158
4 Two-Particle Systems in the Berggren Basis
The restoration of rotational invariance can be effected in two ways. The first one
is to directly project intrinsic wave function onto the space of rotationally invariant
wave functions, where the total angular momentum J is a good quantum number.
While this approach is exact and treats the Pauli principle exactly, it is rather timeconsuming even for well-bound systems, so that it would be very inefficient within
the Berggren formalism.
The second method is to include the rotation of the core directly. Indeed,
rotational invariance of the Hamiltonian is broken if one simply replaces a spherical
core potential by a deformed core potential because the axes of the deformed
potential are fixed. Consequently, if one allows the axes of the deformed potential
to rotate freely in coordinate space, no direction of space is preferred and rotational
invariance is restored. Hence, one generates in this method several rotational states
for the core which are then coupled to the angular momentum of valence particles
to form a many-body wave function of well-defined total angular momentum J .
Multipolar anions are among the most extended quantum systems. They have
been measured experimentally and are being theoretically studied. In the following,
one will study the physics of dipolar and quadrupolar anions using particle-rotor
model formulated in the Berggren basis. The fundamental feature of anions is that
they consist of a valence electron above a neutral core. Consequently, there is no
long-range Coulomb potential acting on the valence electron, which can be bound
to the core through the action of the dipolar and/or quadrupolar potentials, and
higher multipoles of the Coulomb potential. Clearly, the valence electron can only
be weakly bound or resonant, so that the Berggren basis is perfectly adapted for
their study.
One of the most important characteristics of dipolar anions is their permanent
dipole moment. The minimal value which allows for an electron to be bound to the
molecule is called the critical dipole moment and is denoted by μ c . It has been first
determined by Fermi and Teller [27] for a point-dipole (μ c = 1.625 D) and then
generalized to an extended dipole with an infinite moment of inertia [28]. However,
high-resolution electron photodetachment experiments suggested a larger value for a
critical moment [29–34], in accordance with nonadiabatic calculations (μ c ∼ 2.5 D)
[35] which include the rotational degrees of freedom of the anion [36–44]. In this
case, dipolar anions have only few bound states, and the value of μ c depends on
the moment of inertia of the neutral core. One should add that the molecular core
possesses a permanent quadrupolar moment as well, whose effect has to be added
to the Hamiltonian. The presence of an electron close to the polarized core also
generates an additional induced dipole moment.
The particle-rotor model used along with the Berggren basis is also well adapted
to the study of deformed nuclei in the vicinity of driplines. In this case, this model
provides a simple theoretical formalism to describe wave functions of complex
structure where both deformation and continuum degrees of freedom are coupled.
The particle-rotor model is particularly useful for odd-mass nuclei, having one
nucleon in a weakly bound or resonance state above a deformed well-bound core.
As the nuclear interaction is short-ranged, it is sufficient to use a deformed WoodsSaxon potential to mimic the interaction between the core and the nucleon.
4 Two-Particle Systems in the Berggren Basis
The restoration of rotational invariance can be effected in two ways. The first one
is to directly project intrinsic wave function onto the space of rotationally invariant
wave functions, where the total angular momentum J is a good quantum number.
While this approach is exact and treats the Pauli principle exactly, it is rather timeconsuming even for well-bound systems, so that it would be very inefficient within
the Berggren formalism.
The second method is to include the rotation of the core directly. Indeed,
rotational invariance of the Hamiltonian is broken if one simply replaces a spherical
core potential by a deformed core potential because the axes of the deformed
potential are fixed. Consequently, if one allows the axes of the deformed potential
to rotate freely in coordinate space, no direction of space is preferred and rotational
invariance is restored. Hence, one generates in this method several rotational states
for the core which are then coupled to the angular momentum of valence particles
to form a many-body wave function of well-defined total angular momentum J .
Multipolar anions are among the most extended quantum systems. They have
been measured experimentally and are being theoretically studied. In the following,
one will study the physics of dipolar and quadrupolar anions using particle-rotor
model formulated in the Berggren basis. The fundamental feature of anions is that
they consist of a valence electron above a neutral core. Consequently, there is no
long-range Coulomb potential acting on the valence electron, which can be bound
to the core through the action of the dipolar and/or quadrupolar potentials, and
higher multipoles of the Coulomb potential. Clearly, the valence electron can only
be weakly bound or resonant, so that the Berggren basis is perfectly adapted for
their study.
One of the most important characteristics of dipolar anions is their permanent
dipole moment. The minimal value which allows for an electron to be bound to the
molecule is called the critical dipole moment and is denoted by μ c . It has been first
determined by Fermi and Teller [27] for a point-dipole (μ c = 1.625 D) and then
generalized to an extended dipole with an infinite moment of inertia [28]. However,
high-resolution electron photodetachment experiments suggested a larger value for a
critical moment [29–34], in accordance with nonadiabatic calculations (μ c ∼ 2.5 D)
[35] which include the rotational degrees of freedom of the anion [36–44]. In this
case, dipolar anions have only few bound states, and the value of μ c depends on
the moment of inertia of the neutral core. One should add that the molecular core
possesses a permanent quadrupolar moment as well, whose effect has to be added
to the Hamiltonian. The presence of an electron close to the polarized core also
generates an additional induced dipole moment.
The particle-rotor model used along with the Berggren basis is also well adapted
to the study of deformed nuclei in the vicinity of driplines. In this case, this model
provides a simple theoretical formalism to describe wave functions of complex
structure where both deformation and continuum degrees of freedom are coupled.
The particle-rotor model is particularly useful for odd-mass nuclei, having one
nucleon in a weakly bound or resonance state above a deformed well-bound core.
As the nuclear interaction is short-ranged, it is sufficient to use a deformed WoodsSaxon potential to mimic the interaction between the core and the nucleon.