Solutions to Exercises
181
Exercise IV.
A. Direct integration and Berggren expansion method are not equivalent as 1/r 2
coupling potentials have to be ignored in the asymptotic region. Indeed, one
should match wave functions to analytical outgoing Hankel functions in the
asymptotic region, but one cannot devise analytical asymptotic forms of wave
functions when moment of inertia is finite. Thus, the direct integration method
cannot provide with exact Hamiltonian eigenstates. As the Berggren basis is
complete, this difficulty is absent in the Berggren expansion method.
B. Let us insert u J
c (r) = g c H
+
(eff ) ,0
(kr) in Eq. (4.19) in the asymptotic region
to devise a solution of this equation. One has k c = k ≡
√
E J ∀c because
I = +∞. One then finds that the proposed ansatz is a solution of Eq. (4.19)
at large radius if {g c } c forms an eigenvector of the matrix [ c (( c + 1)δ cc +
a cc ] cc , of associated eigenvalue α = (eff) (( (eff) + 1), so that one can pose
(eff) = (2α)/(1 +
√
1 + 4α) (see Sect. (2.3)). Hence, the asymptote of the wave
functions of the particle-rotor system can be calculated exactly in this case, so
that it is numerically precise to use direct integration to calculate the eigenstates
of the particle-rotor Hamiltonian when I = +∞.
C. As the calculated state is well bound, the direct integration is stable even
if approximate for finite . It becomes unstable for very weakly bound and
resonance states of molecules. The numerical use of the Berggren expansion
method is stable, provided that one uses appropriate contours.
Exercise V.
A. As the potentials are proportional to 1/r 3 for large r, they decrease quickly on
the real r-axis, so that u c (r) ∝ H
+
c ,0 (k c r) in the asymptotic region.
Therefore, one can match wave functions to their exact analytical outgoing
Hankel functions in the asymptotic region and the direct integration method is
applicable.
Thus, both direct integration and Berggren expansion methods are equivalent in
the quadrupolar case, contrary to the dipolar case (see Exercise IV).
B. Direct integration is hereby stable for well-bound states because of the fast
decrease of wave functions on the real r-axis. However, as seen from effected
numerical calculations, direct integration becomes unstable for very weakly
bound and resonance states of molecules, similarly to the dipolar case (see
Exercise IV). Consequently, the Berggren expansion method still has to be used
for the calculation of weakly bound and resonance states in practice, even though
direct integration and Berggren expansion methods are theoretically equivalent.
Exercise VI. Channel wave functions do not change much when varying Hamiltonian parameters, so that only energies follow this change, while the B(E1) reduced
transition probabilities remain practically the same.
181
Exercise IV.
A. Direct integration and Berggren expansion method are not equivalent as 1/r 2
coupling potentials have to be ignored in the asymptotic region. Indeed, one
should match wave functions to analytical outgoing Hankel functions in the
asymptotic region, but one cannot devise analytical asymptotic forms of wave
functions when moment of inertia is finite. Thus, the direct integration method
cannot provide with exact Hamiltonian eigenstates. As the Berggren basis is
complete, this difficulty is absent in the Berggren expansion method.
B. Let us insert u J
c (r) = g c H
+
(eff ) ,0
(kr) in Eq. (4.19) in the asymptotic region
to devise a solution of this equation. One has k c = k ≡
√
E J ∀c because
I = +∞. One then finds that the proposed ansatz is a solution of Eq. (4.19)
at large radius if {g c } c forms an eigenvector of the matrix [ c (( c + 1)δ cc +
a cc ] cc , of associated eigenvalue α = (eff) (( (eff) + 1), so that one can pose
(eff) = (2α)/(1 +
√
1 + 4α) (see Sect. (2.3)). Hence, the asymptote of the wave
functions of the particle-rotor system can be calculated exactly in this case, so
that it is numerically precise to use direct integration to calculate the eigenstates
of the particle-rotor Hamiltonian when I = +∞.
C. As the calculated state is well bound, the direct integration is stable even
if approximate for finite . It becomes unstable for very weakly bound and
resonance states of molecules. The numerical use of the Berggren expansion
method is stable, provided that one uses appropriate contours.
Exercise V.
A. As the potentials are proportional to 1/r 3 for large r, they decrease quickly on
the real r-axis, so that u c (r) ∝ H
+
c ,0 (k c r) in the asymptotic region.
Therefore, one can match wave functions to their exact analytical outgoing
Hankel functions in the asymptotic region and the direct integration method is
applicable.
Thus, both direct integration and Berggren expansion methods are equivalent in
the quadrupolar case, contrary to the dipolar case (see Exercise IV).
B. Direct integration is hereby stable for well-bound states because of the fast
decrease of wave functions on the real r-axis. However, as seen from effected
numerical calculations, direct integration becomes unstable for very weakly
bound and resonance states of molecules, similarly to the dipolar case (see
Exercise IV). Consequently, the Berggren expansion method still has to be used
for the calculation of weakly bound and resonance states in practice, even though
direct integration and Berggren expansion methods are theoretically equivalent.
Exercise VI. Channel wave functions do not change much when varying Hamiltonian parameters, so that only energies follow this change, while the B(E1) reduced
transition probabilities remain practically the same.