172
4 Two-Particle Systems in the Berggren Basis
the lowest resonant states as a function of the quadrupole moment in an oblate
setting. If E i is the energy of ith resonance of a supercritical molecule, then by
changing the electric quadrupole moment continuously beyond Q −
zz,c , one arrives at
E i → E
i ≈ E i+1 . In the immediate vicinity of the critical quadrupole moment in
Figs. 4.9b (Q −
zz,c ) and 4.9c (Q +
zz,c ), one can see the rearrangement of eigenvalues.
Moreover, for |Q ±
zz | << |Q ±
zz,c | the eigenvalues are almost equal for oblate and
prolate configurations, as the corresponding wave functions are hardly sensitive to
details of the potential.
Such a rearrangement of eigenvalues at the critical quadrupole moment can also
be seen by considering the scattering length of the system at different values of Q zz .
At low energies (k → 0), the scattering length is related to the = 0 phase shift
δ 0 (k) of a scattering state through:
lim
k→0
k
tan δ(k)
= −
1
a 0
.
(4.34)
To illustrate the criticality of the system, one may investigate a simple case
where only = 0 diagonal element of the channel-channel coupling potential is
present (see panel (d) in Fig. 4.9). Indeed, the scaled parameter q s,c = qs reaches
a constant at around the critical value of a quadrupole moment [57]. Therefore,
by changing s → s so that the = 0 diagonal element of the potential becomes
more and more important, one can effectively evolve the negative critical quadrupole
moment Q
−
zz,c,0 = −2q s ,c s for = 0 to the value Q −
zz,c of the complete problem.
To this end, one has to decrease s, or conversely increase q s,c , to localize the electron
in an almost pure = 0 bound state.
Exercise V
One will consider the precision of the direct integration method and the
Berggren expansion method for numerical studies of the quadrupolar anions, and
compare results with those obtained in the dipolar anion case (see Exercise IV).
A. Run the particle-rotor code for the quadrupolar anion with both the direct
integration method and the Berggren expansion method. Compare energies and
wave functions obtained in these two methods and show that they provide with
almost identical values.
Explain why the situation is different as compared to the dipolar case.
B. By comparing the numerical stability of direct integration and Berggren
expansion methods with the provided code, explain why it is preferable in
practice to use the Berggren expansion method for the calculation of weakly
bound and resonance states.
4 Two-Particle Systems in the Berggren Basis
the lowest resonant states as a function of the quadrupole moment in an oblate
setting. If E i is the energy of ith resonance of a supercritical molecule, then by
changing the electric quadrupole moment continuously beyond Q −
zz,c , one arrives at
E i → E
i ≈ E i+1 . In the immediate vicinity of the critical quadrupole moment in
Figs. 4.9b (Q −
zz,c ) and 4.9c (Q +
zz,c ), one can see the rearrangement of eigenvalues.
Moreover, for |Q ±
zz | << |Q ±
zz,c | the eigenvalues are almost equal for oblate and
prolate configurations, as the corresponding wave functions are hardly sensitive to
details of the potential.
Such a rearrangement of eigenvalues at the critical quadrupole moment can also
be seen by considering the scattering length of the system at different values of Q zz .
At low energies (k → 0), the scattering length is related to the = 0 phase shift
δ 0 (k) of a scattering state through:
lim
k→0
k
tan δ(k)
= −
1
a 0
.
(4.34)
To illustrate the criticality of the system, one may investigate a simple case
where only = 0 diagonal element of the channel-channel coupling potential is
present (see panel (d) in Fig. 4.9). Indeed, the scaled parameter q s,c = qs reaches
a constant at around the critical value of a quadrupole moment [57]. Therefore,
by changing s → s so that the = 0 diagonal element of the potential becomes
more and more important, one can effectively evolve the negative critical quadrupole
moment Q
−
zz,c,0 = −2q s ,c s for = 0 to the value Q −
zz,c of the complete problem.
To this end, one has to decrease s, or conversely increase q s,c , to localize the electron
in an almost pure = 0 bound state.
Exercise V
One will consider the precision of the direct integration method and the
Berggren expansion method for numerical studies of the quadrupolar anions, and
compare results with those obtained in the dipolar anion case (see Exercise IV).
A. Run the particle-rotor code for the quadrupolar anion with both the direct
integration method and the Berggren expansion method. Compare energies and
wave functions obtained in these two methods and show that they provide with
almost identical values.
Explain why the situation is different as compared to the dipolar case.
B. By comparing the numerical stability of direct integration and Berggren
expansion methods with the provided code, explain why it is preferable in
practice to use the Berggren expansion method for the calculation of weakly
bound and resonance states.