Absorbing boundary conditions for the time-dependent Schrödinger-type equations in ${\mathbb{R}}^3$

Absorbing boundary conditions are presented for three-dimensional time-dependent Schrödinger-type of equations as a means to reduce the cost of the quantum-mechanical calculations. The boundary condition is first derived from a semidiscrete approximation of the Schrödinger equation with the advantage that the resulting formulas are automatically compatible with the finite-difference scheme and no further discretization is needed in space. The absorbing boundary condition is expressed as a discrete Dirichlet-to-Neumann map, which can be further approximated in time by using rational approximations of the Laplace transform to enable a more efficient implementation. This approach can be applied to domains with arbitrary geometry. The stability of the zeroth-order and first-order absorbing boundary conditions is proved. We tested the boundary conditions on benchmark problems. The effectiveness is further verified by a time-dependent Hartree-Fock model with Skyrme interactions. The accuracy in terms of energy and nucleon density is examined as well.

Figure 1

An illustration of the model reduction for one-dimensional Schrödinger equation. $\psi$ is the wave function, initially supported in the computational domain ${\mathrm{\Omega }}_{\text{I}}$.

Figure 2

The solutions computed using fixed boundary condition, zeroth-order, first-order, and second-order ABCs compared with the exact solution.

Figure 3

Comparison of the total electron density in ${\mathrm{\Omega }}_{\text{I}}$ with different ABCs (left). Comparison of the total electron density in ${\mathrm{\Omega }}_{\text{I}}$ with different complex absorbing potential strength (right).

Figure 4

The number of electrons as a function of time.

Figure 5

Projection of the 3D electron density on $x-y$ plane. Time-dependent Schrödinger equation with the fixed boundary condition (first row), the first-order ABC (second row), the second-order ABC (third row), and the exact solution (last row). The color indicated the electron density.

Figure 6

The loss of nucleons and total energy.

Figure 7

Error of the nucleon density in time evolution of ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ collision, projected onto the $x\text{−}y$ plane. From the top to the bottom: (A) $s_1={10}^{-1},s_2=2\times {10}^{-1}$; (B) $s_1={10}^0,s_2=2\times {10}^0$; (C) $s_1={10}^1,s_2=2\times {10}^1$; (D) $s_1={10}^2,s_2=2\times {10}^2$; (E) $s_1={10}^{-2},s_2=2\times {10}^{-2}$; (F) fixed boundary condition.

Figure 8

One-dimensional discrete Green's function versus one-dimensional continuum Green's function, $s=1,h=1$.

Figure 9

Three-dimensional discrete Green's function and three-dimensional continuum Green's function, $h=0.01,s=10$.