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 *R*_{a} for the reactant scattering
coordinate and
for the internal coordinates.
and
are the appropriate reduced masses associated
with *R* and *R*_{a}.

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

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

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

**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
*T*^{c,a}_{FI}(*E*) as

where is defined as

with a reduced mass associated with

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

is a particular product internal state, and = , with the internal energy for the final state

The second term in Eq. (7) can be written in terms of *A*_{FI}(*E*),
which was defined in Ref. [12] as

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

[Proof of Eq. (9) assumes the wave packet can be expanded in outgoing waves around

For a reactive channel

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, *P*_{r}(*E*), from a given initial state *I*. This latter method
for the total reaction probability is based on
,
which
leads to

From the form for

where we used Eq. (10) and the relation between

which corresponds to Eq. (11) with