Nuclear kinetic density from ab initio theory

Background: The nuclear kinetic density is one of many fundamental, nonobservable quantities in density functional theory (DFT) dependent on the nonlocal nuclear density. Often, approximations may be made when computing the density that may result in spurious contributions in other DFT quantities. With the ability to compute the nonlocal nuclear density from ab initio wave functions, it is now possible to estimate effects of such spurious contributions.

Purpose: We derive the kinetic density using ab initio nonlocal scalar one-body nuclear densities computed within the no-core shell model (NCSM) approach, utilizing two- and three-nucleon chiral interactions as the sole input. The ability to compute translationally invariant nonlocal densities allows us to gauge the impact of the spurious center-of-mass (c.m.) contributions in DFT quantities, such as the kinetic density, and provide ab initio insight into refining energy density functionals.

Methods: The nonlocal nuclear densities are derived from the NCSM one-body densities calculated in second quantization. We present a review of c.m. contaminated and translationally invariant nuclear densities. We then derive an analytic expression for the kinetic density using these nonlocal densities, producing an ab initio kinetic density.

Results: The ground-state nonlocal densities of ${}^{4,6,8}\mathrm{He},{}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$ are used to compute the kinetic densities of the aforementioned nuclei. The impact of c.m. removal techniques in the density are discussed and compared to a procedure applied in DFT. The results of this work can be extended to other fundamental quantities in DFT.

Conclusions: The use of a general nonlocal density allows for the calculation of fundamental quantities taken as input in theories such as DFT. This allows benchmarking c.m. removal procedures and provides a bridge for comparison between ab initio and DFT many-body techniques.

Figure 1

Ground-state ${}^4\mathrm{He}$ nonlocal proton and neutron densities calculated using the bare $NN\text{−}{\mathrm{N}}^4\mathrm{LO}$(500) interaction with an $N_{\max }=18$ basis space. An oscillator frequency of $\hslash \mathrm{\Omega }=20.0$ MeV was used for this calculation.

Figure 2

Ground-state ${}^4\mathrm{He}$ nonlocal proton and neutron densities calculated using the SRG-evolved $NN\text{−}{\mathrm{N}}^4\mathrm{LO}\left(500\right)+3N\mathrm{lnl}$ interaction with an $N_{\max }=14$ basis space. An oscillator frequency of $\hslash \mathrm{\Omega }=20.0$ MeV and a flow parameter of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ were used for the calculations.

Figure 3

Ground-state trinv local density comparison for ${}^4\mathrm{He}$. (a) We show calculations with two-body $(NN+3N\mathrm{ind}$ SRG) and two- plus three-body $(NN+3N$ SRG) SRG-evolved interactions. (b) We show calculations with the bare two-body $NN\text{−}{\mathrm{N}}^4\mathrm{LO}$(500) interaction.

Figure 4

Ground-state ${}^8\mathrm{He}$ nonlocal proton and neutron densities calculated using the $NN\text{−}{\mathrm{N}}^4\mathrm{LO}\left(500\right)+3N\mathrm{lnl}$ interaction with an $N_{\max }=10$ basis space. An oscillator frequency of $\hslash \mathrm{\Omega }=20.0$ MeV and a flow parameter of ${\lambda }_{\mathrm{SRG}}=2.0$ ${\mathrm{fm}}^{-}1$ were used for the calculations.

Figure 5

Ground-state ${}^{16}\mathrm{O}$ nonlocal proton and neutron densities calculated using the $NN\text{−}{\mathrm{N}}^4\mathrm{LO}\left(500\right)+3N\mathrm{lnl}$ interaction with an $N_{\max }=8$ importance truncated basis space. An oscillator frequency of $\hslash \mathrm{\Omega }=20.0$ MeV and a flow parameter of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ were used for the calculations.

Figure 6

Ground-state ${}^4\mathrm{He}$ local proton (a), neutron (b), and total densities (c) computed using the $NN\text{−}{\mathrm{N}}^4\mathrm{LO}\left(500\right)+3N\mathrm{lnl}$ interaction with an $N_{\max }=14$ basis space, an oscillator frequency of $\hslash \mathrm{\Omega }=20.0$ MeV, and a flow parameter of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$.

Figure 7

Ground-state ${}^8\mathrm{He}$ local proton (a), neutron (b), and total densities (c) computed using the $NN\text{−}{\mathrm{N}}^4\mathrm{LO}\left(500\right)+3N\mathrm{lnl}$ interaction with an $N_{\max }=10$ basis space, an oscillator frequency of $\hslash \mathrm{\Omega }=20.0$ MeV, and a flow parameter of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$.

Figure 8

Ground-state ${}^{16}\mathrm{O}$ local proton (a), neutron (b), and total densities (c) computed using the $NN\text{−}{\mathrm{N}}^4\mathrm{LO}\left(500\right)+3N\mathrm{lnl}$ interaction with an $N_{\max }=8$ importance truncated basis space, an oscillator frequency of $\hslash \mathrm{\Omega }=20.0$ MeV, and a flow parameter of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$.

Figure 9

Ground-state trinv kinetic density comparison for ${}^4\mathrm{He}$. (a) We show calculations with two-body $(NN+3N\mathrm{ind}$ SRG) and two- plus three-body $(NN+3N$ SRG) SRG-evolved interactions. (b) We show calculations with the bare two-body $NN\text{−}{\mathrm{N}}^4\mathrm{LO}$(500) interaction. Nonlocal densities were computed as previously described in Figs. 1 and 2, respectively.

Figure 10

Ground-state ${}^4\mathrm{He}$ comparisons between the trinv and wiCOM kinetic densities. Proton (a), neutron (b), and total kinetic densities (c) are shown. The nonlocal density was computed as previously described in Sec. 3a. The expectation value of the intrinsic kinetic energy for ${}^4\mathrm{He}$ is $51.91\mathrm{MeV}$.

Figure 11

Ground-state ${}^6\mathrm{He}$ comparisons between the trinv and wiCOM kinetic densities. Proton (a), neutron (b), and total kinetic densities (c) are shown. The nonlocal density was computed as previously described in Sec. 3a. The expectation value of the intrinsic kinetic energy for ${}^6\mathrm{He}$ is $78.27\mathrm{MeV}$.

Figure 12

Ground-state ${}^8\mathrm{He}$ comparisons between the trinv and wiCOM kinetic densities. Proton (a), neutron (b), and total kinetic densities (c) are shown. The nonlocal density was computed as previously described in Sec. 3a. The expectation value of the intrinsic kinetic energy for ${}^8\mathrm{He}$ is $116.30\mathrm{MeV}$.

Figure 13

Ground-state ${}^{12}\mathrm{C}$ comparisons between the trinv and wiCOM kinetic densities. Proton (a), neutron (b), and kinetic total densities (c) are shown. The nonlocal density was computed as previously described in Sec. 3a. The expectation value of the intrinsic kinetic energy for ${}^{12}\mathrm{C}$ is $219.84\mathrm{MeV}$.

Figure 14

Ground-state ${}^{16}\mathrm{O}$ comparisons between the trinv and wiCOM kinetic densities. Proton (a), neutron (b), and total kinetic densities (c) are shown. The nonlocal density was computed as previously described in Sec. 3a. The expectation value of the intrinsic kinetic energy for ${}^{16}\mathrm{O}$ is $301.69\mathrm{MeV}$.

Figure 15

Ground-state $N_{\max }$ convergence results for ${}^4\mathrm{He}$ trinv kinetic neutron density. The nonlocal density was computed as previously described in Sec. 3a.

Figure 16

Ground-state $N_{\max }$ convergence results for ${}^{16}\mathrm{O}$ trinv kinetic neutron density. The nonlocal density was computed as previously described in Sec. 3a.

Figure 17

Ground-state total kinetic density results for ${}^4\mathrm{He}$ (a), ${}^8\mathrm{He}$ (b), ${}^{12}\mathrm{C}$ (c), and ${}^{16}\mathrm{O}$ (d) calculated with the $NN\text{−}{\mathrm{N}}^4\mathrm{LO}\left(500\right)+3N\mathrm{lnl}$ interaction. The nonlocal densities for the nuclei were computed as previously described in Sec. 3a. The DFT kinetic density was obtained by using Eq. (19), ${\tau }_{\mathrm{DFT}}\left(r\right)=(1-\frac{1}{A}){\tau }_{wiCOM}\left(r\right)$.