164
4 Two-Particle Systems in the Berggren Basis
of rotational bands. In the body-fixed frame, the density of the valence particle
(electron) in the state J π is axially symmetric and can be decomposed as:
ρ J (r, θ ) =
K J
ρ J K J (r, θ ) ,
(4.30)
where (r, θ ) stand for the polar coordinates of the particle in the intrinsic frame, and
the K J -components of the density are:
ρ J K J (r, θ ) =
,,
j
2j + 1
2J + 1
J j 0|J K J
K J j 0|J K J
×
u J
j (r) ∗
r
u
J
j (r)
r
Y
K J ∗
(θ, 0) Y
K J
(θ, 0) .
(4.31)
If all K J -components except one vanish in Eq. (4.30), the adiabatic strong-coupling
limit is reached and K J becomes a good quantum number. In this particular case,
ρ J K J can be identified as the intrinsic density of a valence particle (electron) density
in the body-fixed (dipole-fixed) reference frame. To quantify the degree of K J -
mixing, it is convenient to introduce the normalization amplitudes:
n J K J =
2j + 1
2J + 1
J j 0|J K J
2
|u
J
j (r)|
2 dr .
(4.32)
Due to (4.29), the normalization amplitudes n J K J fulfill the condition:
K J
n J K J = 1 .
(4.33)
The excitation energies of the lowest-energy bound and resonance states are
plotted in Fig. 4.3 as a function of J (J + 1). The J π = 0 + , 1 − , 2 + bound states
form a rotational band K J = 0 build on the ground state 0
+
1 . Another K J = 0
rotational band is built upon the 0
+
2 resonance. Majority of the resonances are
strongly K J -mixed (see Fig. 4.3). Consequently, an identification of other rotational
bands in the continuum, based on the concept of intrinsic density, is not obvious. As
shown in Fig. 4.4, there appear clusters of resonances having the same total angular
momentum J within one group g i . In each of those clusters, the dominant channel
wave functions have the same orbital angular momentum of the valence electron
but different rotational angular momenta of the molecule j .
4 Two-Particle Systems in the Berggren Basis
of rotational bands. In the body-fixed frame, the density of the valence particle
(electron) in the state J π is axially symmetric and can be decomposed as:
ρ J (r, θ ) =
K J
ρ J K J (r, θ ) ,
(4.30)
where (r, θ ) stand for the polar coordinates of the particle in the intrinsic frame, and
the K J -components of the density are:
ρ J K J (r, θ ) =
,,
j
2j + 1
2J + 1
J j 0|J K J
K J j 0|J K J
×
u J
j (r) ∗
r
u
J
j (r)
r
Y
K J ∗
(θ, 0) Y
K J
(θ, 0) .
(4.31)
If all K J -components except one vanish in Eq. (4.30), the adiabatic strong-coupling
limit is reached and K J becomes a good quantum number. In this particular case,
ρ J K J can be identified as the intrinsic density of a valence particle (electron) density
in the body-fixed (dipole-fixed) reference frame. To quantify the degree of K J -
mixing, it is convenient to introduce the normalization amplitudes:
n J K J =
2j + 1
2J + 1
J j 0|J K J
2
|u
J
j (r)|
2 dr .
(4.32)
Due to (4.29), the normalization amplitudes n J K J fulfill the condition:
K J
n J K J = 1 .
(4.33)
The excitation energies of the lowest-energy bound and resonance states are
plotted in Fig. 4.3 as a function of J (J + 1). The J π = 0 + , 1 − , 2 + bound states
form a rotational band K J = 0 build on the ground state 0
+
1 . Another K J = 0
rotational band is built upon the 0
+
2 resonance. Majority of the resonances are
strongly K J -mixed (see Fig. 4.3). Consequently, an identification of other rotational
bands in the continuum, based on the concept of intrinsic density, is not obvious. As
shown in Fig. 4.4, there appear clusters of resonances having the same total angular
momentum J within one group g i . In each of those clusters, the dominant channel
wave functions have the same orbital angular momentum of the valence electron
but different rotational angular momenta of the molecule j .