130
4 The Treatment of Few-Body Reactions
and in each box i the potential is assumed to be constant along the propagation coordinates. Accordingly, the equations resulting from such expansion (and integration
over s τ ) are
−
2
2μ
d
2
∂ S 2
τ
− D
i
τ
φ τ v (s τ ; S τ , , τ ) = 0
(4.51)
which are integrated up to the asymptotes where by imposing the scattering boundary
conditions the S vv (E) matrix elements are evaluated (whose square modulus is the
reaction probability) and combined to compute the reactive scattering cross section.
Of the second type is the adiabatic treatment of the overall rotational energy in
which following the formalism of J.M. Bowman [58] the Hamiltonian is then given
by
H
J,,
= H
J =0
e f f + E
J,,
(Q),
(4.52)
where E
J,,
(Q) is the rotational energy calculated at each nuclear configuration,
denoted Q. In many cases, is nearly a good quantum number and the symmetric
top expression may be used, e.g., for a prolate symmetric top
E
J,,
(Q) = ¯
B(Q)J (J + 1) + [A(Q) − ¯
B(Q)]
2
(4.53)
4.3.2 Leveraging on Classical Mechanics
Once abandoned the idea of carrying out accurate quantum calculations, the simplest approach in terms of formalism and computer demand is the use of a classical
mechanics treatment. After all, if you do not need to reproduce in full the rich structure of a quantum calculation trajectory calculations are of great help even when for
numerical reasons their integration is only partially successful in terms of energy
andor angular momentum conservation. This is singled out by the comparison of the
J = 0 exact quantum P
J =0
00 probability computed at v, j = 0, 0 with the corresponding one obtained from classical mechanics for the N + N 2 reaction given in Fig. 4.7
(this reaction will be considered again when comparing the values of the thermal rate
coefficient of the N + N 2 reaction computed using quantum reactive IOSA, semiclassical and quasiclassical techniques, among them and with experimental data. The
figure tells us that trajectory calculations are able, indeed, to reproduce the average
trend (and when using moderate rejection criteria for discarding poor energy conserving trajectories, also the absolute value) of quantum reactive probabilities even
in just above the threshold energy region. On the contrary, they are unable to reproduce the detailed structure of the quantum reactive probability. Therefore, while the
use of quasiclassical probabilities and cross sections in a multiscale application may
not be necessarily safe in some cases, the use of more averaged quantities does not
4 The Treatment of Few-Body Reactions
and in each box i the potential is assumed to be constant along the propagation coordinates. Accordingly, the equations resulting from such expansion (and integration
over s τ ) are
−
2
2μ
d
2
∂ S 2
τ
− D
i
τ
φ τ v (s τ ; S τ , , τ ) = 0
(4.51)
which are integrated up to the asymptotes where by imposing the scattering boundary
conditions the S vv (E) matrix elements are evaluated (whose square modulus is the
reaction probability) and combined to compute the reactive scattering cross section.
Of the second type is the adiabatic treatment of the overall rotational energy in
which following the formalism of J.M. Bowman [58] the Hamiltonian is then given
by
H
J,,
= H
J =0
e f f + E
J,,
(Q),
(4.52)
where E
J,,
(Q) is the rotational energy calculated at each nuclear configuration,
denoted Q. In many cases, is nearly a good quantum number and the symmetric
top expression may be used, e.g., for a prolate symmetric top
E
J,,
(Q) = ¯
B(Q)J (J + 1) + [A(Q) − ¯
B(Q)]
2
(4.53)
4.3.2 Leveraging on Classical Mechanics
Once abandoned the idea of carrying out accurate quantum calculations, the simplest approach in terms of formalism and computer demand is the use of a classical
mechanics treatment. After all, if you do not need to reproduce in full the rich structure of a quantum calculation trajectory calculations are of great help even when for
numerical reasons their integration is only partially successful in terms of energy
andor angular momentum conservation. This is singled out by the comparison of the
J = 0 exact quantum P
J =0
00 probability computed at v, j = 0, 0 with the corresponding one obtained from classical mechanics for the N + N 2 reaction given in Fig. 4.7
(this reaction will be considered again when comparing the values of the thermal rate
coefficient of the N + N 2 reaction computed using quantum reactive IOSA, semiclassical and quasiclassical techniques, among them and with experimental data. The
figure tells us that trajectory calculations are able, indeed, to reproduce the average
trend (and when using moderate rejection criteria for discarding poor energy conserving trajectories, also the absolute value) of quantum reactive probabilities even
in just above the threshold energy region. On the contrary, they are unable to reproduce the detailed structure of the quantum reactive probability. Therefore, while the
use of quasiclassical probabilities and cross sections in a multiscale application may
not be necessarily safe in some cases, the use of more averaged quantities does not