162
4 Two-Particle Systems in the Berggren Basis
spurious effects induced by the regulator:
f (r) = 1 − exp
−(r/r 0 )
6
,
(4.27)
which was introduced in Eqs. (4.24) and (4.25) to avoid a singularity at r → 0. The
parameter r 0 in (4.27) defines an effective short-range scale for the regularization.
Quadrupolar anions are fascinating objects as well. Even less bound than dipolar
anions, quadrupolar anions have been obtained experimentally only recently [50].
It is difficult to know whether an electron is bound through the quadrupole field or
not [51–53]. Consequently, the accumulated knowledge about quadrupolar anions
is mainly of theoretical origin. For example, ab initio calculations have indeed
confirmed that quadrupole binding is much weaker than dipole binding [54, 55].
Moreover, due to the very small binding energies of quadrupolar anions, it is
difficult to interpret experimental data, as in the case of the CS
−
2 molecule [56],
where ab initio calculations have shown that it can exist in a weakly bound linear
configuration.
Quadrupolar anions are also described in terms of pseudo-potentials. However,
as its dipolar moment is hereby equal to zero, the pseudo-potentials will only have
one term, generated by a linear charge distribution (±q, ∓2q, ±q):
V λ (r) =
e
4πε 0
Q ±
s 2
⎧
⎨
⎩
1
r >
−
1
r
for λ = 0
r <
r >
λ 1
r >
for λ = 2, 4, 6 . . .
(4.28)
with r > = max(r, s) and r < = min(r, s). For this simple geometry of the system and
in the adiabatic limit, i.e., for an infinite moment of inertia of the neutral molecule,
it is possible to calculate very precisely the positive and negative critical electric
quadrupole moments of the core required to attach an excess electron in a J π = 0 +
state [57]. In Ref. [57], using the finite-scaling method [58] recently introduced in
atomic physics [59–61], the critical quadrupole moments have been expressed as a
function of the scaled parameter q s = qs, so that Q ±
zz = ±2q s s. The critical values
for the scaled parameter have been found analytically to be: q +
s,c = 3.98251 (ea 0 )
and q −
s,c = 1.46970 (ea 0 ), for prolate and oblate critical quadrupole moments,
respectively. These values are consistent with numerical calculations [62].
4.3.2 Dipolar Anions
Wave functions of an electron coupled to a neutral dipolar molecule [27, 28]
are extreme examples of the giant quantum halos [63–68]. Resonance energies
of dipolar anions, including those associated with rotational threshold states, can
been determined in high-resolution electron photodetachment experiments [29, 31–
34,69]. Theoretically, however, the literature on the unbound part of the spectrum of
dipole potentials, and multipolar anions in particular, are fairly limited [43, 70–77].
4 Two-Particle Systems in the Berggren Basis
spurious effects induced by the regulator:
f (r) = 1 − exp
−(r/r 0 )
6
,
(4.27)
which was introduced in Eqs. (4.24) and (4.25) to avoid a singularity at r → 0. The
parameter r 0 in (4.27) defines an effective short-range scale for the regularization.
Quadrupolar anions are fascinating objects as well. Even less bound than dipolar
anions, quadrupolar anions have been obtained experimentally only recently [50].
It is difficult to know whether an electron is bound through the quadrupole field or
not [51–53]. Consequently, the accumulated knowledge about quadrupolar anions
is mainly of theoretical origin. For example, ab initio calculations have indeed
confirmed that quadrupole binding is much weaker than dipole binding [54, 55].
Moreover, due to the very small binding energies of quadrupolar anions, it is
difficult to interpret experimental data, as in the case of the CS
−
2 molecule [56],
where ab initio calculations have shown that it can exist in a weakly bound linear
configuration.
Quadrupolar anions are also described in terms of pseudo-potentials. However,
as its dipolar moment is hereby equal to zero, the pseudo-potentials will only have
one term, generated by a linear charge distribution (±q, ∓2q, ±q):
V λ (r) =
e
4πε 0
Q ±
s 2
⎧
⎨
⎩
1
r >
−
1
r
for λ = 0
r <
r >
λ 1
r >
for λ = 2, 4, 6 . . .
(4.28)
with r > = max(r, s) and r < = min(r, s). For this simple geometry of the system and
in the adiabatic limit, i.e., for an infinite moment of inertia of the neutral molecule,
it is possible to calculate very precisely the positive and negative critical electric
quadrupole moments of the core required to attach an excess electron in a J π = 0 +
state [57]. In Ref. [57], using the finite-scaling method [58] recently introduced in
atomic physics [59–61], the critical quadrupole moments have been expressed as a
function of the scaled parameter q s = qs, so that Q ±
zz = ±2q s s. The critical values
for the scaled parameter have been found analytically to be: q +
s,c = 3.98251 (ea 0 )
and q −
s,c = 1.46970 (ea 0 ), for prolate and oblate critical quadrupole moments,
respectively. These values are consistent with numerical calculations [62].
4.3.2 Dipolar Anions
Wave functions of an electron coupled to a neutral dipolar molecule [27, 28]
are extreme examples of the giant quantum halos [63–68]. Resonance energies
of dipolar anions, including those associated with rotational threshold states, can
been determined in high-resolution electron photodetachment experiments [29, 31–
34,69]. Theoretically, however, the literature on the unbound part of the spectrum of
dipole potentials, and multipolar anions in particular, are fairly limited [43, 70–77].