122
M.C. Bacchus-Montabonel
In the semiclassical approach, the nuclei are considered to follow a classical trajectory R(t) = b + vt with regard to the impact parameter b and the velocity v [28].
The time-dependent Schrödinger equation reduces thus to:
H
el
r, R(t)
− i
∂
∂t
× Ψ (r, b, v, t) = 0
(6.7)
where r stands for the electronic coordinates. It may be solved for each velocity v
and impact parameter b by expanding the total wave function on the eigenfunctions
ψ adia
mΛ of H el with eigenvalues ε mΛ :
Ψ (r, b, v, t) =
mΛ
a mΛ (b, v, t)ψ
adia
mΛ
r, R(t)
× exp
−i
t
o
ε mΛ
R
t
dt
.
(6.8)
By integration of equation (6.7), the capture probabilities are given by P (b, v) =
mΛ |a mΛ (b, v, ∞)| 2 with summation over all charge exchange channels. The
cross section is then defined by:
σ (v) = 2π
bP (b, v)db.
(6.9)
In this approach, the collision dynamics was treated using the EIKONXS program [29] taking into account radial and rotational coupling matrix elements, as
well as translation effects, although they are expected to be low at these energies.
6.2.3 Thermal Rate Constant
The rate constants k(T ) are calculated by averaging the cross sections σ (E) over a
Maxwellian velocity distribution at temperature T [30]:
k(T ) =
8
πμ
1/2
1
k B T
3/2 ∞
0
Eσ (E) × exp
−
E
k B T
dE.
(6.10)
The rate constant for the reverse ionization process k rev (T ) may be determined easily by means of the micro-reversibility relation from the corresponding charge transfer rate constant k(T ):
k rev (T ) = g exp
−
ΔE
k B T
k(T ),
(6.11)
where g is the ratio of the statistical weights of initial and final states, and ΔE is the
energy gain of the charge transfer reaction.
6.3 Molecular Calculations
The electron spin being conserved in the collision process, we have to determine
the potential energies of the different molecular states involved in the process for all
M.C. Bacchus-Montabonel
In the semiclassical approach, the nuclei are considered to follow a classical trajectory R(t) = b + vt with regard to the impact parameter b and the velocity v [28].
The time-dependent Schrödinger equation reduces thus to:
H
el
r, R(t)
− i
∂
∂t
× Ψ (r, b, v, t) = 0
(6.7)
where r stands for the electronic coordinates. It may be solved for each velocity v
and impact parameter b by expanding the total wave function on the eigenfunctions
ψ adia
mΛ of H el with eigenvalues ε mΛ :
Ψ (r, b, v, t) =
mΛ
a mΛ (b, v, t)ψ
adia
mΛ
r, R(t)
× exp
−i
t
o
ε mΛ
R
t
dt
.
(6.8)
By integration of equation (6.7), the capture probabilities are given by P (b, v) =
mΛ |a mΛ (b, v, ∞)| 2 with summation over all charge exchange channels. The
cross section is then defined by:
σ (v) = 2π
bP (b, v)db.
(6.9)
In this approach, the collision dynamics was treated using the EIKONXS program [29] taking into account radial and rotational coupling matrix elements, as
well as translation effects, although they are expected to be low at these energies.
6.2.3 Thermal Rate Constant
The rate constants k(T ) are calculated by averaging the cross sections σ (E) over a
Maxwellian velocity distribution at temperature T [30]:
k(T ) =
8
πμ
1/2
1
k B T
3/2 ∞
0
Eσ (E) × exp
−
E
k B T
dE.
(6.10)
The rate constant for the reverse ionization process k rev (T ) may be determined easily by means of the micro-reversibility relation from the corresponding charge transfer rate constant k(T ):
k rev (T ) = g exp
−
ΔE
k B T
k(T ),
(6.11)
where g is the ratio of the statistical weights of initial and final states, and ΔE is the
energy gain of the charge transfer reaction.
6.3 Molecular Calculations
The electron spin being conserved in the collision process, we have to determine
the potential energies of the different molecular states involved in the process for all