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Asymptotic product and flux analysis of complex packets

We show here that the asymptotic product analysis method[12] is a special case of a T-matrix (flux-based) method to calculate state-resolved reaction probabilities[7]. We also show that a flux-based method to calculate total reaction probabilities[5,6], as used in Ref. [14], is a special case of both.

Consider a wave packet satisfying the time-dependent Schrödinger equation, where denotes the coordinates of the collisional system. We let correspond to separated reactants consistent with particular internal quantum numbers I. We refer to the reactant channel as channel a'' [e.g. A + BC(I)]. We are interested in either the probabilities to form products in particular quantum states F, or the total reaction probability. We refer to the product channel as channel c'' [e.g., C+AB(F)]. If is chosen to be in product coordinates, we denote the scattering coordinate by R, and use for all internal coordinates. If is chosen to be in reactant coordinates, we use Ra for the reactant scattering coordinate and for the internal coordinates. and are the appropriate reduced masses associated with R and Ra.

The (complex) wave packet may be expanded in time-independent scattering wave functions ,

 (1)

 (2)

where . Throughout we assume the normalization convention for all energy-resolved continuum functions. If the initial wave packet is given by , where denotes a diatomic internal state, and is localized in the reactant region, we can write as

 (3)

where is the same as in Ref. [12]. (The -1 factor in Eq. (3) is a particular choice for an arbitrary unit modulus phase factor. This choice, coupled with a +1 phase factor for below makes consistent with the functions in Ref. [12], and ensures the phase of the S matrix is the same.) = with the internal energy for initial state I.

T-matrix elements are defined as , where is the inelastic scattering wave function in the product channel[15]. (The factor arises because of the normalization we use.)

Following the analysis in Ref. [7], we can write Tc,aFI(E) as

 (4)

where is defined as

 (5)

with a reduced mass associated with s. The surface condition can be chosen, e.g., with s= ra, the reactant molecule separation or, with s = R, the product separation. (More general surfaces and flux operators are possible[7].) In the T-matrix formulation any coordinate system can be used and, depending on the surface condition, can be an asymptotic wave function or a distorted wave which allows an analysis surface closer to the interaction region.

Using a surface condition based on product Jacobi coordinate R and an asymptotic form for , we can prove that the T-matrix method is equivalent to the asymptotic analysis method. is then given by

 (6)

is a particular product internal state, and = , with the internal energy for the final state F. Inserting Eqs. (5) and (6) into Eq. (4), and evaluating the flux at , with being an appropriate asymptotic value of R, we get

 Tc,aFI(E) = (7)

The second term in Eq. (7) can be written in terms of AFI(E), which was defined in Ref. [12] as

 (8)

where the integration of the term in brackets is over , by realizing that = . [This follows from Eq. (2).] The first term in Eq. (7) is simplified by noting

 (9)

[Proof of Eq. (9) assumes the wave packet can be expanded in outgoing waves around R = .] Inserting Eqs. (3), (8), and (9) into Eq. (7), we obtain

 (10)

For a reactive channel Tc,aFI(E) is related to Sc,aFI(E) as , which get us to the same expression for Sc,aFI(E) as derived in Ref. [12].

Thus, we have shown that in the case that the projection of onto final states happens in the asymptotic region, the T-matrix formulation of Ref. [7] and the asymptotic analysis method[3,4,12] yield the same result.

We now show that both methods give the same result as the method from Refs. [5,6], if one only calculates the total reaction probability, Pr(E), from a given initial state I. This latter method for the total reaction probability is based on , which leads to

 (11)

From the form for Sc,aFI(E) above, Pr(E) is given as

 (12)

where we used Eq. (10) and the relation between Tc,aFI(E)and Sc,aFI(E). Using Eqs. (3), (9), and the completeness relation , we get

 (13)

which corresponds to Eq. (11) with s chosen to be R.

Next: Flux and real wave Up: Flux analysis for calculating Previous: Introduction
Anthony J. H. M. Meijer
1998-06-19