358 The European Physical Journal D
states) as test examples for which we correctly treat the
dynamics in a space-fixed (SF) frame of reference. The
state-to-state rotationally inelastic cross-sections are ob-
tained from a full quantum account of angular momenta
coupling during the collision and convergence is tested on
the final stability of the S-matrix elements.
The following section briefly describes the scattering
equations while Section 3 reports the evaluation of the
interaction forces. Section 4 presents our results and com-
pares them with some of the experiments for the test sys-
tems we have looked at in the present study, while Sec-
tion 5 gives our conclusions.
2 The quantum dynamics
In this initial test study we shall treat the N
2
and
H
2
molecular targets as rigid rotors, disregarding from the
time being the effects of vibrational excitations during the
rotational energy transfer collisions. Furthermore, since we
will focus on rather low-energy collisions, we expect the
rotational-to-vibrational mode coupling to be rather inef-
ficient and therefore not to markedly affect pure rotational
inelasticity at these energies [9]. Such effects, however, are
not in general minor ones, especially at higher collision en-
ergies, but will be discussed more in details in future ap-
plications. Furthermore, as reported in the following sec-
tion, the electron interaction with the molecules will be
described by a local, energy-dependent effective potential.
Here again the non-local nature of the exchange interac-
tion is somewhat simplified in our treatment. However,
the present model has already yielded very good results
for other, more complicated targets [10] and thus we ex-
pect that it will be sufficiently realistic also for the present
study.
In the case of the small distortions which are induced
into the target rotations by the impinging particle, the
total scattering wavefunction can be expanded in terms of
asymptotic target rotational eigenfunctions
Hrot(Rˆeq)Yjmj(Rˆeq) =


2
2I j(j + 1)Yjmj(Rˆeq) (1)
with I being the isolated molecule moment of inertia [11]
and (Rˆeq) the space orientation of the molecular bond,
kept at its equilibrium value. Hence, the total scattering
wavefunction is given as
Ψn(E, re,Req) =
∑
f
ui→f (re, E)Yf (Rˆeq) (2)
where |f〉 denotes the |j′mj′〉 final states of the rotat-
ing molecule that are involved in the expansion and the
ui→f (re, E) are the channel components of the scatter-
ing wavefunction which have to be determined by solving
the usual Schro¨dinger equation subject to its scattering
boundary conditions, with re being the scattered electron
vector position from the molecular center of mass (c.o.m.),
with radial component given by
ui→f (re) → δifh(−)(re)− Sifh(+)(re) as (re) ∼ ∞ (3)
here h(±)(re) is a pair or linearly independent free partial
wave solutions defined as
h(±)if ∼ δifk
−1/2
i exp [i(kir ± liπ/2)] . (4)
When they are chosen to be appropriate Riccati-Hankel
functions, then the Sif coefficients become the elements of
the reduced scattering matrix, often additionally labeled
by the total angular momentum of the system: J = j + l,
the latter l being the continuum electron partial wave
component. Usually, one expects that the numerically con-
verged scattering observables can be obtained by retaining
only a limited number of discrete, asymptotic target states
in the expansion (2).
The ui→f are therefore expanded in products of to-
tal angular momentum eigenfunctions and of radial func-
tions ϕJλλ′ (E, re), where J is the magnitude of the total
angular momentum and, λ′ = (j′, l′). The radial functions
are in turn solutions of the familiar set of coupled, second
order homogeneous differential equations (in the case of
local interactions) [11,12]
[
d2
dr2e
I
2
−
1
r2e
l
2 + K2
]
Φ
J(E, re) = UJΦJ (E, re) (5)
where I is the unit matrix, ΦJ is the matrix of radial
functions and atomic units are used throughout this work.
Hence
(l2)λλ′′ = l′(l′ + 1)δλ′λ′′ (6)
(K2)λλ′′ = k2j′δλ′λ′′ = 2(E − Ej′ )δλ′λ′′ (7)
(UJ (re))λ′λ′′ = 2
∑
L
fL(l′j′; l′′j′′; J)VL(re) (8)
where the fL(l′j′; l′′j′′; J) are the well-known, real coef-
ficients of Percival and Seaton [13] and the coupling be-
tween the asymptotic (diabatic) target states is given by
the radial matrix elements which we shall discuss in detail
in the next section. Since L is even and fL(l′j′; l′′j′′; J) is
real, the UJλ′λ′′ is nonzero only if j′ − j′′ is even, i.e. the
matrix is block diagonal with two subblocks that contain
only even values of (l′ + j′) or only odd values of (l′ + j′).
Thus, when one starts from j = 0 it is only necessary to
include those values of the partial wave index l for which
|J − j| ≤ l ≤ J + j and for which l+J is even. When jmax
is the maximum value of j included in the expansion (2),
then the fL’s are zero for L > 2jmax when j′ < jmax
and j < jmax. Thus, the direct coupling between rota-
tional levels will be controlled, as we shall further discuss
below, by the largest multipolar coupling VL included in
equation (8).
The number of channels to be included in the expan-
sion for equation (5) obviously depends on the system and
on the collision energy. Furthermore, for each selected col-
lision energy it also depends on the region of interaction
that is being sampled during the search for the channel
eigenfunctions. In the short-range regions, which corre-
spond to the strongest interactions, one should include
all those channels which become locally open because of