4.3 The Particle-Rotor Model in the Berggren Basis
167
devise solutions of Eq. (4.19) in the asymptotic region of the form u J
c (r) =
g c H
+
(eff ) ,0
(kr), where k, (eff ) , and g c are constants to determine.
C. Explain why the direct integration method is relatively precise here, and why
the Berggren expansion method has to be preferred nevertheless for the study
described in this section.
Excitation energies of resonances are plotted in Fig. 4.5 as a function of the
molecular angular momentum j for different groups of resonances of Fig. 4.4. It is
seen that these states form very regular rotational band sequences in j rather than in
J . Different members of such bands lie close in the complex energy plane and have
similar densities ρ J K J (r, θ ). The rotational resonance structures are governed by a
weak coupling, whereby the orbital motion of a valence electron is decoupled
from the rotational motion of a dipolar neutral molecule.
At the detachment threshold, there appears a transition from the strong-coupling
regime, in which the attached electron follows the rotational motion of the core,
to the weak-coupling regime, where the electron’s rotational motion is almost
decoupled from that of the rotor.
4.3.3 Quadrupolar Anions
In order to benchmark the Berggren expansion method, results obtained in the
adiabatic limit, i.e., for an infinite moment of inertia, are compared with the
analytical results [57] for the quadrupolar anions with critical electric quadrupole
moment Q ±
zz,c = ±2q ±
s,c s. The distance s in Berggren calculations in the adiabatic
limit is fixed at 1.6 a 0 [92], close to the value in CS
−
2 (s = 1.554 a 0 [93]). The
corresponding critical quadrupole moments are thus Q −
zz,c = −2.35152 ea 2
0 and
Q +
zz,c = 6.372016 ea 2
0 . In Berggren expansion method, in addition to max , the
momentum cutoff k max has to be defined. By taking a real contour discretized
with 80 points, and k max = 12 a
−1
0 , one obtains Q −
zz,c = −2.35164 ea 2
0 . The critical
quadrupole moment is well described for the oblate configuration of an attached
electron that is well localized around the two positive charges at the center of the
molecule. For the prolate deformation, the situation is different. Indeed, as shown in
Fig. 4.6, the results of the Berggren expansion method do not approach the analytical
value as closely as for the oblate configuration. For max = 14 (and k max = 12 a
−1
0 ),
one obtains Q +
zz,c = 6.3984 ea 2
0 with the Berggren expansion method. While the
convergence of Q +
zz,c with max (and k max ) is slower than for Q −
zz,c , the Berggren
expansion method results converges for max = 14 and k max = 12, a
−1
0 . The Pauli
blocking at short distances [95–97] reduces the binding in the oblate configuration.
Hence, in general, it is the prolate configuration that is more likely to bind electrons.
Thus, while the oblate configuration results are useful for benchmarking purpose,
their physical interpretation should be dealt with caution.
The analysis of the unbound states of quadrupolar anions is conveniently
performed using the Berggren expansion method. The full excitation spectrum
167
devise solutions of Eq. (4.19) in the asymptotic region of the form u J
c (r) =
g c H
+
(eff ) ,0
(kr), where k, (eff ) , and g c are constants to determine.
C. Explain why the direct integration method is relatively precise here, and why
the Berggren expansion method has to be preferred nevertheless for the study
described in this section.
Excitation energies of resonances are plotted in Fig. 4.5 as a function of the
molecular angular momentum j for different groups of resonances of Fig. 4.4. It is
seen that these states form very regular rotational band sequences in j rather than in
J . Different members of such bands lie close in the complex energy plane and have
similar densities ρ J K J (r, θ ). The rotational resonance structures are governed by a
weak coupling, whereby the orbital motion of a valence electron is decoupled
from the rotational motion of a dipolar neutral molecule.
At the detachment threshold, there appears a transition from the strong-coupling
regime, in which the attached electron follows the rotational motion of the core,
to the weak-coupling regime, where the electron’s rotational motion is almost
decoupled from that of the rotor.
4.3.3 Quadrupolar Anions
In order to benchmark the Berggren expansion method, results obtained in the
adiabatic limit, i.e., for an infinite moment of inertia, are compared with the
analytical results [57] for the quadrupolar anions with critical electric quadrupole
moment Q ±
zz,c = ±2q ±
s,c s. The distance s in Berggren calculations in the adiabatic
limit is fixed at 1.6 a 0 [92], close to the value in CS
−
2 (s = 1.554 a 0 [93]). The
corresponding critical quadrupole moments are thus Q −
zz,c = −2.35152 ea 2
0 and
Q +
zz,c = 6.372016 ea 2
0 . In Berggren expansion method, in addition to max , the
momentum cutoff k max has to be defined. By taking a real contour discretized
with 80 points, and k max = 12 a
−1
0 , one obtains Q −
zz,c = −2.35164 ea 2
0 . The critical
quadrupole moment is well described for the oblate configuration of an attached
electron that is well localized around the two positive charges at the center of the
molecule. For the prolate deformation, the situation is different. Indeed, as shown in
Fig. 4.6, the results of the Berggren expansion method do not approach the analytical
value as closely as for the oblate configuration. For max = 14 (and k max = 12 a
−1
0 ),
one obtains Q +
zz,c = 6.3984 ea 2
0 with the Berggren expansion method. While the
convergence of Q +
zz,c with max (and k max ) is slower than for Q −
zz,c , the Berggren
expansion method results converges for max = 14 and k max = 12, a
−1
0 . The Pauli
blocking at short distances [95–97] reduces the binding in the oblate configuration.
Hence, in general, it is the prolate configuration that is more likely to bind electrons.
Thus, while the oblate configuration results are useful for benchmarking purpose,
their physical interpretation should be dealt with caution.
The analysis of the unbound states of quadrupolar anions is conveniently
performed using the Berggren expansion method. The full excitation spectrum