Role of spatially compact nucleon wave packets in an ab initio description of ${}^3\mathrm{H}$ within high-momentum antisymmetrized molecular dynamics

We found the important role of the spatially compact nucleon wave packets to fully describe the nucleon correlations within the framework of the high-momentum antisymmetrized molecular dynamics (HM-AMD), which is a variational method based on the antisymmetrized molecular dynamics. In HM-AMD, short-range and tensor correlations between nucleons are described by nucleon pairs with high momentum (high-momentum pairs), which are given by putting large centroids of the Gaussian wave packets in opposite signs. In this paper, we further improve momentum distribution of the high-momentum pairs by varying the width parameter of the Gaussian wave packets of nucleons. We show the reliability of this new scheme by applying it to the ${}^3\mathrm{H}$ nucleus with the bare Argonne v8' potential. It is found that the spatially compact nucleon wave packets give the important effect to lower the total energy, which brings sufficient high-momentum components in the nucleus. We also include the spin-parallel configuration of the three nucleons induced by the tensor interaction, which is necessary to converge the HM-AMD results. Finally, comparable results to the other theoretical calculations are obtained for the total energy and Hamiltonian components.

Figure 1

Diagrams of the cluster expansion for ${}^3\mathrm{H}$ composed of a proton $p_1$ and neutrons $n_1$ and $n_2$. Possible short-range or tensor correlations between two nucleons by (a) single and (b) double high-momentum pair(s) for ${}^3\mathrm{H}$. In each panel, “$\uparrow \uparrow \downarrow$” shows the spin directions of the three nucleons in the ordinary spin configuration ($S_z=1/2$), while “$\uparrow \uparrow \uparrow$” represents the spin-parallel configuration ($S_z=3/2$) induced by the tensor interaction.

Figure 2

Total energy $E_{\mathrm{tot}}$ obtained with the double HM-AMD calculation as a function of the Gaussian width parameter ${\nu }^{\prime }$ of the double pair wave functions where $\nu$ of the ${\left(0s\right)}^3$ AMD and single-pair wave functions is fixed to be (a) 0.08, (b) 0.09, (c) 0.10, and (d) 0.12 in units of ${\mathrm{fm}}^{-2}$. The $S_z=3/2$ single-pair wave functions are included in the solid curve (case D in each panel), whereas dashed curve (case C) shows the results without the spin-parallel components. Open circles (case E) show the total energies obtained by the double HM-AMD calculations where both the $S_z=3/2$ single-pair and double-pair wave functions are included. Horizontal dotted (dashed) line denoted as case A (case B) indicates the total energy obtained by the single HM-AMD calculation with $\nu$, where the $S_z=3/2$ single-pair wave functions are not included (included). For comparison, the $F^2\text{−}\mathrm{TOAMD}$ [10, 11] (dot-dashed) and GFMC [20] (dot-dot-dashed) results are also shown by the horizontal lines. Vertical dotted lines in panels (c) and (d) show the ${\nu }^{\prime }$ value, which is equal to $\nu$.

Figure 3

(a) The upper panel shows the $\nu$ dependence of the total energy ($E_{\mathrm{tot}}$) and the expectation values of the kinetic energy ($K$), and the central ($V_{\mathrm{cent}}$), spin-orbit ($V_{\mathrm{LS}}$), tensor ($V_{\mathrm{tens}}$) interactions in the single HM-AMD calculation using the $N_{\mathrm{basis}}=169$ bases without the spin-parallel components. The bottom panel shows the $\nu$ dependence of the nuclear radius. (b) Same as panel (a), but for the ${\nu }^{\prime }$ dependence of the double HM-AMD calculations with $\nu =0.08$, 0.09, 0.10, and $0.12{\mathrm{fm}}^{-2}$ using the $N_{\mathrm{basis}}=889$ bases, which corresponds to case C in Fig. 2.