Nuclear deformation:

If U(R) is the potential shape to be deformed, the deformed nuclear potential can be constructed as

$$\mathbf{V}(\xi,\mathbf{R})=U(R-\delta(\hat{R}')),$$ (28)

where $\hat{R}'$ are the angular coordinates $(\theta,\varphi)$ referred to the intrinsic reference frame. The function $\delta$ is normally expanded in multipoles:

$$\delta(\hat{R}')=\sum_{\lambda}\delta_{\lambda}Y_{\lambda0}(\hat{R}')$$ (29)

As in the case of cluster form factors, it is convenient to expand the coupling potential V in spherical harmonics, giving rise to the radial multipoles $V_{N}^{\lambda}(R)$, whose reduced matrix elements are given by

$$\langle I'\vert\vert V_{N}^{\lambda}(R)\vert\vert I\rangle=-... ...lta_{\lambda}\vert\vert I\rangle}{\sqrt{4\pi}}\frac{dU(R)}{dR},$$ (30)

with the same shape for all nuclear multipoles $\lambda>0$.

The values of $\langle I'\vert\vert\delta_{\lambda}\vert\vert I\rangle$ are model dependent:

• Rotational model:
  

  $$\langle I'\vert\vert\delta_{\lambda}\vert\vert I\rangle=\sqr... ...rt I'K\rangle\langle\chi\vert\delta_{\lambda}\vert\chi\rangle,$$ (31)

  
  
  where K is the projection of the angular momenta I and I' within a rotational model and $\langle\chi\vert\delta_{\lambda}\vert\chi\rangle$ is the expectation value of the operator $\hat{\delta}_{\lambda}$ in the internal state of the deformed nucleus. Even more, if the mass and charge distributions coincide:
  

  $$\langle I'\vert\vert\delta_{\lambda}\vert\vert I\rangle=\fra... ...bda-1}}\langle I'\vert\vert M(E_{\lambda})\vert\vert I\rangle,$$ (32)

  
  
  where R₀ is an average radius. According to (23) and (31) and the previous relation holds also for the matrix elements in the intrinsic state:
  

  $$\langle\chi\vert\vert\delta_{\lambda}\vert\vert\chi\rangle=\... ...a-1}}\langle\chi\vert\vert M(E_{\lambda})\vert\vert\chi\rangle,$$ (33)

  
  
  In the rotational model, it is not necessary to provide FRESCO with the all matrix elements above. The only information required by FRESCO are the deformation lengths $\delta_{\lambda}$, which are assigned to the variables $P(\lambda)$:
  

  $$P(\lambda)=\langle\chi\vert\delta_{\lambda}\vert\chi\rangle=R_{0}\beta_{\lambda},$$ (34)

  
  
  where $\beta_{\lambda}$ the deformation parameters. Note that within the rotational model, the nuclear and coulomb matrix elements are related by:
  

  $$M_{n}(E\lambda)=\frac{3Z\beta_{\lambda}R^{\lambda}}{4\pi}$$ (35)

  
  

• General case: In general, if we are not within a specific model, there is not a simple relation between the matrix elements connecting different states I and I' and so it is necessary to give them explicitly for each pair of excited states. In particular, FRESCO uses the matrix elements:
  

  $$RDEF(\lambda;I\rightarrow I') (-1)^{\frac{I-I'+\vert I-I'\vert}{2}}\langle I'\vert\vert\delta_{\lambda}\vert\vert I\rangle$$ =

  $$(-1)^{\frac{I-I'+\vert I-I'\vert}{2}}\sqrt{2I+1}\langle IK\lambda0\vert I'K\rangle\langle\chi\vert\delta_{\lambda}\vert\chi\rangle$$ =

  $$(-1)^{\frac{I-I'+\vert I-I'\vert}{2}}\sqrt{2I+1}\langle IK\lambda0\vert I'K\rangle\beta_{\lambda}R_{0}$$ = (36)

  
  
  This information is provided to fresco through the namelist &step/, which contains the variables²:
  - ib: final state with angular momentum I'
  - ia: initial state with angular momentum I
  - k: multipolarity $\lambda$
  - str: strength of the coupling factor $RDEF(\lambda;I\rightarrow I')$

Figure 2: Energy levels for the ¹⁹F considered in the CC calculation.