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Next: Flux and real wave Up: Flux analysis for calculating Previous: Introduction

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 $\psi ({\bf x},t)$ satisfying the time-dependent Schrödinger equation, where ${\bf x}$ denotes the coordinates of the collisional system. We let $\psi({\bf x}, t=0)$ 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 ${\bf x}$ is chosen to be in product coordinates, we denote the scattering coordinate by R, and use $\bf r$ for all internal coordinates. If ${\bf x}$ is chosen to be in reactant coordinates, we use Ra for the reactant scattering coordinate and ${\bf r}_a$ for the internal coordinates. $\mu^c$ and $\mu^a$ are the appropriate reduced masses associated with R and Ra.

The (complex) wave packet $\psi ({\bf x},t)$ may be expanded in time-independent scattering wave functions $\psi^+({\bf x},E)$,

\psi({\bf x},t)
= \frac{1}{2\pi\hbar} \int e^{-iEt/\hbar} \psi^+ ({\bf x},E) a_\psi (E) dE,
\end{displaymath} (1)

 \begin{displaymath}\psi^+ ({\bf x},E) = \frac{1}{a_\psi (E)} \int e^{iEt/\hbar}
e^{-i\hat{H}t/\hbar} \psi({\bf x},0) dt,
\end{displaymath} (2)

where $a_\psi(E)\equiv\left\langle\psi^+({\bf x},E)\vert\psi({\bf x},0)\right\rangle$. Throughout we assume the normalization convention $\langle \psi^+ (E) \vert \psi^+ (E') \rangle = 2 \pi \hbar \delta (E-E')$for all energy-resolved continuum functions. If the initial wave packet is given by $\zeta
(R_a) \phi_I({\bf r}_a)$, where $\phi_I ({\bf r}_a)$ denotes a diatomic internal state, and $\zeta$ is localized in the reactant region, we can write $a_\psi(E)$ as

a_\psi (E) =
- \sqrt{ \frac{\mu^a}{\hbar k^a_I} }
\int e^{...
= - 2 \pi \sqrt{ \frac{\mu^a}{\hbar k^a_I}} \bar{g}(-k^a_I),
\end{displaymath} (3)

where $\bar{g}(-k^a_I)$ 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 $\psi_F$ below makes $\psi^+$ consistent with the $\psi^-$ functions in Ref. [12], and ensures the phase of the S matrix is the same.) $\hbar k^a_I$ = $\sqrt{2 \mu^a (E- \epsilon_I ) }$ with $\epsilon_I$ the internal energy for initial state I.

T-matrix elements are defined as $T^{c,a}_{FI}(E)\equiv \left\langle\psi_F({\bf x},E)\left\vert V \right\vert
\psi^+({\bf x},E)\right\rangle / (2 \pi \hbar ) $, where $\psi_F({\bf x},E)$ is the inelastic scattering wave function in the product channel[15]. (The $2 \pi \hbar $ factor arises because of the normalization we use.)

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

T^{c,a}_{FI}(E) = \frac{i}{2\pi}
\left\langle\psi_F({\bf x},...
...eft\vert \hat{F}
\psi^+({\bf x},E)
\end{displaymath} (4)

where $\hat{F}$ is defined as

\hat{F}=\frac{\hbar}{2i\mu^s} \left\{
...{\partial s}
-\frac{\partial}{\partial s}\delta(s-s_0)\right\}
\end{displaymath} (5)

with $\mu^s$ 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, $\psi_F({\bf x})$ 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 $\psi_F$, we can prove that the T-matrix method is equivalent to the asymptotic analysis method. $\psi_F({\bf x})$ is then given by

\psi_F({\bf x},E) = \phi_F({\bf r})e^{ik^c_FR}\sqrt{\frac{\mu^c}{\hbar k^c_F}}.
\end{displaymath} (6)

$\phi_F$ is a particular product internal state, and $\hbar k^c_F$ = $\sqrt{2 \mu^c (E-
\epsilon_F ) }$, with $\epsilon_F$ the internal energy for the final state F. Inserting Eqs. (5) and (6) into Eq. (4), and evaluating the flux at $R_0=R_\infty$, with $R_\infty$ being an appropriate asymptotic value of R, we get

Tc,aFI(E) = $\displaystyle \frac{1}{4\pi}\sqrt{\frac{\hbar}{\mu^c k^c_F}} \left\{
R}\right]_{R_\infty}\right.\right\rangle e^{-ik^c_FR_\infty} \right.$  
    $\displaystyle \left. + ik^c_F
\left\langle\phi_F({\bf r}) \left\vert\psi^+({\bf r},R_\infty,E)
\right.\right\rangle e^{-ik^c_FR_\infty}\right\}.$ (7)

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

A_{FI}(E)= \frac{1}{2\pi}
\int_0^\infty e^{iEt/\hbar}
\psi(R=R_\infty,{{\bf r}},t)\right.\right\rangle_{\bf r}dt,
\end{displaymath} (8)

where the integration of the term in brackets is over $\bf r$, by realizing that $\left\langle\phi_F({\bf r}) \left\vert\psi^+({\bf r},R_\infty,E)
\right.\right\rangle$ = $\left[a_\psi(E)\right]^{-1} A_{FI}(E)$. [This follows from Eq. (2).] The first term in Eq. (7) is simplified by noting

\left\langle \phi_{F}({{\bf r}}) \left\vert
\left[ \frac{\pa...
...\frac{\hbar k^a_I}{\mu^a}}
\end{displaymath} (9)

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

T^{c,a}_{FI}(E) = \frac{-i}{2\pi}
\left[\frac{\hbar^2 k^a_I...
\frac{A_{FI}(E)}{\bar{g}(-k^a_I)} e^{-i k^c_F R_\infty}.
\end{displaymath} (10)

For a reactive channel Tc,aFI(E) is related to Sc,aFI(E) as $S^{c,a}_{FI}(E)=- 2\pi iT^{c,a}_{FI}(E)$, 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 $\psi^+({\bf x},E)$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 $P_r(E) =
\langle \psi^+ ({\bf x}, E) \vert \hat{F} \vert \psi ( {\bf x}, E)^+ \rangle$, which leads to

$\displaystyle P_r(E) =
\frac{\hbar}{\mu^s \left\vert a_\psi (E) \right\vert^2}$ $\textstyle {\rm Im}$ $\displaystyle \left\langle
\int e^{iEt/\hbar} \psi({\bf x},t) dt
  $\textstyle \times$ $\displaystyle \left\vert\int e^{iEt/\hbar}
\left[\delta({s-s_0}) \frac{\partial}{\partial s} \psi({\bf x},t)\right] dt
\right\rangle~~.$ (11)

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

P_r(E) = \sum_F \left\vert S^{c,a}_{FI}(E) \right\vert^2
= ...
...) \right]^2}
\sum_F A_{FI}^*(E) \left[ k^c_F A_{FI}(E)\right],
\end{displaymath} (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 $\sum_F \phi_{F}^\ast({{\bf r}'}) \phi_{F}({{\bf r}}) = \delta({{\bf r}'}
-{{\bf r}})$, we get

$\displaystyle P_r(E) =
\frac{\hbar}{\mu^c \left\vert a_\psi (E) \right\vert^2}$ $\textstyle {\rm Im}$ $\displaystyle \left\langle
\int e^{iEt/\hbar} \psi({\bf x},t) dt
  $\textstyle \times$ $\displaystyle \left\vert\int e^{iEt/\hbar}
\left[\delta({R-R_0}) \frac{\partial}{\partial R} \psi({\bf x},t)\right] dt
\right\rangle,$ (13)

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

next up previous
Next: Flux and real wave Up: Flux analysis for calculating Previous: Introduction
Anthony J. H. M. Meijer