A computational method for multi-channel scattering calculations. Applications to rotational excitation and long-lived states of He-N2

Volume28,numb’er

CHE?&CAL

3

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1 October

1974

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.: A CQMPUX4TIONAL

PHYSICS LETTERS

METHOD

F,OR MULTI&ANNEL SCATTEIkIKG AI'PLICA'TIQNS TO.ROTATIONAL EXCITATION. x4ND LONG~IVED STATEF OF&-NZ”

CALCUkIONS. :

Michael J. RED&iON* and D&id A. MICHA* Quantum Theory Project, Deparrments of Chemistry and Physics. WilhMzson ilol& University of Rorida, Gainesville, HoAda 32611, USA Received 25 March 1974

A computational scheme is reported which uses a reference potential approximation to provide a11efficient numerical solution of the multichannel Volterra integral equation. Applicztions to rotational :~elztic scattering and quasibound states of the He;Nz system are presented, together with a discussion of the effects of varying the number of closed channels.

1. Introduction

Energy transfer in molecular collisions may occur and subject to a great variety of interaction poten+ over wide ranges of relative velocities [l-3]. Cross sections for transfer processes have been measured for many neutral-neutral and ion-neutral reactions [461. As a consequence, there is a continuing need for ‘more general and efficient computational schemes. Two recent developments in the solution of coupled-channei scattering equationb have been the introduction of noniterative integral equations methods [7,8] and the use of reference or piece-wise potentials [9 J, which approximate the interaction to any desired accuracy. This last deyelopment has proved to be very efficient, and we want to incorporate it in the integral equation method. We illustrate our approach with scattering of He by N, represented by-a rigid rotor at low (thermal) energies. We present and compare results for rotational excitation and also for a quasibound, or compound-

state, resonance [lo] _Resonances provide a stringent test of coupledchannel algorithms because of the tendencjr of wavefunctions in closed channek to pow exFonentially and hence to invalidate asymptotic scattering amplitudes. In their own right, resonances are also of,interest in connection with collision-induced spectroscopy and long-lived states in beam experiments [ 111. To learn about them we investigate a res_onance as a function of energy and of the number of coupled channels.

2. Computational me-ihod We present here a simplified description for the one-channel case, which is easily gene:alized to the multkhannkl case. The soiution to the partial-wave Lippmarm-Schwinger equation, [ 121

&(r) =i[(kr) + J

‘. . . a Acknowledgement

for tina& cial support. * Work perforticd in’partii fulfiltient of the requirements, ., for the Ph. D degree, Chemistry Department, U+versity ofFlorida. * Alfred P. St&n Foundation Fellow.. :

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where

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GP(r,r’) V(r’i ti[(r’) dr’,

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GP(i,r’)= - k-l I:,(kr,)hr$k,),

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and where jl qd hyj zre spherical Bessel and Hankel .’fu+ions, &be expressed in &-ms of a reference -

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LETTERS

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md

G,(b) =13A t 1i.B +

where @is,obtained as the solution of the Volt&a tit&a! equation of the second kind.: :

k-l j- j. (kr’) Upx1 (r’) dr’. l7

The constant 1 t C(equal to the Jost function) is obtamed automatically, once eq. (4) has been integrated into the asymptotic region, since _ (5)

0

Sarns and Kouri showed that eq. (4) can be solved non-iteratively by replacing the integrals with numerical quadratures [7] _ An alternative approach !s suggested by the work of Goidon 191, who developed a method fGr solving the-.Schrodingar equation by dividing the radial coordinate axis into segments, and by approximating the potential in each segment by a reference potential for which analytic solutions are known. Wc adopt this ‘procedure, and’replace the potential and non-local wavefunction iri Ihe integrands in eq. (4) with 2 reference potential and the corresponding local solutions [14]. There are several choices of reference potential one n-&t use to solve eq. (4). We have elected to cornbine the ceutrifugal potential with V(r) and to ap-.proximate the resulting effective potential U,(r) by a set o.f.constant steps. Since our reference potential is constant in each interval {a,b), the local wavefuunction Go may be expressed as a linear combination of trigo,, nometric funct&s, x1 and x2: in the.form = AX&j + BXZ(j-)

(a .&r-G b),

(6)

whe.re A tid B are constants for the interval (u,b). Forco’nvenience’we.write eq. (4) in rhe form, .o.Cr) = j. (kr) [ 1 + G,(r)] .- A,“) (kr) F,(r)’ 9,. : w&e :

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G,(a).

The im:egials.II , .....I4 are of the form

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C=

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(We ha/e left the constant VI0 in the integrand to preserve the order of matrix multiplication in the multichannel generalization of these equations.) The quarttities Fi(a) and G/(u) are the accumulated values of Fr and Gr up to r =a. The solution is propagated into the asymptotic region, where iIF vanishes and FI and Gr become constant, by determining,4 and B for each interval from the corrsnuity conditions for @O(P)and d$O/dr at the left boundary of each interval. The wavefunction is zero at the origin, and inspection of the integral equation for d$O/dr, obtained by differentiating eq. (7), proTides the starting conditions for lhe derivative. The method of Cordon [9] is used to estimate the local errors introduced by the potential approximation within 2 step and to select the appropriate size for t!!e r_ext interval. For a typical anisotropic potential of the LennardJones ty?e, the coupledchannels integration takes between 60 and 100 steps to reach the asymptotic region, with the integration time increasing as the square of the number of channels (we have checked this behavior up to 16 coupledchannels). Since we are using a referen:e potential, the step sizes computed at one energy may be used, to a good approximation, at other neiaboringenergies because they are relatively insensitive to changes in the collision energy. Since we do not disgonalize the potential within.each interval, our expansion, eq. (S), does not,represent an exactsolution to the local problem. However, it should be a goad approximation provided the intervals do not become too large. We could diagonalize Up but it is not-clear ihat this extra complication is practical unless one,is interested in many calculations over a small energy range, or is dealing with very strong couplings. We,have’found that it is sometimes necessary to : stnbilizk tllc equations when asymptotically Closed .,I chann,els are includedln..tiie basis. We have simply ‘tiansforrncd G/ (v&ich grows the fastest) into upper triangularform [15,16]:Thc~~mmetry.~f,the~R-ma:

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1 October

CHEMICAL PHYSICS LEFERS

Volume 28, number 3

Table 1 illustrating the effect of closed chnn-neh, with channel indices (lj) for’J=6, at 2.2 X 10m3 eV

wein et al. [20]:

A ii-channel R-matti,

(6,2)

.(6,0)

(4,2)

-1.1628 a) -1.1625b) (-1.1587)c)

-0.0302 -0.0311 (-0.0319)

-0.0301 a) -0.0311 b) (-0.0319)

c)

0.1329 0.1387 (0.14!0)

0.0084 0.0084 (0.0084)

0.0043 0.0040 (0.0043)

-0.0007 -0.0007 (-0.0007)

0.0403 a)

0.0044

0.1856

0.0407 b)

0.004b

0.1880

,O.OOSl -0.0051

(0.0409) c)

(0.0042)

(0.1890)

(-0.0051)

0.0083 a)

,::::::jbiI

-0.0051

:;$g: (-0:0007)

-0.0051 (-0.0051)

;.;;c& (0:0242)

b) g-channel calculations

(5 closed charm&). g-channel results from the program of Kouri et al.

trix obtained by this method is good, and is not very sensitive to the frequency of stabilization. We have applied this computational method to a variety of atom-atom and atom-diatom collision problems at thermal and 1 eV-range collision energies [14], and have verified the method by comparison with results available in the Literature [ 17,181. In table 1, we present a 4channel R-matrix 0btaine.d for J = 6 at 2.2 X l@ eir with a He-N2 potetitial discussed in the next section, using the formalism of Arthuti andDalgarno [19].Atthisenergy,thej=Otoj=2 transition is allowed, and we present results obtained using both a 4-channel (j = 0,2) and a.9channel (j = 0,2,4) basis. Also shown, for comparison, are 9.channel results obtained with the integral equation program of Sams and Kouri [7]. When WCinclude the closed channels, the cross sections are changed by a fraction of one percent.

0.375 P,(cos e)] XQ

-2[1~0.172P~(cosB)]r~), where x = r,/r,

E = 2.94 X 10m3 eV, rm .= 3.52 A:

lar resonance we have studied is one previously reported by von Seggern and Toennies [lo] in their extensive of the energy dependence of &total elascalculation tic cross section for He-N, _The resonance occttr~ for J = I = 4 around 1,.066 X 10e3 e’t’, which is below the j = 2 threshold. Calculations were made tickling rigid rotor states up to j = 6, which required solving up to 14 coupled equations. In fig. 1, we illustrate the energy dependence of the J = 4 partial cross sections obtained with both 4-

bases. The five additional

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f 40. : n L c

z 20. ----___

to a He-N2

quasibound-stare

reso-

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As

es0

8.65

wnenurraer(Crr7) a~

illustration

of the stability of the numerical

method described in the preceding section, we present the results 0f.a study of the basis dependence of the. shape and position of.a quasibound state in tht He-N2 system,.T&e potential used was that suggested by Erle:

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Applications

closed chan-

nels produce a shift in the position of the resonance approximately equal to t!le width of the resonance. but they have a negligible effect on the magnitude ad shape of the curve. This means that proparties depending on integrals over the total cross section w&d be unaffected, by the use of the smaller basis. A cakuiation including 14-channels (j = 0,2,4,6) gave the same result as the 9-channel u = 0,2,4) calculation. Also shown in fig. 1, are the magnitudes of several other partial cross sections in the vicinity of the resonance. They are constant over the ener,ey range Ed-

-0”

3.

(11)

and P2(cos 0) is a Legendre polynomial. The particu-

and g-channel

3) 4channel calculations. cj

V(x,O) = E {[1+

(a,21

0.0403 0.0407 (0.0409)

L974

Fig. i. He-I% quasibound-stateresonance forJ = f = 4.

ShorvnarC’ti,Giesults’obtained with J fourchannel

basis and a nine-chanml basis (&):AIso shown are several other partid cross sections that.wntribute’substati~~~ to the total cross section, and which arc constant for this narrow energy rang=.

(---)

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343

‘V&me

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28. numb&r 3

CHEMICAt; PHYSIC? LE-ITIZRS .’

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compassed by fig. 1, so the shape of the total elastic. cross section in the vicir?lty oQhe iesonance Is deter&ined entirely by theJ = .4 partial cross section. Calculations of the total elastic cross section, inciuding 25 ‘partia! waves with the.4.chanrie! basis; gave a.value of ’ 235 A2 cc the resdnance and about 170 A2 at a slightly lower energy. These numbers are‘h agreement with the results of von Se&ern and Toennies.[lO]. Although the &ift in the position of the resonance caus‘id by using the larger basis is s-mall, it conceivably may be measurable in collision-induced spectroscopy [I 11.

4. Distiussion The calculatipns of the position, shape, and basis dependence of the quasibound-state resonance described in the preceding section sen’r, as a critlc;al test ofthe stability of our computatioiral procedure. AS a qualitative indication of the relative efficiency of various current programs, -we find that, for the He-?<, potentialgiven above, oui program is about 4 times as fast in solving 4 coupled openchannel equations as ,the i)ltcgral,-equation [7].and differential-equation [18,21] programs, based on step-by-step methods, we had available. Our procedure is 3 times faster than that of differential equations with reference potentials [22] at the first calculated enerw (where we however do not diagonalize the potential), but twice slower at -the other energies. We have also found that theR-ma_ ‘tti, obtained by our procedure is more symmetric thar’that bbtained-with differential equation algorithms, which zggests impioved stability. . ..These comparis&S are of course dependent on potentials and collision energies_ Foilony-range interactions and high energies, the reference potential meti;‘ods should.be much more eificient. But 4-channel c& cuiations at 0.2 eV for the short-range Krauss-Mies -He-H2 potential [23]. using our piogram and the pro: &ms of Koari et al. 2nd McCuire took about thg ; same amount of time. we feel that the computation.@ ,I? con+tision, scheme reported here can be used effectively in a large number-bf scattering problems. In using a constant referenckpotential and intcgral.equationm, one obtains exprkssions’that are easily progmmmed, s&e t-hey rcquire only simple functip,ns available from machine ‘!J-y_: ,_344 .,...

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The :luthors wish to ihank‘ R.G:Gordon,

D.J.

K&ri, and P.W. M@ire for useful discussions ,and for providkg copies of their scattering p!ograrns.

Referenxs [l] D. Secrest, Ann. Rev. Phys. Chem.‘24 (1973) 379. [2] R.E,. Levine, in: MTP Intematior+tl Review of Science, Phys. Chem. Vol. 1, ed:W. Byers Brown (University Park Press, Baltimore, 1971) p. 229. [3] D.A. Micha, Advan. Quantum Chem. 8 (19741, to be pubsshed. [4] J. Kinsey, in: MTP Intemitional Review of Science, Phy.;. Chem. Vol. 9, ed. J.C. Polanyi (University Park Press, Baltimore, 1971) p. 173. i5] T. Carrington and J.C. Polanyi, in: MTP International Review of Science, Phys. Chem. Vol. 9, ed. J.C. Polanyi (University Park Press, Baltimore, 1971) p. 135. [S] J. Dylbrinand M.J. Henchman, in: MTP International Review of Science, Phyi. Chem. Vol. 9, ed. J.C. Polanyi (University Park Press, Baltimore, 1971) p. 213. [7] W.N. Sams and D.J. Kouri, I. Chem. Phys. 51 (1969) 480?,4815. [S] B.R. Johnson and D. &crest, J. Math. Phys. 7 (1966)

Comp. Phys. 10 (1971) 81. [lo]

: ‘.

Ackndwledgement

218;‘. [9] R.G. Gordon, J. Chem. Phys. 51 (1969) 14; Meth.

:

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1 dctober 1974

M. van Segg;m and J.P. Toennies, 2. Physik 218 (1969)

ill] ?:: Micha Accounts Chcm Res 6 (1973) 138 [12] J.R. Taylor: Scattering theok (Wiley, New York, 1972). [13] G.F. Druknrev, The theory of electron-atom collisions (Academic Press, New York, 1965). [ 14 ] M.J. Redmon, Ph.D. Dissertation, Chemistry Department, University of Florida (1973). [iSI R.A. White and E.F. Hayes, J. Chem. Phys. 57 (1972) 2985. [ 161 W. Bxtcs and D. Secrcst, I. Chem. Phys. 56 (1972) 640. [17] R.B. Bernstein, J. Chem. Phys. 33 (1960) 795; W.A. Lester and R.B. Bernstein, Chem. Phys. Letters 1 (196:‘) 207, 247. (181 P. McGuire and.D.A. Micha, In&. J. Quantum Chem. 6. (197:!) 111. [ 191 A.M. Arthurs and A. Dalgamo, Proc. Roy. Sac. A256, (1960) 540. [SO] W._ErIewein, M. van Seggek and J.P. Toennies, Z. Physik 211 (1968) 35.. [21] P. McGuire, private communication; [2?] R.G.,:ordon, Program QCPE 187, Quantum Chemistry -Piogr.lm Exchange, Indiana University; Bloomington, __ [idiana. ,‘. [23] M. Kmuss and F.Ij..iies, J. Chem..Phys: 4.2 (1965) : ::y, 1; ., ,.,. ;, ,_ : ._:‘,.,, ,, 2703.

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