Model calculation on branched chemical reactions

Volume 15, number 1

CHEMICALPHYSICS

LETTERS

iSfuiy1972

Received 6 March 1972 Revised manuscripr receiwd 7-6 April 1972

A quantum mechanical model calculation on the reaction CI f IBr + Cl1+ Br or -+ I f ClBr sug@s that the .preferenci: of the less esorhermic prodlict in the energy range E = 3.5kcatlmole may be due to resonances between quantum stattx of the reactants with the hvourcd product configuration.

The reaction of an atom A with a diatomic rnolecule BC with masses ?ttA j: FTZ~f ?>‘zC+ ~2.~ may lead to two ~hern~~~ly different ~O~~g~ra~~oRs,AB + C or AC + B. Accordingly, the triatomic exchange reaction ?AB+C

/” II shortiy:

A+BC\

AC+B

I

fl)

1, III

is the simplest exampfe of a branched reaction. It will give insight into principal mechanisms playing a role in this class of reactions. Recently, reactions of type (1) have been studied exper~m~nt~~y. Amazingly, for CI f Brl [ I,21 and H f- CII [3] it is found that the !ess exothermic con+ figurations Br + Cl1 and Cl + HI are preferred, rather than 1 c BrCl er f + FlCl. This has been qualitatively explained by the a$.sumption that the reactants A f BC may form two stericallp different collision eompiexes, ABC and ACB, Then, if atom B is less electronegative than C, compfex A3C is more stable than ACB, and as a consequence it is predicted that configuration AB f C is preferred rather thar: AC + B f3, 31. This argument, hczvever, neglects dynamical aspects of a reaction and thus may contain only part, or none, of the truth. Indeed, an estimate of the accessible rotzitianal phase space of product-con~~urations shows that the most exotherrnic configuration should be * Extra& ‘from the dissertation OFJ.M. submitted Technische UniversWt Aliinchen.

to the

favoured [4]. In order to investigate the dynamitai effects more completely we present a model calcula. tion which shows that the result of a branched thermal reaction is very strongly determined by quantum mechanical resonances as well, Resonances depending on the potential surface, level spacings and tot31 energy of the whole system are known to occur between reactant and product states in unbranched chemical reactions [5 1. They are indicated by a high transition ~robab~ity between certain reactant and product states. In the case of branched chemical reactions, a quantum state of the reactants A + BC may come into resonance with a quantum staie of products AB c C OYAC f B and correspondingly will tend to populate prodact configurations AB + C or AC + B, respectively. In general one would expect a complex competition of ail mechanisms: effects of sterically different co&&ion complexes; distribution into the rotational phase space of products, resonances of reactant and product quantum states, possibly some more. Theoretically, al3 these phenomena should be handled in a three-dimensional cakulation of an atom-diatom co&ion. However, to our kno~v~~dge, realistic calc~Iations so far have been restricted to the nonreactive 161 and homonuclear reactive case [7], or to systetis i~vo~v~g at least two equal atoms [8]. The difr%zglty of a fully heteronuclear branched reaction is that the three possible configurations are adequateIy described-by three different asymptotic

Volume 1.5. number 1

15 July 1972

CHEMICAL PHYSICS LETTERS

hamiltonians. The couplings of their eigenfunctions (in the Born-Oppenheimer scheme) can then still be evaluated formally [9, lo], but we believe that at present, - for reasons of computer economics -, one still needs model assumptions to reduce the mathematical complexity of a realistic three-dimensional molecular colIision. Therefore, we consider chemical species with zero angular and rotational quantum numbers as being representative for ail particles colliding in a molecular beam. This is suggested becailse only tow impact parameters provide the close contact necessary for a reaction, and because often only about 10% of the total reactive energy is transferred to rotational degrees Of freedom [ 1 l] . Neglecting rotational quantum numbers the triatomic system is represented by the hamiltonian

with coordinates which asymptotically characterize either configuration and simplify computational work. We define arbitrarily distances measured from the asymptotically free atom as positive, so that [cf. (4)] for configurations + +m ; rBc = 0,

(I) 2 A+BC:r,, (II) 2 B+AC:r,* (III) 2 CtAB:

-+ -00 ; rBC z -rAB

rAB x 0 ; rBc + -00 .

(Note that in the collinear case one has rAC = rAB + rBC + const.) The corresponding asymptotic outgoing wavefunctions for configurations X (X = I, 11, III) are given by

Here TF is the transmission cient for an incoming wave Here X denotes configurati0ns I, II or III [cf. (l)] . For example in case X = I, ??zx = ~~~~~ is the reduced mass of molecule BC,

r,=r X

C

-r

-,o

B

=

EC - “BC

is the (radial) vibrational am litude of molecule Z (with equilibrium distance rBC), and

R,

esp (-ikfi RI) qk(r,)

(4) BC

= rA - (mg rB+m c rc)/(~~~B+~~fC)= RA,BC . (9

In case X = II, !II, particle indices have to be eschanged cyclically. Note that no angular variables occur in the kinetic term of(Z) and that its form is independent of the sign or an additive constant in the definition of variables rX or RX. Therefore it represents QJZ_J’ stericai orientation of the three atoms. We consider collisions in collinear arrangements A-BC and A-CB, allowing for products AB +C and AC +B, provided we use an effective potential V,,ff averaged over all possible orientations. Then we may rewrite (2) in the form +veif,

(6)

+ +-c=z

(resp. reflection-)

coeffi-

(‘I)

of reactants with vibrational quantum number TJ scattered into configuration X with vibrational quantum number <. The sum is to be taken over al! JV, o zn channeis of configuration X with internal eners ei$ and Internal (diatomic) wavefunction $(rX). The translational momenturn X-p follows from the energy-conservation relation (hk;)2/Z\iX

+ +

= E + L&-s

(IO)

for total energy E and exothermicity fiY of configuration X. The main difficulty in our model is the specification of the effective potential T/cff. It has to be thought of as a projection of the true potential on a two-dimensional subspace (spanned by rAB and rBC). 111any case, Veff must comprise the neglected rotational degrees of freedom plus all possible conformations of collision complexes. The potential veff used in our work is given in fig. 1. According to (7) it represents three distinct asymptotic configurational “valleys!‘. As we are interested only in the basic dynamical features of branched reactions, we have limited ourselves to a model potential with flat valleys and infinite walls, because they allow for mathematically practical eigenfunctions 137

V&I&

15, number 1

1.5fuly 1972

CkiE!4CAJ1. PHYSICSLETTERS

Table 1 Parameters use& in the cnicuIation of reaction (I 3). uz = atomic wei$lf of free atom; tix = esotherrnicify [ f 1; 0~ = reactive cross section f t J; Awx = esperimental vibrational zncgy of diatomic moteculc [ 151; ek = fitted vibrational energy of diatomic mokcute [cf. eq. (1211, amplitude param-

mr

35.5

0

Br + CII 1+ BrCI

79.9 126.9

7.7 10.2

Cl +

Kg, I:Two-dimensionzti nlodel potentia! Chfffor the branched reaction (1) with paTamtttcrs for rexrion CJ3). “&)

= (Z,h)“?

sin [(rx /g + I/2) @]

ill)

(with vibrational ~nplitude a) with internat eigenvahies

Within this rather crude model we now use parameters given in table 1 in order to si.rnuIate the halogen exchange reaction ~ Br + C11 Cl f Bri

(13) ’ f + B&i,

which is compared with the corresponding experiment of Loesch and Beck [ 11. For this reaction there are no activation barriers. Hence, for a first informative approximation we have joined the three potentid valIeys in the reaction region simpiy by a line2rly interpolated surface (cf.. fig. 1). The total wavefunction $ is then constructed numerically by the ficite-difference boumhy-value method of Diestler and McKay [ 121. This method can be extended to our three-valley-potential topology (instead of the usual one- or two-valley topoiagies) because in it no global {reaction-) coordinates are used which are essential for many other numerical m$hods [ 13 ] but are not available for branched reactions at the time. A typical w~vefunction is shown in fg. 2.. one of the advantages of the Diestler-Me&y

?

4.5 k 60% 7 t 60%

0.76 !.OS

0.66 3.17

1.23 1.31

method is that one needs to know only the asymptotic wavefunctions (81, (9) to construct $ in the interaction region sn(J obtain the transmission coefficients *p:l [cf. (S)] _ From these one evaluates the reaction pr~bab~ities

(14) Numerical drawbacks of the method have been noticed already

by Diestler

f 141. They are mainly

due to the

limited number of grid points in the interaction region (see fig. 2) and !he difficult choice of boundary values which can be optimized only by subtle intuition. In our caicuiations we include 2 3r 3 closed channels and analyze for 6 or 7 of them, The numeric31 error in microreversibility and conservation of flux in general was Iess than 10%. The total ~vavefunct~on $ in fig. 2 afready te!ls qualitatively that in the energy range E = 3.5 kcai/mote t%e less exothermic configuration Br+CII is preferred. {This is not due to the geometric “angles” between the valleys of our potent&t as is shown by test calculations with r?rA = i?zrj = in, = f?lH which give equal weight (within 2%) to either configuration AIB+C and AC-I-B, as expected. More quantitative results are shown in fig. 3. Here we plot the relative total cross section for con~~urations Br-tClf and f+BrC1, NI1

Also shown in fig. 3 are the corresponding results for initially higher excited channels q = 2, 3. It is obvious that for the experimentd situation 111 we find

Volume

15, number

1

indeed

a preference of configuration Br+ClI*. With. energy, however, this preference is weakened, since then resonances with I+SrCI states play a greater role. This holds also for higher vibrational reactant states. Of course we cannot expect detailed quantitative increasing

agreement between our model calculation and experi* Esperimental

energy values are not given in ref. [?I.

aJ

15 July 1972

CHEMICAL PHYSICS LETTERS

.

C!I

because our potential vcfi (see fig. 1) is only hi very rou& approsirnation to the real hypersurface. Instead, fig. 3 indicates - as a qualitative result -- ;1 principal mechanism infIuencing branched reactions, namely resonances between quantum states of reactants and one of the possible products. We feel that via this mechanism it might be possible to “direct” a branched reaction towards a desired product configuration, if one prepares the reactants initial!y In the “Proper” resonance state. Of course the other mechanisms mentioned earlier will also play a role. The mutual inieraction of all effects is to be considered in future work. As an experimental test we should like to suggest that one should measure the exit of the halogene exchange reaciion (13) (or others) over a wider energy range (translament,

tional and internal) in order to search for a situation where different configurations, e.g., I+BrCI, are the favoured

products.

I am very indebted to Professors G.L. Hofacker and E.W. Schlagand to Miss B. Hauffe and Mr. R. Couzelle Fig. 2. Absolute value of wavcfunction I+Afor reasrion (I 3); E = 3.5 kcal/mo!e. At this energy. product configuration RI+ Cl1 is obviously preferred. It is also seen that reactants with low vibrationa! energy (7 = I) are scattered into higher escited products (rwinly E = 3). This is in accordance with csperimcnt [ 1] .

for stimu!ating discussions on brz4led chemical reactions. A grant of the Studienstiftung des deutschen Volkes is gratefully acknowledged. The LeibnizRechenzentrum der Bayerischen schaften provided free computer machine.

Fig. 3. Relative total reaction cross sections for reaction (13) in the energy range l-7 configurations

reflects resonances

of reactants

with different

Akademie der Wissentime on its TK 440

kcsl/mole. Changing preference of proc’uct

product

quantum

states.

Vdltame 15, number i

CHEMICAL

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LETTERS

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6th