A semiclassical, nonperturbative approach to the description of molecular collisionsSuresh C. Mehrotra and James E. Boggs Citation: The Journal of Chemical Physics 62, 1453 (1975); doi: 10.1063/1.430604 View online: http://dx.doi.org/10.1063/1.430604 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/62/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Semiclassical approach to the description of the basic properties of nanoobjects Low Temp. Phys. 34, 838 (2008); 10.1063/1.2981399 Semiclassical description of antiproton collisions with molecular Hydrogen AIP Conf. Proc. 796, 276 (2005); 10.1063/1.2130179 Semiclassical algebraic description of inelastic collisions J. Chem. Phys. 85, 5611 (1986); 10.1063/1.451575 Semiclassical approach to spontaneous emission of molecular collision systems: A dynamical theory offluorescence line shapes J. Chem. Phys. 76, 3396 (1982); 10.1063/1.443464 A semiclassical, nonperturbative approach to collisioninduced transitions between rotational levels for the N2–Arsystem J. Chem. Phys. 63, 4618 (1975); 10.1063/1.431272

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

132.174.255.116 On: Fri, 19 Dec 2014 16:53:58

A semiclassical, nonperturbative approach to the description of molecular collisions

Suresh C. Mehrotra and James E. Boggs

Department of Chemistry. The University of Taxas at Austin. Austin. Texas 78712 (Received 19 July 1914)

Using the effective potential formulated by Rabitz [J. Chem. Phys. 57. 1118 (1912)]. rotational energy transition probabilities for OCS molecules colliding with other OCS molecules are calculated by solving coupled Schrodinger equations under the classical straight line path approximation. These results are compared with similar transition probabilities calculated from second-order perturbation theory. The results are quite different for strong collisions. For weak collisions, the discrepancies are less for low values of J. On the basis of these results, qualitative explanations of rotational energy transfer for weak and strong collisions are offered.

INTRODUCTION

The transfer of rotational energy during molecular collisions is of great interest to both experimentalists and theoreticians. Experimentally, the process has been studied by the measurement of pressure broadening of spectral lines,l molecular beam methods,2 microwave double resonance,3 and indirectly by other techniques. The experimental data were first interpreted by first­order perturbation theory4 and more recently by an ap­proach which, although not strictly a perturbation meth­od, is a first-order treatment.5 The quality of the ex­perimental data has been improving in recent years, and it now appears that the disagreements between the­ory and experiment are beyond the limits of experimen­tal error.1,e It therefore seems worthwhile to attempt a partial treatment of the problem in a more fundamen­tal way to see if light can be shed on the mechanism of the energy transfer and estimates can be obtained of the probable reliability of the previous more complete but more approximate treatments.

An artificial but useful separation of intermolecular events into strong and weak collisions is frequently used in the discussion of energy transfer. 4,5 For the limiting case of a weak collision in which the intermolecular in­teraction is small and of long range, the perturbation techniques in common use4,5 are expected to be highly reliable. For the theoretical treatment of the extreme case of strong collision, the probability of finding a molecule out of its initial state after the collision is as­sumed to be 1; i. e., for a strong collision L,1¢i Pi _/(b)

= 1 for b < bo, where Pi -I is the transition probability for a state i to a state j, b is the impact parameter, and bo is the impact parameter at which the probability of finding the molecule out of its initial state becomes 1 by perturbation theory. This treatment does not pro­vide any information about the individual values of Pi -I' Rabitz and Gordon7 have used a method in which the re­gion 0 ",:; b ",:; bo is divided into many equally spaced inter­vals. Various transition probabilities are calculated and their total is normalized to unity. Agreement with experiments is reasonably good with some exceptions, especially in those cases in which strong collisions are most significant.1,a Since perturbation theory does not converge in the region 0 ",:; b ",:; bo' this method would not seem to be accurate for the treatment of strong colli­sions.

Similarly, some method is needed to calculate the transition probability for those parts of the region b> bo where the transition probability Pi _ i is not much small­er than 1. Furthermore, the recent discovery of inter­stellar molecules and their anomalous rotation distribu­tion provides a strong incentive for the development of a method to treat strong collisions accurately. For an explanation of these anomalies, a detailed understanding of molecular collision is necessary. 9

For the treatment of strong collisions and intermediate cases, a newly developed nonperturbative approach10

may be very useful. This technique has recently been successfully applied to spectroscopic problems ll and to collision theory.12

In the present paper, a nonperturbative approach with the effective potential used by Rabitz13 is used to exam­ine the rotational energy transfer mechanism during dif­ferent types of collisions between OCS-OCS molecules. In the first section, the theory and numerical techniques used in the calculation are explained. In subsequent sections, the results are discussed and certain conclu­sions are presented.

THEORY

Consider the Hamiltonian H of a system of two mole­cules colliding under an interaction potential V,

(1) ~ (2) ~ (~~ H=Ho (Ol)+H o (02)+ VR,Ol'O:!)' (1)

where H~j)(Oi)' (i = 1,2) is the Hamiltonian which de­scribes the internal coordinate of the ith molecule, ni denotes the orientation of the ·ith molecule, and 0 is the orientation of R, where r R I is the distance between the two molecules. In general, the interaction energy V (01, O2, R) can be expanded in terms of spherical har­monics,14

VUD(n 1 ,02,R)= L L A'l'2,(R)(1112J.L1J.L2I ZJ.L) '1'2' "1"2"

~ ( * X Y'l"l (01)Y'2"2 O2) Y ,,, (0) , (2)

where A'l'2,(R) is the expansion coefficient. Under the nonoverlapping condition [R> r1 and R > rg, where rj (i = 1, 2) is the radius of the charge distribution of the ith molecule], the expansion coefficient can be written as

The Journal of Chemical Physics. Vol. 62, No.4, 15 February 1975 Copyright © 1975 American Institute of Physics 1453

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

132.174.255.116 On: Fri, 19 Dec 2014 16:53:58

1454 S. C. Mehrotra and J. E. Boggs: Description of molecular collisions

and

l = II + l2 •

Here l1 and l2 depend on the type of interaction con­sideredj e. g., II = 1, l2 = 1 is for dipole-dipole interac­tion, II = 2, l2 = 2 for quadrupole-quadrupole interaction, etc. The Schrooinger equation for the system can be written as

H II/! ) = iff :t II/!) , (3)

where II/!) describes the wavefunction of the system. Replacing H- Heff, Eq. (3) becomeslS

Heff

ll/!, 1/ 2>=iff :t IIP/1J2 > , (4)

where H ef! is the Rabitz effective Hamiltonian and IIPJ1J2) is the effective state vector. The operator H,(/'

where [Ji ] = (2Ji + 1), and

nWJ J J'J' =(EJ +EJ ) - {EJ• +EJ,) , 1212 1 2 1 2

(10)

where 211n is Planck's constant. Also,

(J1J all T 111211lJ ~J ~ ) = {[Z][ J 1][ J21[J ;l[ J ~1/(411)3')-1 /2

J~) o ' (11)

where (: : :) denotes a 3j symbol.

In Eq. (8), the summation is over all rotational levels of the absorber and the perturberj but for practical purposes, the summation has to be restricted by some lower and upper values of J; (i = 1,2) which are deter­mined by the matrix (il Vet[ I j). This is explained in the next section.

By using Eq. (8), the number of coupled equations is reduced considerably. For example, in the case of J1

= 1, J 2 = 1, and considering only first-order dipole-di­pole interactions, 81 equations are reduced to only 9 equations.

In Table I, the physical parameters for OCS mole­cules used in this calculation are given. A classical straight line path approximation as shown in Fig. 1 is taken, since the de Broglie wavelength associated with

Classical Straight Line 2 Perturber (2) Trajectory

~b Z Absorber (Il

FIG. 1. Classical straight line trajectory used in the calcula­tion.

has the following properties:

<J;J~IHorrl,J1J2)=oJ J.oJ J.(EJ JdEJ J')' (5) 11221222

<J~J~IJ1J2>=OJ J'OJ J', (6) 1 1 2 2

EJ.J,=EJ.+EJ" (i=1,2). & t l f

Here I ,J1 J 2 > represents the effective state vector and EJ i represents the energy of the i th molecule in the ro­tational state J i • The effective state IIPJ J ) of the

1 2 system can be expanded as

IIPJ 1 )= L <lJ'J.(t)eiwJF2tIJ;J~). 1 2 JiJi 1 2

(7)

Substituting Eq. (7) in Eq. (4) and using properties (5) and (6), we get

where I <lJ 1J212 is the effective probability of finding the molecules in states J1 and J 2 ,

OCS is small as compared to the range of the interaction, and the interaction energy of the system is very much less than the translational energy. For example, for an impact parameter b = 5.0 A, the dipole-dipole inter­action energy of OCS molecules is approximately 400 GHz, whereas the translational energy is of the order of 2000 GHz.

Equations (8) are solved by the Adams-Moulton pro­cedure with Runge-Kutta starter. An error analysis is conducted after each step with the Adams-Moulton meth­od. If .4;:;1 is the numerical solution obtained by the Adams-Bashforth formula and...t<';;l from the Adams­Moulton formula, where the vector sign denotes a col­umn matrix for the array.4, then it can be shown15 that an estimate of the single step error is given by

(12)

where y is a numerical factor determined by the order of the Adams-Bashforth and Adams-Moulton formula used. For the fourth-order Adams-Bashforth and fourth-order Adams-Moulton formula, 'Y = 19/270. The step size is automatically adjusted in the program such that max I Ej I is not more than 0.0001. The array ~~ is normalized to unity after each step.

TABLE 1. Parameters used in the calculation.

Dipole momen~ Quadrupole momentb

Rotational constantC

Velocity

0.715 X 10"18 esu 0.88 x 10-26 esu

6081.490 MHz 1.0x105 cm/sec

aJ . S. Muenter, J. Chem. Phys. 48, 4544 (1968). two H. Flygare, J. Chem. Phys. 50, 1714 (1969). CWo C. King and W. Gordy, Phys. Rev. 90, 319 (1953i.

J. Chern. Phys., Vol. 62, No.4, 15 February 1975

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

132.174.255.116 On: Fri, 19 Dec 2014 16:53:58

s. C. Mehrotra and J. E. Boggs: Description of molecular collisions 1455

1.0

- FOR b =5.0 ~ ---- FOR b 06.0 A

- , ~ •. 5 . -- - ----- - - ---n.-

(a)

0 ' , "

-20 -10 0 10 20 30 40 50 Z (/\)

1.0

-----------

_. FOR b oS.OA .... FOR b 0 15.0A

-I .5

n.-

(b)

0 -20 -10 0 10 20 30 40 50

Z(A)

1.0

.. - b = 5.0 A ---- b =8.0 A

3 ,

• . 5 q -n.- ------------

(e)

0 -20 -10 0 10

Z(A) 20 30 40 50

1.0

0

0 - b = 8.0 A "!. 0

T --- b = 15.0 A 0 "!. .5

n.-

.------------._-------

(d)

0 -20 -10 0 10 20 30 40 50

Z(A)

FIG. 2. TDTP graph for (a) P1.1-1.1 at impact parameter b

==5.0 A and b==6.0 A corresponding to strong collision; (b) Pt.1-1.1 at impact parameter b == 8. 0 A and b= 15. 0 A correspond­ing to weak collision; (c) P 1•0 - 1•0 at impact parameter b= 5. 0 A (strong collision) and b=8.0 A (weak collision); (d) P1.20-1.20 at impact parameter b = 8.0 A and b = 15. 0 A.

DISCUSSION

To understand energy transfer during a molecular collision, it is very useful to plot the time dependence of the transition probability P,., for a molecule remain­ing in the initial state i during a molecular collision for different types of collisions. In Figs. 2(a)-2(d), graphs

are shown corresponding to different impact parameters and different J values. We will call these graphs "time dependent transition probability" (TDTP) graphs.

To understand the rotational energy transfer mecha­nism, let us consider two TDTP graphs corresponding to impact parameters b = 6. 0 A and b = 5.0 A [Fig. 2(a)] . At b = 6. 0 A, a minimum occurs atZ = 2.50 A (fordefini­tion of Z, see Fig. 1), at which point the transition probability for remaining in the same state is 0.0061 (very close to 0, as in the extreme case of strong col­lision). At Z = 2. 50 A, the interaction energy is of the order of 250 GHz and coupling with various states other than the initial state becomes significant owing to the increase of the population of such states. The transi­tion probability of finding the molecule in the initial state will therefore increase with time. Corresponding to the impact parameter b = 5. 0 A, a minimum of the TDTP curve [Fig. 2(a)] occurs at Z = - O. 625 A (at a smaller distance due to the stronger collision) and cor­responding transition probability is 0.0278. Between the distances corresponding to Z = - O. 625 A to Z = 7.0 A, the transition probability starts increasing, and so the populations of other states decrease. As a result, the coupling factor becomes less Significant at Z = 7. 5 A, but at this distance the interaction energy is still sufficiently large to excite the molecule from its initial state, leading to a maximum in the TDTP curve. Thus, a minimum in the TDTP curve is due to strong interac­tions and a maximum is due to strong coupling between various states.

It is also interesting to observe in the b = 5.0 A case that whereas the energy transfers from its initial state to other states during the colliSion, the final probability of finding the molecule in its initial state is almost uni­ty. Such types of collisions are said to be adiabatic collisions in which the states of the molecules do not undergo a permanent change due to the collision.

During a weak collision, the TDTP graph suggests the instantaneous transition probability Pi.' (t) decreases at all times during the collision until the intraction en­ergy is no longer sufficient to excite the molecule from its initial state. During a strong colliSion, due to the strong interaction, the instantaneous transition prob­ability P,. i (t) decreases rapidly, and at some time t 1 ,

P,.,(tl) becomes very small in the potential region where the interaction energy is still large. Since the coupling between the ith and jth state at the instant t 1

depends on the matrix element VlJe'w'Jt 1 aJ , this cou­pling increases due to the fact that a, ~ 0 and aJ *- 0, and so energy is retransferred to the initial state from other states. Thus, during a strong colliSion, the en­ergy is transferred and retransferred between the initial state and other states.

Thus we can characterize a strong collision as one for which the TDTP has at least one minimum. A col­lision may be considered to consist of n extreme cases of strong collisions and m weak collisions, where n =- 0, 1, 2, ..• , and m = 0, 1. When m = 0 and n *- 0, the total effect of the collision is the same as in the extreme case of a strong colliSion; whereas when n = 0 and m = 1, we can treat the collision by perturbation technique.

J. Chern. Phys., Vol. 62, No.4, 15 February 1975 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

132.174.255.116 On: Fri, 19 Dec 2014 16:53:58

1456 S. C. Mehrotra and J. E. Boggs: Description of molecular collisions

TABLE II. Calculated values of time independent transition probability Pl,1~Ji,J2'

0,0 0,1 0,2 0,3 1,0 1,1 1,2 1,3 2,0 2,1 2,2 2,3 3,0 3,1 3,2 3,3

0.0454 0.0000 0.0161 f 0.0000 0.8861 0.0000 f 0.0161 0.0000 0.0373 f f f f f

5.00

0.3010 0.0000 0.1474 0.0000 0.0000 0.0007 0.0000 0.1789 0.1474 0.0000 0.0220 0.0000 0.0000 0.1789 0.0000 0.0261

0.0441 0.0022 0.0503 0.0044 0.0022 0.1091 0.0036 0.0240 0.0503 0.0036 0.2999 0.1361 0.0044 0.0240 0.1361 0.1068

e e 0.4823 0.0177 e 0.0000 e e 0.4823 e e e 0.0177 e e e

"Transition probability taking condition (I). ~ransition probability taking condition (II). cTransition probability taking condition (III).

0.2629 0.0000 0.2832 f 0.0000 0.0692 0.0000 f 0.2382 0.0000 0.1921 f f f f f

8.00

0.2893 0.0000 0.1828 0.0000 0.0000 0.0917 0.0000 0.0906 0.1828 0.0000 0.0791 0.0000 0-.0000 0.0906 0.0000 0.0142

0.2581 0.0068 0.1710 0.0021 0.0068 0.0850 0.0118 0.0879 0.1710 0.0118 0.0773 0.0042 0.0021 0.0879 0.0042 0.0130

Iv<i

e e 0.1257 0.0018 e 0.7450 e e 0.1257 e e e 0.0018 e e e

0.0347 0.0000 0.0335 f 0.0000 0.8762 0.0000 f 0.0335 0.0000 0.0223 f f f f f

15.00

0.0349 0.0000 0.0324 0.0000 0.0000 0.8761 0.0000 0.0008 0.0329 0.0000 0.0213 0.0000 0.0000 0.0008 0.0000 0.0001

0.0349 0.0002 0.0330 0.0001 0.0002 0.8743 0.0002 0.0008 0.0330 0.0002 0.0214 0.0001 0.0001 0.0008 0.0001 0.0001

dTransition probability from perturbation theory. eCorresponding transition probabilities are very small. fDenotes corresponding transitions are not allowed.

This concept may be very useful for the further develop­ment of molecular collision theory.

Jr=Jj + 1,

Jrwn = J j - 1, if J f 2: 1

= 0, if J j ~ 1 (i = 1, 2) ,

Iv<! e e 0.0102 0.0000 e 0.9795 e e 0.0102 e e e 0.0000 e e e

Beside the above points, the following interesting ob­servations can be noted from the TDTP graphS shown in Figs. 2(a)-2(d):

(1) The minimum of the TDTP curve shifts to the right with increase of impact parameters.

where Jr and Jrwn are upper and lower limits of J; in Eq. (8) and J j is the initial state of the ith molecule;

(2) For higher values of J 2 , oscillations in TDTP curves are due to the factor efwlJt in the potential term. [See Fig. 2(d).]

To estimate the importance of various interactions, transition probabilities are calculated in the following three ways for the initial state J 1 = 1 and J 2 = 1:

(I) taking dipole-dipole interaction only and choosing boundary conditions on the rotational levels of the absorber and perturber as

(IT) taking dipole-dipole interaction, but boundary conditions on the energy levels are taken as

Jr=Jj + 2,

Jrwn=Ji -2, if J j 2:2

= 0, if J j < 2 .

Here also, Jr and Jrwn are upper and lower limits of J; in Eq. (8). This type of boundary condition is equi­valent to consideration of the effect of second-order dipole-dipole interaction;

TABLE III. Calculated values of time independent transition probability PI,o -J;.J;'

5.0 6.00 7.00

0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2 3,0 3,1 3,2

0.0048 0.0166 0.0257 0.0380 0.0404 0.3907 0.0114 0.1720 0.0774 0.0338 0.1066 0.0857

c 0.8218 0.0301 0.1482 c c c c c c c c

0.0003 0.1571 0.0154 0.0248 0.0292 0.5179 0.0166 0.0097 0.0450 0.0977 0.0131 0.0754

c 0.3963 0.0101 0.5936 c c c c c c c c

0.0000 0.2790 0.0094 0.1626 0.0150 0.2827 0.0074 0.1174 0.0201 0.0614 0.0032 0.0436

"Transition probability taking condition (III). bTransition probability from perturbation theory. cCorresponding transition probabilities are very small.

c 0.2139 0.0040 0.7821 c c c c c c c c

8.00

0.0000 0.2693 0.0054 0.3669 0.0075 0.1278 0.0027 0.1610 0.0087 0.0290 0.0024 0.0204

c 0.1254 0.0018 0.8728 c c c c c c c c

J. Chem. Phys., Vol. 62, No.4, 15 February 1975

10.00

0.0000 0.1614 0.0018 0.6800 0.0021 0.0261 0.0004 0.1156 0.0020 0.0060 0.0012 0.0042

c 0.0514 0.0005 0.9482 c c c c c c c c

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

132.174.255.116 On: Fri, 19 Dec 2014 16:53:58

'\ x

.5 \

5.0

1.0

o .5 "

.5

o 5.0

S. C. Mehrotra and J. E. Boggs: Description of molecular collisions 1457

10.0

(0)

• PI,O--O,I

• PI•0 - 0 •2

.. Pr = PI,O-DoO +~.o-o.I+~.O-O,2 x p.(-I)

15.0 20.0

IMPACT PARAMETER. bCA)

10.0

(d)

• PI,O-I,I

• PI•O- I •2

• PT·PI.0-I.O+'1.0-1.I+P~0-1.2 x p.(O)

• P"t-I"

• PT • P.,t-I,I + P.,I-t,3

15.0 20.0

IMPACT PARAMETER, b CA)

'i:'10 ..., <l

~ (J) UJ F :J 10 .5 ~ ~ <l.

Z 0 i= iii z

0 ex 0: f-

.5

..., <l 0:

Ii: (J) UJ 0 i= :J 10 ex ~ .5 0: <l.

Z 0 t: (J)

z ex 0: f- 0

.5

/x x

/

IX

____ x ____ x

(I)

6 P ap +P a T 1.20_1.207.20_0.21

x p.(O)

. ....----.---------. .------/ l x-x..J:-x-----'-------'--------'

5.0

5.0

(9)

(h)

10.0 15.0 20.0

IMPACT PARAMETER. b (A)

10.0

(i)

(j)

• P I•O - 3•1

• PI,O-3.2

• PT' PI•0 - 3•0 + PI•O- 3•1+ PI•O- 3•2 x P.(2)

(k)

.. PT a P •• 20_ 5,18 "'~.20_3.tI+PI.20-3120 x P.(2)

15.0 20.0

IMPACT PARAMETER. b (Al

FIG. 3. Time independent transition probability vs impact parameter for (a) transition tJ.J1 = -1 with initial J 1 = 1, J2 = 0; (b) tran­sition tJ.Jl = -1 with initial values J 1 = 1, J 2= 1; (c) transition tJ.J1 = -1 with initial values J 1 =1, J 2= 20; (d) transition tJ.J1 = 0 with ini­tial values J 1 = 1, J 2= 0; (e) transition tJ.J1 = 0 with initial values J 1 = 1, J 2= 1; (£) transition tJ.J1 = 0 with initial values J 1 = 1, J 2= 20; (g) transition tJ.J1=1 with initial values J 1=1, J 2=0; (h) transition tJ.J1=1 with initial values Jl=1, J2=1; (i) transition tJ.J1=1 with initial values J 1 = 1, J2 = 20; (j) transition tJ.J1 = 2 with initial values Jl = 1, J2 = 0; (k) transition tJ.Jl = 2 with initial values J 1 = 1, J 2 = 1; (1) transition tJ.J1 = 2 with initial values J 1 = 1, J 2= 20.

J. Chem. Phys., Vol. 62, No.4, 15 February 1975

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

132.174.255.116 On: Fri, 19 Dec 2014 16:53:58

1458 S. C. Mehrotra and J. E. Boggs: Description of molecular collisions

TABLE rv. Calculated values of time independent transition probability Pl.20-Ji.J2.

b in A 8.00 10.0 12.0

Ji. Jf ra lIb ra Ub ra

0,18 0.0000 c 0.0003 c 0.0002 0,19 0.0275 c 0.0094 c 0.0148 0,20 0.0298 c 0.0018 c 0.0456 0,21 0.0570 0.1388 0.0584 0.0504 0.2372 0,22 0.0323 0.0019 0.0201 0.0003 0.0056 1,18 0.0093 c 0.0011 c 0.0012 1,19 0.0005 c 0.0023 c 0.0016 1,20 0.2714 0.6987 0.2817 0.8921 0.2853 1,21 0.0111 c 0.0337 c 0.0059 1,22 0.0062 c 0.0018 c 0.0640 2,18 0.0022 0.0021 0.0015 0.0004 0.0003 2,19 0.0676 0.1428 0.0298 0.0540 0.0079 2,20 0.1032 c 0.0882 c 0.1010 2,21 0.1032 c 0.2266 c 0.0011 2,22 0.0001 c 0.0074 c 0.0188 3,lS 0.0063 0.0137 0.0006 0.0022 0.0004 3,19 0.0626 0.0020 0.0186 0.0006 0.0123 3,20 0.1015 c 0.0156 c 0.0274 3,21 0.1358 c 0.1070 c 0.0480 3,22 0.0164 c 0.0539 c 0.0589

~ransition probability taking condition (III). ~ransition probability from perturbation theory. "Corresponding transition probabilities are very small.

lIb

c c c 0.0203 0.0001 c c 0.9562 c c 0.0001 0.0227 c c c 0.0004 0.0002 c c c

(ill) taking dipole-dipole, dipole-quadrupole, quadru­pole-dipole, and quadrupole-quadrupole interactions with the boundary conditions the same as in (II). The result of these calculations for some impact parame­ters are shown in Table II.

Similar calculations are done for J 1 = 1, J 2 = 0 and for J 1 = 1, J 2 = 20, using only the most general boundary con­dition of Set Ill. The results for some impact parame­ters are shown in Tables III and IV. Corresponding re­sults are compared with second-order perturbation the­ory, which are calculated in the same way as by Rabitz and Gordon. 7 These calculations do not include changes in J for which angular momentum is not conserved, since in the perturbation theory approach such collisions have very small transition probabilities. From the tables, we can see that second-order and quadrupole in­teractions are very Significant for strong collisions. For such collisions, the higher-order interactions will be significant. A larger number of rotational states of the absorber and perturber should also be included in the summation of Eq. (8) for more accurate values of the transition probabilities.

It is illuminating to plot time independent transition probabilities for different transitions vs impact parame­ters. These graphs are shown in Figs. 3(a)-3(1). The transi tion probabilities cal culated from Gordon's per­turbation theory are plotted on the same curve as P g(t:.J 1), where t:.J 1 is the change in rotational state. The follow­ing points can be noted from these graphs:

(A) Transition probabilities corresponding to transi­tions t:.J1 = 0 decrease with increaSing impact parame-ter at large impact parameters as in perturbation theory, but they are lower than the values calculated from per­turbation theory. The differences from perturbation the-

15.0 20.0 ra lIb ra lIb

0.0000 c 0.0000 c 0.0090 c 0.0029 c 0.0691 c 0.0449 c 0.1964 0.0058 0.0297 0.0008 0.0011 0.0000 0.0000 0.0000 0.0007 c 0.0000 c 0.0046 c 0.0002 c 0.2785 0.9871 0.7524 0.9980 0.0070 c 0.0001 c 0.0224 c 0.0001 c 0.0000 0.0000 0.0000 0.0000 0.0365 0.0070 0.0013 0.0012 0.0967 c 0.0447 c 0.1453 c 0.0161 c 0.0004 c 0.0000 c 0.0004 0.0001 0.0000 0.0000 0.0140 0.0001 0.0034 0.0000 0.0366 c 0.0484 c 0.0075 c 0.0002 c 0.0035 c 0.0000 c

ory are larger for higher values of J2 • Moreover, a strong collision is more likely to lead to a transition out of the initial state if the value of J2 is large.

(B) About 20% of the molecules can be excited by the transition during a strong collision.

(C) Resonance behavior is observed in the case of in­elastic colliSions. The resonance is due to the fact that the average interaction energy for a certain impact pa­rameter is matched to the energy required for an in­elastic collision.

(D) bo is higher for higher values of J 1 and is 6%-10% higher than calculated from the perturbation theory.

CONCLUSION

Although we have not attempted to compare our results with experiments (e. g., with self-broadening of OCS) because this requires averaging over the impactparame­ter and the rotation quantum number J 2 , which requires impractical computation time, the above results provide interesting information about rotational energy transfer during collisions. The definition of strong collisions with the help of TDTP curves may be very useful in the treatment of strong collisions. There are many dis­crepancies between experiments and the theory of line broadening of molecules perturbed by inert gases, 1

where strong collisions are important. The results presented in this paper suggest an approach to this prob­lem, and in the future we will try to examine such prob­lems on the basis of these results.

ACKNOWLEDGMENT

This work has been supported by a grant from the Robert A. Welch Foundation. One of the authors (S. C.

J. Chem. Phys., Vol. 62, No.4, 15 February 1975

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

132.174.255.116 On: Fri, 19 Dec 2014 16:53:58

S. C. Mehrotra and J. E. Boggs: Description of molecular collisions 1459

M. ) is thankful to the Ministry of Education and Social Welfare, Government of India, for financial support in the form of a fellowship. The authors are also thankful to Dr. R. E. Wyatt for many useful discussions and to Dr. R. G. Gordon and Dr. H. Rabitz for providing com­puter programs for the calculation of second-order transition probability from the perturbation technique.

lThe most recent review is Krishnaji, J. Sci. Ind. Res. 32, 168 (1973).

2For instance, see J. P. Toennies, Disc. Faraday Soc. 33, 96 (1962); Z. Phys. 193, 76 (1966).

3For instance, see T. Oka, "Collision-Induced Transitions Be­tween Rotational Levels, " in Advances in Atomic and Molec­ular Physics, edited by D. R. Bates and I. Estermann (Aca­demic, New York, 1973).

'Po W. Anderson, Phy. Rev. 76, 647 (1949); C. J. Tsao and B. Curnutte, J. Quant. Spectrosc. Radiat. Transfer 2, 41 (1962).

5J . S. Murphy and J. E. Boggs, J. Chern. Phys. 47, 691

(1967); 47, 4152 (1967); 49, 3333 (1968); 50, 3320 (1969); 54, 2443 (1971); V. Prakash and J. E. Boggs, J. Chern. Phys. 55, 1492 (1971); 57, 2599 (1972).

6D. S. Olson, C. O. Britt, V. Prakash, and J. E. Boggs, J. Phys. B 6, 206 (1973).

7H. Rabitz and R. G. Gordon, J. Chern. Phys. 53, 1815 (1970); 53, 1831 (1970).

8Krishnaji, S. L. Srivastava, and P. C. Pandey, Chern. Phys. Lett. 13, 372 (1972).

3D. M. Rank, C. H. Townes, and W. J. Welch, Science 174, 1083 (1971).

lOR. Wallace, Phys. Rev. A 2, 1711 (1970). 11R . Wallace and B. W. Goodwin, J. Magn. Reson. 8, 41

(1972); 9, 290 (1973). 12B. Corrigall, B. Kupper, and R. Wallace, Phys. Rev. A 4

977 (1971); A. Penner and R. Wallace, ibid. 5, 639 (1972). 13H. Rabitz, J. Chern. Phys. 57, 1718 (1972). 14C. G. Gray, Can. J. Phys. 46, 135 (1968); R. J. Beuhler

and J. o. Hirschfelder, Phys. Rev. 83, 628 (1951). 15S. C. Conte and J. Titas, "An Interpretive Floating Point

Subroutine for the Solution of Systems of Ordinary Differen­tial Equations on the 1103A Computer," NN-91, the Ramo­Wooldridge Corporation, 1958.

J. Chern. Phys., Vol. 62, No.4, 15 February 1975

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

132.174.255.116 On: Fri, 19 Dec 2014 16:53:58