# Nuclear Structure with Discrete Non-Orthogonal Shell Model: new frontiers

D. D. Dao  F. Nowacki  Université de Strasbourg, CNRS, IPHC UMR7178, 23 rue du
Loess, F-67000 Strasbourg, France

(August 27, 2024)

###### Abstract

We present developments and applications for the diagonalization of shell-
model hamiltonians in a discrete non-orthogonal basis (DNO-SM). The method,
and its actual numerical implementation CARINA, based on mean-field and
beyond-mean field techniques has already been applied in previous studies and
is focused on basis states selection optimization. The method is benchmarked
against a full set of $sd$ shell exact diagonalizations, and is applied for
the first time to the heavy deformed 254No nucleus.

###### pacs:

23.20.Js, 23.20.Lv, 27.60.+j, 25.85.Ca

##  I Introduction

In the recent decades, the advent of radioactive beam factories associated
with developments of more sophisticated experimental methods has enabled to
discover new manisfestations of many-body nuclear dynamics in many places of
the nuclear chart. New phenomena like halo systems, two-proton radioactivity,
occurence of new magic numbers, vanishing of shell closures, soft dipole
modes, or even searches for superheavy nuclei have been observed and have
stimulated the continuous developments and improvements of theoretical methods
in order to interpret such phenomena.

Among the various theoretical frameworks available, the Shell-Model (SM) or
Configuration Interaction (CI), either in its no-core or valence space
implementations, has always been one of the most powerful methods in the
description of quantum nuclear systems Caurier et al. (2005); Nowacki et al.
(2021), in particular with the numerical development of efficient
diagonalization codes which have opened the era of the so-called ”Large Scale
Shell-Model calculations” for light and medium-mass nuclei up to $A\sim 150$
Caurier et al. (1999); Nowacki et al. (2016); Hinke et al. (2012); Siciliano
et al. (2020); Rosiak et al. (2018); Naïdja et al. (2017) and have become the
method of choice to explain the observed nuclear phenomena, guide experimental
programs, and not the least, allowed for a deeper understanding of man
astrophysical objects and processes in which exotic nuclei often play the key
role. However, its success was always minored by the exponential growth of the
systems basis involved. At the same time, variational methods with symmetry
breaking and restoration have also shown great success for decades and have
proven to a certain extent to be applicable over the nuclear chart from the
lightest nuclear systems to the most heavy ones Bender et al. (2003); Egido
(2016); Robledo et al. (2018).

Although these methods provide distinct description, the merging of the mean-
field techniques within the shell-model formalism has already been developed
and studied in the literature in the past, starting from the pioneering work
of Ripka Ripka (1965, 1968), later followed by the different VAMPIR
implementations Schmid and Grummer (1987); Schmid (2004). One of the major
achievements up to now was proposed by the Tokyo group with the Monte Carlo
Shell-Model Otsuka et al. (1998); Utsuno et al. (1999); Tsunoda et al. (2014).
And more recently several implementations were used, either in an punctual
manner Gao and Horoi (2009); Hinohara and Kanada-En’yo (2011) or in a more
ambitious scale with the recent development of the TAURUS numerical suite
Bally, B. et al. (2021a); Sánchez-Fernández et al. (2021); Bally et al.
(2019).

In the present work, we present the formalism of the Discrete Non-Orthogonal
Shell-Model (DNO-SM) and its associated numerical implementation CARINA. The
DNO-SM amounts to diagonalize valence shell-model hamiltonians in a non-
orthogonal basis with the use of beyond-mean-field techniques. The detailled
framework is exposed in the next section and applications to $sd$ shell nuclei
in comparison with exact diagonalisations are discussed in section III. The
final section exposes an application to a very heavy system in the 254No case.

##  II Theoretical Framework

###  II.1 Shell Model formulation revisited

####  II.1.1 The diagonalization dilemma

The ultimate question in the Shell Model, once a physically meaningful valence
space $\mathcal{E}$ is equipped together with the associated effective
interaction $\hat{\mathcal{V}}$ for the problem at hand, is to tackle the
secular equation

$\hat{\mathcal{H}}|\Psi\rangle=E|\Psi\rangle$ (1)

where $\hat{\mathcal{H}}$ represents the effective Hamiltonian composed of
$\hat{\mathcal{V}}$ and a one-body single-particle energy $\\{e_{i}\\}$ part

$\hat{\mathcal{H}}=\sum_{i\in\mathcal{E}}e_{i}a^{\dagger}_{i}a_{i}+\frac{1}{4}\sum_{ijkl\in\mathcal{E}}\langle
ij|\hat{\mathcal{V}}|kl\rangle a^{\dagger}_{i}a^{\dagger}_{j}a_{l}a_{k}.$ (2)

$\\{a^{\dagger}_{i}\\}$ and $\\{a_{i}\\}$ are creation and annihilation
operators satisfying the common anti-commutation rules for fermionic systems.

By defining a set of basis states
$\mathcal{B}=\\{|\phi_{m}\rangle,m\in\mathbb{N}\\}$ constructed from the
single-particle spherical oscillator valence space $\mathcal{E}$ with which we
can write the eigenstate $|\Psi\rangle$ as

$|\Psi\rangle=\sum_{m=1}^{\mathrm{dim}(\mathcal{B})}c_{m}|\phi_{m}\rangle,$
(3)

the classic Shell Model resolution of (1) then amounts to addressing the
eigenvalue problem

$\sum_{m=1}^{\mathrm{dim}(\mathcal{B})}\mathcal{H}_{m^{\prime}m}c_{m}=E\,c_{m^{\prime}}$
(4)

by an exact diagonalization of the Hamiltonian matrix
$\mathcal{H}_{m^{\prime}m}=\langle\phi_{m^{\prime}}|\hat{\mathcal{H}}|\phi_{m}\rangle$
in the model space $\mathscr{H}=\overline{\mathrm{Span}\>\mathcal{B}}$.
Although spectacular designs of Shell Model codes have been achieved Caurier
et al. (2005) to reach space dimensions larger and larger, it is still an
inherent problem that makes difficult to extend the Shell Model applicability
into heavier mass nuclei. To deal with this dilemma, we look for a replacement
of $\mathcal{B}$ with a different family of basis states. The existence of
such basis takes the root in the original idea of the Generator Coordinate
Method (GCM) first proposed in Refs. Griffin and Wheeler (1957); Peierls and
Yoccoz (1957). As we shall discuss in the following, it can be however viewed
in an independent status with respect to the GCM, thanks to the work of the
authors in Ref. Broeckhove and Deumens (1979).

####  II.1.2 Discrete non-orthogonal basis

The starting point of the GCM is the hypothesis that one can find a family of
states depending on the continuous (generator) coordinate(s) $q$

$\mathit{\Gamma}=\\{|\Phi(q)\rangle\>|\>q\in\mathbb{R}\\}$ (5)

so that the latter forms a model subspace
$\mathscr{H}_{q}=\overline{\mathrm{span}\>\mathit{\Gamma}}\subseteq\mathscr{H}$
following the nature of the coordinate(s) $q$ that we choose in the generation
of $\mathit{\Gamma}$. The core of our presentation of $\mathscr{H}_{q}$ relies
on the following existence theorem first noticed in Ref. Broeckhove and
Deumens (1979). Suppose $\mathscr{H}_{q}$ is a separable Hilbert subspace,
i.e. $\mathscr{H}_{q}\subseteq\mathscr{H}\subset\mathscr{L}^{2}$ where
$\mathscr{L}^{2}$ denotes the full Hilbert space associated with the space of
square integrable functions, the separability property of $\mathscr{H}_{q}$
implies the existence of a countable family

$\mathit{\Gamma}_{0}=\\{|\Phi(q_{i})\rangle\>|\>i\in\mathbb{N}\\}\subset\mathit{\Gamma}$
(6)

which is in general skew or non-orthogonal set of states with the property
$\mathscr{H}_{q}=\overline{\mathrm{span}\>\mathit{\Gamma}_{0}}$ (cf. the
detailed demonstration given in Appendix of Ref. Broeckhove and Deumens
(1979)). This enables us to tackle now the diagonalization of
$\hat{\mathcal{H}}$ in the subspace $\mathscr{H}_{q}$ represented by the
discrete non-orthogonal basis set $\mathit{\Gamma}_{0}$. Indeed, by expressing
the eigenstate $|\Psi\rangle$ as

$|\Psi\rangle=\sum_{i=0}^{\infty}f(q_{i})\>|\Phi(q_{i})\rangle,$ (7)

the projection of (1) in $\mathscr{H}_{q}$ becomes equivalent to the
generalized eigenvalue problem

$\sum_{i=0}^{\infty}\Big{[}\mathcal{H}(q_{i^{\prime}},q_{i})-E\;\mathcal{N}(q_{i^{\prime}},q_{i})\Big{]}f(q_{i})=0$
(8)

where
$\mathcal{O}(q_{i^{\prime}},q_{i})=\langle\Phi(q_{i^{\prime}})|\hat{\mathcal{O}}|\Phi(q_{i})\rangle$
($\hat{\mathcal{O}}=\hat{\mathcal{H}},\mathbf{1}$) are the Hamiltonian and
norm matrix elements and $f(q_{i})$ the expansion coefficient.

Therefore, instead of using the basis $\mathcal{B}$ spanning the full model
space $\mathscr{H}$, the above presented theorem on the existence (not
necessarily unique) of a discrete non-orthogonal basis set of
$\hat{\mathcal{H}}$ suggests that:

  1. 1.

the diagonalization of $\hat{\mathcal{H}}$ in $\mathscr{H}_{q}$ becomes
“exact” when $\mathscr{H}_{q}=\mathscr{H}$, which means the coordinate(s) $q$
must be chosen so as to “exhaust” in some way the space $\mathcal{E}$;

  2. 2.

the truncation of the infinite countable set $\mathit{\Gamma}_{0}$ could be
done in a variational way such that the finite sum

$|\Psi\rangle\approx\sum_{i=0}^{n}f(q_{i})|\Phi(q_{i})\rangle$ (9)

yields an optimal approximation.

Whether we are able to choose $q$ to fulfil the condition
$\mathscr{H}_{q}=\mathscr{H}$ can be verified a posteriori. What needed is
then an efficient truncation method of $\mathit{\Gamma}_{0}$, which we shall
address now.

####  II.1.3 Truncation method with the minimization technique

The existence theorem as presented previously has enabled us to transform the
classic Shell Model eigenvalue problem (4), formulated in the orthonormal
basis $\mathcal{B}$, into the generalized one (8) through the discrete non-
orthogonal basis $\mathit{\Gamma}_{0}$. As noted earlier by the authors of
Ref. Broeckhove and Deumens (1979), the minimization technique originally
proposed by E. Caurier in Ref. Caurier (1975) provides an iterative
prescription to truncate $\mathit{\Gamma}_{0}$ in a variational way.

It proceeds as follows Caurier (1975): “the first point $q_{0}$ is the one
such that $|\Phi(q_{0})\rangle$ minimizes the energy. The second point $q_{1}$
is chosen in such a way that the energy obtained from diagonalizing the
Hamiltonian in the $2$–dimensional space spanned by $|\Phi(q_{0})\rangle$ and
$|\Phi(q_{1})\rangle$ be a minimum. One proceeds in the same way to determine
the third basis vector $|\Phi(q_{2})\rangle$ etc…”.

The technique was however implemented only in toy model examples of Hydrogen
atom in molecular physics (cf. e.g. F. Arickx and Leuven (1981)) and has never
been considered in realistic nuclear structure calculations. Therefore,
instead of the conventional problem represented by
$(\hat{\mathcal{H}},\mathscr{H},\mathcal{B})$, our current work exploits fully
this technique for the first time in the Shell Model framework formulated in
terms of $(\hat{\mathcal{H}},\mathscr{H}_{q},\mathit{\Gamma}_{0})$. This
resulting model will be from now on referred to as Discrete Non-Orthogonal
Shell Model (DNO-SM).

To be now more precise in the practical realization of the DNO-SM, we assume
that the Projected Constrained Hartree–Fock (PCHF) approach provides us with a
basis generation method which will be our focus in the next subsection.
However, before going further, let us note that, the above proposed technique
suggested to minimize one state at a time. More generally in order to cover
broader physical situations such as nuclear coexistences where the potential
energy surface may exhibit several local minima or when states are not of
collective nature, it may be more preferable to obtain various excited states
by a single minimization. A generalization to deal with the minimization of
many states simultaneously shall be presented.

###  II.2 Projected constrained Hartree-Fock basis

Having introduced the general framework of our approach to the dimensionality
problem encountered in the classic Shell Model, we present now the
construction of the many-body basis in the DNO-SM. The choice of degrees of
freedom here is important to take into account correlations as much as
possible in the generation of the basis. This should be inferred on physical
grounds. Moreover, the many-body basis must conserve important symmetries of
the effective Hamiltonian, in particular, the cases associated with conserved
quantities such as the angular momentum and particle numbers. Such basis could
be built upon the Constrained Hartree-Fock (CHF) method which relies on the
rotational symmetry breaking at the mean field to incorporate deformations. A
projection onto good angular momentum can be applied later before proceeding
to the full diagonalization.

A Hartree-Fock (HF) state
$|\Phi(q)\rangle=\displaystyle\prod_{i=1}^{A}a^{\dagger}_{i}|0\rangle$ for a
nucleus of $A$–particles is obtained from CHF calculations under the
conditions

$\displaystyle\frac{1}{2}\langle\Phi|\hat{Q}_{\lambda\mu}+(-)^{\mu}\hat{Q}_{\lambda-\mu}|\Phi\rangle=Q_{\lambda\mu},$
(10)
$\displaystyle\langle\Phi|\hat{J}_{m}|\Phi\rangle=\langle\hat{J}_{m}\rangle\>\>(m=x,z),$
(11)

where $\hat{Q}_{\lambda\mu}=r^{\lambda}Y_{\lambda\mu}(\theta,\varphi)$ is the
multipole operator expressed in terms of the spherical harmonics
$Y_{\lambda\mu}$ and $\hat{J}_{m}$ the components of the total angular
momentum operator $\hat{\mathbf{J}}$. Here $Q_{\lambda\mu}$ and
$\langle\hat{J}_{m}\rangle$ are desired constraining expectation values. Our
numerical implementations of constrained HF calculations follow the standard
modified Broyden method as described in Refs. Johnson (1988); Eyert (1996);
Baran et al. (2008). Furthermore, to ensure the correct constraining value
along the iterative HF process, we use the augmented Lagrange method proposed
in Ref. Staszczak, A. et al. (2010).

Once this is done, the resulting HF state is projected onto good angular
momentum $J$ through the usual procedure using the projection operator Lamme
and Boeker (1968); Ripka (1968)

$\displaystyle\mathcal{P}^{J}_{MK}(A)=\frac{2J+1}{4\pi^{2}\big{(}3-(-)^{A}\big{)}}\times$
(12)
$\displaystyle\int_{0}^{2\pi}d\alpha\int_{0}^{\pi}d\beta\int_{0}^{\gamma_{\rm
max}}d\gamma\>\mathcal{D}^{J*}_{MK}(\alpha,\beta,\gamma)\>\hat{R}(\alpha,\beta,\gamma),$

where $\gamma_{\rm max}=\big{(}3-(-)^{A}\big{)}\pi$,
$\hat{R}(\alpha,\beta,\gamma)$ and
$\mathcal{D}^{J*}_{MK}(\alpha,\beta,\gamma)$ denote the rotation operator and
the Wigner matrix Messiah (1965). The case of $\gamma_{\rm max}=4\pi$
corresponds to a HF state describing odd systems. Thus we have included the
dependence of the projection operator on the mass number $A$ for convenience.
This provides us a family of PCHF states characterized by the angular momentum
projection onto the intrinsic axis $|K|\leq J$ and the coordinate $q$ for a
given $J$

$\mathit{\Gamma}=\\{\mathcal{P}^{J}_{MK}(A)|\Phi(q)\rangle\>|\>q\in\mathbb{R}\\}$
(13)

with which we can now formulate the DNO-SM’s working equations.

The HF procedure to generate deformed Slater determinants in the construction
of DNO-SM basis is implemented to treat both odd– and even–nuclei without
further assumptions. More specifically, we do not impose any self–consistent
symmetries (e.g. no time-reversal, no parity conservation) at the HF mean
field to exploit at best what is offered by the single-particle valence space
$\mathcal{E}$. The construction of DNO-SM basis for odd nuclei is done via the
constrain of angular momentum components $\hat{J}_{m}\>(m=x,z)$ which, as we
will show later, can provide a very good many-body basis for such nuclei. The
formal aspect of this approach to treat odd-mass nuclei has been pointed out
in Ref. Bertsch et al. (2009). More recently, the theoretical demonstration in
the Hartree-Fock-Bogoliubov framework is described in Ref. Kasuya and Yoshida
(2021).

Since there is no self-consistent symmetries adopted here, the angular
momentum projection demands to perform the integration over Euler angles
$\Omega=(\alpha,\beta,\gamma)$ in full intervals without restrictions. To do
so, we have developed an analytical formula that performs an exact integration
over $(\alpha,\gamma)$ whose derivation is presented in Appendix B. The
integration over $\beta$ is done numerically using the Gauss–Legendre
quadrature rule.

###  II.3 DNO-SM formalism

####  II.3.1 Secular equation in non-orthogonal PCHF basis

Let us start with the ansatz (7) where we specify the eigenstate of
$\hat{\mathcal{H}}$ by $|\alpha JM\rangle$ of good total angular momentum $J$,
its projection $M$ in the laboratory frame and an index $\alpha$ labelling
indices of corresponding energy levels and other quantum numbers. In terms of
the PCHF states
$\mathcal{P}^{J}_{MK}|\Phi(q)\rangle\in\mathit{\Gamma}_{0}\subset\Gamma$, it
is given by

$|\alpha
JM\rangle=\sum_{q,K}f^{(J)}_{\alpha}(q;K)\>\mathcal{P}^{J}_{MK}|\Phi(q)\rangle.$
(14)

To simplify the notation, we omit the dependence on mass number $A$ in the
projection operator $\mathcal{P}^{J}_{MK}$ and $q$ is understood to take
discrete values. The projected equation (8) then becomes

$\sum_{q,K}\bigg{[}\mathcal{H}^{J}_{K^{\prime}K}(q^{\prime},q)-E^{(J)}_{\alpha}\mathcal{N}^{J}_{K^{\prime}K}(q^{\prime},q)\bigg{]}f^{(J)}_{\alpha}(q;K)=0$
(15)

where the Hamiltonian and the norm matrix elements
$\mathcal{O}^{(J)}_{K^{\prime}K}(q^{\prime},q)=\langle\Phi(q^{\prime})|\hat{\mathcal{O}}\,\mathcal{P}^{J}_{K^{\prime}K}|\Phi(q)\rangle$
(with $\hat{\mathcal{O}}=\hat{\mathcal{H}},\mathbf{1}$) are evaluated through
a three-fold integration over Euler angles $\Omega=(\alpha,\beta,\gamma)$

$\displaystyle\mathcal{O}^{(J)}_{K^{\prime}K}(q^{\prime},q)$
$\displaystyle=\frac{2J+1}{4\pi^{2}\big{(}3-(-)^{A}\big{)}}\times$ (16)
$\displaystyle\int
d\Omega\,\mathcal{D}^{J*}_{MK}(\Omega)\>\langle\Phi(q^{\prime})|\hat{\mathcal{O}}\hat{R}(\Omega)|\Phi(q)\rangle.$

The core of (16) is in the evaluation of the kernels
$\langle\Phi(q^{\prime})|\hat{\mathcal{O}}\hat{R}(\Omega)|\Phi(q)\rangle$ for
given pair of Slater determinants. For these calculations, we use the minor
formula as presented in Ref. Watt (1972). This is given in Appendix A. The
matrix element $\mathcal{O}^{(J)}_{K^{\prime}K}(q^{\prime},q)$ obtained from
the Euler angles integration is then presented in Appendix B.

The treatment of the generalized eigenvalue problem (15) has been well
documented in the framework of the generator coordinate method (see e.g. Refs.
Rodríguez and Egido (2010); Bender and Heenen (2008); Burzyński and
Dobaczewski (1995)). We follow the standard technique that begins with the
determination of the natural eigenbasis functions $u^{(J)}_{i}(q;K)$ of the
norm matrix

$\sum_{q,K}\mathcal{N}^{J}_{K^{\prime}K}(q^{\prime},q)\>u^{(J)}_{i}(q;K)=\eta^{(J)}_{i}\>u^{(J)}_{i}(q^{\prime};K^{\prime}).$
(17)

By retaining only positive norm eigenvalues that we denote by
$\\{\eta^{(J)}_{i}>0,\>i\in\mathbb{N}\\}$, the so-called natural state
characterized by the corresponding norm eigenvalue $\eta^{(J)}_{i}$ is defined
as

$\displaystyle|\eta^{(JM)}_{i}\rangle=\frac{1}{\sqrt{\eta^{(J)}_{i}}}\sum_{q,K}u^{(J)}_{i}(q;K)\mathcal{P}^{J}_{MK}|\Phi(q)\rangle,$
(18)
$\displaystyle\langle\eta^{(JM)}_{i^{\prime}}|\eta^{(JM)}_{i}\rangle=\delta_{i^{\prime}i}$

and satisfies the orthogonality condition. This natural basis allows to
transform the projected equation in the non-orthogonal PCHF basis onto the
usual eigenvalue value problem of the form

$\sum_{i^{\prime}}\mathcal{H}^{(J)}_{i^{\prime}i}\>g^{(J)}_{i^{\prime}}=E^{(J)}_{\alpha}\>g^{(J)}_{i}$
(19)

where the Hamiltonian matrix now is expressed between orthogonal natural basis
states $\\{|\eta^{(JM)}_{i}\rangle\\}$

$\displaystyle\mathcal{H}^{(J)}_{i^{\prime}i}=\frac{1}{\sqrt{\eta^{(J)}_{i^{\prime}}\eta^{(J)}_{i}}}\sum_{q^{\prime}K^{\prime},qK}\>u^{(J)*}_{i^{\prime}}(q^{\prime};K^{\prime})\times$
(20)
$\displaystyle\mathcal{H}^{J}_{K^{\prime}K}(q^{\prime},q)\>u^{(J)}_{i}(q;K).$

The nuclear state $|\alpha JM\rangle$ is thus a linear superposition in the
natural basis

$|\alpha JM\rangle=\sum_{i}\;g^{(J)}_{i}\>|\eta^{(JM)}_{i}\rangle$ (21)

with the transformation onto the non-orthogonal PCHF basis expressed through
the expansion coefficient $f^{(J)}_{\alpha}(q;K)$ of (14)

$f^{(J)}_{\alpha}(q;K)=\sum_{i}\;\frac{g^{(J)}_{i}}{\sqrt{\eta^{(J)}_{i}}}\>u^{(J)}_{i}(q;K).$
(22)

One can notice that $f^{(J)}_{\alpha}(q;K)$ is not properly normalized and
does not represent the probability amplitude of finding a given configuration
of $\mathit{\Gamma}_{0}$. To be able to analyze the content of the nuclear
states, we can define the normalized probability amplitude to find a component
$\mathcal{P}^{J}_{MK}|\Phi(q)\rangle$ in the state $|\alpha JM\rangle$ by Ring
and Schuck (1980); Rodríguez and Egido (2010)

$\mathcal{M}^{(J)}_{\alpha}(q;K)=\sum_{q^{\prime},K^{\prime}}\;\big{[}\hat{\mathcal{N}}^{1/2}\big{]}^{(J)}_{K^{\prime}K}(q^{\prime},q)\>f^{(J)}_{\alpha}(q^{\prime};K^{\prime}).$
(23)

The corresponding probability to find the intrinsic angular momentum component
$K$ or the component $q$ in the state $|\alpha JM\rangle$ is respectively
given by

$\displaystyle\mathcal{P}^{(J)}_{\alpha}(K)=\sum_{q}\Big{|}\mathcal{M}^{(J)}_{\alpha}(q;K)\Big{|}^{2},$
(24)
$\displaystyle\mathcal{P}^{(J)}_{\alpha}(q)=\sum_{K}\Big{|}\mathcal{M}^{(J)}_{\alpha}(q;K)\Big{|}^{2},$

with the normalization relation

$\sum_{K}\mathcal{P}^{(J)}_{\alpha}(K)=\sum_{q}\mathcal{P}^{(J)}_{\alpha}(q)=1.$
(25)

####  II.3.2 Truncation of the PCHF basis with the generalized minimization
technique

We come now to the truncation of the discrete family $\mathit{\Gamma}_{0}$
with the minimization technique. The idea about how to choose states of
$\mathit{\Gamma}_{0}$ is quite straightforward. That is, applying to the
present case including the angular momentum projection, based on the selection
of discrete values of the coordinate(s) $q$ which minimizes the energy, i.e.
we let the Hamiltonian itself to choose what is the best state from a
variational viewpoint. Furthermore, it is implicitly understood that the
ground state or an excited state has to be chosen beforehand. Then the
following iterative procedure can be implemented:

  1. 1)

Fix the state $E^{(J)}_{\alpha}$ to be minimized with $\alpha$ indexing energy
levels;

  2. 2)

Define a searching region of the coordinate(s) $q$;

  3. 3)

Start from the first point which can be chosen as the HF minimum;

  4. 4)

Solve the projected Shell Model circular equation (15) over the whole
searching region of $q$ to find the second state and proceed the same way in
next iterations until the convergence of $E^{(J)}_{\alpha}$.

More precisely, the convergence criterion is defined by the absolute energy
gain $\Delta E^{(J)}_{\alpha}(k)=E^{(J)}_{\alpha}(k)-E^{(J)}_{\alpha}(k-1)$ at
the iteration number $k$. If $\Delta E^{(J)}_{\alpha}(k)>\epsilon>0$ we keep
the CHF state, otherwise it is not retained. The minimization will stop when
no more states are found in the searching region.

In practice, we observe that fixing $(\alpha,J)$ will exclude in the
minimization the states of $\mathit{\Gamma}_{0}$ which could be relevant for
an other state $(\alpha^{\prime},J^{\prime})$. Hence, small components of the
wave functions could be missed whereas the overall spectrum remains well
described. The key point is thus, minimizing as many excited states at the
same time as possible will eventually lead to the improvement of one and
another mutually and also the ground state. This idea leads us to generalize
the above iterative procedure in the following way: we let the minimization
process to determine not only the coordinate(s) $q$ but also the state
$(\alpha,J)$ that gains the most energy at a given iteration. This is done by
comparing the energy differences $\\{\Delta E^{(J)}_{\alpha}\\}$ for every
state $(\alpha,J)$ in a given set of nuclear states which we want to describe.
The procedure then continues until we find no more states satisfying the
condition $\mathrm{max}\\{\Delta E^{(J)}_{\alpha}\\}>\epsilon$, which defines
the convergence criterion in this scheme. We will comment on the choice of
$\epsilon$ in the following.

The minimization procedure as such requires an organizational scheme of
partitioning the coordinate(s) $q$, which form in general a multi-dimensional
surface. Although it is possible to determine all of them simultaneously in
principle, it might not be necessary to do so. The reason is that, as a
consequence of the existence theorem, different countable sets
$\mathit{\Gamma}_{0}\subset\Gamma$ could be qualified as basis for
$\mathscr{H}_{q}$. In this work, we define the following organization of the
coordinate(s) $q$. Only the deformation parameters $Q_{\lambda\mu}$ are
determined from the minimization process. The cranking components
$\langle\hat{J}_{m}\rangle$ ($m=x,z$) are not. Instead, they are fixed in
advance. That is, we perform the minimization in the deformation surface
$\\{Q_{\lambda\mu}\\}$ associated to each value of
$\langle\hat{J}_{m}\rangle=J^{(1)}_{m},J^{(2)}_{m},J^{(3)}_{m},...$ ordered
from input.

####  II.3.3 Choice of the coordinate(s) $q$

In order the generate $\mathit{\Gamma}_{0}$, in this present work, we limit
ourselves to quadrupole deformations (axial and triaxial) as the common choice
of generator coordinates for triaxial systems Ring and Schuck (1980), whose
expectation values in a HF state $|\Phi\rangle$ are denoted by $Q_{20},Q_{22}$
respectively

$\displaystyle Q_{20}$
$\displaystyle=\sqrt{\frac{16\pi}{5}}\,\sum_{\tau=p,n}\>e^{(\tau)}_{mass}\>\langle\Phi|\hat{Q}^{(\tau)}_{20}|\Phi\rangle$
(26) $\displaystyle Q_{22}$
$\displaystyle=\sqrt{\frac{8\pi}{5}}\,\sum_{\tau=p,n}\>e^{(\tau)}_{mass}\>\langle\Phi|\big{(}\hat{Q}^{(\tau)}_{2-2}+\hat{Q}^{(\tau)}_{22}\big{)}|\Phi\rangle.$
(27)

In this particular case, we use the usual Hill-Wheeler $(\beta,\gamma)$
parameters Hill and Wheeler (1953) which are related to $(Q_{20},Q_{22})$
through the total quadrupole moment $Q=\sqrt{Q_{20}^{2}+Q_{22}^{2}}$

$\beta=\frac{b^{2}Q\sqrt{5\pi}}{3r_{0}^{2}A^{5/3}},\>\>\gamma=\arctan\big{(}\frac{Q_{22}}{Q_{20}}\big{)}$
(28)

where $b^{2}$ (in fm2) is the harmonics oscillator parameter Brussard and
Glaudemans (1977); Suhonen (2007); Zuker et al. (2015)

$b^{2}=\frac{41.4}{45A^{-1/3}-25A^{-2/3}},$ (29)

and $A$ and $r_{0}=1.2$ (in fm) are the nuclear mass number and radius
parameter.

Within the shell-model formalism, the use of valence spaces and truncation of
the Hilbert space implies the need of effective hamiltonians as well as
effective operators. Effective operators can be derived by Many-Body
Perturbation Theory Coraggio and Itaco (2020) but quadrupole operators are
usually renormalized with the use of an effective charge. The effective charge
have been defined by several authors, we use here the notation defined in Ref.
Suhonen (2007):

$e_{el}^{(p)}=(1+\chi_{p})e,\>\>\>e_{el}^{(n)}=\chi_{n}e$ (30)

with $\chi$ being the electric polarization charge. $\chi$ value is intimately
connected to the valence space used and can be derived by Many-Body
Perturbation Theory Coraggio and Itaco (2020) and is shell dependent but for
$0\hbar\omega$ spaces, the microscopic Dufour-Zuker Dufour and Zuker (1996)
$\chi_{p}=0.31$, $\chi_{n}=0.46$ are standard values. Finally, for the mass
quadrupole operators, we will use

$e_{mass}^{(p)}=e_{mass}^{(n)}=(1+\chi_{p}+\chi_{n})e$ (31)

to be consistent with the deformation parameters defined above.

##  III Benchmark and comparison in sd nuclei

###  III.1 Even nuclei

To assess the quality of the PCHF basis $\mathit{\Gamma}_{0}$ against the
oscillator one in the classic Shell Model, we perform a systematic comparison
of two models in even sd nuclei using the USDB effective interaction Brown and
Richter (2006).

In the first series of calculations shown in Table 2, to see the overall
performance of our minimization method, we compare the ground state and the
first excited states $2^{+}_{1},4^{+}_{1}$ energies as well as the
corresponding $E2$–transition probabilities for Neon, Magnesium, Silicon and
Argon even-even isotopes. These calculations are performed using: i) the
common set of states
$J_{\alpha}\in\\{0_{1},2_{1,2},4_{1},8_{1},12_{1},16_{1}\\}$ whenever
possible; ii) practically the same $(\beta,\gamma)$ mesh for all considered
nuclei (cf. Table 1); iii) the same set of cranking components
$\langle\hat{J}_{z}\rangle\in\\{-2,-4,-8,-12,-16\\}$ and iv) the same number
of integration points in Angular Momentum Projection, namely, $11$ points for
the analytical integration over the angles $\alpha,\gamma\in[0,2\pi]$ and $20$
Gauss–Legendre quadrature points for the integration over the angle
$\beta\in[0,\pi]$. After a first round of minimization to find non-cranked CHF
states over the range $\gamma\in[0^{\circ},60^{\circ}]$ discretized into a
one-dimensional mesh of $N_{g}$ points, the minimization procedure is then
iterated for each above-mentioned cranking components
$\langle\hat{J}_{z}\rangle$.
As we can see in Table 2, the agreement between two models is excellent for
both relative energies and $E2$–transition probabilities, indicating that the
wave functions have already converged for the considered states. The number of
CHF basis states found by the minimization process is relatively small at the
beginning and the end of the shell and increases as we go to nuclei at mid
shell. Such are the cases of deformed open shell nuclei 24Mg, 28Si which
seemingly require more basis states to capture correlations in the mixing than
in others. In the present calculations, a similar trend is exhibited in the
ground state binding energy difference (of order $\sim 0.1-1.0$ MeV) with
respect to the exact result (shown in Figure 1). There are two possible
reasons for this underbinding:
$\bullet$ Our minimization technique builds up the many-body basis based on a
finite set of states preliminarily defined, in the present case, i.e. the set
$J_{\alpha}\in\\{0_{1},2_{1,2},4_{1},8_{1},12_{1},16_{1}\\}$, hence,
indirectly omits basis states possibly relevant for other states not included
in the set. Therefore, the representation of the effective Hamiltonian in the
truncated basis $\mathit{\Gamma}_{0}$ is not fully complete as the
construction is under way. This is in contrast to the full SM diagonalization
in the oscillator basis because all many-body matrix elements of the effective
Hamiltonian are available once the valence space is defined.
$\bullet$ The second reason may be related to the choice of the coordinate(s)
$q$. Here, the question that arises is which $q$ to be used to ensure a priori
the full representation of the effective Hamiltonian.

Figure 1:  Energy difference (in keV) between the DNO-SM and the exact SM diagonalizations in the ground state of even Ne, Mg and Si isotopes from Table 2. Table 1:  Discretization of the $(\beta,\gamma)$ plane for minimization procedure for selected sd nuclei using the USDB effective interaction. The Hartree-Fock minimum energy $(E_{\rm HF})$ and corresponding mass quadrupole parameters $(\beta_{\rm min},\gamma_{\rm min})$ are also given. $\gamma_{\rm min}$ is in degree. The effective charges for protons and neutrons are $e^{(p)}_{mass}=e^{(n)}_{mass}=1.77$ (see EQ.(31)). $\beta$ | $N_{b}$ | $N_{g}$ | nucleus | $\beta_{\rm min}$ |  $\gamma_{\rm min}$ (∘) |  $E_{\rm HF}$ (MeV)
---|---|---|---|---|---|---
$[0.1,0.7]$ | $11$ | $9$ |  20Ne | $0.527$ | $0.0$ | $-36.404$
22Ne | $0.498$ | $0.0$ | $-53.474$
$[0.1,0.53]$ | $11$ | $9$ |  24Ne | $0.313$ | $23.8$ | $-66.225$
$[0.1,0.48]$ | $7$ | $9$ |  26Ne | $0.247$ | $0.0$ | $-77.258$
28Ne | $0.198$ | $0.0$ | $-83.441$
24Mg | $0.499$ | $12.0$ | $-80.965$
26Mg | $0.395$ | $34.0$ | $-98.887$
28Mg | $0.325$ | $0.0$ | $-115.625$
30Mg | $0.212$ | $0.0$ | $-126.735$
28Si | $0.425$ | $60.0$ | $-130.021$
30Si | $0.285$ | $46.8$ | $-148.238$
32Si | $0.221$ | $60.0$ | $-166.344$
32S | $0.0$ | $-$ | $-176.393$
|  |  |  34S | $0.152$ | $60.0$ | $-198.097$
|  |  |  36Ar | $0.199$ | $60.0$ | $-226.611$
Table 2: Systematics comparison of the diagonalization in the model spaces $\mathscr{H}_{q}$ and $\mathscr{H}$ of the Shell Model in even sd nuclei. The absolute ground state energy $E_{\rm gs}$ and relative energies of the first $2^{+}$ and $4^{+}$ are given in MeV. Their reduced transition probabilities $B(E2;2^{+}_{1}\to 0^{+}_{1})$ and $B(E2;4^{+}_{1}\to 2^{+}_{1})$ are in e2.fm4 unit. $N_{q}$ denotes the number of CHF states $|\Phi(q)\rangle$ found with the minimization procedure. nucleus |  | $\mathscr{H}_{q}$ |  | $\mathscr{H}$ |  |  $E_{\rm gs}$ (MeV) |  | $B(E2;2^{+}_{1}\to 0^{+}_{1})$ |  | $B(E2;4^{+}_{1}\to 2^{+}_{1})$
---|---|---|---|---|---|---|---|---|---|---
| $N_{q}$ | $2^{+}_{1}$ | $4^{+}_{1}$ |  | $2^{+}_{1}$ | $4^{+}_{1}$ |  | $\mathscr{H}_{q}$ | $\mathscr{H}$ |  | $\mathscr{H}_{q}$ | $\mathscr{H}$ |  | $\mathscr{H}_{q}$ | $\mathscr{H}$
20Ne |  | $16$ | $1.76$ | $4.14$ |  | $1.75$ | $4.18$ |  | $-40.40$ | $-40.47$ |  | $47.0$ | $46.9$ |  | $56.3$ | $55.3$
22Ne |  | $41$ | $1.37$ | $3.39$ |  | $1.36$ | $3.36$ |  | $-57.32$ | $-57.58$ |  | $48.1$ | $46.9$ |  | $64.0$ | $63.3$
24Ne |  | $39$ | $2.13$ | $4.01$ |  | $2.11$ | $3.99$ |  | $-71.26$ | $-71.72$ |  | $38.8$ | $38.7$ |  | $31.9$ | $31.3$
26Ne |  | $26$ | $2.12$ | $3.72$ |  | $2.06$ | $3.51$ |  | $-80.98$ | $-81.56$ |  | $39.6$ | $38.5$ |  | $37.8$ | $33.8$
28Ne |  | $12$ | $1.55$ | $2.82$ |  | $1.62$ | $2.99$ |  | $-86.13$ | $-86.54$ |  | $34.3$ | $34.1$ |  | $32.7$ | $30.7$
24Mg |  | $42$ | $1.52$ | $4.37$ |  | $1.50$ | $4.37$ |  | $-86.82$ | $-87.10$ |  | $76.1$ | $74.4$ |  | $99.1$ | $97.5$
26Mg |  | $50$ | $1.80$ | $4.36$ |  | $1.89$ | $4.36$ |  | $-104.56$ | $-105.52$ |  | $66.1$ | $65.2$ |  | $27.5$ | $18.0$
28Mg |  | $50$ | $1.40$ | $4.07$ |  | $1.52$ | $4.17$ |  | $-119.71$ | $-120.50$ |  | $62.9$ | $60.2$ |  | $75.7$ | $67.5$
30Mg |  | $21$ | $1.50$ | $3.89$ |  | $1.59$ | $3.89$ |  | $-129.77$ | $-130.47$ |  | $53.0$ | $49.1$ |  | $39.6$ | $32.5$
28Si |  | $71$ | $2.12$ | $4.78$ |  | $1.93$ | $4.61$ |  | $-135.54$ | $-135.86$ |  | $77.4$ | $77.9$ |  | $106.8$ | $109.6$
30Si |  | $46$ | $2.25$ | $5.59$ |  | $2.26$ | $5.33$ |  | $-154.09$ | $-154.75$ |  | $46.9$ | $45.9$ |  | $54.0$ | $15.8$
32Si |  | $43$ | $1.99$ | $5.79$ |  | $2.05$ | $5.88$ |  | $-170.06$ | $-170.52$ |  | $41.1$ | $42.5$ |  | $66.9$ | $65.2$
32S |  | $50$ | $2.05$ | $4.55$ |  | $2.16$ | $4.65$ |  | $-182.10$ | $-182.45$ |  | $48.0$ | $46.9$ |  | $69.2$ | $66.8$
34S |  | $36$ | $2.01$ | $4.66$ |  | $2.13$ | $4.83$ |  | $-202.08$ | $-202.50$ |  | $38.3$ | $36.0$ |  | $51.7$ | $48.0$
36Ar |  | $12$ | $1.81$ | $4.46$ |  | $1.82$ | $4.49$ |  | $-230.22$ | $-230.28$ |  | $50.4$ | $50.6$ |  | $63.2$ | $62.9$

###  III.2 Detailed analysis of 24Mg

In the second series of calculations, we investigate the specific case of
24Mg, which is a typical example of deformed nuclei in the sd shell. This
nucleus was extensively studied in the past within various approaches
Rodríguez and Egido (2010); Bender and Heenen (2008); Yao et al. (2009); Gao
and Horoi (2009); Yao et al. (2010). It is well established that it has a
triaxial shape in its ground state with a rotational band built on top.
Moreover, the triaxiality manifests itself in the existence of the so-called
$\gamma$–band that was experimentally observed. In order to examine whether
our method is able to correctly describe these features, we explore different
minimization schemes as proposed in II.3.2.

To fix ideas, let us first mention that the present calculations of 24Mg as
shown in Figure 3 are performed using the same $(\beta,\gamma)$ discretization
as in Table 1 and the cranking components
$\langle\hat{J}_{z}\rangle\in\\{0,-2,-3,-4,-5\\}$. Results reported in the
figures 3b), 3c) and 3d) are obtained from the ground-state and many states
minimizations in comparison with the full mesh diagonalization 3e) and the
exact SM diagonalization 3f). The searching domain for minimizations is also
the dimension of the full mesh calculation which is a priori $7\times 9\times
5=315$. In practice, our Hartree-Fock calculations found $277$ converged
solutions.

From Figure 3 the first thing we note is that Angular Momentum Projection from
the HF minimum gives already very good intraband transitions. In contrast, the
interband transitions $BE(2;2^{+}_{2}\rightarrow 2^{+}_{1})$ or
$BE(2;6^{+}_{1}\rightarrow 5^{+}_{1})$ which reflect the $K$-mixing of the two
bands are significantly underestimated. In minimizing the ground state alone,
the interband transitions are considerably improved and consistent with the
exact SM result using only $6$ basis states. This means we have already the
right wave functions with the mixing of a few number of CHF states, even
though about $\sim 600$ keV is still missing in the ground state binding.
Carrying out the same minimization of the ground state until no more CHF basis
vectors are found in the full mesh, we obtain $39$ states that perfectly
reproduce the exact SM result. This calculation also confirms that in the
ground-state minimization, $6$ CHF vectors is enough to have good solutions in
this nucleus.

Now to see whether we can improve further by minimizing many states at the
same time, as shown in the figure 3d), we find $55$ CHF states that provide an
overall better energy spectra than the ground-state minimization but does not
change much the picture. The excited states in the figure 3d) are slightly
more compressed. This effect, as shown in Figure 5, can be traced back to the
addition of cranked CHF states which are known to stretch down the relative
spectrum, although in the present calculation the effect is not as huge as
noted in e.g. Ref. Borrajo et al. (2015).

The first difference between the two minimization schemes are in the absolute
binding energies as reported in Table 4. Minimizing all the states provides
solutions with lower bindings which are, in addition, in excellent agreement
with the full mesh diagonalization. The second difference that we observe is
in the state decomposition in the $(\beta,\gamma)$ plane. In Figures 2a), 2b)
and 2c), for illustration, we show the contributions of CHF basis vectors into
the ground state wave function in the potential energy surface of
$\langle\hat{J}_{z}\rangle=0$. By comparing the figures 2a) and 2b), it is
worth pointed out that the two minimizations produce two sets of basis vectors
which give energy spectra and transitions of the same quality. The
contributions of CHF states in the figure 2b) are somehow fragmented into
other configurations in the figure 2a). The effect is even more visible in the
full mesh calculation shown in the figure 2c1), which means there is a
redistribution of contributions in the final wave function when one uses
”redundant” basis states. Now, what we observe in the figure 2c2) is the key
point: the dominant CHF configurations in the full mesh diagonalization are
lying in the same region as in the minimization. In Table 3, we report some of
the most dominant configurations in the ground state as shown in Figure 2. The
most important configurations are around the Hartree-Fock minimum and they
appear in both minimization calculations. In the full mesh calculation, apart
from a scaling effect due to the presence of many other states, it is the same
CHF configurations which contribute the most. This clearly shows that our
minimization method picks up the ”right” physical configurations of the most
importance. This comparison also explains the reason why the minimization
using $6$ states gives already a good description.

We summarize now the essential points which can be drawn from the present
discussion:
i) There can be different sets of CHF states that provide the same description
of the relative energy spectra and transitions.
ii) There exists specific CHF states that contribute more significantly than
others. And the minimization tells us where to find them.
iii) One must pay attention to the interpretation because of i), it is
possible to replace some set of basis states with an other, although it could
be at the price of taking into account more basis states than necessary.
iv) It is more advantageous to perform the minimization than the full mesh
diagonalization since basis states are interdependent in the sense of iii). In
cases where such calculation is feasible like in the present one, the full
mesh diagonalization should expose the limit one obtains in the minimization.
By the way, it is worth noting here that in a calculation from Monte-Carlo
Shell Model of Ref. Honma et al. (1996) employing a stochastic sampling method
to choose basis states, $800$ Hartree-Fock states stochastically chosen give
$-86.91$ MeV of ground state binding for 24Mg (versus the exact value $-87.08$
MeV) with the USD interaction.
v) Finally, let us comment on a technical point regarding the choice of
$\epsilon$ defining the convergence criterion of the method. For the present
study, we use $\epsilon=1$ keV. Such value seems to be too strict but is
necessary to push the method to its limit and to see whether it reproduces the
exact SM result. For that goal, this value is sufficient to avoid linear
redundancies in the selection of CHF states. The redundancies among non-
orthogonal states are very well known in the generator coordinate method and
resulting from the nearly zero norm eigenvalues of the overlap matrix. In the
minimization process, they manifest themselves through very tiny contributions
in energy of order $\lesssim 10^{-1}$ keV. We therefore exclude basis states
yielding those negligible contributions.

$\beta$ | $\gamma$ |  |  $\mathcal{P}^{(J)}_{\alpha}(\beta,\gamma)$ ($\%$)
---|---|---|---
| a) ground-state | b) Many-state | c) Full
$0.499$ | $12.0$ |  | $5.62$ | $7.43$ | $1.16$
$0.540$ | $17.74$ |  | $14.35$ | $15.58$ | $2.97$
$0.540$ | $11.85$ |  | $4.71$ | $14.64$ | $1.30$
$0.414$ | $17.74$ |  | $7.28$ | - | $1.44$
$0.351$ | $23.63$ |  | $5.25$ | - | $1.23$
$0.414$ | $11.85$ |  | $3.88$ | $7.54$ | $0.66$
$0.414$ | $29.52$ |  | - | $3.39$ | $0.61$
$0.477$ | $17.74$ |  | - | - | $1.38$
$0.477$ | $23.63$ |  | - | - | $1.29$

Table 3:  Contributions of CHF basis states (calculated from (24)) into the
ground state wave function in three calculations shown in Figure 2a), 2b) and
2c) respectively. Figure 2:  Structure of the ground state depicted in the
potential energy surface of $\langle\hat{J}_{z}\rangle=0$. The yellow circles
(defined with the same scale) in a), b) and c1) represent the contribution of
CHF basis vectors used in the ground-state minimization, the many-state
minimization and the full mesh diagonalization respectively. In c2) the red
circles represent the same CHF basis vectors of c1) but using a scaling factor
of $5$ times larger (see discussion in texts).

$J^{\pi}_{\alpha}$ | $0_{1}^{+}$ | $2_{1}^{+}$ | $4_{1}^{+}$ | $6_{1}^{+}$ | $8_{1}^{+}$
---|---|---|---|---|---
Figure 3c) | $-86.828$ | $-85.288$ | $-82.430$ | $-78.508$ | $-75.028$
Figure 3d) | $-86.861$ | $-85.353$ | $-82.489$ | $-78.619$ | $-75.283$
Figure 3e) | $-86.831$ | $-85.322$ | $-82.483$ | $-78.580$ | $-75.249$

$J^{\pi}_{\alpha}$ | $2_{2}^{+}$ | $3_{1}^{+}$ | $4_{2}^{+}$ | $5_{1}^{+}$ | $6_{2}^{+}$ | $7_{1}^{+}$
---|---|---|---|---|---|---
Figure 3c) | $-82.713$ | $-81.647$ | $-80.744$ | $-78.905$ | $-77.274$ | $-74.576$
Figure 3d) | $-82.748$ | $-81.695$ | $-80.869$ | $-79.018$ | $-77.351$ | $-74.661$
Figure 3e) | $-82.761$ | $-81.724$ | $-80.843$ | $-78.992$ | $-77.370$ | $-74.677$

Table 4:  Comparison of binding energies (in MeV) obtained from the
minimization of the ground state, of many states and the full mesh
diagonalization in Figures 3c), 3d) and 3e) respectively. Figure 3:  24Mg
spectrum calculated with the USDB interaction using the DNO-SM compared to the
classic SM diagonalization. Black numbers are relative energies (in keV) and
blue ones the $B(E2)$ values (in e2.fm4). $K$ denotes the dominant
wave–function component of different members of the band. Figures on top show
the evolution of the energy spectra between a) the Angular Momentum Projection
of the Hartree-Fock minimum, b) and c) the minimization of the ground state,
and d) the minimization of all considered states. Figures at bottom present e)
the DNO-SM diagonalization in the full $7\times 9\times 5$
$(\beta,\gamma,\langle\hat{J}_{z}\rangle)$ mesh (see texts) and f) the exact
SM diagonalization. The ground state binding energy $(E_{\rm gs})$ is also
given for comparison.

Figure 4:  Convergence of the binding energies ($E$) (top left) where the
horizontal lines represent the exact SM values, the binding energy difference
($\Delta E$) (top right) with respect to the exact SM result and the $B(E2)$
transitions (bottom) as a function of CHF states in the many-state
minimization shown in Figure 3d).

Figure 5:  Evolution of the ground state band (top) and the $\gamma$-band
(bottom) in 24Mg as cranked CHF states are added in the many-state
minimization of Figure 3d).

To look further now into the convergence process, we present in Figure 4 the
evolution of absolute binding energies, the energy difference with respect to
the exact SM result and the transitions as a function of the number of CHF
states found in the many-state minimization of Figure 3d). The intraband and
interband transitions converge after about $\sim 5$ iterations. In binding
energies, the excited state $8^{+}_{1}$ approaches the exact binding value
more rapidly than the ground state. This is the effect of cranked CHF
configurations and is consistent with the slight compression of the energy
spectra observed in Figure 3d), Figure 5 and Table 5.

Figure 6:  Structure of the first three yrast states $0^{+}_{1}$, $2^{+}_{1}$
and $4^{+}_{1}$ in the sextant $\gamma\in[0,60^{\circ}]$ of the
$(\beta,\gamma)$ plane.

In Figures 6 and 7, the deformation structure of the yrast and the
$\gamma$-band first three states is visualized in the potential energy
surfaces calculated with all cranking components
$\langle\hat{J}_{z}\rangle\in\\{0,-2,-3,-4,-5\\}$. As already shown in Figure
2, we see now in the full decomposition, the most important states emerge from
the triaxial region around the Hartree-Fock minimum.

Physically, the triaxiality developing here is often interpreted as
originating from the $K$-mixing nature of the ground state Ring and Schuck
(1980). This feature can be understood from the wave function analysis in
$K$-quantum numbers as presented in Figure 8 where contributions from
particular $K$-components are calculated by (24). Indeed, by construction
single-$K$ wave functions are fully preserved by the axial contraint. The
small $K$-mixing we observe in this analysis hence reflects the deviation from
a pure axial picture and therefore signal the presence of triaxial
configurations which we have previously analyzed. This mixing via the $K=0,2$
components also explains the weak interband transitions observed in the
B$(E2)$ values.

Figure 7:  Structure of the first three states in the $\gamma$-band
$2^{+}_{2}$, $3^{+}_{1}$ and $4^{+}_{2}$ in the sextant
$\gamma\in[0,60^{\circ}]$ of the $(\beta,\gamma)$ plane.

Figure 8:  Decomposition of states into $K$–quantum numbers in 24Mg

As a conclusion, the analysis of the wave function structures of 24Mg shows
quantitatively the intimate connection between the development of triaxiality
and $K$–mixing presences from a microscopic calculation using an underlying
effective interaction. The ground state $K$–mixing is identified as the origin
of the so-called $\gamma$-band assigned in this nucleus from geometrical
models Ring and Schuck (1980). This is highly non-trivial but expected given
the quality of the USDB interaction in this mass region.

###  III.3 Odd nuclei: example of 25Mg

The above systematic benchmark has shown our DNO-SM’s efficiency compared to
the classic SM in even nuclei. As previously mentioned, the present framework
is also capable to deal with odd nuclei in the same way without further
treatments of the odd particle at the Hartree–Fock level. We find that the
cranking Hartree–Fock can provide an excellent approach to set up the initial
set of PCHF states from which we construct the basis $\mathit{\Gamma}_{0}$
through the diagonalization-minimization process. To illustrate this, we
present a DNO-SM calculation of 25Mg whose spectrum appears in Figure 9
compared to the classic SM one. This nucleus is known to have for instance two
distinct bands: the ground state band of $K=5/2$ and an excited band of
$K=1/2$. Employing the USDB effective interaction, this assignment is clearly
seen in the wave function structure of each member of the bands as shown in
Figure 10. The minimization is carried out with a discretization of $7$ points
$\beta\in[0.1,0.51]$, $8$ points $\gamma\in[1.5^{\circ},60^{\circ}]$ and with
cranking components $\langle\hat{J}_{z}\rangle=-1/2,-3/2,-5/2,-7/2,-9/2$. The
spectrum of two bands is reproduced remarkably well with a rms error of $44$
keV, using $61$ CHF states. The ground state binding energy is found to be
$-93.89$ MeV versus $-94.40$ MeV in the exact SM result.

Figure 9:  25Mg spectrum

Figure 10:  Decomposition of states into $K$–quantum numbers in 25Mg

In summary for the systematic comparison we have performed between the classic
exact SM and our newly developed DNO-SM, we give now a brief account of its
main features as well as related questions that remain to be answered

(i) Our DNO-SM formalism is composed of three elements: the existence theorem
of discrete non-orthogonal bases exposed in (6), the truncation scheme
provided by the minimization technique and the diagonalization technique from
the GCM theory. While the latter has been already well documented in the
literature, the first element is the most important piece from the formal
point of view (thanks to Ref. Broeckhove and Deumens (1979)) because it
provides us a formal language to express the Shell Model in a natural way with
an emphasis on the different representations of the effective Hamiltonian.
(ii) Our minimization technique has proved to be a very efficient truncation
scheme in the build-up of our non-orthogonal many-body basis for the Shell
Model. In all considered nuclei, a very few number of HF states is needed to
obtain solutions with very good agreement with the exact SM result.
(iii) The procedure constructs the full representation of the effective
Hamiltonian in an iterative way where we observe that DNO-SM solutions are
closer and closer from the above to the exact diagonalization limit:
$\mathscr{H}_{q}\longrightarrow\mathscr{H}$, which means our DNO-SM is an
approximation (given some particular choice of the coordinate(s) $q$) to the
classic SM. This point merits to be mentioned because it points out the need
to design an theoretical approach to choose the coordinate(s) $q$ if one wants
to obtain the full representation of the effective Hamiltonian. Whether this
question can be formally demonstrated a priori without numerical calculations
is an interesting problem to investigate.
(iv) The wave functions and excitation energies converge faster than the
absolute energy. As depicted in Figure 1, there’s some missing energy in the
ground state ranging around $50-900$ keV depending on specific cases. However,
it has practically no impacts on the wave functions because transition
probabilities are already very well reproduced (cf. Table 2). The physical
origin of this missing binding energy is very likely related to (iii), i.e.
the choice of $q$. Here we would like to mention that the multipolarity
decomposition of the effective Hamiltonian designed by A. Zuker et al. (see
e.g. Ref. Caurier et al. (2005)) could be instrumental to investigate both
(iii) and (iv).
(v) Lastly, the present framework is capable to treat equally well even- and
odd-mass nuclei where the cranking method is found to be really efficient in
the construction of the many-body basis.

To conclude, it is worth noting that the energy difference we have found using
the PCHF basis is of the same order and consistent with what have been
reported in similar studies of Refs. Wang et al. (2018); Bally, B. et al.
(2021a); Sánchez-Fernández et al. (2021); Bally, B. et al. (2021b) where more
sophisticated trial wave functions of Hartree-Fock-Bogoliubov (HFB) types with
pairing correlations included explicitly are used (see e.g. Figure 8 of Ref.
Sánchez-Fernández et al. (2021)). Finally, let us mention that our definition
of $(\beta,\gamma)$ is the same as in Refs. Bally, B. et al. (2021a); Sánchez-
Fernández et al. (2021) up to a factor of effective charge
$e^{(p)}_{mass}=e^{(n)}_{mass}$. For example, in 24Mg, omitting the effective
charge would yield $\beta=0.499/1.77\approx 0.282$ which agrees well with Ref.
Bally, B. et al. (2021a) (cf. Table 1 therein). We would like however to keep
this effective charge factor for defining mass quadrupole moments to be able
to compare with $\beta_{2}$–values extracted from experimental electric BE(2)
transitions which yield, e.g. $0.605$ in 24Mg.

##  IV A first Shell Model calculation of ${}^{254}\text{No}$

The quest for superheavy elements is a subject under intensive experimental
investigations and usually, the predictions for the shell stabilization of the
superheavy elements predictions rely essentially on ”standard” mean-field
calculations. In the following, we will apply the DNO-SM method to illustrate
its applicability in the context of superheavy systems and we propose here for
the first time, a shell-model type of description of ${}^{254}\text{No}$
superheavy nucleus, using the DNO-SM. The shell-model valence space is spanned
by the full Z=82-126 proton major shell and the full N=126-184 neutron major
shell beyond ${}^{208}\text{Pb}$, namely, the single proton orbitals
0$h_{\frac{9}{2}}$, 1$f_{\frac{7}{2}}$, 0$i_{\frac{13}{2}}$,
1$f_{\frac{5}{2}}$, 2$p_{\frac{3}{2}}$, 2$p_{\frac{1}{2}}$, and the single
neutron orbitals 1$g_{\frac{9}{2}}$, 0$i_{\frac{11}{2}}$, 0$j_{\frac{15}{2}}$,
2$d_{\frac{5}{2}}$, 3$s_{\frac{1}{2}}$, 1$g_{\frac{7}{2}}$, 2$d_{\frac{3}{2}}$
(the valence space is illustrated in in Fig. 11) . As effective interaction,
we use the modified Kuo-Herling realistic interaction (see Caurier et al.
(2003) for details) which was applied with great success along the N=126
isotones Hauschild et al. (2001); Caurier et al. (2003). The single particle
energies are borrowed from ${}^{209}\text{Bi}$ and ${}^{209}\text{Pb}$ spectra
for protons and neutrons respectively. The electric polarization charge here
is taken to be $\chi_{p}$ = $\chi_{n}$ = 0.5 .

Figure 11:  Valence space above 208Pb

${}^{254}\text{No}$ is a deformed nucleus whose spectroscopy has been
intensively investigated the recent years. In addition to the observation of
its rotational Yrast structure, a side $K=3+$ band and two long lived isomers
have been observed Herzberg et al. (2006); Clark et al. (2010). Figure 12
shows the potential energy surface obtained for the mass ($\beta$,$\gamma$)
deformation parameters (obtained from the mass quadrupole moments defined in
section II.3.3. The deformation landscape shows a clear prolate axial minimum
around $\beta\sim 0.2$ extending moderately towards non-axial shapes.

Figure 12:  Potential Energy Surface (PES) of 254No calculated with the Kuo-
Herling effective interaction where the yellow diamond is the axial HF minimum
with the mass quadrupole parameter $\beta=0.2$.

All the considered states are calculated with the minimization technique over
a restricted region around the corresponding HF minimum. The spectrum of 254No
resulting from these calculations is shown in figure 13 as the function of the
number of HF states retained by the minimization procedure. With a relatively
small number of basis states, we observe a fast and good convergence of the
low-lying members of the Yrast band but also for the higher lying isomer
$8^{-}$ isomer. As already shown in the previous section, the DNO-SM allows
analysis of the states under study in terms of intrinsic quantities, namely,
deformations $(\beta,\gamma)$ and the intrinsic angular momentum. The whole
low-lying spectrum is presented in figure 14. The various states are shown and
grouped in 3 structures: a $K^{\pi}=0^{+}$ Yrast rotational band, a
$K^{\pi}=3^{+}$ multiplet and the $K^{\pi}=8^{-}$ state. There is a excellent
agreement for the reproduction of the Yrast rotational sequence, and the
$8^{-}$ isomeric state. The $3^{-}$ state banhead of the $K=3$ multiplet is
lying a little bit too low, by the interband spacing is also very well
reproduced. The formalism allows to extract the fractionnal spherical
occupancies of the orbitals in the valence space. The structure of the three
$0^{+}_{1}$, $3^{+}_{1}$ and $8^{-}_{1}$ states is shown in Table 5. with a
possible large mixing of spherical orbital our description is richer than
single quasi-particle estimates which are often used to assign exited and
isomeric states to a single excited configurations. This is reflected in the
partial occupancies of the whole proton and neutron orbitals involved in the
valence space. Nevertheless, on can point out that both for protons and
neutrons, the fillings proceeds through the ”largest” orbitals and that
excited $3^{+}_{1}$ and $8^{-}_{1}$ states mainly differ from the ground state
by an additionnal proton particle filling the $1h_{9/2}$ orbital, assigning
these states as having a ”proton” nature. We may recall here that we obtain an
excellent reproduction of the experimental data with no adjustement of the
effective interaction, designed more than two decades ago. Nevertheless, in
order to confirm the present outcome from our calculations, a broader
systematic study has been developped, and in particular we would like to
connect the slight energy shift of the $K^{\pi}=3^{+}$ multiplet to a specific
single particle monopole drift for a better reproductive and predictive
description of the overall region.

Figure 13:  Evolution of the lower part of 254No spectrum with respect to the
number of HF states found by the minimization procedure. Figure 14:
Comparison of DNO-SM calculation using 15 HF states with the experimental
spectrum.

Figure 15:  $K$-quantum number content of the isomeric $3^{+}$ band and the $8^{-}$ state. proton orbits | $1h_{9/2}$ | $1i_{13/2}$ | $2f_{7/2}$ | $2f_{5/2}$ | $3p_{3/2}$ | $3p_{1/2}$ |
---|---|---|---|---|---|---|---
$0^{+}_{1}$ | $5.66$ | $7.99$ | $3.44$ | $1.58$ | $0.76$ | $0.57$ |
$8^{-}$ | $6.52$ | $7.82$ | $3.28$ | $1.20$ | $0.79$ | $0.39$ |
$3^{+}$ | $6.50$ | $7.98$ | $3.31$ | $1.14$ | $0.72$ | $0.35$ |
neutron orbits | $1i_{11/2}$ | $1j_{15/2}$ | $2g_{9/2}$ | $2g_{7/2}$ | $3d_{5/2}$ | $3d_{3/2}$ | $4s_{1/2}$
$0^{+}_{1}$ | $7.28$ | $9.67$ | $5.45$ | $1.11$ | $1.16$ | $0.87$ | $0.46$
$8^{-}$ | $7.29$ | $9.04$ | $6.07$ | $1.12$ | $1.15$ | $0.88$ | $0.45$
$3^{+}$ | $7.31$ | $9.94$ | $5.43$ | $0.99$ | $1.07$ | $0.83$ | $0.43$
Table 5: Occupancies of the spherical orbitals for the ground state and
$8^{-}$ and $3^{+}$ states.

##  V Conclusion and perspectives

As a summary, in this paper we have exposed the formalism of the DNO-SM which
amounts to diagonalize shell-model hamiltonians in a non-orthogonal basis with
the use of beyond-mean-field techniques. Particular effort has been put into
the proper selection of optimal basis state used for the diagonalisation. We
benchmarked the method over a large set of $sd$ shell nuclei and could
reproduce the energies, and transitions probabilities of the exact
diagonalisations with minimal cost of very few basis states. For the first
time, we applied the DNO-SM method to a superheavy system 254No which is
obviously far beyond the capabilities of actual diagonalisations codes. The
DNO-SM formalism has also recently been very useful in the interpretation of
several experimental studies Calinescu et al. (2021); Rezynkina et al. (2022);
Reygadas et al. (2022); Rocchini et al. (2022) and has been to be extremely
promising both for instrinsic interpretation of shell-model diagonalisations,
and setting new frontiers for nuclear structure studies within the shell-model
framework.

##  VI Acknowlegments

The authors would like to dedicate the present work to the memory of the late
Etienne Caurier, who was at the initiative of the present developments.

##  Appendix A Matrix elements in the PCHF basis

Considering the two-body Hamiltonian $\mathcal{\hat{H}}$ defined in (2), in
the PCHF basis $\mathcal{P}^{J}_{MK}|\Phi(q)\rangle\in\mathit{\Gamma}_{0}$, it
is represented by the set of matrix elements
$\langle\Phi(q^{\prime})|\mathcal{\hat{H}}\mathcal{P}^{J}_{K^{\prime}K}|\Phi(q)\rangle$
given by (16). The calculation of this matrix element requires an evaluation
of the hamiltonian kernel which, for two arbitrary Slater determinants
$|\Phi^{\prime}\rangle$ and $|\Phi\rangle$, takes the forms (cf. e.g. Ref.
Watt (1972))

$\langle\Phi^{\prime}|\mathcal{\hat{H}}\hat{R}(\Omega)|\Phi\rangle=\sum_{\begin{subarray}{c}p\in\Phi^{\prime}\\\
q\in\Phi\end{subarray}}(-)^{p+q}M_{pq}(\Omega)\,\langle
p|\hat{E}\hat{R}(\Omega)|q\rangle+\sum_{\begin{subarray}{c}p<q\in\Phi^{\prime}\\\
r<s\in\Phi\end{subarray}}(-)^{p+q+r+s}M_{pqrs}(\Omega)\langle
pq|\hat{\mathcal{V}}\hat{R}(\Omega)|rs\rangle$ (32)

where $M_{pq}(\Omega),M_{pqrs}(\Omega)$ are respectively first– and
second–order minors of the $A\times A$ matrix
$N(\Omega)=D^{\prime\dagger}\cdot R(\Omega)\cdot D$ with a rectangular matrix
$D_{ip}=\\{C^{(p)}_{i},\,p\in\Phi,i\in\mathcal{E}\\}$ representing the single
particle HF states in the Slater $|\Phi\rangle$. $C^{(p)}_{i}$ represents here
the expansion coefficient of the single-particle Hartree-Fock state
$|p\rangle$ in the single-particle Harmonic Oscillator basis
$\\{|i\rangle\\}$. The matrix element of the one-body single-particle energy
$\hat{E}$ is given by

$\langle
p|\hat{E}\hat{R}(\Omega)|q\rangle=\sum_{i_{1},i_{2},i\in\mathcal{E}}C^{\prime(p)}_{i_{1}}C^{(q)}_{i_{2}}\,e_{i_{1}i}\,R_{ii_{2}}(\Omega).$
(33)

Whereas for the two-body term, it is written in terms of the antisymmetrized
matrix element $\langle
JT(i_{1}i_{2})|\hat{\mathcal{V}}|JT(i_{3}i_{4})\rangle$ of good angular
momentum and isospin $J,T$

$\langle
pq|\hat{\mathcal{V}}\hat{R}(\Omega)|rs\rangle=\sum_{\begin{subarray}{c}JMM^{\prime}\\\
TT_{z}\\\
i_{1}i_{2}i_{3}i_{4}\end{subarray}}C^{TT_{z}}_{\frac{1}{2}\tau_{p}\frac{1}{2}\tau_{q}}C^{TT_{z}}_{\frac{1}{2}\tau_{r}\frac{1}{2}\tau_{s}}C^{JM}_{j_{1}m_{1}j_{2}m_{2}}C^{JM^{\prime}}_{j_{3}m_{3}j_{4}m_{4}}\mathcal{D}^{J}_{MM^{\prime}}(\Omega)\,C^{(p)}_{i_{1}}C^{(q)}_{i_{2}}C^{(r)}_{i_{3}}C^{(s)}_{i_{4}}\>\langle
JT(i_{1}i_{2})|\hat{\mathcal{V}}|JT(i_{3}i_{4})\rangle.$ (34)

Here $C^{JM}_{j_{1}m_{1}j_{2}m_{2}}$ is the Clebsch–Gordan coefficient.

##  Appendix B Derivation of the analytical integration over the Euler angles
$\alpha,\gamma$

Appendix A provides the necessary elements we need to derive an analytical
formula for the integrations over $\alpha,\gamma$ in (16). For that goal, we
just need to perform the derivation with $\hat{\mathcal{O}}=\mathbf{1}$, the
same procedure holds for other operators such as the Hamiltonian or transition
operators. In this case, we have the norm matrix element

$\displaystyle\mathcal{N}^{J}_{K^{\prime}K}=\frac{2J+1}{4\pi^{2}\big{(}3-(-)^{A}\big{)}}\int
d\Omega\>\mathcal{D}^{J*}_{MK}(\Omega)\>\mathrm{det}\,N(\Omega)$ (35)

where we have used the equality
$\mathrm{det}\,N(\Omega)=\langle\Phi^{\prime}|\hat{R}(\Omega)|\Phi\rangle$.
Let us rewrite this quantity in an explicit way with a summation over all
permutations $\sigma$ of the permutation group of $A$–particles $S_{A}$

$\displaystyle\mathrm{det}\,N=\sum_{\sigma\in
S_{A}}\mathrm{sgn}(\sigma)\prod_{\lambda=1}^{A}N_{\lambda,\sigma(\lambda)}$
(36) $\displaystyle=\sum_{\sigma\in
S_{A}}\mathrm{sgn}(\sigma)\prod_{\lambda=1}^{A}\Big{(}\sum_{i_{1}i_{2}\in\mathcal{E}}D^{\prime}_{\lambda
i_{1}}R_{i_{1}i_{2}}D_{i_{2}\sigma(\lambda)}\Big{)}$
$\displaystyle=\sum_{\begin{subarray}{c}i_{1}(1)i_{2}(1)\\\ \cdots\\\
i_{1}(A)i_{2}(A)\end{subarray}}\prod_{\lambda=1}^{A}D^{\prime\dagger}_{\lambda
i_{1}(\lambda)}\prod_{\lambda=1}^{A}R_{i_{1}(\lambda)i_{2}(\lambda)}$
$\displaystyle\phantom{=}\Big{(}\sum_{\sigma\in
S_{A}}\mathrm{sgn}(\sigma)\prod_{\lambda=1}^{A}D_{i_{2}(\lambda)\sigma(\lambda)}\Big{)}.$

The norm matrix element thus becomes

$\displaystyle\mathcal{N}^{(J)}_{K^{\prime}K}=\frac{2J+1}{4\pi^{2}\big{(}3-(-)^{A}\big{)}}\sum_{\begin{subarray}{c}i_{1}(1)i_{2}(1)\\\
\cdots\\\ i_{1}(A)i_{2}(A)\end{subarray}}$ (37)
$\displaystyle\prod_{\lambda=1}^{A}D^{\prime\dagger}_{\lambda
i_{1}(\lambda)}\Big{(}\sum_{\sigma\in
S_{A}}\mathrm{sgn}(\sigma)\prod_{\lambda=1}^{A}D_{i_{2}(\lambda)\sigma(\lambda)}\Big{)}\times$
$\displaystyle\int
d\Omega\>\mathcal{D}^{J*}_{MK}(\Omega)\prod_{\lambda=1}^{A}R_{i_{1}(\lambda)i_{2}(\lambda)}(\Omega).$

In this form, the integration over Euler angles $\Omega=(\alpha,\beta,\gamma)$
is isolated and can be subject to a direct evaluation using the rotation
matrix in spherical oscillator basis $R_{i_{1}i_{2}}(\Omega)=e^{-i\alpha
m_{1}}d^{j}_{m_{1}m_{2}}(\beta)e^{-i\gamma m_{2}}$. Hence, one can write

$\displaystyle\int
d\Omega\>\mathcal{D}^{J*}_{MK}(\Omega)\prod_{\lambda=1}^{A}R_{i_{1}(\lambda)i_{2}(\lambda)}(\Omega)$
(38)
$\displaystyle=\int\displaylimits_{0}^{2\pi}d\alpha\,e^{i\alpha\big{[}K^{\prime}-\sum_{\lambda=1}^{A}m_{1}(\lambda)\big{]}}\int\displaylimits_{0}^{\gamma_{\rm
max}}d\gamma\,e^{i\gamma\big{[}K-\sum_{\lambda=1}^{A}m_{2}(\lambda)\big{]}}$
$\displaystyle\phantom{=}\int\displaylimits_{0}^{\pi}d\beta\,\sin\beta
d^{*J}_{K^{\prime}K}(\beta)\prod_{\lambda=1}^{A}d^{j(\lambda)}_{m_{1}(\lambda)m_{2}(\lambda)}(\beta)$

from which it is trivial to calculate the integrals over $\alpha,\gamma$. The
result simply reads

$\displaystyle\int
d\Omega\>\mathcal{D}^{J*}_{MK}(\Omega)\prod_{\lambda=1}^{A}R_{i_{1}(\lambda)i_{2}(\lambda)}(\Omega)=2\pi\gamma_{\rm
max}\times$ (39) $\displaystyle\delta_{\Delta K^{\prime},0}\delta_{\Delta
K,0}\int\displaylimits_{0}^{\pi}d\beta\sin\beta\,d^{*J}_{K^{\prime}K}(\beta)\prod_{\lambda=1}^{A}d^{j(\lambda)}_{m_{1}(\lambda)m_{2}(\lambda)}(\beta)$

where we denote $\Delta
K^{\prime}=K^{\prime}-\sum_{\lambda=1}^{A}m_{1}(\lambda)$ and $\Delta
K=K-\sum_{\lambda=1}^{A}m_{2}(\lambda)$. This form is of course not practical
as it involves summations over $A!$ permutations. Our next step is to recast
it into the familiar expression as in (35) with only the integration over
$\beta$ being left to evaluate. To do so, notice that

$\delta_{n,0}=\frac{1}{N}\sum_{k=1}^{N}e^{i\frac{2\pi k}{N}n}\text{ if
}\left\\{\begin{aligned} N\in\mathbb{N},\>N\geq 2,\>\>\\\
n\in\mathbb{Z},\>e^{i\frac{2\pi}{N}n}\neq 1.\end{aligned}\right.$ (40)

It is obvious that $\Delta K^{\prime},\Delta K\in\mathbb{Z}$ regardless the
odd or even mass number $A$ so that applying this identity allows us to obtain

$\displaystyle\int
d\Omega\>\mathcal{D}^{J*}_{MK}(\Omega)\prod_{\lambda=1}^{A}R_{i_{1}(\lambda)i_{2}(\lambda)}(\Omega)=\frac{2\pi\gamma_{\rm
max}}{N_{\alpha}N_{\gamma}}\times$ (41)
$\displaystyle\sum_{k_{1}=1}^{N_{\alpha}}\sum_{k_{2}=1}^{N_{\gamma}}\int\displaylimits_{0}^{\pi}d\beta\sin\beta\,\mathcal{D}^{*J}_{K^{\prime}K}\big{(}\frac{2\pi
k_{1}}{N_{\alpha}},\beta,\frac{2\pi k_{2}}{N_{\gamma}}\big{)}\times$
$\displaystyle\prod_{\lambda=1}^{A}R_{i_{1}(\lambda)i_{2}(\lambda)}\big{(}\frac{2\pi
k_{1}}{N_{\alpha}},\beta,\frac{2\pi k_{2}}{N_{\gamma}}\big{)}$

where $N_{\alpha},N_{\gamma}\in\mathbb{Z}$ are chosen according to the
condition (40). The final expression of the norm matrix element thus reads

$\displaystyle\mathcal{N}^{J}_{K^{\prime}K}=\frac{2J+1}{2}\cdot\frac{1}{N_{\alpha}N_{\gamma}}\sum_{k_{1}=1}^{N_{\alpha}}\sum_{k_{2}=1}^{N_{\gamma}}\int\displaylimits_{0}^{\pi}d\beta\sin\beta\>\times$
(42) $\displaystyle\mathcal{D}^{*J}_{K^{\prime}K}\big{(}\frac{2\pi
k_{1}}{N_{\alpha}},\beta,\frac{2\pi
k_{2}}{N_{\gamma}}\big{)}\times\mathrm{det}\,N\big{(}\frac{2\pi
k_{1}}{N_{\alpha}},\beta,\frac{2\pi k_{2}}{N_{\gamma}}\big{)}.$

The same reasoning leaves us with the Hamiltonian matrix element where
$\alpha,\gamma$ are exactly integrated out

$\displaystyle\mathcal{H}^{J}_{K^{\prime}K}=\frac{2J+1}{2}\cdot\frac{1}{N_{\alpha}N_{\gamma}}\sum_{k_{1}=1}^{N_{\alpha}}\sum_{k_{2}=1}^{N_{\gamma}}\int\displaylimits_{0}^{\pi}d\beta\sin\beta\>\mathcal{D}^{*J}_{K^{\prime}K}\big{(}\frac{2\pi
k_{1}}{N_{\alpha}},\beta,\frac{2\pi
k_{2}}{N_{\gamma}}\big{)}\>\langle\Phi^{\prime}|\hat{\mathcal{H}}\hat{R}\big{(}\frac{2\pi
k_{1}}{N_{\alpha}},\beta,\frac{2\pi k_{2}}{N_{\gamma}}\big{)}|\Phi\rangle.$
(43)

$N_{u}(u=\alpha,\gamma)$ will be chosen to ensure the conditions

$e^{i\frac{2\pi}{N_{u}}\Delta K_{v}}\neq 1\quad\forall v=0,1,2\text{ with
}\left\\{\begin{aligned} &\Delta
K_{0}=K-\sum_{\lambda=1}^{A}m_{i}(\lambda)\quad\forall i\in\mathcal{E}\\\
&\Delta
K_{1}=K-m_{i}-\sum_{\lambda=1}^{A-1}m_{i^{\prime}}(\lambda)\quad\forall
i,i^{\prime}\in\mathcal{E}\\\ &\Delta
K_{2}=K-M(i_{1},i_{2})-\sum_{\lambda=1}^{A-2}m_{i}(\lambda)\quad\forall
i,i_{1},i_{2}\in\mathcal{E}\\\ \end{aligned}\right.$ (44)

with $i_{1},i_{2}$ designating two single-particle harmonics oscillator states
coupled to a total angular momentum $M=m_{i_{1}}+m_{i_{2}}$.

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