J. Phys. Chem. 1991, 95, 8082-8086
8082
A Coupled-Channel Approach to Molecular Photodissociation Using Decay Boundary COndltlOnS h v i d A. Micha Departments of Chemistry and of Physics, University of Florida, Gainesville, Florida 3261 I (Received: March 29, 1991)
Energies, widths, and decay probabilities of molecular states undergoing photodissociation are obtained within a treatment based on the solution of coupled differential equations with decay boundary conditions. Photodissociation phenomena are described by analogy to prdissociation introducing a quantized electromagnetic field. The formulation considers electronically diabatic phenomena and is applicable to the absorption of visible or ultraviolet photons; it can also be applied to photodissociation following the absorption of multiple infrared photons, insofar as it is a nonperturbative formulation. A prccedure is described to calculate properties of dissociative molecular states from coupled-channel calculations at a set of real energies.
1. Introduction Studies of molecular dissociation induced by visible or ultraviolet photons provide insight into the molecular structure and spectra of reactants, transition dipoles, and the interaction forces between the dissociation products. These studies are particularly informative when they provide state-testate rates or cross sections for angular distributions and state populations of product molecules. Experimental work on photodissociation in molecular beams' and on dissociation in molecular gases using light pulses* are two important source of data on statetestate photodissociation rates. Experimental results on detailed photodissociation rates have been recently reviewed by several author^.^ The theoretical interpretation of these data, and the prediction of experimental results, requires a quantal theory allowing for the coupling of the reactant initial and excited states and for the coupling of the fragments states as the molecule breaks up. The available theoretical methods for calculating photodissociation cross sections have been reviewed in several recent publications? We describe in what follows a different theoretical approach where we expand the time-independent wave function of the molecule plus photon system in a basis of states, to obtain a set of coupled-channel differential equations, and introduce decay boundary conditions suitable for breakup of the molecule. We construct physical solutions to the differential equations, to derive an expression for dissociation rates, and to outline a procedure for calculating energies, lifetimes, and decay probabilities of dissociative states. We use the decay boundary conditions by analogy with a procedure followed in theoretical studies of predissociation.' In predissociation phenomena, a molecule AB*, excited by absorption of photons or of kinetics energy in collisions, undergoes deexcitation into a transient species ABt; after some time, the transient species breaks up into fragments A and B. Photodissociation occurs when a photon t#~ is absorbed by AB in a stationary state; this may then tum into a transient species which again breaks up into A and B fragments. These events are described by
-. ++
AB* -.AB? A B (1) + 4 4 AB^--- A B (2) and can be treated by the same approach, as we shall show, AB
(1) (a) Parker, D. H.; Beinstein, R. B. Annu. Reu. Phys. Chem. 1989,40, 561. (b) Levine, R. D.; Bemstein. R. B. Moleculur Reaction Dynamics and Chemicul Reacrfuiry, 2nd 4.;Oxford University Press: N e w York, 1987. (2) (a) Khundhar, L. R.; &wail. A. H. Annu. Rev. Phys. Chem. 1990, 4I, 15. (b) Hamilton, C. E.; Kinsey, J. S.;Field, R. W. Annu. Reo. Phys. Chem. 1986,37,493. (3) (a) Lcone, S. Ado. Chem. Phys. 1982, 50, 255. (b) Hall, G. E.; Houston, P. Annu. Reo. Phys. Chem.. in press. (c) Simon, J. P. J . Phys. Chem. 1984,88, 255. (4) (a) B a h t Kurti, G. G.; Shapiro, M. Adu. Chem. Phys. 1985,60,403. (b) Schinke, R. Annu. Reu. Pbys. Chem. 1988, 39, 39. (c) Beswick, J. A,; Jortner, J. Adu. Chem. Phys. 1981. 47, 363. (5) (a) Atabck, 0.;Lefevre, R. Phys. Rev. 1980, A22, 1817. (b) Lefevre, R. J . Chem. Phys. 1990, 92, 2869.
0022-3654/91/2095-8082S02.50/0 , I
I
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provided one considers AB + 4 to be the equivalent of an excited system. These phenomena may be electronically adiabatic, hence involving a single electronic state, or they may be electronically diabatic. Infrared multiphoton dissociation is usually adiabatic, while dissociation by a single visible or UV photon is electronically diabatic. Here we concentrate on the formulation for singlephoton dissociation. Many examples of diabatic photodissociation may be found in the cited reviews. A relatively simple example is furnished by the excitation of HCl(XIZ+)to the transient state HCI(A'II) with breakup into H(ZSl/2)+ Cl(2P3/2,12). Another example is that of CH31dissociating into CH3 plus f('P3/L1/z) where however the description is complicated by coupling of the relative motion of the fragments to several normal-mode vibrations of the transient electronically excited polyatomic. In all these cases, calculations of dissociation rates and cross sections for the breakup of the transient species require first a calculation of the energies and lifetimes of their states and then of probabilities for their decay into the final product states. Photodissociation phenomena can be described with a first-order perturbation approach, from the integral of the electric dipole operator between the initial bound state and a final dissociative state, independently ~alculated,6~~ or more accurately by solving a set of coupled differential equations to obtain wave functions with outgoing waves. Several procedures have recently been developed to accomplish this, such as using an artificial channel8 to transform the photodissociation problem into an artificial scattering problem or solving inhomogeneous scattering equations with the inhomogeneous term derived from the initially bound molecular state.g These accurate methods are examples of coupled-channel procedures based on expansions in internal states of the molecule and the numerical solution of the resulting coupled differential equations that have proven to be very suitable for small molecules, containing a few atoms or a few active degrees of freedom.4 The procedure developed in this contribution uses instead an expansion in molecule plus photon states that leads to the coupling of energetically open and closed channels. It is hence closely related to the ones developed for scattering resonances. Resonances can be described within several formalisms of scattering and in particular by treatments based on the coupling of discrete and continuum molecular stateslo or on the partition of the set of asymptotic internal (target plus projectile) states into openand closed-channel sets." A coupled-channel approach along these lines was introduced by us for molecular systems some time ago and was applied to (6) Shapiro, M.; Bcrsohn, R. Annu. Reu. Phys. Chem. 1982, 33, 409. (7) Kulander, K.; Light, J. C. J . Chem. Phys. 1980, 73, 4337. 18) Shamro. M. J . Cbem. Phvs. 1972. 56. 2582, (9) B a d , Y: B.; Freed, K. F.;Kouri, D. J. i.Chcm. Phys. 1981,74,4380. (IO) Fano, U. Phys. Reu. 1961, 124, 1866. (11) Feshbach, H. Ann. Phys. (N.Y.) 1958, 5, 357; 1962, 19, 287.
0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 8083
Molecular Photodissociation
+
the adiabatic resonance scattering of Ar H2.I2 The present approach differs from that one because we now use molecular states in the expansion and decay instead of scattering boundary conditions. A recent approach has been described to directly calculate energies and lifetimes of long-lived states from the closed channel terms of the total wave function and could possibly be used also to describe photodis~ociation.~~ The next section describes the differential equations and asymptotic conditions for a general basis set expansion. In the case of photodissociation, the basis contains photon states leading to a moleculefield representation of states and operator^.^^*'^ We also summarize here the properties of the wave functions for complex energies and channel wavenumbers,I6 which appear in dissociation studies. The following section derives the asymptotic conditions that apply for decay, making use of the decay boundary conditions first introduced by Siegert,I7 and gives expressions for decay rates. The asymptotic conditions and differential equations derived here are presented in a form suitable for calculations with available computer programs incorporating open and closed channel.'* Section 4 presents a procedure to calculate energies and widths of long-lived states and from them decay amplitudes for dissociation. We compare in the Conclusion the present approach with the ones previously available in the literature. 2. Molecule-Photon Wave Functions The total wave function 0 for a molecule interacting with photons is a function of the relative position vector R of the centers of mass of the molecular species A and B, of all the internal variables of the AB system (electronic and nuclear), and of the variables describing the photon. Introducing polar coordinates for the relative position, R = (R,0,@), we separate the relative distance R from all the remaining variables X. The total wave function can be expanded in a basis set (u,(X;R), Y = I-N, constructed as products of internal states of AB times photon states times angular momentum eigenfunctions of the relative orientation angles. These basis functions are shown here to depend parametrically on the distance R. The channel label Y is a collection of electronic, vibrational, rotational, orbital, and photon quantum numbers. At a given total energy E, and indicating with E, the asymptotic energy of the state u,, the number N of channels can be separated into No energetically open channels with E, I E plus N, closed ones with E, > E. The wave function 0 is thereby separated into factors dependent on the relative distance R and on the remaining variables X. It is convenient to introduce a matrix notation where u(X;R) is a 1 X N row matrix. The expansion is then
\k(R,X) = u(X;R)RIO(R)C
(3)
where O(R) is a N X No rectangular matrix of radial scattering functions, and C is a No X 1 row matrix of coefficients chosen to impose asymptotic conditions. The radial functions satisfy a set of coupled differential equations ((2M)-l[(h/i)(d/dR)I
+ C(R)I2 + H(R) - EI}@(R)= 0
C(R) = (ul(h/i) du/dR) H(R) = (uIHRIu)
(5)
(6)
where M is the reduced mass for relative motion, E is the total energy, I is the identity matrix, G(R) is the momentum coupling matrix, and H is the matrix of the potential energies and couplings, all of dimension N X N. We are using a bracket notation where (I) indicates integration over the variables X for fixed R, and a matrix within a bra is the adjoint of the one within a ket. The Hamiltonian operator H R contains all the kinetic and potential energy terms for the internal and rotational motions, for fixed radial distance R. In the case of photodissociation, it must include, in addition to the molecular Hamiltonian HM,the energy HFof the electromagnetic field and the coupling H M Fof this field with the molecular system. The formulation of photodissociation breakup can be done by analogy with predissociation by introducing a quantized photon field, thereby working in the so-called molecule-field repre~entati0n.I~The total Hamiltonian is then HR = HM + HF + H M F (7)
where aq,,is the annihilation operator of a photon of wavevector
4 a_ndenergy ho = hqc (c is the speed oflight) and polarization
u, 6 is the quantized electric field,I5 and D is the dipole operator of the molecular system, a function of electron and nuclear variables. The internal states must contain the field states and are functions of a set X which also contains the photon variables. Assuming, to simplify the treatment, that only photons of a single frequency are initially present and are absorbed in the excitation, the internal states take the form
u,(X R ) = uJXAB; R)lE4")
(10)
where a contains electronic, orbital, and vibrational-rotational quantum numbers, XAB refers to electronic, orbital, vibrational, and rotational variables, and the ket is a field state with n photons. In dissociation by a single visible or UV photon, n = 0, 1 and there are two or more a states; in dissociation by infrared multiphoton absorption, n can be larger but there usually is only one electronic state involved. Consequently one has, in addition to the original molecular potential energy surfaces, the same ones raised by one or more photon energy quanta and, in addition to the potential energy couplings, couplings mediated by the transition dipole between 0- and 1-photon states or n- and (n f 1)-photon states for n L 1. Given all these potentials and couplings, one can proceed to solve the coupled differential equations as before, and one can impose the same asymptotic conditions as in predissociation, for a total energy E = ElAB+ ho corresponding to the initial molecular state a = l and a single photon to begin with. The (YY') matrix elements of H(R) take the form
Hv/
( H M ) Q+~ ( H ~ ) n n h t+ (HMF),J
(11)
(4)
(12) (a) Micha, D. A. fhys. Rev. 1967, 162, 88. (b) Ace. Chem. Res. 1973, 6, 138. (13) Sun, Y.; Du, M. L.; Dalgarno, A. J . Chem. fhys. 1990, 93, 8840. (14) George, T. F.; Zimmerman, I. H.; Yuan, J.-M.; Laing, J. R.; De Vries, P. L. Ace. Chem. Res. 1977, I O , 449. ( I 5) Loudon, R. The Quantum Theory of Light; Clarendon Press: Oxford, 1973. (16) Newton, R. Scattering Theory of Waves and Particles, 2nd ed.; Springer-Verlag: Berlin, 1982. (17) (a) Siegert, A. F. J. Phys. Rev. 1939,56,750. (b) Micha, D. A. J . Math. Phys. 1967, 8, 1716. (18) (a) Redmon, M. J.; Micha, D. A. Chem. fhys. Lea. 1974.28.341. (b) Johnson, B. R. J. Compur. fhys. 1973, 13,445. (c) Light, J. C.; Walker, R. B. J . Chem. Phys. 1976,65,4272. (d) Allison, A. C. Ado. At. Mol. Opt. Phys. 1988, 25, 323. (19) Brumer, P.; Shapiro, k.Adv. Chem. fhys. 1985, 60, 371.
The radial wave functions can be expressed as linear combinations of the two independent solutions F(+)(R)and F-)(R)of the differential equations, which are irregular solutions at the origen R = 0, with the asymptotic forms fl*)(R) N exp(fikR) (15) where k indicates here a N X N diagonal matrix containing the wavenumbers of each channel. Indicating asymptotic channel energies with E, H VQQ(R) nhqc, R m (16)
+
-
8084 The Journal of Physical Chemistry, Vol. 95, No. 21, 199'1 we have, depending on whether channels are energetically open or closed k, = [(2M/hZ)(E - E,)]'/', k, = ilk,I,
E
E L E,
< E,
(17) (18)
[(2M/h2)lE - E,1]'/'
lkYl
respectively. Hence, for the closed channels we have the exponentially decreasing and increasing asymptotic solutions
c * ) ( R ) N exp(r1kJR)
(19)
Micha previously bound states of VI, which have now energies above the dissociation threshold EZAB and constitute open channels. The open-channel states contain asymptotically only outgoing waves. Bound states with energies below the threshold remain bound and constitute closed channels, decreasing exponentially at large distances. The superposition of open-channel states describes dissociative states. A dissociative state n is then characterized by a wave function containing only outgoing waves. To impose the asymptotic conditions for decay into product fragments, the wave function 9 can be written in the form
The radial solutions can now be written as the combinations
\k(R,X) = u(X;R)R-l +id)(R)
@ ( R )= (i/2)[F(-)(R)k-'A(+) - F(+)(R)k-'A(-)]
$id)(R) = 9(R)CLd) (30) where +:d) is an N X 1 matrix, and the mixing coefficients in C can be chosen to impose that
(20)
where the coefficient matrices A(*) can be obtained from the Wronskian matrices W[F,@] = Fa' - (F)"iP
(21)
with the upper index t indicating a transpose matrix and the prime a derivative with respect to R. Choosing for convenience the starting conditions for integration of the differential equations to be, at a small radius R1
@ ( R , )= 0
(22)
where we are using a matrix block notation with e.g. the subindexes co indicating a NE X No matrix, we find that the coefficient matrices are given in terms of the irregular solutions at the starting radius, insofar as A(*) = W[F(*),@],,
(24)
It will be useful in what follows to consider the behavior of the coefficients in the complex energy plane with Im (E) < 0. This means that the channel wavenumbers become complex and can be written in the form k=K-iX
(25)
where K and X are N X N diagonal matrices and A, 2 0; the asymptotic solutions are then c*)(R)
N
exp(firJZ) exp(*XJ?)
(26)
which are also exponentially increasing and decreasing, but with reversed behavior compared to the closed-channel case; in effect, a function that decreases exponentially below the energy threshold of a closed channel becomes exponentially increasing above the threshold. This suggests a connection between bound states and decay states, which we shall consider in the next section. Inspecting the coupled differential equations and the boundary conditions for the irregular solutions, one concludes t h a P F(-)(R,k*) = F(+)(R,k)*= F(+)(R,-k*)
(27) because these three matrix functions satisfy the same differential equations and boundary conditions. From the Wronskian expression for the coefficients it then follows that A(-)(k*) = A(+)(k)* = A(+)(+*) (28)
The asymptotic behavior of the wave function 9 is therefore specified in the plane of each complex channel wavenumber. 3. Decay Boundary Conditions and Dissociative States The selection of our boundary conditions can be easily understood with an example involving only two moleculefield states. The first one, with a = 1 and a single photon, is given by V,,(R) + h w and can hold bound states. The second one contains no photons and equals Vz2, a re ulsive potential. When their asymptotic energies satisfy E l A h w > EZAB, coupling of these potentials by the dipolar photon interaction leads to decay of the
!+
(29)
$id)(R) = F(+)(R)D, (31) at all distances larger than a value R,, beyond which the interaction potentials are negligible. In this expression, the N X 1 matrix D,,contains the decay amplitudes of the transient species AB*(n) into all open channels. These decay boundary conditions are also known as the Siegert boundary conditions1' and have been successfully used to describe predi~sociation.~ Noting that at large R the matrices of irregular solutions are diagonal, and imposing continuity for all R > R,, we can separate blocks corresponding to open and closed channels to write (i/2)(F&,)b-'Ac) - Fg)k;lA$))Ci:)
= Fg)Don (32)
(i/2)(F&)&-'&+) - @2)h-1g;))Ci:) = F$DCn
(33)
For real energies E, the solutions FL)(R) grow exponentially as R increases past R,. The coefficient matrix which multiplies the growing exponentials, must consequently be mathematically negligible if the equalities are to be satisfied. Numerical procedures can be followed to avoid exponentially increasing terms in the radial solutions and to ensure that the equalities are satisfied.lE Using Wronskians or simply matching (-) and (+) components at R,, we find the set of coupled algebraic equations
e),
Ag)C#) = O,, A$+)C$ = 0,, (-i/2)k,,-IA&)C$$ = Do, (+/2)&-lA&)C$
= D,,
(34)
The first equation clearly requires that the determinant det (Ac)) = 0 (35) as the condition for the existence of nontrivial solutions for the mixing coefficients CL!). These can be obtained from the first equation and the normalization condition (cg))cg) = 1
(36)
which imposes the physical condition that the probabilities of decay into all the open channels must add up to one. The second equation serves to verify that the block of co radial coefficients is numerically negligible. Once the mixing coefficients have been found, the third equation directly shows how one obtains the decay amplitudes into the open channels. The fourth equation gives us the normalization of closed-channel amplitudes. The determinantal equation can be satisfied for certain values of the energy of the system det (Ag)[k(E)]] = 0
-
E,, = 6, - iFJ2
(37)
where 6,is the energy of the long-lived state and I?, is its width. This determinantal equation is similar to the one giving bound states for the potential Vll(R) + h w ip the absence of coupling by the photon. Then the determinantal equation would arise from
The Journal of Physical Chemistry, Vol. 95, NO. 21, 1991 8085
Molecular Photodissociation the requirement that the asymptotic wave functions decrease exponentially, and its roots would be real eigenvalues, equal to the eigenvalues for VI, raised by the photon energy, hw. This suggests that the would give good starting values in the search for the roots E,. Returning to the channel wavenumbers, we have that
e,
kv(En)' = (Knv - iA,,)' = ( 2 M / h 2 ) ( & ,- irJ2
- E,)
(38) The requirement of outgoing waves means that we want solutions with K, > 0, and the requirement that probability densities decay with time means that r, > 0, from which we conclude that A,, > 0. We therefore find that the spacial component of the wave function increases asymptotically as exp(A,,.R) for decay of the nth state into the uth open channel. 4. Calculation of Properties of Photodissociative States We consider the time evolution of the molecular states when the photon flux is turned off at a time t = 0, after which the photodissociation fragments are detected. The full moleculefield state n, including the time dependence due to decay, is given for t>Oby
N
Tn(R,X,t) = *n(R,X) exp(-E,t/h)
(39)
u(X;R)R1 exp(ik$)D,
(40)
exp(-iE,t / h )
where k, is a diagonal matrix with the elements k, = K,, - iA,, obtained by solving the determinantal equation. The procedure to be followed to construct this wave function consists of the following: (a) Integration of the coupled-channel differential equations starting with chosen initial values of functions and derivatives a t a small distance R1, as done in scattering problems;'* (b) extraction of the matrices of coefficients and &*) from the asymptotic forms of the numerical solutions, by matching functions and derivatives at a large R,, for example using Wronskians, to the expressions containing the irregular solutions fl*)(R); (c) calculation of the roots of the determinantal equation, det (A:)) = 0, to obtain the decay energies and wavevectors; (d) calculation of the mixing coefficients C$ at the complex decay energies and from them the decay matrix D,. Properties of the dissociative states can then be derived from the wave function T,,(t). For example, the time-dependent population of fragments in state u following breakup of the state n is given, at a distance R, by N,,(R,t) = ~ ( T n ( f ) ~ u uwhere ) R ~ zthe , bracket indicates an integral at a fixed distance R. The population is then
e)
Npu(R,t) = R-' exp[4un(R,t)IIDunl2
(41)
4vn(Rt)l = 2Aufl- rnt/h
(42)
which shows a maximum when the exponent satisfies d 4 = 0, leading to dR/dt = I',/(2hAUn) = u,, (43) R Ro + U,,(t - to) (44) This is the equation for the trajectory of the decay fragments in state u, which are found to move outward with relative velocity up,. The population change with time is given by
= -(r,/h)" (45) indicating a rate of decay r,,/h,or decay time h/I',,, as expected. Photodissociation cross sections can be derived from expectation values of the flux operator. We can calculate the decay flux in the radial direction for the n u breakup from the flux operator, to find asymptotically dN,"/dt
-
4;)
N
(~K,,/M)R' exp(2An,.R) exp(-~nt/h)lD,n12 (46)
showing that the flux decreases exponentially in time while growing exponentially with the radial distance. To obtain a differential cross section, we consider a measurement lasting a time T, during which fragments are detected coming from within a sphere of
radius u and a solid angle dQ from the dissociation center. Multiplying this flux times the element of area R2 dQ, averaging within a conical volume bounded by 0 I R I a, and further averaging over the interval 0 It I T, we find the decay rate dR$
= (h~,,/M)(2X,,a)-'[exp(2X,,a) - I ] X (r,,T/h)-'[l - exp(-r,T/h)]lD,,,12
dR (47)
The usual expression for the differential cross section follows from dR$) = Fphdu$ where F h is the photon flux, when r,T/h The welf-known result is then
(48)
