148
4 The Treatment of Few-Body Reactions
Another typical characteristic of the LHH character of this reaction and of its
strong exoergicity is the typical Franck–Condon like nature of the product vibrational
distributions that have a single peak at the product state energetically equivalent to
the v state of the reactant (if v = 0) and are bimodal if v = 1.
Finally, Reaction (d), that is a typical light heavy-light (LHL) asymmetric endoergic system with a dominant noncollinear MEP, shows in the density of its adiabats
both the light nature of the entrance channel diatom HF and the heavy nature of its
exit channel diatom LiF. A peculiarity of this reaction is the fact that the third channel leading to LiH is highly endoergic (and therefore closed at thermal energies) and
its behavior is exothermic like because of the much larger zero point energy of the
reactant HF with respect to that of the product LiF to compensate for the endoergicity
of the process.
As already mentioned, the above-described dynamical studies do not only offer
a rationale for a more complete understanding of the mechanisms of elementary
chemical processes but they can also offer the numerical procedures able to cover
the last mile to the calculation of the signal measured by beam experiments (see
next chapter for a clear example). They are, in fact, able to estimate on an ab initio
fashion the signal of the experimental apparatus without making model assumptions
provided that its geometry is fully specified.
In particular, for Reaction (d), the S matrix elements (S
J
v jl,v j l ) obtained, as already
discussed, by projecting the asymptotic value of the propagated wavefunctions of the
investigated atom–diatom system onto the corresponding final states of the channel
of interest, bear the information necessary to construct the related measured integral
cross section (the fixed energy E state v j to state v
j
partial (fixed J ) probabilities
P
J
v j,v j (E) obtained by squaring the S
J
v jl,v j l elements and properly summing them
over l and l
) allow, in fact, to evaluate the corresponding integral cross section
σ v j,v j (E) =
π
k
2
v j
P v j,v j (E) =
π
k
2
v j
J
(2J + 1)P
J
v j,v j (E).
(4.72)
and have been used to cover the last mile aimed at reproducing the intensity of the
product beam detected by the experimental apparatus.
4.5 Problems
4.5.1 Qualitative Problems
1. Open Channels for H + H 2 : How many open rovibrational channels are there
for scattering energies at E = 0.7, 0.9, 1.1, 1.3, 1.5 eV for the H + H 2 system
assuming J = 0?
4 The Treatment of Few-Body Reactions
Another typical characteristic of the LHH character of this reaction and of its
strong exoergicity is the typical Franck–Condon like nature of the product vibrational
distributions that have a single peak at the product state energetically equivalent to
the v state of the reactant (if v = 0) and are bimodal if v = 1.
Finally, Reaction (d), that is a typical light heavy-light (LHL) asymmetric endoergic system with a dominant noncollinear MEP, shows in the density of its adiabats
both the light nature of the entrance channel diatom HF and the heavy nature of its
exit channel diatom LiF. A peculiarity of this reaction is the fact that the third channel leading to LiH is highly endoergic (and therefore closed at thermal energies) and
its behavior is exothermic like because of the much larger zero point energy of the
reactant HF with respect to that of the product LiF to compensate for the endoergicity
of the process.
As already mentioned, the above-described dynamical studies do not only offer
a rationale for a more complete understanding of the mechanisms of elementary
chemical processes but they can also offer the numerical procedures able to cover
the last mile to the calculation of the signal measured by beam experiments (see
next chapter for a clear example). They are, in fact, able to estimate on an ab initio
fashion the signal of the experimental apparatus without making model assumptions
provided that its geometry is fully specified.
In particular, for Reaction (d), the S matrix elements (S
J
v jl,v j l ) obtained, as already
discussed, by projecting the asymptotic value of the propagated wavefunctions of the
investigated atom–diatom system onto the corresponding final states of the channel
of interest, bear the information necessary to construct the related measured integral
cross section (the fixed energy E state v j to state v
j
partial (fixed J ) probabilities
P
J
v j,v j (E) obtained by squaring the S
J
v jl,v j l elements and properly summing them
over l and l
) allow, in fact, to evaluate the corresponding integral cross section
σ v j,v j (E) =
π
k
2
v j
P v j,v j (E) =
π
k
2
v j
J
(2J + 1)P
J
v j,v j (E).
(4.72)
and have been used to cover the last mile aimed at reproducing the intensity of the
product beam detected by the experimental apparatus.
4.5 Problems
4.5.1 Qualitative Problems
1. Open Channels for H + H 2 : How many open rovibrational channels are there
for scattering energies at E = 0.7, 0.9, 1.1, 1.3, 1.5 eV for the H + H 2 system
assuming J = 0?