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Preferential orientations for vibrational-translational energy transfer
Volume.6, number 4
CXEMICAL PHYSICS LETTERS
PRE,FEtiENTIAL
:
ORIENTATIONS
FOR
ENERGY
._
TRANSFER
F. A. GIAWURCO
Istituto
di
Ch. Fisica.
VIBRATIONAL-
15 August 1970
TRANSLATIONAL
*
and U. LAMANNA **
v. Risorginento
35, Plea, Italy
Received 4 May 1979 An approximate angle-dependent Fotential is suggested for atom-diatom inelastic collisions in which vfbrational excitations take place. Calculations are performed via a partial wave expansion of the total wsvefunction and in the close coupling treatment. Results for some model cases are examined and the approximations discussed.
1. INTRODUCTION
ln Stidies of vibrational-translational (V-T) energy transfer, a head-on collision is often assumed and oblique collisions are accounted for at the end of calculations by introducing a “sterlc factor “. lf a more realistic model is considered for subreactive excitation processes, where the forces act between atoms of the colliding molecules and therefore orientation has to be taken into account, the necessity of ccxsidering combined vibrational-rotational-translational (VR-T) energy transfer arises [l]. The main problem of the latter, more complete, treatment has been that the number of involved reaction channels may be very large, indeed infinite, and if the coupling between these channels is strong then it is not possible to consider them. as isolated but all must be treated together during the collision. This strong coupling situation complicates the analysis enormously, and to overcome this it has been suggested that an estimate based upon general unitarity arguments and averaging processes is one of the possible ways out [Z]. At that stage it becomes very important to choose a suitable interaction potential which allows for both a reduction of numerical work and a reasonable reproduction of such experimental data as resonances, total cross sections ahd relaxation times of gas mixtures. The aim of this note is to present some model calculations relative to different target-projec* Work supported hy a grant from the Italian National. Resenrch.CounciI
**-Istituto
di Ch.
(C.N.R.).
Analilica,
v.Amendola
173. Bari.
Italy.
tiie orientations during the encounter, and to examine some qualitative conclusions which can be drawn from them. 2. DESCRlPTION OF TRE MODEL
In previous papers [3] a numerical method was cross sections of processes involving the exchange of vibrational and translational energy on collision of molecules with internally structureless particles. This method, which neglects the effect of coupling between rotational and vibrational states and assumes a spherically symmetric scattering potential, yielded satisfactory results in studies of vibrational relaxation times and showed a definite improved agreement with experiments when the molecular vibrators were described in terms of Morse oscillators [4]. Cbviously, since the molecular target was assimilated to a “breathing sphere” interacting via a (6,12) potential with the imcomiag projectile, such a model could not possibly reveal the role of resonances in these collisions. This is quite an important point, for it has been assumed by many researchers that chemical reactions, for instance, might proceed mainly through intermediate resonant complexes in vibrationally excited configuraused for calculating
tions 151.
Moreover, in the relaxation of certain molecules it has been noted that there exists an anomalous temperature. dependence of vibrational relaxation times or transition probabilities which can be correlated to the orientation dependence of the interaction energy [6].
Volume 6, number4
CHEMICAL PHYSICS LETTERS
Therefore, as a first step towards the selection of a reliable anisotropic potential, we have attempted to explicitly introduce the effect of such orientation when computing the inelastic cross sections of some simple diatomic molecules collisionally excited to their first vibrational level above ground state. This was done by finding an “observableRtotal cross section which was parametrically dependent upon the angle y between the diatomic bond distance R anci the projectile-to-molecularcenter-of-mass distance Y. Such a procedure should allow for qualitative considerations along series of atom-diatom encounters. The total interaction potential can, then, be written as: V(Y,Y, R ) - V, V,(+-,v) Q(R) ,
(Ii
where the multiplicative constant T/Oensures that the scattering potential for the ground state molecule is given by the assumed Lennard-Jones empirical function [7], V!(R) is the short-range interaction (see below) and V (Y, y) is the angledependent part of the potenka4 . Our way of constructing Vl(r, y) was to assume separate interactions of a (6,121 L. -J. type between the Ai atoms of the molecular target and the incoming projectile B, i.e.:
where the above approximate relations are based on assumptions ‘described elsewhere [4, ‘i]! Vi(R) being equal to exp a(R - IL! with Q evaluated at the classical distance of cIosest approach [3!. 3. RESULTS AND DISCUSSION Sample calculations were performed for some homonuclear diatomic molecules coiliding with D and with a structureless Ei2 psojectiIe. The collision translational energy was of 2.00 eV and, as previously tested in the close-coupling approximation [3], three vibrational levels above the ground state (0 state) needed to be coupled to bring the 0 - 1 cross sections to convergency. Table 1 assembles all the parameters of both the molecular Morse oscillators [lo] and the interaction (6,12) Lennard-Jones potentiak [ll] for the following computed cases: Ei2(XL .ZE 02(X3Z-), N2(X1 G), arrd CI2(X’Ci). In tabIe 2 the leve 7 separations of those molecules are quoted for preference, whereas table 3 presents th calculated cross sections QD2 (in units of 3 rag) for the orientations examined. Our three-body interaction potential V(r, R) can, quite generally, be expanded as foIIows: V( r ,R)
It is then trivial to give r in terms of the various ri , R and of a prefixed y [8]. Then the total wave function is expanded in terms of a sum of products of the complete orthonormal set of unperturbed molecular wavefunctions representing the motion of the collidiw particle, these latter functions can be further expanded in terms of the particle partial waves. This finally yields, in the usual way, an infinite set of coupled differential equations, to be solved under the boundary conditions imposed by the scattering problem [9]. The matrix elements of the interaction term give rise to the inelastic scattering and are of the form (the P,‘s are normalized Legendre polynomials): ‘ &&Y)
= F _k&(R)pl(COSQ)
x V(r,y,R)PI,(cos8)ddsinedR =(47&~~(R)V(r,
Y,R!c&R)~R
;
(3)
I5 August L9’70
= t;
~(~,R)P+cosy),
(4)
with the usual meaning of the symbols; only terms with I even contribute for our cases, because of symmetry. The Vl (r,R) can be obtained from the complete potential by integration: ii Vl (7, @ = $(2Z +l)/ V( r ,hfjP, (cos,?sintd,, . 0 (6) The VO term can give us some feeling about the shape of the isotropic part of the fuI1 three-body potential. As it turns out in our cases, it is strongly repulsive at relatively short distances and only slowly becomes weakly attractive at large values of r, as the exampie of fig. 1, where the case of H2 colliding with D is plotted, cleariy shows. A possibly useful way of depicting the overall behaviour of our calculated cross sections is to plot them as functions of y, with its values varying between O” and f IWO: the different resuIts associated with different molecules could then be compared with similar findings for transition probabilities computed via a Morse-Qpe interaction potential [8] and could also give us a sort of “angular” cone for effective collisional excitations. _ 327
Volume 6, number 4
:
CHEMICAL PHYSICSLETTERS
15August1970
Table 1 a).Morseoscillator~meters b)Lemu&d-Jonee pafor molecular~vibrators (au) riumtera forfuteraction potential (au) V(R)= D{&-ztz(R-Re)] -2exp[-M-q&) WI = 4w/rf2-(m61 Target
'%
o : Profec- c x103 Q tile 1.3988 0.1744 1.0093 D 0.1162 5.5936 D.
Re
Ni
2.0748 0.3091 1.2894 H2
02
22824
CL2 -3;7580
0.1323 6.4357
6.1916 1.2712 Ii2 0.0910 1.0892 H2
0.2039 6.0416 0.3074
6.9565
Table2 -Energyseparations (in ev) among a f&wlow-lyiog vibrathud levels -of the examined Morse oscillators AE
’
H,
Cl2
o--G
0;5030
0.2621
0.1735
0.0690
o-2
0.9761
0.5207
0.3441
0.1371
0 -3.
1.4192
0.7759
0.5117
0.2041
Teble3 @L computedvalues ;:orvarious orientations andwith colUsionalenergyEV=2. OeV; four coupledstates P WOunW -a) D a) H2 n 0 loo go
cl2 1I.634-~ 0.432-= -1 0.114. 0.033-l
O2 0.960-3 o.800-3
N2 0.206-' o.l72-3
o.484-3 0.208-3
o.lo2-3 0.052-'
H2 o.G3-3 o.390-3 0.295-3 0.175-3
i.
Fig.1. -potential term fortheH2.D collisional process(see text); --symmetric Lennard-Jones (6,12) potential.forthe same
case.
found for y = 90’ configurations. In fact, for the H2 case the “efficiency” of the 0 - 1 excitation is still noticeable at y = 30°, whereas the Cl2 target 0.02-3 o.093-3 O.OOSS-1 o.06C3 shcws for the same angle a drop to about 5 X lo-2 0.00016'' 0.0061-3 0.0025-3 0.021-3 60° of its value at y = O”. As indicated in table3, o.oooo7-1 0.00076-3 o.ooo5-3 o.oo63-3 60° both 0, and N2 molecules exhibit similar behaso0 o.oooo35-1o.ooo5-3 o.ooo34-3 0.0049-3 viours although intermediate between the cases h) of figs. 2 and 3. 5.6500 3.6450 2.1030 1.0800 6-E In classical terms those findings seem to suga) StructureIess projectiles. gest that for target molecules with high moments b) Excitation croessections caicuIatedwitbacphericaDy of inertia the present approximation of keeping y symmetric potential . (see text). 'constantduringtheencounter is still a reasonable one; for onljl a very small momentum transfer Figs.. 2 and 3 show such plots for II3 colliding takes place with the much lighter projectile. Moreovex, when the pppulated rotational levels with D. and 912 c.olliding with a- structureless H2 .projectfle.‘- It is clear from them that both the are t.ho& with small J values, the incoming particle at thermal energies can be described as imvibratoi: structure and the L.-J. parameters pacting with a very @owly rotating target (y u affect the angular dependence of QoL; however, in all easesexamined, the collinear co@sions co@.). : ObviouSlyi the physical va&tity-of such an apprcduc+!d the-highest kelastic-cross sections, proach depends upon the extent- to which the whereas prnctically negligible Q9L values were ._ -; .32g, ,:I;‘_. ‘. -,_’ 300 4o"
.: .
.
Volume 6. number 4
CHEMICAL PHYSICS LETTERS
15 August 1970
Fig. 2. Variations d the Qol cross sections for the H2,D collisionsl excitation process when the angle of impact y changes. EC011= 2.00 eV.
Fig. 3. Influence of the changes of y (a&&e of impact) upon the Qol cross sections for the vibratiqnal excitation nf Cl2 by structureless H2. Ec.,ll = 2.00 eV. 329
Volume 6, number 4
.
CHEMICAL PHYSICS LETTERS
(T-I?)
15 August 1970
‘4IEFERENCES [i] L. F.HerzfeId and T.A.Litovits, Absorption and - dispersion of ultrasonic waves (Academic Press,
New York, 1959). [2] R.B.Bernstein. A.Ddgarnc, M;S.W.Massey.snd I. C.PercivsI, Proc. Roy. Sot. A247 (1963) 427. [3] F.A.Gianturco and RMarriott, J. Phys. B2 (1969) 1332 and references therein. [4] A. J-Taylor and F .A.Gisnturco. Chem. Phys. Litters 4 (1969) 376. [5] P.G.Burke, D.Scrutton. J.H.Tait and A. J.Taylor, J. Phys. B2 (1969) 1155. [ 61 H.K.Sbin. J. Chem. Phys. 49 (1968) 3964. [7l R.Marriott. Proc. Phys. Sot. (London) 83 (1964) 159. [S] H.K.Sh& Intern. J. Quantum Chem. 2 (1968) 265. [9] N.F.Mott and M.S.W.Maesey, The theory of atomic collisions (Oxford Univ. Press, London, 1947). [lo] D-Steele. E.R.Lippincott and J.T.Vanderslice, Rev. Mod. Phys. 34 (1962) 239. [ll] J. 0. Hirschfelder, -F. C. Curt&s and R. B. Bird. Molecular theory of gases and liquids (Wiley, New York, 1954), [12] R.B.Bernstein, R.E.Roberts and C.F.Curtiss. J. Chem. Phys. 50 (1969) 5163.
give useful- information along series of similq molecules coRXing with an appropriate _ projectile.. Studies with a more complete treatment of nonreactive @elastic collisions involving both rotational and vibrational excitation of simple molecules are of course very much in need at this point and would provide a sound check of the present results.
ACKNOWLEDGEMENTS All calcu1ation.s were performed in double precision on a 360/65 IBM machine. We are grateful to Professor R. Moccia for his many helpful suggestions as this work was in progress.
-_
._
‘_
.._..
. .
-.
.-
.
__.
. .
L
.,’ _I
.330’
.I
-
;..
_:.
._-
-.
CXEMICAL PHYSICS LETTERS
PRE,FEtiENTIAL
:
ORIENTATIONS
FOR
ENERGY
._
TRANSFER
F. A. GIAWURCO
Istituto
di
Ch. Fisica.
VIBRATIONAL-
15 August 1970
TRANSLATIONAL
*
and U. LAMANNA **
v. Risorginento
35, Plea, Italy
Received 4 May 1979 An approximate angle-dependent Fotential is suggested for atom-diatom inelastic collisions in which vfbrational excitations take place. Calculations are performed via a partial wave expansion of the total wsvefunction and in the close coupling treatment. Results for some model cases are examined and the approximations discussed.
1. INTRODUCTION
ln Stidies of vibrational-translational (V-T) energy transfer, a head-on collision is often assumed and oblique collisions are accounted for at the end of calculations by introducing a “sterlc factor “. lf a more realistic model is considered for subreactive excitation processes, where the forces act between atoms of the colliding molecules and therefore orientation has to be taken into account, the necessity of ccxsidering combined vibrational-rotational-translational (VR-T) energy transfer arises [l]. The main problem of the latter, more complete, treatment has been that the number of involved reaction channels may be very large, indeed infinite, and if the coupling between these channels is strong then it is not possible to consider them. as isolated but all must be treated together during the collision. This strong coupling situation complicates the analysis enormously, and to overcome this it has been suggested that an estimate based upon general unitarity arguments and averaging processes is one of the possible ways out [Z]. At that stage it becomes very important to choose a suitable interaction potential which allows for both a reduction of numerical work and a reasonable reproduction of such experimental data as resonances, total cross sections ahd relaxation times of gas mixtures. The aim of this note is to present some model calculations relative to different target-projec* Work supported hy a grant from the Italian National. Resenrch.CounciI
**-Istituto
di Ch.
(C.N.R.).
Analilica,
v.Amendola
173. Bari.
Italy.
tiie orientations during the encounter, and to examine some qualitative conclusions which can be drawn from them. 2. DESCRlPTION OF TRE MODEL
In previous papers [3] a numerical method was cross sections of processes involving the exchange of vibrational and translational energy on collision of molecules with internally structureless particles. This method, which neglects the effect of coupling between rotational and vibrational states and assumes a spherically symmetric scattering potential, yielded satisfactory results in studies of vibrational relaxation times and showed a definite improved agreement with experiments when the molecular vibrators were described in terms of Morse oscillators [4]. Cbviously, since the molecular target was assimilated to a “breathing sphere” interacting via a (6,12) potential with the imcomiag projectile, such a model could not possibly reveal the role of resonances in these collisions. This is quite an important point, for it has been assumed by many researchers that chemical reactions, for instance, might proceed mainly through intermediate resonant complexes in vibrationally excited configuraused for calculating
tions 151.
Moreover, in the relaxation of certain molecules it has been noted that there exists an anomalous temperature. dependence of vibrational relaxation times or transition probabilities which can be correlated to the orientation dependence of the interaction energy [6].
Volume 6, number4
CHEMICAL PHYSICS LETTERS
Therefore, as a first step towards the selection of a reliable anisotropic potential, we have attempted to explicitly introduce the effect of such orientation when computing the inelastic cross sections of some simple diatomic molecules collisionally excited to their first vibrational level above ground state. This was done by finding an “observableRtotal cross section which was parametrically dependent upon the angle y between the diatomic bond distance R anci the projectile-to-molecularcenter-of-mass distance Y. Such a procedure should allow for qualitative considerations along series of atom-diatom encounters. The total interaction potential can, then, be written as: V(Y,Y, R ) - V, V,(+-,v) Q(R) ,
(Ii
where the multiplicative constant T/Oensures that the scattering potential for the ground state molecule is given by the assumed Lennard-Jones empirical function [7], V!(R) is the short-range interaction (see below) and V (Y, y) is the angledependent part of the potenka4 . Our way of constructing Vl(r, y) was to assume separate interactions of a (6,121 L. -J. type between the Ai atoms of the molecular target and the incoming projectile B, i.e.:
where the above approximate relations are based on assumptions ‘described elsewhere [4, ‘i]! Vi(R) being equal to exp a(R - IL! with Q evaluated at the classical distance of cIosest approach [3!. 3. RESULTS AND DISCUSSION Sample calculations were performed for some homonuclear diatomic molecules coiliding with D and with a structureless Ei2 psojectiIe. The collision translational energy was of 2.00 eV and, as previously tested in the close-coupling approximation [3], three vibrational levels above the ground state (0 state) needed to be coupled to bring the 0 - 1 cross sections to convergency. Table 1 assembles all the parameters of both the molecular Morse oscillators [lo] and the interaction (6,12) Lennard-Jones potentiak [ll] for the following computed cases: Ei2(XL .ZE 02(X3Z-), N2(X1 G), arrd CI2(X’Ci). In tabIe 2 the leve 7 separations of those molecules are quoted for preference, whereas table 3 presents th calculated cross sections QD2 (in units of 3 rag) for the orientations examined. Our three-body interaction potential V(r, R) can, quite generally, be expanded as foIIows: V( r ,R)
It is then trivial to give r in terms of the various ri , R and of a prefixed y [8]. Then the total wave function is expanded in terms of a sum of products of the complete orthonormal set of unperturbed molecular wavefunctions representing the motion of the collidiw particle, these latter functions can be further expanded in terms of the particle partial waves. This finally yields, in the usual way, an infinite set of coupled differential equations, to be solved under the boundary conditions imposed by the scattering problem [9]. The matrix elements of the interaction term give rise to the inelastic scattering and are of the form (the P,‘s are normalized Legendre polynomials): ‘ &&Y)
= F _k&(R)pl(COSQ)
x V(r,y,R)PI,(cos8)ddsinedR =(47&~~(R)V(r,
Y,R!c&R)~R
;
(3)
I5 August L9’70
= t;
~(~,R)P+cosy),
(4)
with the usual meaning of the symbols; only terms with I even contribute for our cases, because of symmetry. The Vl (r,R) can be obtained from the complete potential by integration: ii Vl (7, @ = $(2Z +l)/ V( r ,hfjP, (cos,?sintd,, . 0 (6) The VO term can give us some feeling about the shape of the isotropic part of the fuI1 three-body potential. As it turns out in our cases, it is strongly repulsive at relatively short distances and only slowly becomes weakly attractive at large values of r, as the exampie of fig. 1, where the case of H2 colliding with D is plotted, cleariy shows. A possibly useful way of depicting the overall behaviour of our calculated cross sections is to plot them as functions of y, with its values varying between O” and f IWO: the different resuIts associated with different molecules could then be compared with similar findings for transition probabilities computed via a Morse-Qpe interaction potential [8] and could also give us a sort of “angular” cone for effective collisional excitations. _ 327
Volume 6, number 4
:
CHEMICAL PHYSICSLETTERS
15August1970
Table 1 a).Morseoscillator~meters b)Lemu&d-Jonee pafor molecular~vibrators (au) riumtera forfuteraction potential (au) V(R)= D{&-ztz(R-Re)] -2exp[-M-q&) WI = 4w/rf2-(m61 Target
'%
o : Profec- c x103 Q tile 1.3988 0.1744 1.0093 D 0.1162 5.5936 D.
Re
Ni
2.0748 0.3091 1.2894 H2
02
22824
CL2 -3;7580
0.1323 6.4357
6.1916 1.2712 Ii2 0.0910 1.0892 H2
0.2039 6.0416 0.3074
6.9565
Table2 -Energyseparations (in ev) among a f&wlow-lyiog vibrathud levels -of the examined Morse oscillators AE
’
H,
Cl2
o--G
0;5030
0.2621
0.1735
0.0690
o-2
0.9761
0.5207
0.3441
0.1371
0 -3.
1.4192
0.7759
0.5117
0.2041
Teble3 @L computedvalues ;:orvarious orientations andwith colUsionalenergyEV=2. OeV; four coupledstates P WOunW -a) D a) H2 n 0 loo go
cl2 1I.634-~ 0.432-= -1 0.114. 0.033-l
O2 0.960-3 o.800-3
N2 0.206-' o.l72-3
o.484-3 0.208-3
o.lo2-3 0.052-'
H2 o.G3-3 o.390-3 0.295-3 0.175-3
i.
Fig.1. -potential term fortheH2.D collisional process(see text); --symmetric Lennard-Jones (6,12) potential.forthe same
case.
found for y = 90’ configurations. In fact, for the H2 case the “efficiency” of the 0 - 1 excitation is still noticeable at y = 30°, whereas the Cl2 target 0.02-3 o.093-3 O.OOSS-1 o.06C3 shcws for the same angle a drop to about 5 X lo-2 0.00016'' 0.0061-3 0.0025-3 0.021-3 60° of its value at y = O”. As indicated in table3, o.oooo7-1 0.00076-3 o.ooo5-3 o.oo63-3 60° both 0, and N2 molecules exhibit similar behaso0 o.oooo35-1o.ooo5-3 o.ooo34-3 0.0049-3 viours although intermediate between the cases h) of figs. 2 and 3. 5.6500 3.6450 2.1030 1.0800 6-E In classical terms those findings seem to suga) StructureIess projectiles. gest that for target molecules with high moments b) Excitation croessections caicuIatedwitbacphericaDy of inertia the present approximation of keeping y symmetric potential . (see text). 'constantduringtheencounter is still a reasonable one; for onljl a very small momentum transfer Figs.. 2 and 3 show such plots for II3 colliding takes place with the much lighter projectile. Moreovex, when the pppulated rotational levels with D. and 912 c.olliding with a- structureless H2 .projectfle.‘- It is clear from them that both the are t.ho& with small J values, the incoming particle at thermal energies can be described as imvibratoi: structure and the L.-J. parameters pacting with a very @owly rotating target (y u affect the angular dependence of QoL; however, in all easesexamined, the collinear co@sions co@.). : ObviouSlyi the physical va&tity-of such an apprcduc+!d the-highest kelastic-cross sections, proach depends upon the extent- to which the whereas prnctically negligible Q9L values were ._ -; .32g, ,:I;‘_. ‘. -,_’ 300 4o"
.: .
.
Volume 6. number 4
CHEMICAL PHYSICS LETTERS
15 August 1970
Fig. 2. Variations d the Qol cross sections for the H2,D collisionsl excitation process when the angle of impact y changes. EC011= 2.00 eV.
Fig. 3. Influence of the changes of y (a&&e of impact) upon the Qol cross sections for the vibratiqnal excitation nf Cl2 by structureless H2. Ec.,ll = 2.00 eV. 329
Volume 6, number 4
.
CHEMICAL PHYSICS LETTERS
(T-I?)
15 August 1970
‘4IEFERENCES [i] L. F.HerzfeId and T.A.Litovits, Absorption and - dispersion of ultrasonic waves (Academic Press,
New York, 1959). [2] R.B.Bernstein. A.Ddgarnc, M;S.W.Massey.snd I. C.PercivsI, Proc. Roy. Sot. A247 (1963) 427. [3] F.A.Gianturco and RMarriott, J. Phys. B2 (1969) 1332 and references therein. [4] A. J-Taylor and F .A.Gisnturco. Chem. Phys. Litters 4 (1969) 376. [5] P.G.Burke, D.Scrutton. J.H.Tait and A. J.Taylor, J. Phys. B2 (1969) 1155. [ 61 H.K.Sbin. J. Chem. Phys. 49 (1968) 3964. [7l R.Marriott. Proc. Phys. Sot. (London) 83 (1964) 159. [S] H.K.Sh& Intern. J. Quantum Chem. 2 (1968) 265. [9] N.F.Mott and M.S.W.Maesey, The theory of atomic collisions (Oxford Univ. Press, London, 1947). [lo] D-Steele. E.R.Lippincott and J.T.Vanderslice, Rev. Mod. Phys. 34 (1962) 239. [ll] J. 0. Hirschfelder, -F. C. Curt&s and R. B. Bird. Molecular theory of gases and liquids (Wiley, New York, 1954), [12] R.B.Bernstein, R.E.Roberts and C.F.Curtiss. J. Chem. Phys. 50 (1969) 5163.
give useful- information along series of similq molecules coRXing with an appropriate _ projectile.. Studies with a more complete treatment of nonreactive @elastic collisions involving both rotational and vibrational excitation of simple molecules are of course very much in need at this point and would provide a sound check of the present results.
ACKNOWLEDGEMENTS All calcu1ation.s were performed in double precision on a 360/65 IBM machine. We are grateful to Professor R. Moccia for his many helpful suggestions as this work was in progress.
-_
._
‘_
.._..
. .
-.
.-
.
__.
. .
L
.,’ _I
.330’
.I
-
;..
_:.
._-
-.