Fully converged three-dimensional collision-induced dissociation calculations with Faddeev—AGS theory

3 June 1982

CHEMICALPHYSICSLETTERS

Volume 89, number 1

FULLY CONVERGED THREE-DnfENSIONAL

COLLISION-INDUCED

DISSOCIATION CALCULATIONS WITH FADDEEV-ACS

THEORY

hI.1. HAFTEL Naval Research Laboratory.

Ir’axhmpton. D C 203 75. USA

and T-K- LIhl

l

Plr)wkalrsches Instinrt der Univerhit

Berm, D-5300 Bonn, West Cerfnany

Rrccived 16 September 1981; in liial form 7 April 1982

The quantum-mechanlcat catculetion oi the colhsion-induced dissocia.tlon cross wwon in I rhrrcdimensional model system is reported. The obtamed results resemble those from some collmear models but mdlcate ckarl~ the futW of

muluple-scattering approGmations ekccpt at hyperthermal enerprs.

I. Introduction The quantal description of collision-induced dissociation (CID) has received considerable attention recently [I-S]. Assorted methods ranging from closedcoupling techmques to rime-dependent wave packet propagation to hyperspherical coordinates have been used in an assault on this problem which has provided a long-standing challenge to chemical dynamicrsts

[Z-S].

Unfortunately

most of the expended

effort has been directed at the

fully converged solution of onedunennonal models of varying degrees of sophistication and although the results promise much one cannot be satisfied that collinear models can adequately represent the dynamics of CID. Extending the aforementioned methods to three-dunenslonal systems will require much effort (both human and machine) and will not come easily since each has its particular stick of deficiencies e g. “overcompleteness” in closedcoupling calculations. Therefore it appears that we must be patient for a httle while longer before one or the other of these “difierentialsquation” and wheels really turn”.

methods can be coaxed into service to elucidate in CID how the “cogs

Having mentioned that we now declare that there exists already a fully developsd technology for solving exactly the three-body scattering problem in three dimensions albeit for nuclear systems. This is the Faddeev-AGS method employing integral equations in momentum space [6-g] _The outstandmg feature of this approach IS that it is designed primarily for three-body systems which interact via pairwise potentials; these rnteractions should preferably be representable by separable forms supporting few bound states. In ttus case one extracts from the theory a computationally feasible solution involving integral equations in one scalar variable. Such a situation may not be realized often III general CID processes but two come to mind, He-He? and perhaps He-Hz, as chemical systems for which the Faddeev-AGS approach appears to be tailor-made [9]. And even if this technique and other integral-equation variants [IO] are proven eventually to be as shackled by a retinue of defects as the others now obviously are, our application of Faddeev-AGS techniques, herein repcrted, will at the very least have made l

Research supported

in part by a rellowship

from the Alexander van I-iumboldr-Sliftung and in

pan. by Ihe NaGoFWlScience

Foundation through Grant No. PHY-7819365.Pemiinent address;Departmentoi Physicsand AtmosphencScience,Drexel Ururersity,Philadelphia, Pennsylvanta 19104, USA. 0 009~2614,W/OOOO-0000/S 02.75 0 1982 North-Holland

31

4 June 1982

CHEhllCAL PHYSICS LETl-ERS

Volume 89, number 1

avaIlable valuable “bench marks” for other fully converged or approximate calculations. Thus in this paper we present the details of the first fully converged quantum-mechanicalcalculation of the cross section of CID in a system of three “helium-like” atoms; the atoms are all taken to have the physical attribures of He but interact pnrwise through different sets of Lennard-Jones 12-6 potentials which are the usual potentials modified by scaling factors and support single vibrational diatomic states at “physically unreasonable” energres. These local potentials. in turn, are replaced by s-wave separable potentials of the form I’(/& k’) = -hg(k) g(k’) ,

(1)

where g(k), from the requirements of our computer code, is grven by the ratio of two polynomial expansions whose coefficients are determined by a least-squares fit of the unitary-pole approximation to the original potential, the strength parameter A is directly related to the diatom’s internal energy EB [9,1 I]. In section 2 we write down the integral equation satisfied by rhe physical on-shell atom-diatom dissociation amphtude as derived from Faddeev-ACS theory for the given separable potentials, indicate how elastic atom-diatom scattering am-

plitudes can be extracted from an integral equation of similarstructure to that for dissociationand then sketch the derailsin obtaining expressionsfor the total cross sections from these amplitudes. Section 3 contains our numerical results and conclusions. The techniques devised to solve the integral equations are crucial for quantitatively accurate results but discussion of these cannot be appropriarely included in the text. We refer the reader to EbenhBh’s paper [8] for the numerical details.

2. Atom-diatom

cross sections through Faddeev-ACS

theory

From ref. [ 1l] we can write down the CID amplitude dlii for three identical spinlessatoms interacting through diatomic potentials of the type given in eq. (1) as

with (3) afrer changmg the definition of the three-body kinematic parameters in order to match those encoded in the computer program we used. In eq. (3). f. is the three-body partial-wave number, 9 the Jacobi momentum of the incident atom with respect to the cm. of the diatom,pf,, 9fa are the final-state c.m. three-body Jacobi momentum vectors with respect to the pair Q (e.g. (lr = 1 refers to the pair 2-3), while PL is the usual Legendre function. The partial-wave amplitudes T, satisfy

where 1 g([;(f kdq,q.)=$

-1

32

ql. +q”) tqq’_Y)p)g([;(;

/ : -Lll

q’2 tq2 tqq’x)]‘/2)

. - q’ - q’7 - qqjc

Q(x)&

9

(5)

CHEMICAL PHYSICS LETTERS

Volume 89. number 1

4 June 1982

(6)

ma Ec.m. As

= p:

+qf,E,

is the internal energy of the diatom.

9

(7)

so that XL satisfies an equation just like eq. (4) except that in the inhomogeneous term the factor gr does not appear. Thus we are able to extract elastic scattering cross sections simultaneously with the dissociatrve cross section from solving a similar integral equation_ To obtain the total cross sections we must now “square” the transition amplitudes, multiply each in turn by the appropriate phase space factor and the energy- and momentum-conserving delta functions, then divide by the relative flu?t of the incident atoms wth respect to the diatom [ l?-1. Following through on this procedure we fmd that the elastic cross section is o, = %%)(n4/90)

sd4 a(93 - 9;)IX(90,9)12

,

09

where q. is the magnitude of the momentum of the incident atom with respect to the c.m. of the diatom and we have already performed one integration to remove the momentum-conserving delta function. Eventually DE =$(fi’/m)a4

~dn,IX(9,,9)I’

=~@‘/m)n’

LFo (= + I)lXL(qo,qo)l?-

(9)

after we integrate over the PL functions. For dissociative scattering we have

(10) Then oD can be broken up into “direct” and “exchange” terms such that oD=u-;+t7;,

_

where

~e,\=~(~2/~)G4/q~)~d~~91~(E-~f-~:)~(~O~~I,~,)[~*(Qo~~~.q~)+~*(40~~3,~j)l-

(13)

These expressions for od, and OF can be further simplified. Thus

(15) 33

Volume 89, number 1

CHEMICAL PHYSICS LETTERS

4

June 1982

Through tedious algebra, we can show that (16) where p/z C.lll.

=4X3-l/z

c&

Re

Wc,_-9”N

d9’9’

I 0

T;(9Oz+.~‘)

“2+9w 9 TL(wp,

?I

qN’&

E,,,.

- 92 - 9’%43

dq

,

(17)

1[3(E,,.-qQ)]“2-9’~’

where Re KfcrS IO rhe real part of the expresaon that follows, TL* represent fiid momenta with p2 + 9” = Ec_mS = $2 f 9’1_ Fhdly OD = g

(fi'/m)(7r6/qo)

7 L

(PL + I)(&

+c?Z) .

is the complex

conjugate

of

TL , and ~,~,p’,q’

(18)

We are now ready to evaluate the elastic and dissociative cross sections. To do this we must solve eq. (4). A number of problems stand in the way of a direct numerical attack on this integral equation_ There is a pole in the r function, a square-root branch point corresponding to the physical two-body threshold and moving singularities in the kernel function of eq. (5). Ebenh6h has discussed the steps which can be taken to remove these troublesome features [8]. We follow his method. This accomplished, the relatively straightforward solution of eq. (4) can then be carried out to various appro_ximations. For example, it can be solved by matrix inversion for as many L values as are needed to achieve a “fully converged” answer. Or if we take in each partial wave the inhomogeneous term only as the transition amplitude we have the “Born” approximation. Taking the result which ensues trom a single iteration of eq. (4) gires us the “fiist iteration” solution, In practice. our quoted fully converged numbers were obtained from matrix inversion truncated at L = 7-,with the first iterate contribution for 3 6.

3. Numerical results and discussion Our calculated values of oE and oD corresponding to systems possessing various diatomic internal energies are displayed in table 1 and pictorially, for one particular example, in fig. 1. Some trends in the tabulated numbers clearly stand out. For example the variation of the cross sections with incident energy manifests characteristics already observed in quasiclassical trajectory descriptions of Ar-Ar2 [ 131. However what is even more surprising is that the features of our results at E, = 1 K are almost a qualitative reproduction of the Barg-Askar “vertical scattering” He-Ne, model and of the Kaye-Kuppermann collinear H-HZ model [4,5]. Thus, our dissociation cross section also shows a noticeable dip soon after rising sharply from the physical threshold (there is no displacement of the theoretical threshold as in Kulander’s model [3]). Barg and Askar attribute this localized minimum to a Feshbach resonance between a level of their Ne2 diatom and an internal He-Ne, triatomic state; we are presently checking to see if a similar explanation holds in our case. Summarizing the above discussion, we can say that the close correspondence between one-dimensional and thr~ediiensional models of three inert (or weak van der Waals) atoms suggests that the collisional behavior of some three-dimensional systems may be qualitatively described by one-dimensional models after all. We also notice that the dissociation cross sections decrease when the system diatom increases in its internal energy. This offers a hint that in a system with more diatomic viitational levels excitational enhancement may *ECU= as obsemed

34

in experimental

CID.

4 June 1982

CHEhUCAL PHYSICS LETTERS

Volume 89. number 1

Table 1 Born. first-iterate and fully converged vah~es of the total elastic and dissociative atom-diatom cross sections for different &atomic internal energies and at a number of c-m. energies. Numbers such as 2.46-7 should be read as 2.46 X IO-’ K (KI

G.m. (K)

Elastic cross section

Dissociatire cross section (AZ)

(A*) Born

fist Iterate

full

Born

fist iterate

full

226000 65.8

1030 39.0

2160 87.5

40100 93 7

27.5 83.6

1 1

1 5

8590 19.7

I 1 1 1 1

33 2 5.1 19 50

1090 121 75.8 279 0.77

3490 680 970 98.1 162

546 65 1 28. I 15 6 0.85

100 100 100 100

13 2 5.7 19

30.6

127

12-t

3.46-7

x41-3

J.Ql-2

30.4

125

12.8

1.124

29.1

120

26.2

10-I

12.5 136

4.49-3 0 310-l

0.410-I 0x01

2.92-3 1.84-3 0.600-I

30.9

15.9 59.1 40.3 2.83

543 75.9 230 123 9 06

x9-

1

0.326 8.87 12.3 23.7 2.18

As to the accuracy of the various appro.ximations, we take a look behind the numbers. The fully converged results satisfy unitarity through the generalized optical theorem to the accuracy allowable by our program IS]_ Convergence 1sgenerally better for higher E,m_ for a futed EB and also for lower EB at a futed Ecm__ ‘(True” convergence is reached only when E,, . B EB. The Born approximation appears to yield an acceptable order-of-magnitude estimate of the elastic cross section but in dissociative collisions it fails for many sets of energy pairs. The fmt

iterate

approximation

is hopelessly

“out

of it” and cannot be trusted as a wable approximation

until we en-

Lg. 1. Total elastic and dissociative cross sections at various c m. energies for the system with 1 K dutomic internal cnerpy. The subscripts E and D label elastic and dissociane respectively while the superscripts8. 1. F refer to the Born, firtitentc and fully converged rcsult~s. respectwety. 35

Volume 89, number 1

CHEMICAL PHYSICS LE-JTERS

4 June 1982

counter high or “hyperthermal” energies. Thus at other than such high incident energies the multiple-scattering series expansions [obtained by successive iteration of eq. (4)] must be approached with caution. How close are our fully converged numbers to the true ones with the original Lennard-Jones potentials? Our experience with the unitary-pole approximation indicates that it is very good fcr the systems with small EB and leads us to conclude that almost all entries under “full” in table 1 have a maximum error of IO!?&a number of them may be within 5% of the true values. We offer these numbers as useful tests for other methods.

Acknowledgement TKL is grateful

to W. Sandhas

and H. Habsrretti

for their kind hospitality

and for useful discussions. He ac-

knowledges with appreciation the facilities made available to him by the Physikalisches Institut der UniversitLt BOM and the kindness of E. Danne, who also typed this manuscript.

References [I] DJ. Diestier. III- Atom-molecule collision theory, ed. R.B. Bernstein (Plenum Press, New York, 1979) pp. 655-667.

[ 71 L.\V. Ford, D.J. DIesUer and A F. Wsgner. J. Chem. Phys. 63 (1975) 2019. E.-W. Knapp, D J. Diestler and Y.-W. Lii. Chem. Phys. Letlers 49 (1977) 379; E.-W. Knapp and D.J. Diestler, J. Chem. Phys. 67 (1977) 4969. 131 K C. Kulander. J. Chem. Phys. 69 (1978) 5064: NucL Phyr A353 (1981) 341~. [aI J. hlanz and J. R6melt.Chem. Phys. Letters 77 (1981) 172; JA_ bye and A. Kupp+rmz~~~. Chem. Phys. Letlers 78 (1981) 546. [S] C.-D. Barg and A. Askar, Chem. Phys Letters 76 (1980) 609. [6] E.W. Schmid and H. Ziegelmann, The quantum mechamcal three-body problem (Pergamon Press, Oxford. 1974). E. Alt, P. Grassberger and W. Sandhas, Nucl. Phys. B2 (1967) 167. 171 C. Stalk and J.A. Tjon, Phys Rev. Letters 35 (1975) 985. [ 81 W. Ebenhiih. Nucl. Phys A191 (1972) 97. 191 H.S. Huber and T.K. Lim, J. Chem. Phys. 68 (1978) 1006; K. Dulfy and T.K. Lim, J. Chem. Phys. 70 (1979) 4778. [lo] V.F. Kharchenko and V.E. Kuzmichev, Proceedings of the 9th International Conference on the rewlody Problem, Eugene, Oregon (1980). Vol. I, p. 173. [ 1 l] bl.1. Htitel and T.K. Lbn, J. Chem. Phys., subnutted for publication. [ 121 C J. Joachain. Quantum collision theory (North-Holland, Amsterdm, 1975) ch. 15. [13] h1.J. Delle Donne, R.E. Howard and R.E. Roberts, J. Chem. Phys. 64 (1976) 3387; R.E. Howard, R.E. Roberts and hl J. Delle Donnc, J. Chem. Phys. 65 (1976) 3067.

36