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Evaluation of dominant reaction probabilities for collinear hydrogen-transfer reactions
Volume
107. number
CHEMICAL
6
EVALUATION
OF DOMINANT
FOR COLLINEAR
REACTION
10 March 1984;
15 June 1984
LET-l-E&3
PROBABILITLES
HYDROGEN-TRANSFER
J. MANZ and H.H.R. SCHOR * Lehrstuhl /lur Theoretische Chenzie, Technische Reozived
PHYSICS
REACTIONS
Uniwrsir~t
hfiincherr.
D-8046
Carching.
in final form 25 April 1984
Dominant reaction probabilities for mBinear hydrogen transfer with approximate evaluated by propagating only few open and closed channels along the hyperspheriul economical method is applied to the F + DBr(0) 4 FD(u’) + Br reaction
1. Introduction Hydrogen
transfer
between
two heavy
E tr,(i)
= Et-(f)
(1)
with little change of kinetic
energy: (2)
-
rough propensity rule (2) has been confiimed experimentally for the systems Cl + HI [l-l], Cl + HBr [I ,4,5], Br + HI [1,4], Cl + DZ [2,4], F + HCl [6-g], O+HCl [lO,ll],Cl+OH [11,12],F+HBr [6,9,13151, F + HE [6,9,13,14], F- + HCl, F- + HBr, F+HI[l6],F+DBr[lS]andBr+HAt [17],andtheoretically, using three-dimensional classical trajectories, for the system Cl + HI [2,18], Cl + Dl [18], F + HCl [8,9],F+HBr [9,15],F+HI [9],F+DBr [15],F+ HF, F + DF [ 191 agd Br + HBr [20], or using the classical, semiclassical or quantum collinear models for the systems Cl + HI [21], Cl + HBr [21-251, H + MuH [26-301, F + HF [3 I], I + HI [28-30,32-341, Cl- + HBr, Cl- + DBr [35], I + Mu1 [28,30,34], Cl + MuCl, Cl + HCl and Cl + DC1 [36,37] _The rule (2) implies,for example, product energy triangle plots with ridgesextendingroughly according to (2) [4,8,14,18], The
P(f+i)>O.Ol.
P(f
03.00 0 Elsevier Science Publishers Physics
Publishing
Division)
+i)
_
and have negli(4)
The suggested boundary 0.01 between dominant and negligible probabilities [(3),(4)] is of course somewhat arbitrary, depending on the system. The dominant
de Quimica. ICEX, Universidade Federal de hIinas Gerais, 30 000 Belo Horizonte. BraziL
(North-Holland
( 3
The others (“?“) are hardly populated gible reaction probabilities:
* Permanent address: Deparmmento
0 009-2614/84/S
conservation of kinetic ener_w are radial coordinate. This accurate and
approximate vibrational adiabaticity of nearly thermoneutral reactions with little product rotational excitation [IO-12,19,20], or strong vibrational population inversion of exothermic reactions [l-5,7-9,13-18]. These effects are amplified in the collinear wcrId with frozen rotations [21-24,26-371. The physical reason for the rule (2) is storage of kinetic energy in the heavy atoms, with little effect of the hydrogen “spectator” [21,23,24,3 1,331. The kinematic rule (3) also serves as a reference, i.e. interesting deviations from (2) call for interpretations in terms of dynamical effects such as multiple encounters [ 1,2,9,15,17-201, compe!ing (direct versus migratory) reaction meckanisms [ 14,15, 38,391, or complex formation, rearrangement and electronically diabatic transitions [40] _Violations of (2) may also be due to relaxation of nascent product distributions [9,14,15,41]. In the case of (2), two groups of product states may be distinguished. Those (“f’) which fulfd (2) usually have dominant reaction probabilities:
atoms:
A+HB(i)+AH(f)+B, often proceeds
Cernloq
B.V.
549
CHEMICAL PHYSICS LEmERS
Volume 107, number 6 transitions
(3) are selected automatically in experiments, e.g. by laser preparation of reactants i in molecular beams and subsequent infrared chemiluminescence detection of domimnt product states f. Such expe~ents yield the reZ#mit ~fo~ation, namely the dominant elements of a sin&e cohmm i of the total transition probability matrix P(f + i). The situation is completely different for present quantum tie-~dependent propagation techniques [21,22,27,29;32-34,36,42,43], which always yield the full reaction probab~ty matrix Pjf f i). Most of this theoretical ~fo~ation is irrelevffrztfor studies of state-selective reactions (I) with specific energy release (2). As a simple but illustrative exampie, consider the exotbermic collinear reaction F 4 DBr(O) -+ F + DBr(0) ,
CW
+FD(;=6)+Br,
Gb)
--f FD&’ = O-5) + Br t
&I
with 8 channels R = 1-7 and 8, corresponding to products FD(u’ = O-6) + Br and reactants F + D&(O), respectively, at thermal collission energies_ Traditional propagation techniques provide the full 8 X 8 matrix for aU transitions rz+ n’. But only two transitions [(Sa), (Sb)] are dominant according to (3) (see section 2); the remaining 64 - 2 = 62 transitions are irtelevant, including the negligl%lechannels (SC) as well as all elastic and inelastic cbnnels of the reverse reaction FDfu)+Br+FD@‘)+Br
_
15 June I984
tonian approach and the Iz- or &-conserving, coupled states, centrifugal sudden, and in~nite-order sudden approx~ations (see the reviews refs. f45--503 and references therein). The decoupling of open channels with different internal energies has also been suggested for ineta’stic collisions by Secrest (see section 4 2 of ref. 14611, but to the best of our knowledge tkis-Japer presents the fist systematic app~cation to rea +.AIS. The method is presented in section 2 with application to the collinear F 4 DBr reaction (5). Conclusions are in section 3.
Dominant reaction probabilities for the collinear F + DBr{O) reaction (5) axe evaluated using the best extended LEPS surface of Jonathan et al. with parameters given in table 3 of ref. [ 15). A single 1 *A1 model surface is appropriate since the reactants Ff2P3,$ + DBr(l C) yield 99% DF(” E) -+Br(*P& and only 1% electronically excited DF(l Z) + Br*(zP, a) products [51] _The surface is plotted using mass~we~hted coordinates x and y defined in fig. I _The skew angle is
C-W
In conclusion, present propagation techniques yield
complete but mostly irrelevant ~fo~tion on coliinear reactions (1) at the expense of an enormous waste of computer time and storage. The purpose of this paper is to present a more economical propagation technique for the evaluation of dominant transition probab~ties (3) of collinear hydrogen-tra~fer reactions (1). The method is a s&nple extension of the two-channel approximation j23-25,28--31,34,36,37] and of the familiar numberof-cl~~d~ha~els convergence test [21,22,26,2729, 33,3*C,36]. It coupIes channels with similar internal
eneq;ies, but decouples channels with very different internal energies. Therefore, the present approach is an internal-energy decoupling scheme, complementary to the angle-momentum recoupling 1441 and de~up~g tec~ues, ~clu~g the effectivity550
Fig. 1. Contour plot of the best extended LEPS potentialenergy surface of Jonathan et al [ 15J for the F + DBrfu) + FD(o*) + Br readion. Contours for E = -SSO, -450. -350 and -250 kJ mol-’ are plotted using mass-weighted coordinates x = (~F,DB&zDB#” rFaBr andy = rDBr. where qz,D& and ~~~ are the d&.tanees tram F to ihe center of mass of DBr and from D to Br. respectively. The skew angle is ~l?m= 20.1’ _The sadcUepoint :: is at x* = 9.322, y* = 1.420 (r fitF = 1973 A 1; the haxrkx Mght is 1.26 W moT' . S-
matricesare propagated along the h~rsp~eri~ from Q = 5.5 A to rf s= 15.5 b.
radius r
CHEMICAL
Volume 107, number 6
PHYSICS
15 June 1984
LETTERS
for IIlaWSWZF= 19.00, mD = 2.014 and “Br = 79.91 amu. All calculations units with conversion 1 hartree = 2625.501 me* The method
are carried out using atomic factors 1 bohr = 0.52917706 A, kJ mol-l and 1 amu = 1822.8496
employsS-matrix
propagation
along
I
I
I
I
I
I
I
I
I
Delves’ radial coordinate r = (x2 +y2)l” [42,43]. At fixed r, the angular Schriidinger equation is solved for
(p,/2001. The C ,,(T) are plotted in fig. 2. The
0000
I
I
I
I
I
I
8888 0000
I
VIN-ICICICI~CI000000000 000000000 000000000 000000000 000000050
Illlddddddidd
P-o1cccII-P-ecoc~cl0000000000 0000000000 0000000000
0999999999
II
1~000000000
mcnmommoomo~ ~ot-4~.-l-c-4P4~P4-l
00000000000 00000000000 I
6 Fig. 2. Vhational energies en(r) versus hyperspheriul radius r for the F + DBr(u) + FD(u’) + Br reaction. Asymptotically, the En@ -+ -) approach the diatomic FD(u’) or DBr(u) levels. For total energy Etot = -360 kJ mol-1 (dashed line - - -), channels n 4 8 and n 3 9 are open and closed, respectively.
The dominant reaction probabilities are evaluated accurately (within *O.OOl) by S-matAx propagation including only thannels n = 4-l 1 (continous lines -) but excluding closed channels with energy E > Emax = -325 kJ mol-’ (n > 12) asweUasopenc~e~n=1,2,3(slashedlines+and t respectively). Channel n = 4 is incbded in the full propagation rangeri
J 5.5 krf
= 15.5 A.
I dd
8888
the vibrational energies C,,(r) and wavefunctions @,(cp; r) using Johnson’s technique [52] with grid size L\lp=
I
I
jdddddddddd
I
I
88 88 88
dddd
Volume 107, number 6
CHEMICAL PHYSICS LEITERS
grid size along r is evaluated dynamically such that relative average changes of energies &Jr) and coupling matrices (@,@)I@,& + Ar)? are kept below 0.002 per box [53]. Altogether, 2000 boxes were generated el5.5 8.. At t-f, the in the range ri z5.5 Adrdrf propagation stops with converged asymptotic Cetiro limiting values [54]. The resulting transition probabilities are listed in table 1. Obviously, the rule (2) is confirmed, i.e. transitions (Sa), (5b) to n’ = 7,8 are dominant according to criterion (3). Table 1 also includes the familiar number-of-closedchannels (N,) convergence test. In accord with previous investigations [22,25-27,32-34,36,37], very few propagated closed channels yield accurate reaction probabilities. In the present case, Iv, = 3 is sufficient, i.e. one should keep three closed channels II = 9, 10, 11 above the initial state F + DBr(O), n = 8, and one may drop all closed channels n > 12. We now generalize this convergence test to open channels: i.e. we propagate dropping successively irrelevant open charu& PD(u’ = 0, 1,2, ___)+ Br, tz = 1, 2,3, . . . The resultsare aizo listed in table 1. Obviously, No = 5 (4) open channels al: sufficient for *O-O01 (3.01) accuracy of the domimdt transition probabilities (5a), (Sb). Simultaneously, one gains a considerable amount of computer time and storage which scale roughly according to N3 and N2, respectively, where N is the number of open plus closed channels, N = No + NC. This profit is at the expense of only irrelevant reaction probabilities (see table 1). As a special case of this method, one may try and drop all closed channels (N, = 0) and keep only two open channels (No = 2), (5a), (Sb), n = 7,8, according to rule (2). This limit corresponds to the single- and two-channel approximations for symmetric and asymmetric hydrogen-exchange reactions (1) 123-25, 28-3 1,34,36,37]. The results are included in table 1. Qualitatively, the elastic channel, eq. (5a), is still preferred, but clearly the two-channel model is quantitatively insufflcient. In contrast, the present method yields accurate results rmd is much more economical than full propagation of all open channels.
3. Conclu!Gons The rough propensity 552
rule (2) suggests dominant
15 June 1984
(3) and negligible (4) channels of hydrogen-transfer reactions (1). The present evaluation of dominant reaction probabilities for collinear hydrogen-transfer reactions simply drops negligible closed attd opetl channels from S-matrix propagation. Irrelevant information on marginally populated product levels or non-reactive reverse processes is lost, but computer time and storage is gained. This economical technique is used in ref. [54] for a systematic investigation of the collinear F + DBr(0) reaction and the relation to experimental fiidings [9,13-15,39,41,51]. It may be anticipated here that the results of table 1 are typical for thermal energies. They agree with similar results for Cl + HBr(0) [3-l] but differ from all other previous results on collinear hydrogen-transfer reactions due to lack of lowenergy (quasi-)degenerate reactant and product levels [22-371. The present method may be generalized as follows: (1) There is no restriction to S-matrix propagation. It should also be advantageous to drop irrelevant channels in wavefunction 122,321 or R-matrix 125,36,37] propagations. (2) In the present case, the discrimination of irrelevant channels follows the physical propensity rule (2). A!tematively, one might employ an automatic routine to search for irrelevant channels n’ for a given reactant state n. Certainly, very small coupling coefficients (+,#)I Qnr(r + Arj for all values of r would predict reasonable candidates of irrelevant channels. Once they are detected in a few sample calculations, computer time and storage will be gained in large scans of total energies. Moreover, this unbiased discrimination of dominant and negligible channels could, in principle, reveal new propensity rules. (3) The propagation along r may be split into boxes [55,56, with different numbers of relevant channels. One expects that vibrationally non-adiabatic transitions should occur only in limited regions along the potential ridge [57]. Thus, one may anticipate converged results from two- or few-channel calculations with r-dependent selection of coupled channels. (4) Finally, tie suggest a combination of the present internal-energy and the well-known angular-momentum decoupling schemes [45-SO] (see also ref. [58] )_ Both techniques may be considered as dimensionalityreducing approximations 1591, and the combination of both reduction schemes should imply valuable gain of computer time and storage. Perhaps the most prom-
Vofume 107, number 6
CHEMICAL PHYSiCS LETTERS
ising suggestion in this direction is a combination of both reduction schemes with the variation-iteration method of Thomas [60]: this would reduce the computed results not only to the dominant sub-matrix, but directly to the desired sub-vector of dominant reaction probabilities.
Acknowledgement We would like to thank Dr. J. Remelt for stimulating discussions and ~omputat~onai advice: P. Gertitschke for valuable computational assistance; Dr. V. Aquilanti for sending us his preprints [30,57] prior to publication; Dr. J. Schtifer for pointing out ref. [60]; Professor G.L. Hofacker for continuous support and hospitality; and the Deutsche Forschungsgeme~schaft, the Fonds der chemischen Industrie and the CNPq (Brazil) for generous financial support. The computations were cwried out on the CYBER 175 of the Bayerische Akademie der Wissenschaften.
References
Ill K.C. Anlauf. P.J. Kuntz, DH. hfayiotte, P.D. Paceg and J.C. Polanyi, Discussions Faraday Sot. 44 (1967) 183. Wongand K.B. WoodalL J. Chem. Phys. 49 (1968) 5189. f31 LT. Cowley, D.S. Home and J.C. Polanyj, Chem. Phys. Letters 12 (1971) 144. f41 D.H. hfaykrtte, J-C. PoJanyi and K.B. Woo&U. J. Cftem. Phys. 57 (1972) 1547. ISI J.R. Airey, J. Chem. Phys. 52 (1970) 156. f61 N. Jouathan, Chl. MeUiar-Smith, S. Okuda, D.H. Skter and D. Timlin, MoL Pbys. 22 (1971) 561. I71 LJ. Hirsch and J.C. Pdanyi, J. Chem. Phys. 57 (1972) 4498. (81 A.hf.G.Din8,L.J. Rirsch,D.S.Perry, J.C.Poiarryiand J.L. Schreiber, Discussions Faraday Sot. 55 (1973) 252. [9] P. Beadle, h1.R. Dunn, N&H. Jonathan, J.P. Liddy and J.C. Naylor, J. Cbem. Sot. Faraday Trans. II 74 (1978) 2170. [lo] D. Arrto’ldi and J. Wolfrum, Chem. Phys. Letters 24 (1974) 234. [Ill JS. Butler, J.W. Hudgeus, MC. Lin and G.K. Smith, Cbem. Phys. Letters 58 (1978) 216. [ 121 B.A. BlackweB, J.C. Polanyi and J J. SIoan, Discussions Faraday Sot. 62 (1976) 111; Chem. Phys. 24 (1977) 25. 1131 J9. Sung and D.W. Se&r, Chem. Phys. Letters 48 (1977) 413. 1141 I;.Tamagake, D-W. Setser and J.P. Sung, J. Chem. Phys. 73 (1980) 2203.
121 K.G. AnJauf, J.C. Polanyi,W.H.
15 June 1984
[15] N.B.H. Jorx~than, P.V. Sellers and A.J. State. Mol. Phys. 43 (1981) 215. [16) J.C. Weisshaar. T.S. Zwier and S.R. Leone, J. Chem. Phys. 75 (1981) 4873. [i7j J.R.Grover, D.E. MaJJoyand J.B. MitcheJJ, J. Chem. Phys. 76 (1982) 362. [18] CA. Parr. J.C. Polanyi and W.H. Wang, J. Chem. Phys. 58 (1973) 5. [I91 R.L. Wilkins, J. Chem. Phys. 59 (1973) 698;60 (1974) 2201. [20] J.M. White and D.L. Thompson, J. Chem. Phys. 61 (1974) 719. [Zl ] hf. Baa. J. Chem. Phys. 62 (1975) 305. [22] J.A. Kaye and A. Kuppamann, Chem. Phys. Letters 92 (1982) 574. [23] V.K. Babamov. V. Lopez and R.X. hlarcus, J. Chtm. Phys. 78 (1983) 5621. [24 J V.R. Babamov, V. Lopez and R.A. 51;~~~s. Chem. Phys. Letters 101 (1983) 507. [2S] N. AbuSaibi, D.J. Kouri. V. Lopez, V.K. Babamov and R-A. Marcus, Chem. Phys. Letters 103 (1984) 358. 1261 J. !&nz and J. Romelt, Chem. Phys. Letters 76 (1980) 337. [27] J. !&nz. E. Pollok and J. RomeIt, Chem. Phys. Letters 86 (1982) 26. PSI V. Aquhnti, S. CavalJi and A. Lag&. Chem. Phys. Letters 93 (1982) 179. J. Romeit. Chem. Phys. 79 (1983) 197. ;:;t V. AquiJanti. S. CawaBi, G. Grossiand A. lagan& Hypedine Interactions to be pubiished. 1311 V.R. Babamovnnd R.A. Marcus. J. Chcm. Phys. 74 (1981) 1790. 1321 J.A. #aye and A. Kuppermann. Chem. Phys. Letters 77 (1981) 573. 1331 J. Mauz and J. Remelt, Chem. Phys. Letters 81 (1981) 179. i-1 C. Hi&r. J. %nr, W-H. ilfiJJer and J. Rbmeh. J. Chem. Phys. 78 (1983) 3850. 1351 S. Chapman, Chem. Phys. Letters 80 (1981) 275. f361 D.R. Bondi, J.N.L. Connor, B.C. Garrett and D.C. TruhJar, J. Chem. Phys. 78 (1983) 5981. 1371 D.K. Bondi. J.N.L. Connor, J. Yanz and J. RBmeh. hiof. Phys. 50 (1983) 467. I381 C.-C. Mei and C,B. Moore, J. Chem. Phys. 70 (1979) 1759. f391 E. Wiirzberg and P.L. Houston, J. Chcm. Phys. 72 (1980) 5915. 1401 JJ. Sloart, D.G. Watson, JX. WIUiamson and J.S. \V%ht, J. Chem. Phys. 75 (1981) 1190. f41 I 5. Brandt, L-W. Dickson, L.N.Y. Swan and J.C. Polanyi, Chem. Phys. 39 (1979) 189. I421 G. Hauke, J. Manz and J. Rijmeh. J. Chem. Phys. 73 (1980) 5040. f431 J. Remelt, Chem. Phys. Letters 74 (1980) 163. 1441 K. hlcLenithan and D. Secrest, J. Chem. Phys. 80 (1984) 2480.
553
CHEMICAL
Volume 107. number 6
(451 H. Rabitz, in: Dynamics of molecular collisions. Modem Theoretical Chemistry, Vol. 1, Part A, ed. W.H. Miller (Plenum Pres. New York, 1976) p. 33. [46] D. Seaest, in: Atom-molecule collision theory, ed. RB.
Bernstein
(Pknum
York, 1979) P. 265. collision theory, ed.
Press. New
[47] D.J. Kouri. in: Atom-molecule RB. Bernstein (Plenum, Press, New York, 1979) p. 301. [48] D. !&crest, in: Atom-molecule collision theory, ed. R.B. Bernstein (Plenum Press. New York, 1979) p. 377. [49] RE. Wyatt, in: Atom-molecule collision theory. ed. R.B. Bernstein (Plenum Press, New York 1979) p. 477. [SO] D. Seaest. in: Lzture notes in mathematics: numerical integration of difrerential eqxatbns and large linear systems, ed. J. Hinze (Springer, Berlin 1982) p. 1. [Sl] J.W. Hepburn, K. I_& R.G. Macdonald. F-l. Northrup and J.C. Polanyi, J. Chem. Phys. 75 (1981) 3353. [SZ] B.R. Johnson, J. Chem. Phys. 67 (1977) 4086.
5.54
PHYSICS LETTERS
ISJune 1984
1531 E.B. Stechel, R.B. Walker and J.C. Liiht, J. Chem.
Phys. 69 (1978) 3518.
[ 541 P. Gertitschke, J. Manz, J. Remelt and H.H.R. Schor, to be published. J. hianz. Mol. Phys. 28 (1974) 399. 1561 J. Manz. Mol. Phys. 30 (1975) 899. [57] V. Aquilanti, S. Cavalli. G. Grossi and A. Laga&. J. Mol. Structure 107 (1984) 95. [SS] P. Botschwina. private communication on forthcoming scattering calculations for Na* + Hz quenching processes [SS]
(1984); P. Botschwina. W. Meyer. 1-V. Hertel and W. Reiland. J. Chem. Phys. 75 (1981) 5438. [SV] J.C. Light, in: Atom-molecule collision theory, ed. R,B. Bernstein (Plenum Press, New York, 1979) p_ 239. [60] L.D. Thomas, J. Chem. Phys. 70 (1979) 2979.
107. number
CHEMICAL
6
EVALUATION
OF DOMINANT
FOR COLLINEAR
REACTION
10 March 1984;
15 June 1984
LET-l-E&3
PROBABILITLES
HYDROGEN-TRANSFER
J. MANZ and H.H.R. SCHOR * Lehrstuhl /lur Theoretische Chenzie, Technische Reozived
PHYSICS
REACTIONS
Uniwrsir~t
hfiincherr.
D-8046
Carching.
in final form 25 April 1984
Dominant reaction probabilities for mBinear hydrogen transfer with approximate evaluated by propagating only few open and closed channels along the hyperspheriul economical method is applied to the F + DBr(0) 4 FD(u’) + Br reaction
1. Introduction Hydrogen
transfer
between
two heavy
E tr,(i)
= Et-(f)
(1)
with little change of kinetic
energy: (2)
-
rough propensity rule (2) has been confiimed experimentally for the systems Cl + HI [l-l], Cl + HBr [I ,4,5], Br + HI [1,4], Cl + DZ [2,4], F + HCl [6-g], O+HCl [lO,ll],Cl+OH [11,12],F+HBr [6,9,13151, F + HE [6,9,13,14], F- + HCl, F- + HBr, F+HI[l6],F+DBr[lS]andBr+HAt [17],andtheoretically, using three-dimensional classical trajectories, for the system Cl + HI [2,18], Cl + Dl [18], F + HCl [8,9],F+HBr [9,15],F+HI [9],F+DBr [15],F+ HF, F + DF [ 191 agd Br + HBr [20], or using the classical, semiclassical or quantum collinear models for the systems Cl + HI [21], Cl + HBr [21-251, H + MuH [26-301, F + HF [3 I], I + HI [28-30,32-341, Cl- + HBr, Cl- + DBr [35], I + Mu1 [28,30,34], Cl + MuCl, Cl + HCl and Cl + DC1 [36,37] _The rule (2) implies,for example, product energy triangle plots with ridgesextendingroughly according to (2) [4,8,14,18], The
P(f+i)>O.Ol.
P(f
03.00 0 Elsevier Science Publishers Physics
Publishing
Division)
+i)
_
and have negli(4)
The suggested boundary 0.01 between dominant and negligible probabilities [(3),(4)] is of course somewhat arbitrary, depending on the system. The dominant
de Quimica. ICEX, Universidade Federal de hIinas Gerais, 30 000 Belo Horizonte. BraziL
(North-Holland
( 3
The others (“?“) are hardly populated gible reaction probabilities:
* Permanent address: Deparmmento
0 009-2614/84/S
conservation of kinetic ener_w are radial coordinate. This accurate and
approximate vibrational adiabaticity of nearly thermoneutral reactions with little product rotational excitation [IO-12,19,20], or strong vibrational population inversion of exothermic reactions [l-5,7-9,13-18]. These effects are amplified in the collinear wcrId with frozen rotations [21-24,26-371. The physical reason for the rule (2) is storage of kinetic energy in the heavy atoms, with little effect of the hydrogen “spectator” [21,23,24,3 1,331. The kinematic rule (3) also serves as a reference, i.e. interesting deviations from (2) call for interpretations in terms of dynamical effects such as multiple encounters [ 1,2,9,15,17-201, compe!ing (direct versus migratory) reaction meckanisms [ 14,15, 38,391, or complex formation, rearrangement and electronically diabatic transitions [40] _Violations of (2) may also be due to relaxation of nascent product distributions [9,14,15,41]. In the case of (2), two groups of product states may be distinguished. Those (“f’) which fulfd (2) usually have dominant reaction probabilities:
atoms:
A+HB(i)+AH(f)+B, often proceeds
Cernloq
B.V.
549
CHEMICAL PHYSICS LEmERS
Volume 107, number 6 transitions
(3) are selected automatically in experiments, e.g. by laser preparation of reactants i in molecular beams and subsequent infrared chemiluminescence detection of domimnt product states f. Such expe~ents yield the reZ#mit ~fo~ation, namely the dominant elements of a sin&e cohmm i of the total transition probability matrix P(f + i). The situation is completely different for present quantum tie-~dependent propagation techniques [21,22,27,29;32-34,36,42,43], which always yield the full reaction probab~ty matrix Pjf f i). Most of this theoretical ~fo~ation is irrelevffrztfor studies of state-selective reactions (I) with specific energy release (2). As a simple but illustrative exampie, consider the exotbermic collinear reaction F 4 DBr(O) -+ F + DBr(0) ,
CW
+FD(;=6)+Br,
Gb)
--f FD&’ = O-5) + Br t
&I
with 8 channels R = 1-7 and 8, corresponding to products FD(u’ = O-6) + Br and reactants F + D&(O), respectively, at thermal collission energies_ Traditional propagation techniques provide the full 8 X 8 matrix for aU transitions rz+ n’. But only two transitions [(Sa), (Sb)] are dominant according to (3) (see section 2); the remaining 64 - 2 = 62 transitions are irtelevant, including the negligl%lechannels (SC) as well as all elastic and inelastic cbnnels of the reverse reaction FDfu)+Br+FD@‘)+Br
_
15 June I984
tonian approach and the Iz- or &-conserving, coupled states, centrifugal sudden, and in~nite-order sudden approx~ations (see the reviews refs. f45--503 and references therein). The decoupling of open channels with different internal energies has also been suggested for ineta’stic collisions by Secrest (see section 4 2 of ref. 14611, but to the best of our knowledge tkis-Japer presents the fist systematic app~cation to rea +.AIS. The method is presented in section 2 with application to the collinear F 4 DBr reaction (5). Conclusions are in section 3.
Dominant reaction probabilities for the collinear F + DBr{O) reaction (5) axe evaluated using the best extended LEPS surface of Jonathan et al. with parameters given in table 3 of ref. [ 15). A single 1 *A1 model surface is appropriate since the reactants Ff2P3,$ + DBr(l C) yield 99% DF(” E) -+Br(*P& and only 1% electronically excited DF(l Z) + Br*(zP, a) products [51] _The surface is plotted using mass~we~hted coordinates x and y defined in fig. I _The skew angle is
C-W
In conclusion, present propagation techniques yield
complete but mostly irrelevant ~fo~tion on coliinear reactions (1) at the expense of an enormous waste of computer time and storage. The purpose of this paper is to present a more economical propagation technique for the evaluation of dominant transition probab~ties (3) of collinear hydrogen-tra~fer reactions (1). The method is a s&nple extension of the two-channel approximation j23-25,28--31,34,36,37] and of the familiar numberof-cl~~d~ha~els convergence test [21,22,26,2729, 33,3*C,36]. It coupIes channels with similar internal
eneq;ies, but decouples channels with very different internal energies. Therefore, the present approach is an internal-energy decoupling scheme, complementary to the angle-momentum recoupling 1441 and de~up~g tec~ues, ~clu~g the effectivity550
Fig. 1. Contour plot of the best extended LEPS potentialenergy surface of Jonathan et al [ 15J for the F + DBrfu) + FD(o*) + Br readion. Contours for E = -SSO, -450. -350 and -250 kJ mol-’ are plotted using mass-weighted coordinates x = (~F,DB&zDB#” rFaBr andy = rDBr. where qz,D& and ~~~ are the d&.tanees tram F to ihe center of mass of DBr and from D to Br. respectively. The skew angle is ~l?m= 20.1’ _The sadcUepoint :: is at x* = 9.322, y* = 1.420 (r fitF = 1973 A 1; the haxrkx Mght is 1.26 W moT' . S-
matricesare propagated along the h~rsp~eri~ from Q = 5.5 A to rf s= 15.5 b.
radius r
CHEMICAL
Volume 107, number 6
PHYSICS
15 June 1984
LETTERS
for IIlaWSWZF= 19.00, mD = 2.014 and “Br = 79.91 amu. All calculations units with conversion 1 hartree = 2625.501 me* The method
are carried out using atomic factors 1 bohr = 0.52917706 A, kJ mol-l and 1 amu = 1822.8496
employsS-matrix
propagation
along
I
I
I
I
I
I
I
I
I
Delves’ radial coordinate r = (x2 +y2)l” [42,43]. At fixed r, the angular Schriidinger equation is solved for
(p,/2001. The C ,,(T) are plotted in fig. 2. The
0000
I
I
I
I
I
I
8888 0000
I
VIN-ICICICI~CI000000000 000000000 000000000 000000000 000000050
Illlddddddidd
P-o1cccII-P-ecoc~cl0000000000 0000000000 0000000000
0999999999
II
1~000000000
mcnmommoomo~ ~ot-4~.-l-c-4P4~P4-l
00000000000 00000000000 I
6 Fig. 2. Vhational energies en(r) versus hyperspheriul radius r for the F + DBr(u) + FD(u’) + Br reaction. Asymptotically, the En@ -+ -) approach the diatomic FD(u’) or DBr(u) levels. For total energy Etot = -360 kJ mol-1 (dashed line - - -), channels n 4 8 and n 3 9 are open and closed, respectively.
The dominant reaction probabilities are evaluated accurately (within *O.OOl) by S-matAx propagation including only thannels n = 4-l 1 (continous lines -) but excluding closed channels with energy E > Emax = -325 kJ mol-’ (n > 12) asweUasopenc~e~n=1,2,3(slashedlines+and t respectively). Channel n = 4 is incbded in the full propagation rangeri
J 5.5 krf
= 15.5 A.
I dd
8888
the vibrational energies C,,(r) and wavefunctions @,(cp; r) using Johnson’s technique [52] with grid size L\lp=
I
I
jdddddddddd
I
I
88 88 88
dddd
Volume 107, number 6
CHEMICAL PHYSICS LEITERS
grid size along r is evaluated dynamically such that relative average changes of energies &Jr) and coupling matrices (@,@)I@,& + Ar)? are kept below 0.002 per box [53]. Altogether, 2000 boxes were generated el5.5 8.. At t-f, the in the range ri z5.5 Adrdrf propagation stops with converged asymptotic Cetiro limiting values [54]. The resulting transition probabilities are listed in table 1. Obviously, the rule (2) is confirmed, i.e. transitions (Sa), (5b) to n’ = 7,8 are dominant according to criterion (3). Table 1 also includes the familiar number-of-closedchannels (N,) convergence test. In accord with previous investigations [22,25-27,32-34,36,37], very few propagated closed channels yield accurate reaction probabilities. In the present case, Iv, = 3 is sufficient, i.e. one should keep three closed channels II = 9, 10, 11 above the initial state F + DBr(O), n = 8, and one may drop all closed channels n > 12. We now generalize this convergence test to open channels: i.e. we propagate dropping successively irrelevant open charu& PD(u’ = 0, 1,2, ___)+ Br, tz = 1, 2,3, . . . The resultsare aizo listed in table 1. Obviously, No = 5 (4) open channels al: sufficient for *O-O01 (3.01) accuracy of the domimdt transition probabilities (5a), (Sb). Simultaneously, one gains a considerable amount of computer time and storage which scale roughly according to N3 and N2, respectively, where N is the number of open plus closed channels, N = No + NC. This profit is at the expense of only irrelevant reaction probabilities (see table 1). As a special case of this method, one may try and drop all closed channels (N, = 0) and keep only two open channels (No = 2), (5a), (Sb), n = 7,8, according to rule (2). This limit corresponds to the single- and two-channel approximations for symmetric and asymmetric hydrogen-exchange reactions (1) 123-25, 28-3 1,34,36,37]. The results are included in table 1. Qualitatively, the elastic channel, eq. (5a), is still preferred, but clearly the two-channel model is quantitatively insufflcient. In contrast, the present method yields accurate results rmd is much more economical than full propagation of all open channels.
3. Conclu!Gons The rough propensity 552
rule (2) suggests dominant
15 June 1984
(3) and negligible (4) channels of hydrogen-transfer reactions (1). The present evaluation of dominant reaction probabilities for collinear hydrogen-transfer reactions simply drops negligible closed attd opetl channels from S-matrix propagation. Irrelevant information on marginally populated product levels or non-reactive reverse processes is lost, but computer time and storage is gained. This economical technique is used in ref. [54] for a systematic investigation of the collinear F + DBr(0) reaction and the relation to experimental fiidings [9,13-15,39,41,51]. It may be anticipated here that the results of table 1 are typical for thermal energies. They agree with similar results for Cl + HBr(0) [3-l] but differ from all other previous results on collinear hydrogen-transfer reactions due to lack of lowenergy (quasi-)degenerate reactant and product levels [22-371. The present method may be generalized as follows: (1) There is no restriction to S-matrix propagation. It should also be advantageous to drop irrelevant channels in wavefunction 122,321 or R-matrix 125,36,37] propagations. (2) In the present case, the discrimination of irrelevant channels follows the physical propensity rule (2). A!tematively, one might employ an automatic routine to search for irrelevant channels n’ for a given reactant state n. Certainly, very small coupling coefficients (+,#)I Qnr(r + Arj for all values of r would predict reasonable candidates of irrelevant channels. Once they are detected in a few sample calculations, computer time and storage will be gained in large scans of total energies. Moreover, this unbiased discrimination of dominant and negligible channels could, in principle, reveal new propensity rules. (3) The propagation along r may be split into boxes [55,56, with different numbers of relevant channels. One expects that vibrationally non-adiabatic transitions should occur only in limited regions along the potential ridge [57]. Thus, one may anticipate converged results from two- or few-channel calculations with r-dependent selection of coupled channels. (4) Finally, tie suggest a combination of the present internal-energy and the well-known angular-momentum decoupling schemes [45-SO] (see also ref. [58] )_ Both techniques may be considered as dimensionalityreducing approximations 1591, and the combination of both reduction schemes should imply valuable gain of computer time and storage. Perhaps the most prom-
Vofume 107, number 6
CHEMICAL PHYSiCS LETTERS
ising suggestion in this direction is a combination of both reduction schemes with the variation-iteration method of Thomas [60]: this would reduce the computed results not only to the dominant sub-matrix, but directly to the desired sub-vector of dominant reaction probabilities.
Acknowledgement We would like to thank Dr. J. Remelt for stimulating discussions and ~omputat~onai advice: P. Gertitschke for valuable computational assistance; Dr. V. Aquilanti for sending us his preprints [30,57] prior to publication; Dr. J. Schtifer for pointing out ref. [60]; Professor G.L. Hofacker for continuous support and hospitality; and the Deutsche Forschungsgeme~schaft, the Fonds der chemischen Industrie and the CNPq (Brazil) for generous financial support. The computations were cwried out on the CYBER 175 of the Bayerische Akademie der Wissenschaften.
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