132
4 The Treatment of Few-Body Reactions
The second case is the already mentioned use of SC-IVR formulation of the S
matrix which, while preserving the ability of discretizing internal energy, leverage is
still entirely based on the regaining of the whole classical trajectory information [61].
4.3.3 Semiclassical Treatments
As mentioned above and already discussed in the first chapter, there are quantities of
classical nature (such as the classical action) allowing to regain quantum-like effects
from the outcomes of trajectory calculations. One can, in fact, formulate the S matrix
elements in a semiclassical, SC, fashion by working out of classical quantities the
features needed to build the semiclassical wavefunction of the considered system.
At the root of the SC approach to chemical processes is the Jeffreys, Kramer,
Brilluoin, and Wentzel (JWKB [62–65]) solution of the one-dimensional l = 0
Schrödinger equation (2.27) once it is written as
d
2
dr 2 +
p
2
2
ψ(r ) = 0
(4.54)
by assuming p(r ) = [2μ(E − V (r ))]
1/2 to be real. The solution of Eq. (4.54) is
ψ(r ) = Ae
±i p/ if p(r ) is independent of r . This is not true, as is usually the case.
In the case of p(r ) varying slowly with r one can make the position ψ(r ) = Ae
±i S/
(with S =
p(r )dr being the classical action integral) whose second derivative
is ψ
(r ) =
−(S
/)
2
± i S
/
ψ
(r ). By expanding the JWKB wavefunction in
series of (S(r ) = S o (r ) + S 1 (r ) +
2 S 2 (r )) and equating to zero in succession its
terms one gets also higher order solutions. It has to be noted here that the derivation
does not place the requirement for p(r ) to be real. Accordingly, together with the
general classically allowed solution
ψ(r ) = A[ p(r )]
−1/2 e
±i S/
(4.55)
for p(r ) positive one can write also the classically forbidden one
ψ(r ) = A[ p(r )]
−1/2 cos
1
r
r o
p(r
)dr
+ α
,
(4.56)
where [ p(r )]
−1/2 relates the amplitude of the wavefunction (or better its square modulus) to the time spent to cross the element dr . Accordingly, the JWKB wavefunction
diverges at any classical turning point.
This can be avoided by resorting to uniform approximations in which the wavefunction is imposed a shape dictated by the locations in r of the classical turning
points of the potential. For example, one can assume that the wavefunction depends
4 The Treatment of Few-Body Reactions
The second case is the already mentioned use of SC-IVR formulation of the S
matrix which, while preserving the ability of discretizing internal energy, leverage is
still entirely based on the regaining of the whole classical trajectory information [61].
4.3.3 Semiclassical Treatments
As mentioned above and already discussed in the first chapter, there are quantities of
classical nature (such as the classical action) allowing to regain quantum-like effects
from the outcomes of trajectory calculations. One can, in fact, formulate the S matrix
elements in a semiclassical, SC, fashion by working out of classical quantities the
features needed to build the semiclassical wavefunction of the considered system.
At the root of the SC approach to chemical processes is the Jeffreys, Kramer,
Brilluoin, and Wentzel (JWKB [62–65]) solution of the one-dimensional l = 0
Schrödinger equation (2.27) once it is written as
d
2
dr 2 +
p
2
2
ψ(r ) = 0
(4.54)
by assuming p(r ) = [2μ(E − V (r ))]
1/2 to be real. The solution of Eq. (4.54) is
ψ(r ) = Ae
±i p/ if p(r ) is independent of r . This is not true, as is usually the case.
In the case of p(r ) varying slowly with r one can make the position ψ(r ) = Ae
±i S/
(with S =
p(r )dr being the classical action integral) whose second derivative
is ψ
(r ) =
−(S
/)
2
± i S
/
ψ
(r ). By expanding the JWKB wavefunction in
series of (S(r ) = S o (r ) + S 1 (r ) +
2 S 2 (r )) and equating to zero in succession its
terms one gets also higher order solutions. It has to be noted here that the derivation
does not place the requirement for p(r ) to be real. Accordingly, together with the
general classically allowed solution
ψ(r ) = A[ p(r )]
−1/2 e
±i S/
(4.55)
for p(r ) positive one can write also the classically forbidden one
ψ(r ) = A[ p(r )]
−1/2 cos
1
r
r o
p(r
)dr
+ α
,
(4.56)
where [ p(r )]
−1/2 relates the amplitude of the wavefunction (or better its square modulus) to the time spent to cross the element dr . Accordingly, the JWKB wavefunction
diverges at any classical turning point.
This can be avoided by resorting to uniform approximations in which the wavefunction is imposed a shape dictated by the locations in r of the classical turning
points of the potential. For example, one can assume that the wavefunction depends