Correlated-basis method for shell-model calculations

We present a basis selection method for truncated shell-model calculations. In this method, the correlated basis is constructed with the eigenvectors of the Hamiltonian that is diagonalized in each partition of the shell model. A truncation scheme is established by naturally taking the low-lying correlated-basis vectors in different partitions, which is equivalent to the $jj$-coupling scheme of the shell model when all the correlated-basis vectors are considered. The results are compared with standard shell-model calculations. The convergence properties of the correlated-basis method are discussed.

Figure 1

Energies of low-lying states in ${}^{28}\mathrm{Si}$, calculated with the correlated-basis shell model at different truncations. $D_{\text{cor}}$ and $D_{jj}$ are the dimensions of the correlated-basis truncation and the “full” dimension of the $jj$-coupling scheme, respectively.

Figure 2

Convergence of the low-lying $0^{+}$, $2^{+}$, $4^{+}$, and $6^{+}$ states in ${}^{28}\mathrm{Si}$. $D_{\mathrm{cor}}$ and $D_M$ are the dimensions of the correlated-basis truncation and the “full” dimension of the standard $M$ scheme, respectively.

Figure 3

Convergence of the squared overlaps of the $0_1^{+}$, $2_1^{+}$, $4_1^{+}$, and $6_1^{+}$ wave functions between the calculations of the correlated basis and $M$ scheme for ${}^{28}\mathrm{Si}$. Note that different scales in horizontal axes have been chosen to give the details of convergence.

Figure 4

Amplitude distributions of the ground-state wave function of ${}^{28}\mathrm{Si}$ expanded in the correlated basis and the $M$ scheme. $\varepsilon$ is the unperturbed energy (diagonal matrix elements) of a basis vector (see the text). Note that different scales in vertical axes have been chosen to give more details of the wave functions.

Figure 5

Convergence of correlated-basis calculations with different truncations determined by $\Delta {\epsilon }_{ic}$ (red triangles) or ${\chi }_{ic}$ (black dots) for the lowest $0^{+}$ and $2^{+}$ states in ${}^{28}\mathrm{Si}$.

Figure 6

Convergence of the low-lying $2^{+}$ and $4^{+}$ states in ${}^{106}\mathrm{Sn}$. $D_{\mathrm{cor}}$ and $D_M$ are the dimensions of the correlated-basis truncation and the “full” dimension of the standard $M$ scheme, respectively.

Figure 7

Calculated energies of the lowest $0_1^{+}$ and $2_1^{+}$ states in ${}^{28}\mathrm{Si}$ as a function of the matrix truncation $n$. The shell-model calculation (open squares) is fitted by an exponential function $A{e}^{-n/\gamma }$ (solid line).