Faraduy Discuss. Chem. Soc., 1983, 75, 131-140 A Quantum-mechanical Internal-collision Model for State-selected Unimolecular Decomposition BY BRUCE K. HOLMER Theoretical Chemistry Institute, 1 I02 University Avenue, Madison, Wisconsin 53706, U S A . AND MARK S. CW~LD Theoretical Chemistry Department, University of Oxford, 1 South Parks Road, Oxford OX1 3QZ Receiued 29th December, 1982 A quan turn-mechanical theory for state-selected unimolecul ar decomposition is developed in terms of the cumulative effect of a succession of internal collisions. An application to the model decomposition of C 0 2 to CO('Z+) + O('D) is developed within the restrictions of a forced harmonic oscillator approximation for the internal scattering matrix. Preliminary results over a restricted energy range yield a distribution of lifetimes over two orders of magnitude, 0.640 ps, with evidence of a statistically determined behaviour on average but with wide fluctuations and with significant quantum effects on the branching ratios to different product channels.1. INTRODUCTION poses the question as to whether the statistical outcome arises from a statistical distribution of decomposing states or whether the decomposition of individual states is itself statistical or rnode-specifi~.~~~ Tn other words, are there close similarities or sub- stantial differences between the lifetimes and fragment state distributions arising from different quantum states in a narrow total energy region? Our approach to the problem is similar to that of Waite and Miller: although there are some differences in the formdation of the theory and the chosen applications are quite different.The central idea for both theories is the physically appealing one that decomposition arises from a sequence of internal collisions between the latent decomposition fragments. The quantity that expresses this idea is an internal S matrix which is introduced in such a way as to allow the imposition of different boundary conditions for open and closed channels. This results in fact in a unified description of bound, resonant and scattering states, so that one might hope in the future to extrapolate from knowledge of the bound states into the continuum. The present computational advantage of the scheme is that one can take over known and tested approximations for single-colIision processes and apply them in the multiple collision or resonant context. Waite and Miller adopt a semiclassical perturbation 10s approximation to describe leakage from the Hhon-Heiles potential.We adopt a forced harmonic oscillator model closely akin to the TTFlTS approxi- mation,1° and apply it to a system chosen roughly to mimic the decomposition of Results are presented for total energies between the CO(u = 2) and CO(u = 3) The success of existing statistical theories of unimoIecular decomposition co* to CO(%+) + O('D).132 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL dissociation thresholds, over which range there are 3 open and 20 accessible closed channels. 28 resonances with lifetimes between 0.6 and 60 ps were detected, but there may be as many as 15-20 others with much longer or much shorter lifetimes.In analysing their behaviour we ask first whether the magnitude of the lifetime is consistent with free-energy transfer between all channels. The answer is yes for the levels with widths in the middle of the range, but not for all levels because the range of lifetimes covers two orders of magnitude. Secondly do the resonant eigenstates show strong interchannel mixing? The answer is definitely yes, to the extent that it is difficult to give any meaningful assignment to most resonances. Finally, are the branching ratios between open channels statistically determined ? At first sight the answer appears to be no, but closer inspection suggests a pattern consistent with quantum-mechanically determined fluctuations about a statistical classical mean.The conclusions are that the decomposition dynamics for this model are statistically determined only in order of magnitude and to the extent that quanta1 interferences are ignored. Test calculations with the mean internal energy transfer reduced by an order of magnitude show clearly mode-specific behaviour. 2. THE INTERNAL COLLISION MODEL It is assumed that the dominant internal collisions occur within a radius R,, which lies inside the outermost classical turning point for all relevant closed channels (see fig. 1 later). This justifies the introduction of an internal S m a t r i ~ , ~ s, defined such that thejth component of the ith internal scattering state behaves as y/ji)(R)RzR, k j - *(Si je - i ~ j ( R ) - S.CJ .eiV,(R)) (1) where qj(R) is defined as the JWKB phase integral R qj(R) = kj(R)dR + n14 a/ a, being the inner turning point for channel j and [ glj[ therefore gives the probability of transition from channel i to channel j in a single collision. Multiple collisions are introduced by constructing linear combinations of the scattering solution + ( i ) with components ylf) and choosing the coefficients B a i consistent with the existence of closed channels, denoted collectively (yk(R)). The correct JWKB behaviour for such channels is Vk(R) Z Ckkk' Sin (&(R) where, with bk as the outer turning point, R @k(R) = lk kk(R)dR - ~ 1 4 . This is conveniently written for later manipulation as (5)B. K. HOLMER AND M. S . CHILD 133 where ctk is the complete JWKB phase integral Consistency between eqn (I), (4) and (5) now implies the following conditions on the coefficients B a i , Bake-i,"k = - 2 B,, s,, ei% (9) i The number of such equations is the number of closed channeIs.The remaining conditions are imposed by requiring that different solutions +(") are incoming in particular channels, a = I say. Thus with x replaed by l, the solution @ is defined such that BZi = &Ii (10) and the sum in eqn (9) is conveniently divided into open (i = l') and closed ( I = k') channel parts : (11) BIke-jak = - 2 BI,, skrkeiak - 2 61,. S,,eiak. El (I + C) = k' I' This is equivalent to the matrix equation (12) (1 3) where b, and Gl have components - b - B e-ia I k - [k k ) g l k = - SIkeiuk and C is a closed-channel matrix with elements ZVk = eiak'gk&ak.It follows from eqn (12) that bI = ml(i + E)-'. yif)(R) Rz" kF+(6,,. e b P ) - SI,.ebi(W} (15) (16) (17) This means on back-substitution from eqn (I), (13) and (15) in eqn (4) that where the true scattering-matrix elements S,, are given by sIl, = ,!Tile - 2 ei4 T',. (I + ~ ) - 1 ,JlkO eirk. hk' The first term corresponds to direct and the second to resonant scattering. This is the form appropriate to a scattering application of the theory. For analysis of the decay properties of the resonant states it is more convenient to replace the incoming boundary conditions [eqn (lo)] by the Seigert I1 boundary condition Bmj = 0 for all open channek i = 1. det (e2ir + see) = 0 (18) (19) The terms dli, in eqn (1 1) then vanish, leaving an eigenvalue equation where 9' is the closed-channel part of s.energies l1 The solutions of eqn (19) are complex E = E, - i r,l2 (20)134 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL specifying the positions, En, and widths, r,, of the resonant states. Furthermore the corresponding eigenvectors, conveniently labelled B,, with components Bnk, determine the outgoing amplitude in open-channel wl, because according to eqn (1) and (4), with Bli = 0, yln)(R) - kl - +TnleiW) (21) with The partial decay probability into channel I is therefore giving a partial decay width It may also be noted that if all channels are closed, sCc includes the entire internal S matrix, the unitarity of which ensures that the eigenenergies are real, because the eigenvalues of a unitary matrix are complex numbers with modulus unity and the complete phase integrals ctk are real at real energies.Thus the theory encompasses bound states as well as direct and resonant scattering. 3. UNITARISED FORCED HARMONIC OSCILLATOR APPROXIMATION As noted by Waite and Miller the strength of present formulation lies in our ability to approximate the internal scattering matrix s, because an exact numerical solution for 5: would be only slightly less time consuming than for the full S matrix given by eqn (17). Waite and Miller use a semiclassical perturbation 10s approxi- mation, but an attractive alternative to handle interactions between the stretching modes of a molecule is the impulse approximation9Pl0 whereby the motion of one oscillator (treated as harmonic) is linearly forced by motion of the other.The s matrix is given in this approximation by where the two parameters E and S are given by E = (2phc0)" F(t ')e'wt'dt '1 I I-, m t and S = (2phco)-' ,/ F(t)F(t')sinco(t - t')dt'dt --co -03 p and co being, respectively, the reduced mass and frequency of the oscillator and F(t) the time evolution of the forcing term. A practical problem in implementing the theory is that s must be unitary in order that the eigenvalues should be real for a fully closed-channel problem. s given by eqn (25) is, however, an infinite matrix, so some truncation is required. Furthermore in the way that the theory is applied the forcing function F(t) differs for each transition, because the time dependence of F(t) arises from the channel velocity which may differ widely from one channel to another.B.K. HOLMER AND M. S. CHILD 135 A convenient unitarisation procedure appropriate to the search for complex eigen- values by solution of eqn (19) is to recognise that the unitary of S for real E generalises to S(E)S(E*)~ = I (28) when E is complex. Thus if so@) is the approximation obtained by truncating s ( E ) , an appropriate form for the theory is Sapp = w(E*)] [T(E*)S(o)(E)T(E*)]3[T(E*)] (29) where T(E*) = (fS'"(E*f]")*. This is preferable to the R matrix procedure adopted by Waites and Miller,B in that it more accurately preserves the magnitudes of individual s matrix elements. 4. MORSE-DRIVEN HARMONIC-OSCILLATOR SYSTEM Any system chosen for implementation of eqn (25)-(27) in the context of uni- molecular dissociation must give rise to both open and closed channels and must involve coupling to a harmonic oscillator.A simple example is the ccupled harmonic1 Morse oscillator sys tern, with hamiI tonian where = r - y(x + xeq); y = m c l h + m,). This reduces under the substitutions x = (h*k-*pBC-*){, r = q/a to (33) where or equivalently where s = '7 - pt. The extraction of an approximate linearly forced harmonic oscillator from this Hamiltonian folIows established The first step is to ignore the cross-term pspc in eqn (35) and to soIve the resulting Hamiltonian equations for the time evolution of136 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL The result may be written u(t) = (1 -!')/{I -f3cos[(l - f ' ) + f i t ] } for O <f< 1 = (f- l)/{f%osh[(J'- l)%t] - 1) forf > 1 (37) where ij2 = 2dP2 (m -t l)/m, f = (E,/D) (38) Em being the Morse energy, measured from the minimum of the channel potential in question.The ranges 0 < f < 1 andf > 1 therefore correspond to closed and open channels, respectively. The next step is to linearise the potential in eqn (34) and to substitute for u(t) from eqn (37) to find F(t); thus V/hm z - l( $) = - (F(t) (39) where F(t) = - 2dD[u2(t) - ~ ( t ) ] . (40) This is the function that determines the parameters E and 6 given by eqn (26) and A cruder approximation (27). F(t) = - 2dPG(l - It]/z), T < It] = 0, z > Itl. was, however, adopted for the present exploratory purposes, with G =f+(1 +f3) z = arc cosf'/[fi(l -f)+] 0 < f < 1 = h[f+ + (f- l)+]/[fi(f- I)+] f'> 1 (42) and f + = ($2 +hW f" = E,"/(m + 110.(43) (44) Heref, and Emv refer to the vth channel The resulting expressions for E and 6, relevant to the calculation of &, are E == 8d2p2(G/T)' (1 - COST)' (45) and 6 = b2pZG2[72 3 + 12(cosz - sinT/z) - 3sinzI. (46) 5. NUMERICAL APPLICATION Parameter values for a test calculation were chosen roughly to model the uni- molecular dissociation of CO, to CO (%+) + O('D) : D, = 43 980 ern-', kw, = 1868 cm-I and h ~ o o - 2 1 7 0 ern-'. The first two imply a Morse exponent a = 3.461 and anharmonicity hcu,x, = 19.84 cm'l. Thc corresponding dimcnsion- less parameters appearing in section 4 are d = 20.3, p = 0.0772, d = 0.860 and m = 0.485.B. K. HOLMER AND M. S . CHILD 137 Detailed calculations were performed for 4340 < < 6510, with E measured from the dissociation threshold for CU (v = 0).Over this range there are three open channels, u = 0, 1,2, and 19 accessible closed channels, i.c. with potential minima < E. The closed-channel potentials and zero%-order eigenvaIues are shown in fig. 1 and typical internal collision probabilities lS,,,l* for u = 0, 1, 2 are illustrated in fig. 2. The latter vary very little over the energy range, because its span of 2000 cm-l is small compared with the maximum kinetic energy of 35 000-40 000 crn-l. Mean 6500 6000 5 500 5000 -4500 -0.25 0 0.25 0.5 0.75 1.0 1.25 20 40 R - RJA 3 I E -+ k Fig. 1. ChanneI potentials and zeroth-order eigenvalues between the CO ( u = 2) and (u = 3) dissociation thresholds. Calculated resonance energies €, and widths r,, are indicated to the right of the diagram.The symbol * indicates levels for which information is given in table 1 and fig. 3. energy transfers of roughly 10 CO vibrational quanta are therefore typical for tran- sitions to and from the open channels. This corresponds to E E 10 in eqn (25) and (26). The maximum kinetic energy in the highest channels is, however, reduced by an order of magnitude; eqn (42) and (45) show that this reduces E to roughly unity, with a corresponding reduction in the mean energy transfer to one CO vibrational quantum. Solutions of eqn (19) for the resonance positions and widths were obtained by scanning the real energy rangein steps of 2 cm-' followed by a Newton-Raphson search in the neighbourhood of the apparent resonances, a procedure that yielded the 28 resonances with energies En and widths I?, indicated to the right of fig. 1.The range of widths is 2.6-55.6 cm-' with a mean width r z 9 cm-'. It is evident, however, that the number of detected resonances is considerably less than the number of zeroth- order states. Further resonances with either substantially larger or substantially smaller widths may therefore be expected. Detailed resuIts for six channels are given for illustrative purposes in table 1 and fig. 3. Table 1 shows the resonance energy En, with r,, and branching ratios Pno,138 0.1 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL - I.= 2 I I I I I , t k Fig. 2. Internal single-collision transition probabilities I Slkl * for scattering into the open channels 1 = 0, 1 , 2 at E = 5577.22 cm-'.The pattern varies very little with energy. 1 and k refer to the CO vibrational label u. Pnl and Pn2 between the three open channels; fig. 3 gives the square moduli IBnkI2 of the Seigert state eigenvector components. The first point of interest is that the lifetimes 7, x hjr,, (47) consistent with the linewidths shown, span the physically plausible range 0.1 < zJps < 10. This gives some confidence in the practical relevance of the model. Secondly the magnitude of I' given by the crude formula r x ( ~ Q ~ E ) P (48) where 6 is the classical Morse oscillator frequency which governs the rate of internal collisions, may be used to estimate the probability P of predissociation per collision. With hG taken as 600 cm-l, which is the mean zorse energy separation relevant to the states shown in fig.1 and r taken as the mean r = 9 cm-', this results in P x 0.09, which may be compared with the statistical probability of 0.13 for scattering from one of a total of 23 channels into one of three which are open. Thus the mean behaviour is approximately statistical, although there is a substantial spread from P x 0.006 to P x 0.6 implied by the range of observed level widths. TabIe I. Decomposition characteristics of selected resonances a 4378.88 22.36 0.41 3 0.300 0.288 b 5218.89 55.68 0.463 0.016 0.536 C 5577.22 2.54 0.286 0.099 0.616 d 5828.94 11.72 0.837 0.059 0.104 e 6249.15 4.92 0.091 0.275 0.634 f 6503.67 0.57 0.062 0.593 0.345B. K. HOLMER AND M. S. CHILD 139 0.2 0.2 0.2 N - U c Qi 0.2 Further evidence of broady statistical behaviour is shown by a roughly uniform distribution over the closed channels for a11 calculated resonances, as illustrated in fig.3. A few cases [e.g. fig. 3(e)] show a predominance of one particular channel, but a more uniform pattern is the norm when all 28 calculated resonances are included. By contrast, the pattern of branching ratios given in table 1 appears to deviate - - ( f 1 I 1 1 L.1 I 1 . 1 1 - ( e l 1 1 . t , l . . . ! . . I - ( d 1 1 . . I I . . ! I 1 1 . 1 . IC) - I , 1 t l I 1 I , 0.2 1 k Fig. 3. Closed-channeI eigenvector components lBnk12 for the selected resonances (a)-(f) detailed in table 1 . markedly from the statisticaI result P,, = P,, = Pn2 = 1/3. This may, however, be a quantum effect. If ITnl12 in eqn (23) is replaced by lL12 = 2 p n k I 2 l L 1 * (49) k which involves ignoring cross-terms in BnL3*nk.&kS1*nk‘, this has a classical interpre- tation as the decomposition probability ISkr\* to channel I, weighted by the strength lBnkI2 of channel k in the eigenvector.Branching ratios calculated by means of eqn (49) are much closer to the statistical values. For the resonance at En = 5577.2 cm” for example, the values Pno = 0.29, P,, = 0.10, Pn2 = 0.62 given in table 1 are re- placed by 0.37, 0.35 and 0.28. The overall conclusion is that the resonances investigated have physicaIly realistic lifetimes and decompose at the mean in a broadly statistical manner. The range of140 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL lifetimes spans two orders of magnitude however, and non-statistical quantum corrections to the open-channel branching ratios are apparent.Finally it should be noted that owing to resolution difficulties the calculation revcaled only 28 out of possibly 47 resonances and the behaviour of the missing resonances may well be difl'erent. A few test calculations were also performed with much smaller energy transfers, by reducing the coupling parameter p from 0.0772 to 0.007 72. The resulting resonances, being only slightly shifted from the zeroth-order eigenvalues, were then much more easily detected. Correspondingly the resonance eigenvectors showed one or two dominant components. Their widths were roughly comparable with the stronger- coupling model for states in the lowest channel, D = 3, (0.4 < . T,/crn-l .: 23.6) but fell off sharply for more buried resonances. Finally the decomposition ratios typically gave a weight of 0.9 to the highest open channel, u I- 2. This shows that an order-of- magnitude change in the mean energy transfer per internal collision causes a complete change from statistical to highly selective decomposition. J. P. Robinson and K. A. Holbrook, Unimolecular Reactions (Wiley-Interscience, New York, 1972). W. Forst, Theory of Unimoleculur Reurtions (Academic Press, New York, 1973). K. A. Reddy and M. V . Berry, Chem. Phys. Lett., 1979, 66,223. R. Naaman, D. M. Lubman and R. N. Zare, J. Chem. Phys., 1979, 71, 4192. R. B. Hall and A. Kaldor, J . Chem, Phys., 1979, 70,4027, B, A. Waite and W. H. Miller, J , Chern. Phys., 1980, 73, 3713; 1981, 74, 3910, ' B. D. Cannon and F. F. Crirn, J + Chem. Phys., 1981,72, 1752. ' B, A, Waite and W. H. Miller, J. Chem. Phys., 1982, 76, 2412. M. S. Child, Molecular Collision Theory (Academic Press, London, 1974). lo F. E. Heidrich, K. R. Wilson and D. Rapp, J. Chem. Phys., 1971,54, 3885. I i A. J . F . Siegert, Phys. Rev., 1939, 56, 750.