Description of nuclear systems with a self-consistent configuration-mixing approach: Theory, algorithm, and application to the ${}^{12}\text{C}$ test nucleus

Background: Although self-consistent multiconfiguration methods have been used for decades to address the description of atomic and molecular many-body systems, only a few trials have been made in the context of nuclear structure.

Purpose: This work aims at the development of such an approach to describe in a unified way various types of correlations in nuclei in a self-consistent manner where the mean-field is improved as correlations are introduced. The goal is to reconcile the usually set-apart shell-model and self-consistent mean-field methods.

Method: This approach is referred to as “variational multiparticle-multihole configuration mixing method.” It is based on a double variational principle which yields a set of two coupled equations that determine at the same time the expansion coefficients of the many-body wave function and the single-particle states. The solution of this problem is obtained by building a doubly iterative numerical algorithm.

Results: The formalism is derived and discussed in a general context, starting from a three-body Hamiltonian. Links to existing many-body techniques such as the formalism of Green's functions are established. First applications are done using the two-body D1S Gogny effective force. The numerical procedure is tested on the ${}^{12}\mathrm{C}$ nucleus to study the convergence features of the algorithm in different contexts. Ground-state properties as well as single-particle quantities are analyzed, and the description of the first $2^{+}$ state is examined.

Conclusions: The self-consistent multiparticle-multihole configuration mixing method is fully applied for the first time to the description of a test nucleus. This study makes it possible to validate our numerical algorithm and leads to encouraging results. To test the method further, we will realize in the second article of this series a systematic description of more nuclei and observables obtained by applying the newly developed numerical procedure with the same Gogny force. As raised in the present work, applications of the variational multiparticle-multihole configuration mixing method will, however, ultimately require the use of an extended and more constrained Gogny force.

Figure 1

Separation of the many-body space $\mathcal{S}$ spanned by the finite single-particle basis, into $\mathcal{P}\oplus \mathcal{Q}$.

Figure 2

Diagrams representing the matrix elements $\langle {\varphi }_{\alpha }|:{\widehat{V}}^{2N}:|{\varphi }_{\beta }\rangle$, according to the difference in excitation orders $\mathrm{\Delta }M=|M_{\alpha }-M_{\beta }|$.

Figure 3

Diagrammatic representation of the direct (left) and exchange (right) parts of the average potential ${\mathrm{\Gamma }}^{2N}\left[\rho \right]$. The double line denotes the correlated one-body density.

Figure 4

Resummation of ring (top) and ladder (bottom) diagrams in $G^{2N}\left[\sigma \right]$.

Figure 5

Modification of the subspaces $\mathcal{P}$ and $\mathcal{Q}$ via the optimization of orbitals.

Figure 6

Convergence procedure of the MPMH configuration mixing method.

Figure 7

Detailed convergence procedure of the MPMH configuration mixing method.

Figure 8

HFB axial PEC (left) and triaxial PES (right) for the ${}^{12}\text{C}$ nucleus.

Figure 9

Schematic representation of the two truncation schemes.

Figure 10

Correlation matrices. (a) Neutron correlations obtained with scheme 1 at macroiteration $N=1$ (left) and $N=15$ (right). (b) Proton-Neutron correlations obtained with scheme 1 at macroiteration $N=1$ (left) and $N=15$ (right). (c) Neutron correlations obtained with scheme 2 at macroiteration $N=1$ (left) and $N=14$ (right). (d) Proton-Neutron correlations obtained with scheme 2 at macroiteration $N=1$ (left) and $N=14$ (right).

Figure 11

Absolute value of the neutron source term in MeV, obtained with scheme 1 at macroiterations $N=1$ (left) and $N=15$ (right).

Figure 12

Absolute value of the neutron source term in MeV, obtained with scheme 2 at macroiterations $N=1$ (left) and $N=14$ (right).

Figure 13

Evolution of the neutron one-body density along the convergence process. The difference between the correlated density and a pure Hartree-Fock density $\mathrm{\Delta }\rho =|\rho -{\rho }_{\text{HF}}|$ is represented in a matrix form in the original Hartree-Fock basis. Truncation scheme 1. See text for explanation.

Figure 14

Comparison between the neutron density matrices given by the first and second variational equations (27) and (28), respectively, at different stages of the convergence process. Truncation scheme 1.

Figure 15

Evolution of the neutron one-body density along the convergence process. The difference between the correlated density and a pure Hartree-Fock density $\mathrm{\Delta }\rho =|\rho -{\rho }_{\text{HF}}|$ is represented in a matrix form in the original Hartree-Fock basis. Truncation scheme 2. See text for explanation.

Figure 16

Comparison between the neutron density matrices given by the first and second variational equations (27) and (28), respectively, at different stages of the convergence process. Truncation scheme 2.

Figure 17

Difference $\mathrm{\Delta }\varepsilon ={\varepsilon }_{\text{HF}}-\varepsilon [\rho ,\sigma ]$ in MeV between SPEs taken as eigenvalues of the Hartree-Fock field and SPEs taken as eigenvalues of the improved mean field $h[\rho ,\sigma ]$. Results obtained with truncation schemes 1 (left) and 2 (right) for protons (up) and neutrons (down). The Fermi levels are marked by a dashed line.

Figure 18

Excitation energy $E^{*}\left(2_1^{+}\right)$ in MeV (top) and transition probability $B(E2:2_1^{+}\to 0_1^{+})$ in Weisskopf units (bottom) for the first excited $2^{+}$ state.