Time-dependent density functional theory with twist-averaged boundary conditions

Background: Time-dependent density functional theory is widely used to describe excitations of many-fermion systems. In its many applications, three-dimensional (3D) coordinate-space representation is used, and infinite-domain calculations are limited to a finite volume represented by a spatial box. For finite quantum systems (atoms, molecules, nuclei, hadrons), the commonly used periodic or reflecting boundary conditions introduce spurious quantization of the continuum states and artificial reflections from boundary; hence, an incorrect treatment of evaporated particles.

Purpose: The finite-volume artifacts for finite systems can be practically cured by invoking an absorbing potential in a certain boundary region sufficiently far from the described system. However, such absorption cannot be applied in the calculations of infinite matter (crystal electrons, quantum fluids, neutron star crust), which suffer from unphysical effects stemming from a finite computational box used. Here, twist-averaged boundary conditions (TABC) have been used successfully to diminish the finite-volume effects. In this work, we extend TABC to time-dependent modes.

Method: We use the 3D time-dependent density functional framework with the Skyrme energy density functional. The practical calculations are carried out for small- and large-amplitude electric dipole and quadrupole oscillations of ${}^{16}\mathrm{O}$. We apply and compare three kinds of boundary conditions: periodic, absorbing, and twist-averaged.

Results: Calculations employing absorbing boundary conditions (ABC) and TABC are superior to those based on periodic boundary conditions. For low-energy excitations, TABC and ABC variants yield very similar results. With only four twist phases per spatial direction in TABC, one obtains an excellent reduction of spurious fluctuations. In the nonlinear regime, one has to deal with evaporated particles. In TABC, the floating nucleon gas remains in the box; the amount of nucleons in the gas is found to be roughly the same as the number of absorbed particles in ABC.

Conclusion: We demonstrate that by using TABC, one can reduce finite-volume effects drastically without adding any additional parameters associated with absorption at large distances. Moreover, TABC are an obvious choice for time-dependent calculations for infinite systems. Since TABC calculations for different twists can be performed independently, the method is trivially adapted to parallel computing.

Figure 1

Quadrupole moment $Q\left(t\right)$ (a) and strength function (b) for the isoscalar $E2$ excitation of ${}^{16}\mathrm{O}$ with $E^{*}=3\mathrm{MeV}$ and $L=40\mathrm{fm}$. The inset illustrates the geometry of the problem: the total density of ${}^{16}\mathrm{O}$ (center), the absorption zone (red), and the region of zero density (black). The low-energy part of the spectrum contains very little strength and is not shown.

Figure 2

Isovector $E1$ strength for ${}^{16}\mathrm{O}$ with $E^{*}=1\mathrm{MeV}$ and two box sizes: $L=32$ and 40 fm.

Figure 3

Isoscalar $E2$ power spectrum for ${}^{16}\mathrm{O}$ at $E^{*}=20\mathrm{MeV}$ and $L=32\mathrm{fm}$. The inset shows the time evolution of the mass quadrupole moment $Q$. See [57] for animations.

Figure 4

Angular-averaged density distribution corresponding to the isoscalar $E2$ mode at $E^{*}=20\mathrm{MeV}$ at two snapshots: $t=0$ and $t={10}^4\mathrm{fm}/c$. The gas densities are indicated by dotted lines: ${\rho }_{n,\mathrm{gas}}=1.15\times {10}^{-5}{\mathrm{fm}}^{-3}$ and ${\rho }_{p,\mathrm{gas}}=1.42\times {10}^{-5}{\mathrm{fm}}^{-3}$.

Figure 5

Isovector $E1$ power spectrum for ${}^{16}\mathrm{O}$ at $E^{*}=22\mathrm{MeV}$ and $L=32\mathrm{fm}$. The inset shows the time evolution of the dipole moment $D$. See [57] for animations.