Volume 30, number 1 CHIEMICAL PHYSICS LETTERS 1 January 1975

1. Introduction

There has been recent interest in developing theo- retical descriptions of molecular collisions involving the dynamic coupling of two or more potential ener- gy surfaces, where colliding species contain internal

nuclear degrees of freedom [l-25.5. In our laboratory we have been interested in constructing state selected transition amplitudes (S-matrix elements) for atom (ion)-diatom collisions from a semiclassical approach [3-81 and a purely quantum approach [ 1,2] ., In this paper we report the results of calculations within the semiclassical approach for the three-dimensional reac-

tion H+ + D2 + HD’ + D, which involves an electronic transition. from the ground singlet state to the first ex- cited singlet state of the $ system. A purely quan- tum calculation on this reaction in thrke dimensions is at:present an insurmountable task, so that the cal- culation of S-matrix elements is possible only through

a semiclassical approach. Our procedure combines the techniques of previous semiclassical cakulations for

three-dimgnsikal reactions on a single surface [26] with those for collinear reactions involving two inter- acting surfaces [3-81. The method by which we impie- ment on a computer the semiclassical theory for elec- tronic transitions has been presented in detail for col- linear H’ + C2 + HD+ + D [S] z and in this paper we merely indicate those additional features necessary to

extend the method to three-dimensional collisions. In section 2 we present briefly the basic theory and meth- od of implementation, in section 3 \ve present results and in section 4 we make some concluding remarks.

2. Theory and method of implementation

* Acknowledgement is made to.the National Science Founda- tion (Grants GP-38815 and GP-33998X), the Air Force Of- fice of ScientiIic Reseaxh (Contract F44620-74E-O073), ’ the Research Corporation, and the Donors of the Petroleum Research,Fund, admini&ed by the American Cbemiczd Society, for partial support of this research.

*pr esent address: Battellc Memorial Institute, Columbus; Ohio43201,USA.

The quantity we wish .to calculate is the S-matrix element,for the transition from an initial state jl~1j~l~) to a final state(2n2j21z) Fcr the initial state, the integer 1 specifies the lower adiabatic surface car- responding to the ground sir&t eIectronic state, “1 andjl. specify the quantuti numbers for the vibration a.nd rotation of D,, respectively. and [I specifies the initial orbital angular momentum quantum number. For the final state, 2 specifies the upper adiabatic sur- face correspondQ to the first excited sin.glet state, n2

‘. . . 49

.-

~E~~~~L.~~~I~ALTREATMENToF ELECTR~NI~TRR~TSITIONS INMOLECULARCOLLISLONS:

THREE-DIMENSIONALH+ + DZ+HD+ +D"

Ying-Wei LINS, Thomas,F. GEORGE and Keiji MOROKUMA Department of Chernisfry, The Univcrrity of Rochester, Rociresrer, New York 11627, USEI

Received 20 August 1974

A semiclassical study is carried out for the threedimensional reaction H” + Dz - HD* + D, which invo:vcs an electronic transition from the pound to the first excited singlet state of the Hi system. The potential energ sur- faces corresponding to the two singlet states exhibit an avoided intersection, and they arc annalytically continued into the complex nuclear coordinate plane. S-mntris elements in the classical limit are computed from complex-valued classical trajectories which propagate on the analytically continued surfaces and switch surfaces continuously at the complex intersection. These elements are computed for the transition from the+,sround rotational and vibrational states of Dz to the ground rotational and first exited vibratior.al states of HD. for zero total angular momentum and for energies in the range from 4.0 eV to 5.0 eV relative to the bottom of the Da well on the loa-er surface.

Volume 30, number 1

.,‘.

CHEMICAL PHYSICS LECERS 1 January 1975

and j2 are the quantum numbers for-HDf end 2, is thC, find orbital angular momentum quantum number. The input necesLqry to calculate. the S-matrix eIement comeS from sel’ected iomplex-valued classical trajectories which begin with reactants ‘-I’ + DZ on surface 1 and

rearrange to products HD+ + D on surface 2. For a gi~n trajectory we “quantize” the initial h&ndary conditions in terms nf a set of action variables “I, jl ,, and I, with the appropriate integer values. These vsti-

a&s correspond to I$, ~3 and 5 in ref. [26] The final boundary conditions are “quantized” in terms of an-

ether set. of action variables tz2, j2 and I2 correspond- ing to ?15,13 and 13 in ref..[2Gj. As in ref. 1261 , the actual integration of a trajectory is carlied out in car- tesian coordinates, and transformations are rnaaje be-. twe& these coordinates and the actio;,-angJe variables at the ,beginning and end of the trajectory.

The two potential energd surfaces are obtained with- in the diatotiics-in-molecules method as used initially for the reaction H’+ D2 --, 19-D+ + D by Tully and Preston [25] - Ezch surface aorresponds to z branch of a double-valued potential ener,y function, and they have an avoided intersection which becomes an actual

intersection in-the asymptotic regions of reactants and products. We anaIyticaJIy continue the. surfaces into the complex nuclear coordinate plane where they ac- tually intersect, and the complex stirfaces of intersec- tion are defined by the branch points of the potential energy function [27,29] . Accmplex-valued trajectory then propagate.? in complex time on these analyti+ly

continued surfaces and switches suifaces continuously

at theti,intersection, i.e.,,at a branch point. (In prac- tice, a trajectory need only cross a branch cut Jvith-

-.“OUL having to pass through rhe branch point itself.) -4 given trajectory can switch back and forth between surfaces in a branching pattern determined by the complex time path of integration. Although there are an’infinitc number ofbranching patterns since the complex surface of intersection is of infinite extent,

only a finite number of p,atterns are significant [5].

The details of the method by which we contrbl the .complex time path of integration for crossing branch

cuts are the same as jr-i the collinear collision in ref. [5] , with the bn.Jy essential difference that each surface de- pen_ds on three rathtir’tfian. two cotiplex coordinates..

.The three coordinates.are given,as’XI; R-j and R3 ,. wheri.R3 is the D-D distance, and I?, and R, are the .,Y-D di$ances; We sh&l’be considering transitions

:.:. ., ‘. ,.gJ ,: .;:‘. ; ., ‘:

from D, in its ground vibrational state. In this ctiSe the brancll pJints in the.reactant regio&ve lsrge imagi- nary corr.ponents in the c&-respondiri~ time plane, so that the contributions from trajectories making transi- tions at these points are negligible due to strong ex-

ponential damping. We therefore need consider the crossing of branch cuts only in the product region. We

can draw a similar branch point stnxture as in fig. 1

of ref. [S] , wllere now there are four branch points in

the complex R2 (or i,) plane corresponding to given values of RI and R, (or R,). As in ref. [5], only two of the four branch points are significant (the other two

.have largcimaginary components in the time plane), and each of the ttio moves on a complex surface of intersectiin, where the surfaces are defined by the

equation R2 =R,(R,, R3) (orR1~=Rl(Rz, k3)) and its compl:x conjugate. Tile complex surfaces of inter-

section are fou,nd for n set of values ofR1, R2 and R; and then fitted to a rational-fraction, which is used in the integration routine for a trajectory.

The clussical limit of the S-matrix element for the (I~z~j~!~) ? (2n2j212) transition is Siven for total an- gular momentum I and energy E as

where D is

D =p’ &W3 W&Q)/%,, qj’il 4/l )I - (2)

[a(~t,i,,,)/a(4n,4i141,)1 is the determinant of a three- dimensional jacobian, where qnI, qi, and 41, are the

angJes cor!jugate to I;,, il and II, respectively. P is a

probability factor which itself is a product of probabil- ity factors, P = prp2., .p,v, where N is the number of times the trajectov passes near a transition point, i.e., a branch point which contributes to a significant branching pattern, and does not make a trarisition be- tween sur5ces. pk is the localized probability for the trajectory riot to tiake a transition at the kth transi- tion point and is computed jn tJle manner discussed in

ref. 151. The phase Q is given as

r2

t s .?,Tdt,- .: .’ (3)

Cl where tl ?.-‘a and t2 * t do. qJ1(g_rZ) is the initial

Volume 30. number 1 CHEhlICAL PHYSICS LETTERS 1 Sznuary 197.5

(final) value of the angle conjugate to the action vari- abIe for J, XI (;k;) is the initia1 (finaI> value of the car- tesian’coordinate for relatiite motion of reactants (pro- ducts) and Y,(Yi) is the momentum conjugate to X, (X2.)_ T is the total nuclear kinetic energy whicil’is expressed in the cartesinn &iables for the initial a& rangement, and the time integral is taken over a corn. plex time path corresponding to the branching pattern. There is a double summation in eq. (2): where one summation is over significant blanching patterns, and ‘3r a given pattern the other sumsnation is over all pas- sible trajector&s which propagate from (111~j~ I, > to (2rt2j212). These trajectories are distinguished by dif- ferent values of qnr, qf, and 91, nt.the beginning of the integration. These values as we!l as 0 all turn out to be complex., and this reflects the fact that electron- ic transitions between adiabatic potential energy sur- faces are intrinsically clossicatly forbid&r2 processes

181.

A,21’4 (with the branch cut tying along the negatii--e real timk a&is), sirxe the direction of&‘.is inc~re along

the positive real time axis than the direction of AL This ckoice,of dr! seems to keep the trajectop nioving forward in real time as weil as keeping II-ET/ down to a reasonable size. Since 6’ depends .?n coordinates, this has the indirect effect of keeping the coordinates close to their real axes and thus sta~lizin~ the trajectory.

3. Resutts

Upon integrating a complex-valued classical trajec- tory, each semiclassical theorist tends to develop a “bag of tricks” for choosing the colnplex time path to ensure that the motioti is stabilized and does not’ “wander off’ into the complex plane. One useful trick centers on identifying the vibrational motion and peri- odically forcing it to the real vibrational coordinate axis [5,29-311 . However, this thrick is not always useful deep in the interaction region, where the strong mising of nuclear degrees of freedom precludes the ident~~cat~on of’3 vibrational notion. We woutd like to mention a trick which is useful for stabilizing a tra- jectory in the strong interaction region, especially, for three-dimensional trajectories. We focus on a given sui- face.

In carrying out the semiclassical calculations We re- gard the two D atoms as distinguishable, and we com- pute trajectories where just one of the D atoms reacts with the approaching W. Due to the identity ofthe two D atoms, the periodic portions of ql; and qir can each be reduced from a period of 2a to T. The correct result for 1S2tl~zfz,hzijlf, (J, E)12, computed from tra- jectories with initial a&es qj, and qlL each in the range from 0 to ;i, is then obtained by multiplying the abso- lute square of the S-matrix element given in eq. (4) by four [26,X!] .

We have focused on the process

v(t) .=, YIRI(t), fQ,(I), &Jr)] = Rev(t) + i!m V(t),

in the energy range from E = 4.0 eV to E = 5.0 eV, where E is measured relative to the bottom of the DI! weif. Here we find only one significant branching pat- tern (in order to obtain S-matrix elements to three sig. nificant figures) which corresponds to’3 trajectory prop- agating bn surface 1 into fhe product interaction region, where, the absolute sep3ration between D and the center of mass of HDe is about 5 bohr, and then switching sur- faces at each signiftcant transition point it encounters. This means that it swit’ches surfaces twice during each vibrational period of’HD+. For this pattern P= I in eq. (2). In the asymptotic product region this switching of adiabatic surfaces. is equivalent to rmaining on the same diabatic surface and thus maintaining the correct HW electronic configur&ori. For this branching pattern there are eight tmjectories for each value of lI = I, =I, which correspond to eight sets ot‘vnlues of the angles

(Q,*, ,4j& 4/J.

OR which a trajectory is ~ro~a~~tin~, and choose com- plex time steps for the integrator to keep IrrV f&n becoming too large. This can be accomplish--d by choosing the time step At to be

L?Lz Rev-V =Tjz-- ‘,, i4)

where (

dv/dt= (F-- vi)/&,. ,. (5j,

A\t, is the prcvious.time step and vi is f&e value of V at the b&ginning of the time step At;. We find t& b&t r&&s when we choose the actual time step as At’ =

d’

i”o&sing on. the ‘chse &xe f, = lz = J = 0, we 1x.w computed 1S2100 1 uoc(O, @I2 For E = 4.G;4.5 and 5.0 eV. The results a& shown in fable 1 .in the column

-. ‘51.

Volurk30, numb&l :- . .., . CHEhlICAL .PHYSICS LElTEliS ,. ,. -:, ‘1. Jqlmy 1975 ..~ ,_ ‘) . ;. :

“.Tab]&‘] ., “. -..:_ .’ . . . ‘. :. : ,’ ,._, .. ., _’

x10-6, ;., . .

’ ,!Szl& I,-,& (0, ml2 fork = 4.b,4.5 and 3.0 eV computed ,’

,I,_.,:,

r_‘.. 24 I, .’ * ;.. ; ,, _. :, ‘,

: from .Gght Frajectorie$ and.& from jus~four.~~ the eight t& ‘. : .:.’ . . ‘_, : ‘- _ I .’ , pc.$~.. m. ,DKii~&~~~e -9 rv ‘.+.- i ras.313 a+ &zi+& h+n eq; (1) -03: s~rnrn~~ and then squaring, whereas“&e.classi& resulti are

. -; . . . obtainkd bisqtiarirtg and then adding

‘. .‘, .,, : . . . .

Primiiiv~ ; .anrsicd .: %L : H+ + D2 <‘fjD’c f-J : .. .’

. . . ,’ .’ 6 E =4.0eV ‘--

E I-.8- :. 4. :. -, g... .*_..‘6:’ ‘,, ... :.. _; .“ : -4 ._., .

-u -4.0 .’ 5.44.x Id”

: 1.40 x IO-4 -2.71 x”10-4 ,” B :

45 4.68 x I$+ 1.35.x lo4 ‘, 3.38 x 1~~; 2 ,. .’

1. ‘.

-5.0 1.81 x lo-.J. 2.65 x.10”’ 4.76 x Ict4. :

i;r : * 6’

‘: :. -

labelled “8”. under. tliti heading “piti@ve”‘. To illus-

* i’-:~:,~.,~ ...

.,.

trate the rrk.ner in which tJle (qnl,, qj, , qr,) values are ‘~&ted for tt;e.eight.trajectories, we have’lkted the real

parts of ((7&, .qf,, qf ) for E = 4.0 eV in.tabJe 2. WF. a!so have computed S2100,1~o0 (0: /?)I2 from just .. .’

1

fouitrajectories (nos. 1: 2, 5 and 6 in,.table 2), which :

0 2 4 6 B

were the f&t four to be found, a~ld these results are L! ,. shown in the column lsbelled !‘4”. Clearly there is a’ Fig. 1. I&~o~~c,&, 4.3)1* as a function of i = Icalculated

significant change In results when zli the necessary from a single trajectory for each 1.

.’ ti-ajectories are not computed. The “clzzicaI” res$ts, ‘.

are obtain’ed from the ‘!pkrLi’tive” resAts for the eight trajectories, it.& clear that the amplitude for reattion

irajecto’i-ies.by Ieaving out the in!erference.terms, i.e.? (6) decreeses shar@~ as J increastis, so that the calcula- begiking with kq. (1) we compute CID-1/2exp(i#/fi)f2 tion of, the cross section for reaciion (6) should not re- for the single significant branching Fattern instead of quire _a large number of J states.

IC@‘1f2exp(i#/h)P. To test the’eccu’kcy. and method of cur.calculations,

: Iti’drder to inve.$ig?te the.&pecdencc of rehction 1 v&,have tised the principle of microsco$c reversibility.

(6) on S, we have comput?cI IS 2~o~,~o~~(~j 4.0)12’at E = We have time-reversed several trajectories in the man--

4.10 eV as a fuunction of! = J, where we have restricted. ,’ ner discusied in ref. [4] ,-and the rksults from the re-_

thesummation over trajectories in eq. I!> to just a versed tra,!ectories-agree with those from the~f&&rd single trajectory. The results. are show&in fig. 1;’ Al- &aje&ories within the accuracy of the computatibn. though we1ar.e ignoring the effects from the o&er seven’ We should m&ion that transitions to the ground

vibrational state, of HDf, unlike t‘ransitions to the first

‘TibIe 2 and higher excited vibration’al states, require spdcial ‘, : me periodic portions of t&e red paItS Of 9hIs 4j, yd 411 for care due to transition $oitits occurring close’in time,

tfle eight trajectories psed in tabIc’1 at 4.0 eY We have dsalt wjth this’p@lem in.detail in refs. [5,6] _-_ .’

44 *. qi, Qr, ‘, ,. for colline?r collisions, and our arguments should carry ,, : over direcay to three-d~ension~~collisions.

1,:. ” 3.08’. 1.57 -:- 1.22 ,. ‘I .d, 2’ ..’ .: 2173. 1 ,’ 1.87 . . i.85’ ,, .- .,

- : : ,. ,’ .‘,

3 2.87. ,‘..’ o.io ‘. : ,O.IG ’ 41 Conclu&n- ‘_ :’ 4,: . 3.16 1.47 ,I,, ., 2.23.,

‘.( ‘_ 1. ., .’ . 5 & .‘: .,

_:’ .. .1:47 : ., ,2..19 ‘. We havD ~etio,r$trat&&t the semicl+$l theoj ,,‘,, _.‘. ,- .

6 ; _l,!cl? ,, ). ‘.:..., 1.54 .’ ,’ ;. 1.29.’ ‘, .~f_~Iectrolljc transitions as,~eveloped in refs, [6,8] ~$0 7. ,’ 1.051 2.61 ; 0.50. y .,:.;- ,’ : ,.‘, be applied to a three-dimensional collision,procesS. It .- . . _: 8: ‘, .’ ,’ .I’10 .. 0.4s : ‘:. ,_ .” ,x70 ‘.. ; .’ ; is somewhat time-consuming’to find all-the trajectories - -.. : _;:

.,- ,.. ‘.- ‘.. .. . . ‘,’ .’ ,./‘, _, ..(._ ;. ‘,. :sj: =_, .I :. ..: _’ ., :’ . . j . . 1.

: ..‘. ; . ;. ,- . . ::‘. ,. .

Voluhu 30, number I CHEXIICAL PHYSICS LETTERS I January’ 197.5

nece.ssary to corncute the S-matrix elements, and for. [IO] A.D. Wilson.and R.D. Levine, hfol. Phys. 27 (1974) 1197.

this reason we have computed only the selected results [ll] J.C. Tully, J. Chem. Phys. 60 (1974) 3C42.

shown. rr~ tabre I and rig. ,I. The ener,g 4.0 eV is at i!.q c& g&+.~>Q~< a& p&: Pwt,p K m7+.?. Ph....- CD. m.T&>: > arsru a, a, x_a>zh>. 2 >>J& vv *a z 7

least 2.0 eV above the reaction threshold, and we 650.

would have to consider more than a significant branch- 1131 R.K. Preston aqd R.J. Cross, J. Chem. Phys. 59 (1973)

3616. ing pattern if we went to energies below 4.0 eV. Never- [14] J.C. Tully, Ber. Bunscnges. Physk. Chem. 77 (1973) 557.

theless, we feel that more extensive calculations to ob- [15] J. Durup, Chcm. Phys. 2 (1973) 226.

tain cross sections at 5.0 eV, 4.5 eV, 4.0 eV and .below [16] R. Diiren, J. Phys. B 6 (1’973) 1801..

are within the capability of modern computers. The 117) I.B. Dclos, J. Chem. Phys. 59 (1973) 2365.

sample calculations reported here will hopefully spur [18] H. Nakamura, Mol. Phys. 26 (1973) 673.

[19] E.R. Fisher and E. Bauer, J. Chem. Phys. 57 (1972) on further semiclassical calculations on the three-dimen- 1966:

sional Hf + D2 + HD+ + D reaction and other collision systems.

References

[l] 1.1% Zimmerman and T.F. George, Cham. Phys. 7 (1975), to bc published.

(21 I.H. Zimmerman and T.F. George, J. Chem. Phys. 61 (Sept. 15,1974), to be pubIished.

[3] Y.-W. Lin, T.F. George and K. Morokuma, J. Phys. B, to be published,

(41 Y.-W. Lin and T.F. George, J. Chem. Phys. 61 (Aug. 15, I974), to be published.

[S] Y.-W. Lin, T.F. George and K. Morokuma, J. Chem. Phys. 60 (1974) 4311.

[6] T.F. George and Y.-W. Lin, J. Chem. Phys. 60 (1974) 2340.

[7] Y.-W. Lin, T.F. George and K. Morokuma, Chem. Phys. Letters 22 (1973) 547.

[8] W.H. Miller and T.F. George, J. Chem. Phys. 56 (1972) 5637.

[9] J.C. Tully, J. Chem. Phys. 61 (1974) 61.

PO1

1211

WI

[231

1241

PI

Ml

[271

L=l

I291

WI

1311 [321

MA. Gonzalez, G. KarI and P.J.S. Watson, I. Chem. Phys. 57 (1972) 4054. E.E. Nikitin and S. Ya. Umanski, Discussions Faraday sot. 53 (1972) 7. R.D. Levine and R.B. Bernstein, Chcm. Phys. Letters 15 (1972) 1. Y. Haas, R.D. Levine and G. Stein, Chcn. Phys. Letters 15 (1972) 7. B.R. Johnson and R.D. Levine, Chem. Phys. Letters 13 (1972) 168. J.C. Tully and RX. Preston, J. Chem. Phys. 55 (197I) 562. J.D. Doll, T.F. George and W.H. MilICr, I. Chem. Phys. 58 (1973) 1343. K. Morokuma and T.F. George, J. Chem. Phyr. 59 (1973) i959. T.F. George and K. Jlorokuma, Chem. Phys. 2 (L9i3) 129. W.H. Miller and T.F. George, J. Chem. Phys. 56 (1972) 5668. T.F. George and W.H. hliller, I. Qiem. Phys. 57 (L972) 2458. J.D. Doll and W.H. Miller, I. Chem. Phys. 57 (1972) 5019. W.H.‘MiIler, J. Chem. Phys. 54 (1971) 5386, appendix B.