Foraday Discuss. Chem. SOC., 1982, 73, 3 1 1-324 Model Studies of Resonances and Unimolecular Decay of Triatomic Van der Waals Molecules BY BRUCE K. HOLMER AND PHILLIP R. CERTAIN Theoretical Chemistry Institute, Department of Chemistry, University of Wisconsin-Madison, 1 101 University Avenue, Madison, Wisconsin 53706, U.S.A. Received 14th December, 198 1 The wave equation appropriate to describing the resonance states of atom-diatomic Van der Waals molecules in body-fixed coordinates is presented. The Van der Waals bond is described by a square-well attraction whose depth can depend on the internal state of the diatomic. With this simple model, the analytic structure of the scattering matrix is expressed by an exact partial fraction expansion in terms of simple poles due to the bound, antibound and resonance states of the complex.The expansion leads to partial fraction expansions for the collisional time delay and the spectroscopic transition probabilities, and the results are compared to the Breit-Wigner parametrization. The square well is idealized further to a delta-function attraction, and the scattering matrix for coupled channels is obtained. Poles corresponding to shape and Feshbach resonances are identified, and an expression for the decay probabilities of a resonance into the open channels is obtained for the model. 1. INTRODUCTION Experimental studies of unimolecular decay processes are undergoing rapid development at the present time because of the selective excitation capabilities of lasers.' The energized state of the molecule thus produced can be viewed as a resonance state; i.e.a state in which the molecule has sufficient energy to dissociate but the dissociation process is slow compared to some relevant time scale such as the time required for the excitation energy to become randomized over the available degrees of freedom of the molecule. It is generally assumed that this latter process is very rapid, and that statistical theories can be applied to predict the rate constants and branching ratios for decay into the various open channels3 Triatomic Van der Waals molecules of the type Ar-H,, He-1, and He-LiH are particularly attractive candidates for theoretical study because of the simplicity of their internal structure and the possibility of detailed information about their energy surfaces.If the diatomic molecule is excited, either vibrationally or rotationally, the excitation energy is usually sufficient to break the weak atom-diatomic Van der Waals bond, thus giving very simple examples of unimolecular decay through resonance states. The quantities of interest are the energy of the resonance state, its lifetime and branching ratios to various decay products (channels). Molecular-beam spectroscopy is the richest source of experimental information about the unimolecular decay of Van der Waals molecule^.^ Crudely speaking, the positions of spectral lines give the resonance state energies, linewidths give lifetimes, and fluorescence of the decay products (either spontaneous or laser-induced) gives information about branching ratios.Molecular-beam scattering experiments can also be used to study unimolecular decay.5 Here the existence of the resonance state is revealed by characteristic features in the scattering cross-section as the relative energy of the colliding molecules312 RESONANCES AND UNIMOLECULAR DECAY is varied i n the vicinity of the resonance energy. The observcd cross-sections often can be fitted by the Breit-Wigner formula,6 which contains the resonance energy and lifetime as parametcrs. Th~oretically,~ the resonance states of triatomic Van der Wads molecules can be associated with poles of the atom-diatom scattering matrix which occur for complex values of the total energy W = E -- ih/22. (1) Here E is the energy of the resonance state and z is its lifetime. In addition, the branching ratios for decay into the open channels corresponding to vibration- rotation states of the diatomic can be determined by examining the residues of the scattering matrix elements at the pole positions and by requiring unitarity.The advantage of studying the complex poles of the scattering matrix, rather than simply computing it for physical valucs of the energy, is that if its analytic properties are simple enough, or if the resonance energies occur close enough to the real axis, all of its properties for real energies can be deduced from a knowledge of the pole positions and residues.* There are several numerical techniques for determining the poles of the scattering r n a t r i ~ . ~ ’ ~ The scattering matrix, however, can exhibit singularities in addition to resonance poles, such as poles and branch cuts arising from the form of the potential- energy surface and branch cuts due to inultiple thresholds.If these features occur close enough to the real energy axis, or close to the resonance poles, they can have observable effects on the scattering matrix evaluated for real energies and can compli- cate the interpretation of experimental data. In addition, since the scattering matrix is not directly observed in experiments, but rathcr is only one of several contributions to the observables, the shape of the resonance in the observable may be different from the shape of the resonance in the scattering matrix itself. Before we can apply numerical methods to the unimolccular problem with con- fidence that the complex energies so obtained have a clear physical significance, it is necessary to have some knowledge of the behaviour of the scattering matrix in the complex energy plane and how its singularities are related to observable quantities, A great deal is known about this in but we seek concrete examples that can be studied in detail.The purpose is partly tutorial, for the authors and, hopc- fully, for the readers. In the present paper we assume an idealized form for the atom- diatomic potential function ; namely, a hard-sphere repulsion and either a square-well or delta-function attraction, where the strength of the attraction can depend on the internal state of the diatomic. Although these potentials are not expected to account for experimental data, many of the features of the scattering matrix are expected to carry over into problems with realistic potentials.I n our work, wc generate two ‘‘ experimental ” quantities, the scattering time delay function and the spectroscopic transition moment, and study how their behaviour for real energies is related to the analytic structure of the scattering matrix, Before specializing the discussion to simplified potcntials, we outline in the next section the general formulation of the triatomic Van der Waals molecule wave equa- tion and define the scattering matrix. In the following section, we introduce the spherically symmetric square-well potential and study its propertics for states of vanishing total angular momentum. This is the simplest possible case to consider, and the singularities of the scattering matrix for this casc will persist in modified form in problems with non-spherical potentials or non-zero angular momenturn.In the last section, we allow the square well to become a delta function and study some aspects of thc problem of determining branching ratios for decay prodiicts.B . K . HOLMER AND P. R . CERTAIN 313 2. THE WAVE EQUATION Previous treatments of Van der Waals molecules have developed the wave equation within a space-fixed axis system.".'2 In our work, however, which closely follows that of Pack,13 we shalI adopt a body-fixed system, since this allows us greater flexibility in dealing with a range of anisotropic potentials. The internal coordinates are r, the diatomic bond length; R, the Van der WaaIs bond length measured from the centre of mass of the diatomic; and 0, the angle between the diatomic bond and the Van der WaaIs bond.With these coordinates, stationary-state eigenfunctions of total angular momentum (quantum numbers J , M ) have the form lyJ2f0 = i x:(R,r,e)DhM (.PY) (2) 51= -J where DhaTf is a representation coefficient of the rotation from space-fixed to body- fixed coordinates. The Euler angles, c1, /?, y, are determined by specifying that the body-fixed z-axis coincides with the Van der Waals bond, and that the diatomic lies in the body-fixed xz-plane. The 0 is an index which is used Iater to complete the labelling of the wavefunctjon. The internal wavefunctions 22 satisfy a set of coupled wave equations where is the angular momentum squared pf the diatomic expressed in body-fixed coordinates and is the quantum number ofj, and J,.In addition R*(J,Q) = [J(J+ 1) - Q(ln 5 I)]' (6) Finally, V(R,r,B) is the intermolecular potentiaI between the atom and the diatomic. The diatomic Hamiltonian possesses bound-state eigensolutions { E ~ , yn(r,O)) where n = {zi,l,Q) is a collective index for the vibrational quantum number u and the rota- tional quantum numbers, I and st, which will be explicitly indicated as the need arises. For fixed total energy E, it is convenient to define channel momenta for the open channels ( E > E J , and decay constants K , = -ik, for the closed channels (E < &,J.314 RESONANCES A N D UNIMOLECULAR DECAY The scattering matrix is defined by considering a process in which an incoming wave in the nth channel is scattered into all other open ~hanne1s.l~ Thus, we seek solutions of the asymptotic (R + co) form where Gi?(R) = k,4((b,,.exp(-i[k,,R - ( J + l’)n/23} -Si,,(E)exp( + i[k,,R - ( J + l’)n/2]}). (1 1) Here SnJ.n is an element of the scattering matrix and fi = (u,I,-Q}. Two important properties of the scattering matrix are unitarity SJ’(E)S’(E) = 1 (12) and symmetry in the channel indices s;l;n = s;;.If the asymptotic form of xg is assumed to be valid for all R, the functions Gin must satisfy the set of coupled equations. ) G?iw(R) J(J + 1) + /’(I’ + 1) - 2R2 ($ + k:T - R2 n-(J,n’)n+(J,n’ - 1) GEYpgy - 1(R) A-f-(J,sz’>n-(J,a’ + ’) Gi?r,n, [ R2 R2 + subject to the boundary conditions (1 1). intermolecular potential evaluated in the basis of internal wavefunctions qn(r,O).not appropriate for treating resonance states. it is adequate and so we turn now to the square-well potential. Here (n’l Vln”) is the matrix element of the For strongly asymmetric potentials the asymptotic form of the wavefunction is For the models we consider, however, 3 . THE SQUARE-WELL POTENTIAL In this section we shall study a square-well potential of the form V(R,r,e) = - - h2 co2(r,6) 0 < R < a 2P = o R > a. (1 5) Representatives from this class of potentials have been studied for a variety of purpose^.'^-'^ We shall make the further restriction that co2(r,0) is diagonal in the basis vn(r,e). This leads, of course, to a dramatic simplification in the coupled eqn (14) since it removes the coupling between wavefunctions with differing values of v and 1.At this stage, resonance states are characterized by quantum numbers J, u and I and the decay channels by the orientation quantum number R. We take one step further, however, and consider states with vanishing total angular momentum, J = R = 0. Even at this vastly simplified level, where only potential scattering is possible, the scattering matrix has a richness which merits detailed study.B . K . HOLMER AND P. R. CERTAIN 315 With all of the simplifications just described, the wave equation becomes R2 U(R)) G,(k,R) = 0 + where U(R) = -wz for 0 < R < a and vanishes otherwise, and we have dropped all superfluous indices. From this equation, we can obtain explicit expressions for the scattering matrix, the time delay and the spectroscopic transition probability.In potential scattering, the scattering matrix, S,(k), is determined by the phase shift, q,(k), zliz.'* where q,(k) is determined from the asymptotic form of the wavefunction Sdk) = exp[2iM)l (17) The collisional time delay T~ is defined by lo although in our numerical work we find it more useful to use the virial-theorem expression zl(k) = $lorn G:(k,R) + R- '("1 d R . dR Resonance states are characterized by large amplitudes in the region of the potential well and hence relative maxima in T ! . Finally, the spectroscopic transition probability for a bound-free, P-branch transi- tion is proportional to (21) r Z k k > = li," G,+l(lc,R)~(R)G,(k,R)dR where M(R) is the electric-dipole transition-moment function and G , + ,(lc,R) is a bound-state solution of eqn (16) with E = - h 2 d / 2 p .The transition probability to resonance states is expected to be large because of the large amplitude of G,(k,R). Previous studies l8 have focused on this amplitude, but we have chosen to study 2, because of the ambiguity associated with defining a unique amplitude and because 2, is closer to experimentally determined quantities. We choose the transition-moment fun'ction to have the form M(R) = M,( 1 - R/u)exp( -&/a) (22) for R < a and to vanish * for R > a. considerations of each quantity. With these definitions, we now turn to detailed A . THE SCATTERING MATRIX The scattering solutions of eqn (16) are (23) Gl(k, R) = A(k'R/k'a)j,(k'R) O<Rc<a = A[XkRj,(kR) + YkRn,(kR)] R > a * For a dipole transition operator M(R) which does not vanish for R > a, additional singularities will occur in the integrand of eqn (21).316 RESONANCES AND UNIMOLECULAR DECAY where krZ = kZ + wz,j,(x) and q ( x ) are spherical Bessel functions,’’ and X = -kajl,(k’a)n, +,(ka) -+ k’aj, + ,(k’a)n,(ka) Y = -kraj, + ,(k’a)j,(ka) t kujr(k’a)j, + ,(ka).Matching the asymptotic form of eqn (23) with eqn (I 8) gives X - i Y X + iY‘ Sl(k) = - Thus it is clear that, for real k, S,*(k)S,(k) = 1 . For complex k the poles of S,(k) correspond to the vanishing of the denominator ; hence where hll)(x) = j,(x) + in,(x) is the spherical Hankel f~ncti0n.l~ We define the com- plex energy which satisfies eqn (26) as k;hj”(k,a)jl + ,(kia) - k, j l ( k ~ a ) h ~ ~ l ( k n a ) = 0 (26) (27) W, = - - ki = E, - irJ2.For potentiah which vanish for R > a and are not too singular at R = 0, the only types of singularities which are possible in S,(k) are poles, and these may be classified as bound-state, antibound-state or resonance The bound-state poles occur for positive pure imaginary momenta (k, = kn), and correspond to wavefunctions which are exponentially decreasing for R --+ co. The antibound states (or virtual states) occur for negative pure imaginary momenta (k, = - i ~ i ) and correspond to wavefunctions which are purely exponentially increasing for R -+ co. (For potentials which do not vanish for R > a, it becomes probIematica1 to define “ purely ex- ponentially increasing ”.) Finally, the resonance-state poles occur in the fourth quadrant of the complex k-plane (k, = an - ifin), and correspond to purely outgoing wavefunctions. For every resonance pole, there is also a pole at -k* = -a, - ip,.For potentials which vanish for R > a, the S,(k) can be expressed exactly by an infinite product of pole terms h2 2111 k + k, S,(k) = exp(-2ika) II ~ n k - k,’ In this expression, all poles, not just the resonance poles, are included. We shall see that this representation for S, gives rise to partial fraction representations for zl and Z l in terms of the poles of S,. Numerical results for the poles of S,(k) are shown in tables 1 and 2 for two dif- TABLE 1 .-POLES OF S,(k) [SEE EQN (17) AND (18)] FOR THE SQUARE-WELL POTENTIAL DESCRIBED IN SECTION 3 Here ma = 1312 and I = 1.Only the two lowest resonance poles are shown. ~~ ~ pole Re(ka) Im(ka) bound states 1 0.0 5.2282 2 0.0 1.2147 antibound states 3 0.0 -3.5281 4 0.0 -0.6542 resonance states 5 6.541 5 - 1.2440 6 10.5482 - 1.4862B . K . HOLMER AND P . R. CERTAIN 317 TABLE 2.-POLES OF Si(k) FOR THE SQUARE-WELL POTENTIAL (SEE TABLE 1) Here ma = 36/5 and I = 2. Only the three lowest resonances are shown; there are no antibound states in this case. pole Re(ka) Im(ka) bound state resonance states 1 0.0 5.1706 2 0.6062 - 2.1761 3 2.0821 - 0.38 14 4 7.8143 - I .2660 ferent choices of well depths and angular momenta. These results are used below 10 represent the energy dependence of the time delay and transition probability. The maximum in the centrifugal barrier occurs at R = a, corresponding to (ka)' = 1(2 + l), so only resonance 3 in table 2 is a sharp tunnelling resonance.The remaining reson- ances in tables 1 and 2 occur above the barrier, with the exception of resonance 2 in table 2 which actually has an energy below threshold, Re(ka)2 < 0. B . THE T I M E D E L A Y Differentiation of eqn (28) for S,(k) yields 2 i k,a ka -t 7 (ka)[(ka)2 - (k,aY]' h - 3- If a resonance pole is close to the real axis, its contribution to the sum in eqn (29) reduces to the Breit-Wigner formula and all the other poIes contribute to a slowly varying background. Eqn (29) is valid, however, regardless of the position of the poles, and is exact if all poles are included. The low-energy behaviour of z,(k) for a cut-off potential such as ours is 2o q ( k ) - k"-' , k+O.(31) Since this behaviour is not explicit in eqn (29), the sum is slowly convergent for small k. We can remove this difficulty, however, by recognizing that since kz,(k) has only simple poles, then k-2'+'zl(k) also does and may be expressed as a partial fraction series of pole terms. This leads to the alternate expansion where the sum still contains all poles of S,(k). It is interesting that eqn (29) and (32) are very similar to results that are obtained in studies of the completeness of the set of Siegert wavefunctions." Numerical results for 7 l are shown in fig. 1 and 2 for the two potentials whose scattering matrix poles are given in tables 1 and 2. It is seen that eqn (32) is an excelIent representation of the exact behaviour over the energy range of the graphs.For larger values of (ka) the truncated pole representation fails and, in general, omission of low-energy poles in eqn (32) leads to a poor fit, since each term in the sum31 8 RESONANCES AND UNIMOLECULAR DECAY varies as kZ1-j for large k. is included in fig. l(a). For comparison the Breit-Wigner time delay, eqn (30), Since it does not include the background terms due to the 0 n c +a ._ $ -1.c W s v -2.0 - - _ _ _ _ _ _ _ _ - - - - - _ -- .- A. _ _ - - - _ _ - - - I , 1 1 1 0 1 , I 0 50 100 I L t : I d I I 1 I I , I I I 0 50 100 (ka)‘ FIG. 1.-Scattering time delay t&) [see eqn (1911 for the square-well potential described in section 3. Here wa = 13/2 and I = 1. (a) (-) Exact; (- - -) pole representation, eqn (32); (- - - -) 3reit-Wigner representation.All poles in table 1 are included. (b) Same as (a) except hara-sphere term, - 2/ka, added to Breit-Wigner representation. bound and antibound states, its threshold behaviour is incorrect. This can be approximately corrected by inclusion of hard-sphere time delay, -22/ka, as shown in fig. l(b) and 2(b). Fig. 2(a) shows the individual contributions of the poles to zl. FIG. 2.-Scattering time delay 71 [see eqn (19)] for the square-well potential. I = 2. to table 2. Here o a = 36/5 and (a) IndividuaI contributions of pole terms in eqn (32); the numbering of curves corresponds (6) 7&): (---) Exact; (- - -) pole representation, eqn (32); (- - - -) Breit-Wigner represen tation with hard-sphere correction.B . K . HOLMER A N D P . R . CERTAIN 319 Each contribution has the correct behaviour for small k, but diverges for large k.In fig. 2 it is seen that the incorrect large k behaviour of the contribution from the sharp resonance 3 is cancelled by the contribution from the " unphysical " resonance 2. c. SPECTROSCOPIC TRANSITION PROBABILITY The normalized bound-state wavefunction GI + ,(tc,R) is 3 G,,,(lc,R) = [:] Ni'+3hiyl(i~a) [t]jl+l(kwR), 0 < R < a where Eqn (21), (23) and (33) combine to give the following expression for the transition moment fi2Zl(rc,k) 4 N 2 -- pa2M,2 nka X 2 + Y 2 X i i"3hSyl(i~a) - 0 x2( 1 - x)exp(-x)j,+l(k,ax)jl(k'ax)dx)2. (36) Partial fraction expansions similar to eqn (29) and (32) can also be obtained for Zl(rc,k). The poles in Z,(rc,k) are due to the normalization factor, l/(X2 + Y2), and therefore the residues of 2, can be easily obtained.* If we let A,,(K> be the residue of [h2Zz,(~,k)]/pa2kf~ at the pole k = k,, of the scattering matrix, then h2-Wc,k) - A,,(rc)(ka)2' + pa2M,2 - (k,a)2'[(ka)2 - ( k , , ~ ) ~ ] (37) where the bound, antibound and resonant state poles are included, and the low- energy behaviour Z,(K,k) - k 2 ' + l , k -+ O (3 8) is explicitly included in the sum.The corresponding Breit-Wigner formula is l 8 where Z:(K,k) is a slowly varying function of k and the sum includes only resonance poles. Often the parameters in the Breit-Wigner representation are used to fit data; however, we shall use the calculated poles and residues without adjustment. The terms in eqn (37) and (39) are complex, although 2, is real, so in calculations only the real parts of the terms are included.Numerical results for Z,(lc,k) are shown in fig. 3 and 4 for A1 = - 1 transitions to the states in tables 1 and 2 from the lowest bound states of each potential. The * See previous footnote.320 RESONANCES AND UNIMOLECULAR DECAY 0 50 1 FIG. 3 .--Spectroscopic transition probability Z&c,k) [see eqn (2111 for the square-well potential described in section 3, Here wa = 13/2 and I = 1 for a A1 = - 1 transition from the ground state. (a) Individual contributions to eqn (37); the numbering corresponds to table 1. (b) Z I ( ~ , k ) : (-) exact; (- - -) Breit-Wigner representation, eqn (39); the poIe representation, eqn (37), is indistinguishable from the exact result. 0 10 20 30 (kaS FIG.4.-Spectroscopic transition probability Z&,k) [see eqn (21)] for the square-well potential described in section 3. Here ma = 36,"s and 1 = 2 for a A1 = -1 transition from the ground state. (-) Exact; (- - -) Breit-Wigner representation using only pole 3 in table 2; the pole remesent- ation, eqn (37), is indistinguishable from the exact result. individual pole contributions to eqn (37) are shown in fig. 3(a), where no hint of structure in the spectrum is seen. These individual terms superimpose, however, to produce the spectrum in fig. 3(6), where the pole representation is indistinguishabIe from the exact result. The sharp feature is due to the antibound state 4 in table 1 , which is quickIy damped by states 2 and 3. The Breit-Wigner representation, whichB.K . HOLMER AND P. R. CERTAIN 321 does not include these contributions, shows only the broad resonance. In fig. 4 the only visible feature is from the sharp tunnelling resonance, which is well described by the Breit-Wigner representation. In summary, we see that the pole representation of the time delay and the transition probability is very accurate so long as all contributing poles are included. Further- more, interference between bound and antibound states can produce sharp features not only in the scattering behaviour but also in the spectrum. It would be desirable to develop pole expansions which incorporate both the low- and high-energy k- dependence, thereby reducing interference effects between terms. 4. THE DELTA-FUNCTION POTENTIAL In order to illustrate several aspects of the coupled-channel problem, namely, the persistence of singularities from the uncoupled probIern and the calculation of branch- ing ratios into the various decay channels, we consider the delta-function potential 22*23 h2 Y(R,r,O) = -- i,(r,Q)d(R - a) 2P where the strength of the attraction, i(r,O), may depend on the internal state of the diatomic.In order to preserve simplicity, we assume that 3.(r,O) is independent of 8 and couples only a small number of vibrational states. In addition, we consider the rotationless case J = I = R = 0. Then the coupled wave eqn (14) reduce to subject to the boundary conditions where At-,, is a matrix element of A(r,U) in the basis of internal states. I n addition to the boundary conditions, the singularity in the potential introduces a cusp condition a t R = a where the first two terms express the difference in the derivative of the wavefunction at R = a.Before doing this, however, it is useful to consider the uncoupled problem to identify the singularities in the simpler case. The scattering eqn (41) and (42) may be solved for S = (S,,,). A . POTENTIAL SCATTERING Here we consider the wave equation which has the solution G(R) = Asin(kR) R < a = exp(-ikR) - S(k)exp(ikR) R > a. (45)322 RESONANCES AND UNIMOLECULAR DECAY Requiring G(R) to be continuous at R = a yields an expression for the constant A , while the cusp condition gives the expression for S(k) [k - 3, sin (ka) exp(-ika)] S(k) = [k - 2 sin (ka) exp(ika)] ' It is evident that S(k) is unitary for real k, but for given by solutions of the equation or The solutions of this equation are easily determined.k = 3, sin (ka) exp(ika) 2ika = 3,a[exp(2ika) - 11. complex k it can have poles (47) For l a > 1, there is a single bound-state pole for k lying on the positive imaginary axis. When 3,a = 1, this pole moves to the negative imaginary axis and becomes an antibound state for 0 < Aa < 1 . Thus, in contrast to the square-well potential, the bound and antibound states do not co-exist. An infinite number of resonance poles occur in the lower half of the k- plane in the vicinity of Re(ka) = mn, rn = 1,2,3 . . . B. THE COUPLED-CHANNEL CASE In order to obtain an explicit expression for the scattering matrix S, it is convenient to introduce matrix notation for eqn (41); thus, we define G = (G,"*(R)}, A = {A,#,,,}, k = {B,,~k,~} (48) whence the wave equations become - d2 + k2 + AB(R - a) ( dR2 (49) Further it is convenient to partition these equations according to the open (kf.> 0) and closed (k:. = -& < 0) channels. Thus, we have two sets of coupled equations, in an obvious notation d2 ki + X,,B(R - a) G,(R) + A,,B(R - a)Go(R) = 0 where the boundary conditions are Go(0) = G,(O) = 0 Go(R) = k;3[Ui-)(R) - U$+)(R)S] R > a Gc(R) = Ub-'(R)A R > a. (51) Here Ui*)(R) and Uh*)(R) are diagonal matrices with elements exp(&ik,R) and exp( *Q), respectively. A is a matrix of constant coefficients. The cusp condition for the closed-channel equations can now be written -(Dc - Acc)Gc(a) + ACOGO(4 = 0 (52) where D, is a diagonal matrix with elements IC, exp(lc,a)/sinh (rc,a). used to simplify the cusp condition for the open channeis to give This result may be (Do - Ao)tJi+)(a)k;3S - (Do - Ao)tUi-)k;3 = 0 (53)B .K . HOLMER AND P. R. CERTAIN 323 where Do is a diagonal matrix with elements k,exp(-iik,a)/sin (k,a), and A, = 1, + U D C - L)-'lL. (54) Provided (Do - &) has an inverse, the expression for S is Despite its apparent complexity, if the coupling matrix A is diagonal, S is readily seen to reduce to a diagonal matrix with elements given by the potential scattering result, eqn (46). Thus for small off-diagonal coupling, all of the scattering resonances obtained in the uncoupled case will appear near their original positions. In addition, the matrix (D, - AcJ-l is singular near the bound-state eigenvahes of the uncoupled case, and this gives rises to the expected Feshbach or compound-state resonances in S .In addition, there are branch point singularities at each threshold (xL, I- 0). The poles of S are given by the generalized cigenvalue equation W = E - ir/2 (56) where C is the eigenvector. The physical significance of C is revealed if the derivation of S is reviewed and it is realized that when eqn (56) is satisfied, there is a solution of the coupled equations of the form (Go is now a column vector) G,(R) = k,aUL+'(R)C (57) that is, a solution in which there is a purely outgoing wave in each open channel. This is the Siegert definition of the resonance wavefunctjon,24 and thus an intuitive expression for the probability for decay into channel u i s This allows a partiaI width for each channel to be defined as 2 4 3 2 5 so that This interpretation is superficial, however, in the same sense as is our choice of the final wavefunction in the study of the transition probabiIity in section 3 .That is, in an experiment it is not the resonance wavefunction which describes the decay, but rather a wave packet constructed by superposition of stationary states defined in the real energy axis in the vicinity of the resonance pole. A deeper analysis is required for an unambiguous interpretation of the decay probabilities. The authors thank the National Science Foundation (U.S.) for support of this They are also grateful to Prof. K. W. McVoy for an enlightening discussion work.and for many references. ' B. D. Cannon and F. F. Crim, J. Chern. Phys., 1981, 75, 1752. ' R. D. Levine, Quanfum Mechanics of Mclleculur &re Processes (Oxford University Press, Oxford, 19691, pp. 268-292. D. W. Noid, M. L. Koszykowski and R. A. Marcus, Annu. Rec. Phys. Chem., 1981, 32, 267. D. H. Levy, Annu. Rev. Phys. Chern., 1980, 31, 197.324 RESONANCES AND UNIMOLECULAR DECAY In fact, we are not aware that such experiments involving Van der Waals molecules have been reported. For a theoretical discussion, see J. A. Beswick, G. Delgado-Barrier and J. Jortner, J. Chem. Phys., 1979, 70,3895, For a molecular-beam study involving unimolecular decay in a " normal " molecule, see J. M. Farrar and Y . T. Lee, J, Chern. Phys., 1975, 65, 1414. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 313. R. W. Numrich and K. G. Kay, J. Chem. Phys., 1979, 70, 4343; 1979, 71, 5352. See the Symposium issue Complex Scaling in the Spectral Theory of the Hamihonian, Inr. J. Quantm. Chem., 1978, 14. * R. G. Newton, J. Math. Phys., 1960, 1, 319. lo J. R. 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