem can be presented as follows. Using the localized description of the one-particle functions illustrated in fig. 1 (X = 0, S), the ground states of HCO+ and HCS+ have the electronic configurations ...(oCO)*(oCH)*(

l~C0)~,

...(oCS)2(oCH)2(2xCS)4.

(la)

(da)

...(oCS)2(oCH)2(2x’CS)2(2n”CS)2(3rr”*CS)’. (4b)

Excited states of rr+x* type can also be obtained from the bent structures such as 2A” ...(oCO)2(oCH)2(x”CO)‘(pO)2(o)1(2&’*CO)’

(lb)

Dissociative recombination implies capture of an electron which will result in the formation of excited states of the neutral species HCO or HCS where the

...(oCS)2(oCH)2(2rc”CS)‘(pS)2(o)‘(3n”*CS)1. (5b)

These states are valence states, which are important for the electronic spectra of the neutral radicals and have already been discussed as such [ 18-201. Introduction of an electron in the o system leads mainly to %+ in which the additional electron occupies a CH* antibonding orbital. Then the electronic configurations of these compounds are ...(~$0)~(crCH)*(

1~&0)~(oCH*)’

...(c~CS)*(~CH)~(~~~CS)~(~CH*)‘.

Fig. 1. Localized description of HCX+ (top) and HCX ground state (bottom) in terms of bond and lone pair orbitals (X ~0, S).

, (5a)

,

(6a) (6b)

Such electronic states in which the CH bond should be very weak are higher in energy, at the equilibrium geometries of the ions, than those involving excitations in the 7csystem. They are of prime interest for dissociative recombination reactions since they cor-

293

D. Talbi et al. /Dissociative recombination ofHCO+ and HCS

relate directly with the products, the CH and CH* leading respectively to a carbon lone pair in CO and CS and to the ground state of the hydrogen atom. The recombination process can be seen, at least for the first step, as a weak interaction between the electron and a positive core, namely the HCO+ or HCS+ ion. This is typically a Rydberg situation and, therefore, electronic configurations such as ...(oC0)2(oCH)2(

lxC0)4(Ri)’

...(oCS)2(oCH)2(2xCS)4(Ri)’

9 )

Va) (7b)

where Rj is a diffusive orbital of s or p character (i= s, x, y, z), must be considered. The importance of these configurations has been pointed out by Bruna et al. [ 181 in a previous study of the HCO electronic spectrum. All configurations (2)- ( 7) will be treated, but special attention will be devoted to the mixing of (6) and (7 ) which will be shown to be the factor governing the dissociative recombination process. 2.1. Determination of the molecular orbitals As a correct description of several potential surfaces of high energies was needed, essentially those of the Rydberg states and those of the dissociative states, it was decided to use the orbitals of the positive ions in all calculations. This common set of orbitals gives a well-behaved wavefunction for all Rydberg states and guarantees reliable positioning of these surfaces relative to those of the corresponding positive ions. In addition, transition density matrix elements can be easily calculated within the same set of orthonorma1 one-particle functions. The SCF orbitals of the positive ions are expanded in a double-zeta (DZ) basis set: H( %/2s), C( 9s, 5p/ 4s 2p), 0(9s, 5p/4s, 2p), S( 13s, 9p/6s, 4~). Exponents and contraction coefficients are those of McLean et al. [ 2 1,22 1. Polarization functions have been added on all atoms: a,(H) =0.9, q,(C) ~0.7327, q,(O)=O.85, (r,,(S)=O.81 and ~q,~(S)=O.45. In addition, the basis set was augmented by diffuse functions to account for the Rydberg components. These functions are centered on the heavy atoms with exponents: a,(C) =0.023, a,(C) =0.021, a,(O) =0.032, a,(O)=0.028 and cr,(S)=O.O23, a,(S)

= 0.020, all taken from the compilation by Dunning andHay [23]. The canonical orbitals obtained at the end of the SCF procedure are not appropriate functions for representing the electronic structures in terms of (2)( 7). Thus, the orbitals have been unitary transformed so as to represent the Lewis-type structure of the molecules illustrated in fig. 1. The localization procedure which yields the projected localized orbitals (PLOs), is the same as that used in a previous study of the ground state of HCO [24]. The basic principles are summarized in the following. (i) A set of non-orthogonal localized orbitals is generated that matches the Lewis structure of the molecule as closely as possible. The set will contain inner shells, bonding and lone-pair orbitals; inner shells are pure atomic functions, bonding orbitals are in-phase combinations of two hybrid atomic orbitals, one for each atom of the bond, whereas lone pairs are single atomic orbitals made orthogonal to the other hybrids on the same center. This is achieved by an hybridization procedure based on a maximum overlap criterion [ 25,261. To complete the valence space we have the antibonding orbitals which are the outof-phase combinations of the same hybrids that describe the corresponding bonding functions. Core correlating functions are obtained as virtual orbitals of atomic calculations. (ii) The transformation that produces the desired localized functions consists in projecting the appropriate non-orthogonal orbitals {X1}into the SCF orbital space {@j}according to @:=,$,

@j<@jlX>

Y

i= 1, 2, .... L;

N>L,

where i runs over the N orbitals of the subset and i covers the L localized functions to be determined. A particularly useful transformation is obtained when L < N. It allows for extracting the best match with the L non-orthogonal functions from any number N of orbitals in a given subset. It is used here for virtual orbitals to generate the antibonding and the Rydberg functions. The N-L remaining orbitals which are the orthogonal complement of the L localized functions are then submitted to a similar transformation to generate the core correlating functions. Finally, the whole projected space is orthogonalized with a symmetric S- ‘I2 transformation.

D. Talbi et al. /Dissociative recombination of HCO+ and HCS

294

the valence orbitals occupied in the HCO+ and HCS+ ions; in the second one, we have the antibonding orbitals which complete the valence space and the full Rydberg space available in the basis set. The C12 nparticle spaces are organized so that all states resulting from the addition of the extra electron are treated on the same footing. Valence correlation effects are included in the internal space and the most important first-order contributions are taken into account. The partitioning of the n-particle space, as it appears in the final wavefunction reported here, is the result of a series of CI calculations, not given in this paper, where the orbitals were incorporated one at a time in the active space of CI 1 and the effects analyzed [ 291. A variety of these calculations pointed to the necessity of including all but the core correlating functions in the external space, which amount to considering a total of 11027 CSFs for the neutral systems and 2390 CSFs for the positive ions. All calculations were done using the ALCHEMY program system to which an original modelizationlocalization package was interfaced.

A major advantage of the method is that the n-particle space is defined in terms of orbitals whose chemical signification is clearly identified. Configurations can be selected according to chemical evidence, and the type of arbitrariness associated with energy related selection processes can be avoided. The CI expansion is directly obtained in a configuration space which contains the quasi-diabatic representation of the problem [ 27 ] needed for further determination of the rate constant [ 28 1. 2.2. CI wavefunctions We will report the results of two CI wavefunctions described below. (i ) Wavefunctions CI 1, expanded in the n-particle spaces of table 1, give, for both HCO and HCS, a full correlated treatment of the orbitals most active for the dissociation process and take into consideration the first series of Rydberg orbitals. They retain the SCF description of the valence oC0 and oCS bonds and sp0 and spS sigma lone pairs. With n-particle spaces of dimension 2792 for the neutral systems and 1379 for the positive ions, they are economical for a large exploration of the potential surfaces and have been used accordingly. (ii) Wavefunctions C12 are expanded in the n-particle spaces of table 2. Here, the internal space is partitioned in two subspaces. The first one contains all Table 1 Wavefunction (X=0, S)

CIl; correlated

Symmetry

treatment

configuration

total number

Calculations have been focused essentially on the higher excited states of HCO and HCS since the

of the CH bond, the x system and the first series of Rydberg

Core a’

Electronic set

A’ A”

3. Results and discussion

a’ core a” core

orbitals

for the HCX systems

distributions

1

set 2

set 3

CH XL, x& CH* lr& x;>

R,RxR, RZ

a’ complement a” complement

7 (6) 6 (5) 5 (4)

0 (0)

classes b.c)

l(1) 2 (2)

of CSFs: 2792 (1379)

a’ For HCO: a’ core= 1SO 1SC sp0 CO. For HCS: a’ core= 1sS 1SC 2sS 2p,S 2p,S spS CS; a” core = 2pS. b, Core orbitals are doubly occupied in all CSFs. All configurations corresponding to the electronic distributions the spin and symmetry requirements are considered. h1 Electronic distributions of the positive ions are given in parentheses.

specified and satisfying

295

D. Talbi et al. /Dissociative recombination of HCO + and HCS+

Table 2 Wavefunction C12; designed for an even-handed treatment of the dissociative valence state and the Rydberg states of the HCX systems (X=0, S) Symmetry

A’ A”

Core a)

a’ core a” core

configuration classes c.d)

Electronic distributions set 1

set 2

set 3

spX CX CH a& nEx

a’& CH*R,R,R,R:R:R; 6; R,R;

17a’ MOs b, 7a” MOs

10 (10) 9 (9) 8 (8) 10 (9) 9 (8)

1 (0) 2 (1) 3 (2) 0 (0) l(1)

0 (0) 0 (0) 0 (0) l(1) l(1)

total number of CSFs: 11027 (2390) a) For HCO: a’ core= Is0 1sC. For HCS: a’ core= 1sS 1sC 2sS 2p,S 2p.S; a” core=Zp$. b, For HCO: 2a’ core correlating functions have been discarded. For HCS: 5a’ and la” core correlating functions have been discarded. ‘I Core orbitals are doubly occupied in all CSFs. All configurations corresponding to the electronic distributions specified and satisfying the spin and symmetry requirements are considered. d, Electronic distributions of the positive ions are given in parentheses.

ground and lower valence states had already been a subject of attention [ 18-20,241. 3.1. Potential surfaces by CIl Primary investigations were done using the geometries of the positive ions determined by one of us in preceding studies: CH=2.06 au, CO=2.09 au for HCO+ and CH=2.04 au, CS=2.79 au for HCS+ [ 301. Our calculations show unequivocally that all the Rydberg states are linear with 2Z more stable than *I-I: HCO:

HCS:

‘C+ (R,)

- 113.2084 au,

*IX+ (Rx)

-113.1805au,

2I-I

-113.1659au;

2C+ (R,)

-435.8267 au,

2x+ (Rx)

-435.8083 au,

2rI

-435.7992 au.

The 211 Rydberg states undergo a Renner-Teller splitting with the bending of the CH bond but, contrary to the valence states where the *A’ state is stabilized in bent equilibrium structures for the ground states of HCO and HCS (see (3a) and (3b) ), both

components of the doublet are here less stable in nonlinear geometries for the two systems. A first evaluation of the energy profiles for the CH clivage is presented in figs. 2a and 2b for the 2C+ and *II states together with the ‘C+ surfaces of the positive ions. Several important points can be seen from this first approach. The first is the strong avoided crossing between the dissociative configuration, hereafter referred to as D, and the Rydberg contiguration of R, asymptote. Interaction takes place between 2.4 and 2.8 au for HCO and between 2.5 and 3.0 au for HCS as illustrated by the change in the coefficients of the quasi-diabatic states in the CI wavefunction of the lowest 2Xf state, hereafter referred to as RD state (see table 3 ). Both RD states have a minimum but it is much deeper for HCS than for HCO, which let us anticipate a better stability of the former system with respect to dissociation in indirect processes (table 3 ). The second point of interest is the crossing of the Table 3

2.4 2.8

0.89 0.48

0.37 0.83

2.5 3.0

0.90 0.49

0.37 0.80

D. Talbi et al, /Dissociative recombination of HCO+ and HCS+

296 a.u.

HCO+

a.u. HCO+

-3.20

a 2 2

4

3

a.u.

3

4

a.u.

dcii

dcH HCS+

a.u

a.u HCS+

/’

-5.60 -I I -5.70

I

-5.00

-5.90

-6.00

b

2

3

4

a-u.

dCH

i 2

3

4

a.u

dCH Fig. 2. (a) Adiabatic potential surfaces by CII for CH bond breaking in HCO. Symbols are: R, Rydberg state of x-type asymptote, R,, Rydberg state ofy-type asymptote, R, Rydberg state of s-type asymptote, D dissociating asymptote; ‘II HCO linear lowest state. (b) Adiabatic potential surfaces by Cl1 for CH bond breaking in HCS. Symbols are: R, Rydberg state of x-type asymptote, R,, Rydberg state ofy-type asymptote, R, Rydberg state of s-type asymptote, D dissociating asymptote; ‘II HCS linear lowest state.

Fig. 3. Adiabatic potential surfaces by C12 for CH bond breaking in HCO. Symbols are: R>,Rydberg state of y-type asymptote, R, Rydberg state of s-type as asymptote; 217HCS linear lowest state. (b) Adiabatic potential surfaces by C12 for CH bond breaking in HCS. Symbols are: R, Rydberg state of y-type asymptote, R, Rydberg state of s-type asymptote; TJ HCS linear lowest state.

291

D. Talbi et al. /Dissociative recombination of HCO + and HCS +

vanished and the RD state which is mainly Rydberg (R,,) at short distance and dissociative at large CH separation is now entirely repulsive. In HCS, interaction between the Rydberg and dissociative configurations is also shifted towards shorter CH distances, but the energy barrier that opposes to dissociation is maintained and high enough to allow for a vibrational bound level in the potential well. Owing to the fact that predissociation is already effective at the *II*X+ crossing point, which is the node of the conical intersection of the potential surfaces [ 19 1, it was not deemed necessary to develop a thorough investigation of nonlinear geometries. On top of the RD state, we have a series of bound Rydberg states which are, in both systems, of the same geometry as that of the positive ions (CH = 2.06 au, CO=2.09 au for HCO and CHc2.03 au, CSc2.79 au for HCS). Franck-Condon factors have been calculated between these states; all have been found in the range 0.99-l .OOwhich implies that there will be no vibrational restriction if electronic de-excitation occurs. The general trends shown by CIl are reinforced at the C12 level. Both direct and indirect processes of dissociative recombination are then expected to proceed much easier for HCO+ than for HCS+. The adiabatic potential curves presented here show that the difference between the two systems is fundamental enough to be already seen unequivocally at the qualitative level. Evaluation of reaction rates and theoretical HCO’/CO and HCS’/CS ratios confirm our conclusions and will be reported in a forthcoming paper [28].

repulsive curves with those of the positive ions. It occurs close to the minimum of the HCO+ curve. In HCS+ the crossing is found at a larger value of the CH elongation and is therefore more distant from the minimum; it should result in a much slower dissociation for a direct process starting from the positive ion. 3.2. Potential surfaces by CI2 The n-particle space of CIl does not include the configurations needed for the treatment of correlation effects in the o bonding orbitals of the diatomic resulting from the CH clivage. These effects are expected to be important for relaxation of the geometry when breaking occurs, as is important the role of semiinternal correlation on the concomitant electronic redistribution. Such effects are taken into account in CI2. However, when doing the actual calculations, the first point to be noted is that some of the energy curves are strongly modified whereas the bond lengths of the diatomic fragments CO and CS are not changed significantly for CH distances in the range 2.0-5.0 au even if they tend naturally towards the diatomic distances for larger separations. The results are illustrated in figs. 3a and 3b. Tables 4-9 contain the energies and wavefunctions for the states of interest. The most important changes with respect to CI 1 are the drastic modifications of the energy profiles of the lowest *E+ states. In HCO, interaction between the Rydberg and dissociative configurations is shifted towards shorter CH distances (2.0-2.4 au instead of 2.4-2.8 au) as an effect of CO correlation. The potential barrier has Table 4 The potential energy curve of the 1‘Z+ (RD) state of HCO (au) Energy

Components of the wavefunction RS

1.50 1.80 2.06 2.20 2.40 3.00 4.00 5.00

- 113.1429

0.96

- 113.2356 - 113.2522 -113.2614 - 113.2793 -113.3214 - 113.3500 - 113.3595

0.96

0.87 -

R,

D -

-

0.33 0.63 0.49 -

-

0.24 0.62 0.76 0.90 0.93 0.94

D. Talbi et al. /Dissociative recombination ofHCO+ and HCS

298

Table 5 The potential

energy curve of the

1*Z+ (RD) state of HCS (au) Energy

Components R,

1.50 1.80 2.04 2.30 2.50 2.80 3.00 4.00 5.00

Table 6 The potential

energy curve ofthe

0.95 0.95

-435.1557 -435.8470 -435.8617 -435.85’10 -435.8451 -435.8565 - 435.8662 -435.8940 -435.9017

2 ‘X+ (Rydberg)

d C”

0.95 0.95 0.23

Table 7 The potential

energy curve of the 2’C+

Components

Energy

d CH

1.50 1.80 2.04 2.30 2.50 2.80 3.00 4.00 5.00

-113.1064 -113.2122 - 113.2456 - 113.2464 - 113.2326 - 113.0748 -113.0186

(Rydberg)

RX

D

-

-

-

-

0.68 0.47 0.37 -

0.61 0.80 0.86 0.92 0.93

state of HCO (au)

R, 1.50 1.80 2.06 2.20 2.40 3.00 4.00 5.00

of the wavefunction

0.41 0.95 0.96 0.95 0.91 0.90

of the wavefunction RX

D

0.90 0.86 0.67 -

0.33 0.45

1 -

state of HCS (au)

Energy

-435.7273 -435.8227 -435.8426 -435.8436 -435.8330 -435.8008 -435.7783 -435.6797 -435.6133

Components

of the wavefunction

R,

RX

-

0.90 0.91 0.91

0.91 0.93 0.93 0.90 0.90

0.86 0.27

D

0.40

-

D. Talbi et al. /Dissociative recombination ofHC0

+ and HCS

299

Table 8 The potential energy curves of the 1 *lT (valence), 2 ‘l-l (Rydberg) states of HCO and 1 ‘E+ state of HCO+ (au) d CH

1.50 1.80 2.06 2.20 2.40 3.00 4.00 5.00

HCO+ 1 ‘E+

HCO 1 *l-J

2%

- 113.2245 -113.3135 - 113.3257 - 113.3200 -113.3041 - 113.2355 -113.1303 - 113.0584

-113.0951 -113.1911 -113.2091 - 113.2064 -113.1943 -113.1368 - 113.0489 - 112.9959

- 113.0757 -113.1915 -113.1779 -113.1233 -113.0401 - 112.9893

Table 9 The potential energy curves of the 1 ‘II (valence), 2 2H (Rydberg) states of HCS and 1 ‘Z+ state of HCS+ (au) d CH

I .50 1.80 2.035 2.30 2.50 2.80 3.00 4.00 5.00

HCS’ 1 ‘E+

HCS 12H

221-1

-435.8730 -435.9626 -435.9763 -435.9647 -435.9467 -435.9140 -435.8913 -435.7933 -435.7317

-435.7197 -435.8132 -435.8296 -435.8206 -435.8045 -435.7742 -435.7529 -435.6589 -435.5947

Acknowledgement

We would like to acknowledge stimulating discussions with Dr. G. Berthier. This work was partly supported by CNRS ATP “Physic0-Chimie des MolCcules Interstellaires”. References [ 1 ] W. Klemperer, Nature 227 ( 1970) 1230. [ 21 E. Herbst and W. Klemperer, Astrophys. J. 188 ( 1974) 255. [ 31 U. Wahlgren, B. Liu, PK. Pearson and H.F. Schaefer, Nature Phys. Sci. 264 (1973) 4. [4] P.J. Bruna, Astrophys. Letters 16 (1975) 107. [ 51 W.P. Kraemer and G.H.F. Diercksen, Astrophys. J. Letters 205 (1976) L97.

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D. Talbi et al. /Dissociative recombination ofHCO+ and HCS

[ 171 T.J. Millar, Mon. Not. Roy. Astron. Sot. 202 (1983) 683.

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[24] F. Pauzat, S. Chekir and Y. Ellinger, J. Chem. Phys.85 (1986) 2861. [25] G. de1 RC, Theoret. Chim. Acta 1 (1983) 188. [ 261 V. Barone, J. Douady, Y. Ellinger, R. Subra and G. de1 Rt, J. Chem. Sot. Faraday Trans. II 75 ( 1979) 1597. 1271 M.C. Bacchus-Montabonel and P. Vermeulin, Comput. Phys. Commun. 30 (1983) 163. [28 I D. Talbi, A.P. Hickman, F. Pauzat, Y. Ellinger and G. Berthier, Astrophys. J., to be published. 1291 A.D. McLean and Y. Ellinger, Chem. Phys. 94 (1985) 25. [ 301 G. Berthier, F. Pauzat and Tao Yuanqi, J. Mol. Struct. 107 (1984) 39.