New truncation scheme for a time-dependent density-matrix approach applied to the ground state of ${}^{16}\mathrm{O}$

The ground state of ${}^{16}\mathrm{O}$ is calculated by using a time-dependent density-matrix approach derived from a new truncation scheme of the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy for reduced density matrices, where a three-body density matrix is approximated by an antisymmetrized product of two-body density matrices. The new scheme is compared with a simpler truncation scheme previously used for the calculation of the ground state of ${}^{16}\mathrm{O}$ where the three-body density matrix is neglected and only two-particle–two-hole elements of the two-body density matrix are considered. It is shown that the results obtained from the two truncation schemes agree well with the exact solution.

Figure 1

Occupation probability of the proton $1p_{3/2}$ state as a function of $t/T$ calculated in TDDM with $C_{\alpha \beta \gamma {\alpha }^{\prime }{\beta }^{\prime }{\gamma }^{\prime }}$ given by Eqs. (7) and (8) (solid line). The dotted line depicts the results in the original truncation scheme of TDDM where the three-body correlation matrix is neglected but all components of $C_{\alpha \beta {\alpha }^{\prime }{\beta }^{\prime }}$ are included. The dot-dashed line shows the results of the simplified version of TDDM where only the 2p-2h and 2h-2p elements of the two-body correlation matrix are included. The exact solution is shown with the square at $t/T=1$.

Figure 2

Same as Fig. 1 but for the proton $1p_{1/2}$ state.

Figure 3

Same as Fig. 1 but for the proton $1d_{5/2}$ state.

Figure 4

Correlation energy in TDDM as a function of $t/T$. The meaning of the three lines is the same as in Fig. 1. The green (gray) solid line depicts the result where $C_{{\mathrm{php}}^{\prime }{\mathrm{h}}^{\prime }},C_{{\mathrm{pp}}^{\prime }{\mathrm{p}}^{\prime \prime }{\mathrm{p}}^{\prime \prime \prime }}$, and $C_{{\mathrm{hh}}^{\prime }{\mathrm{h}}^{\prime \prime }{\mathrm{h}}^{\prime \prime \prime }}$ are calculated by using $C_{{\mathrm{pp}}^{\prime }{\mathrm{hh}}^{\prime }}$ and $C_{{\mathrm{hh}}^{\prime }{\mathrm{pp}}^{\prime }}$ (see text). The exact solution is shown with the square.

Figure 5

Sum of the absolute value of Eq. (16) as a function of time steps. The meaning of the four lines is the same as in Fig. 4. Time step 4000 corresponds to $T$.