Effective quadrupole-quadrupole interaction from density functional theory

The density functional theory of nuclear structure provides a many-particle wave function that is useful for static properties, but an extension of the theory is necessary to describe correlation effects or other dynamic properties. We propose a procedure to extend the theory by mapping the properties of a self-consistent mean-field theory onto an effective shell-model Hamiltonian with quadrupole-quadrupole interaction. In this initial study, we consider the $\mathit{sd}$-shell nuclei ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, ${}^{28}\mathrm{Si}$, and ${}^{36}\mathrm{Ar}$. The method is first tested with the USD shell-model Hamiltonian, using its mean-field approximation to construct an effective Hamiltonian and partially recover correlation effects. We find that more than half of the correlation energy is due to the quadrupole interaction. We then follow a similar procedure but using the SLy4 Skyrme energy functional as our starting point and truncating the space to the spherical $\mathit{sd}$ shell. The constructed shell-model Hamiltonian is found to satisfy minimal consistency requirements to reproduce the properties of the mean-field solution. The quadrupolar correlation energies computed with the mapped Hamiltonian are reasonable compared with those computed by other methods. The method also provides a well-defined renormalization of the quadrupole operator in the shell-model space, the “effective charge” of the phenomenological shell model.

Figure 1

${}^{28}\mathrm{Si}$ potential energy landscape obtained from the constrained Hartree-Fock solution of the USD Hamiltonian.

Figure 2

Occupation numbers $\langle {\hat{n}}_j\rangle$ of the spherical orbitals in the deformed mean-field solution as a function of the spherical single-particle energy ${\epsilon }_j^{(0)}$ in four $\mathit{sd}$-shell nuclei (${}^{20}\mathrm{Ne}$,${}^{24}\mathrm{Mg}$, ${}^{28}\mathrm{Si}$, and ${}^{36}\mathrm{Ar}$). Results for the effective Hamiltonian (11) (filled squares) are compared with results for the USD interaction (open circles). Also shown are results for the effective interaction (22) discussed in Sec. 7 (filled triangles).

Figure 3

Correlation energies $E_{\mathrm{corr}}$ [see Eq. (15)] versus mass number $A$ in deformed $N=Z$ $\mathit{sd}$ shell nuclei. The symbols are as in Fig. 2. Quadrupolar correlations are responsible for more than 50% of the full correlation energies (with the exception of ${}^{36}\mathrm{Ar}$).

Figure 4

${}^{28}\mathrm{Si}$ potential energy landscape obtained in the absence of pairing with the SLy4 interaction.

Figure 5

Ratio $|(j\left|\right|\hat{Q}\left|\right|j^{\text{'}}){}_{\mathrm{SLy}4}/(j\left|\right|\hat{Q}\left|\right|j^{\text{'}}){}_{\mathrm{WS}}|$ of the mass quadrupole reduced matrix elements in ${}^{28}\mathrm{Si}$ versus $|(j\left|\right|\hat{Q}\left|\right|j^{\text{'}}){}_{\mathrm{WS}}|$. Filled circles, diagonal elements; open triangles, off-diagonal elements.

Figure 6

Quadrupole moment of the SCMF ground state of ${}^{28}\mathrm{Si}$, using the SLy4 Skyrme energy functional and different truncations in the spherical basis. The horizontal axis is the maximum single-particle energy in the truncated space. Each filled circle corresponds to a spherical single-particle state of protons or neutrons. The untruncated moment of −89.2 fm${}^2$ is shown by the arrow on the right.

Figure 7

Left, $\langle {\hat{h}}^{(0)}\rangle {}_{\xi }$ versus ξ in ${}^{28}\mathrm{Si}$. Right, $\langle {\hat{h}}_0^{(2)}\rangle {}_{\xi }$ versus ξ. The density matrix ${\mathit{\rho }}_{\xi }$ corresponds to the single-particle (Hartree) Hamiltonian ${\mathbf{h}}_{\mathrm{eff}}={\mathbf{h}}^{(0)}+\xi {\mathbf{h}}_0^{(2)}$.

Figure 8

Left, deformation energy versus coupling parameter $g$ in ${}^{28}\mathrm{Si}$ when effective Hamiltonian (11) is solved in the Hartree approximation. Right, quadrupole moment $\langle \hat{Q}\rangle$ versus $g$ in the Hartree approximation for the same effective Hamiltonian. The filled triangles describe the SCMF solution of the SLy4 energy functional (note that the triangle in the left panel is a fit).

Figure 9

Deformation energy versus quadrupole moment of ${}^{28}\mathrm{Si}$ using the Hartree approximation for the effective Hamiltonian (11) in the $\mathit{sd}$ shell. The triangle shows the SCMF solution for the SLy4 energy functional, where the quadrupole moment is calculated in the truncated space ($\mathit{sd}$ shell).

Figure 10

Quadrupole moment in the SCMF method versus mass number $A$ for $4N$ $\mathit{sd}$-shell nuclei. Results are shown for two effective CISM Hamiltonians: the $\hat{Q}·\hat{Q}$ interaction, Eq. (22) (filled triangles), and the ${\hat{h}}^{(2)}·{\hat{h}}^{(2)}$ interaction, Eq (11) (filled squares). They are compared with results for the USD Hamiltonian (open circles).