Semi-classical pressure broadening calculations using different potentials

SpectrochimicaActa, Vol. 42A, No.

9, pp. 1027-1028, 1986.

0584-8539/86 $3.00+0.00 Pergamon Journals Ltd.

Printed in Great Britain.

Semi-classical pressure broadening calculations using different potentials A. PICARD-BERSELLINI,*M. BROQUIER* and C. BOULETt *Laboratoire de Photopbysique Mol6culairedu CNRS, B~timent 213, Universit6 de Paris-Sud, 91405-Orsay Cedex, France and t'D6partement de Physique Atomique et Mol6culaire, Universit6 de Rennes I, Campus de Beaulieu, 35042-Rennes Cedex, France

(Received 17 February 1986; accepted 13 March 1986) Abstract--We have made a comparison between the pressure broadening coefficientsof OCS-Ar and semiclassical calculations using different potentials. of closest approach r, the real trajectory is approximated by its parabolic trajectory. Then one introduces the influence ofF~so which derives from the potential Vlso in the equation of motion around the

We have performed semi-classical pressure broadening calculations. The translational motion of all particles is treated classically. The internal states of the particles are treated quantum mechanically. The semi-classical differential broadening cross section [1], where b is the impact parameter and S is the collision matrix, is: 1

S(fi, f'i',b,v)= ( - lYf +J~ x/((2.h + 1)(2j~+ 1)) m,...l,..-~--;.'(j~Lm'IM/j~m~)(J'[LmyM/j~m') s M

{A- < ~ ISlj;m; >*}

(1)

The following terms ( / ) are the Clebsch-Gordan coefficients. Then the linewidth is given by: 1

So = ( - 1)J'+J~ (2j, + 1-----~~" (jflm'rO/j,m;)(jrlmfM/y,m,){l -
Isljfmf

> * }

(2)

where distance of the closest approach. They obtain for r(t):

7if =nb

21tbdb

v f (v)dv Re{S(fi, f T , b, v)},

FlSOt2 r(t) -~ r, + v,t + - 2#

(3) where So represents the diagonal part of the S matrix. For the calculation of the classical S matrix we have used the formalism of SMITH et al. [2], where V = Vi + V=: V~represents the isotropic part of the interaction; V. denotes the rest of the interaction. The classical trajectories are determined by V~. The scattering matrix terms taken between the vibration rotation states [vjm> are given by:





=

(6)

We have used two potentials. LIu and MARCUS [4] used this potential: V (r,O) = Vo(r)+ V2(r)P2(cosO )

(7)

vo(r) =

4e[(~)t2 --(~)61

(8)

l)2(r)

4e ~t r

(9)

=

"

(4) DREYFUS [5] used this potential:

l

= f ; . dt ei.$J't

O" 12

O" 6

x , (5) where {R(t)} is a function of the impact parameter b and the velocity u. It describes the classical trajectory in the center of a reference mass. For the trajectory description we have used the description of BONAMYet aL [3]. Around the distance

+ P1 (cos 0)4e ~ a

+ P24e[6(~) 1027

12

(10) O" 6

-l'(~) ],

1028

A. PICARD-BERSELLINIet al.

OscuLatingparabota I I l I |

~ \ \ \ \

/

rc

Realtrajectory

~

(~r(£)

Fig. 1. Geometry of the collision (in the particular type ofeollision represented here, the repulsive forces are most important).

Table 1

(HWHM) of an isolated line is then given by nV

OCS-Ar Potential Llv and MARCUS DREYFUS

Transition P(3) P(4) P(5) P(6) P(7)

146.7 143.1 139.8 136.5 132.6

where

135.7 133.6 131.3. 129 126.5

oyi = Re

where ,o:<,

I

2 l+q\

(r o is when

d"c

] t/6

f;

2n R ,

S o d R c.

(14)

o

where R is the distance between the center of mass of the molecule and the perturber; 0 is the angle between the radiation perturber internuclear axis and the molecular axis; ~ is the energy when VIso is minimum; o is the value when Vlso = 0. The calculation of the terms of the ~/matrix is the same as the calculation of the terms of the matrix of the anisotropic part of the potential. Then it is necessary to diagonalize the r/matrix and thus we can calculate the semi-classical differential broadening cross section S D [Eqn. (1)]. The integral over the impact parameter b is:

,_,
(13)

(1,)

.2)

2e ]

b= 0). The half width at half maximum

Then we have calculated the pressure broadening cross section for different transitions and we have used the two potentials of Ltu and MARCUS [4] and DREYFUS [5] (Table 1). We can conclude that the semi-classical calculation and the sudden approximation [6] give the same result for low J. The IOS approximation is accurate for low J lines but does not properly predict the dependence on line number. These results confirm a previous calculation of pressure broadened linewidths for OCS in Ar performed by GREEN [7]. On the other hand it seems difficult to conclude which of the potentials is best because the two potentials seem to be in good agreement with experimental results.

REFERENCES [1] C. BOULET,Thesis, array (1979). [2] E.W. SMITH,M. GIRAUDand J. COOPER,J. chem. Phys. 65, 1256 (1976). [3] J. BO~AMY,L. BON^MVand D. ROBERT,J. chem. Phys. 67, 4441 (1977). [4] W.K. LIUand R. A. MARCUS,J. chem. Phys. 63, 272, 290 (1975). [5"] C. DREYFUS,Thesis, Paris (1980). [6] M. BROQU1ER,A. PICARD-BERSELLINI,B. J. WHITAKER and S. GREEN,J. chem. Phys. 84, 2104 (1986). [7] S. GREEN,J.Q.S.R.T. 33, 299 (1985),