Two-dimensional pseudospectral Hartree-Fock method for low-$Z$ atoms in intense magnetic fields

The energy levels of the first few low-lying states of helium and lithium atoms in intense magnetic fields up to $\approx {10}^8–{10}^9$ T are calculated in this study. A pseudospectral method is employed for the computational procedure. The methodology involves computing the eigenvalues and eigenvectors of the generalized two-dimensional Hartree-Fock partial differential equations for these two- and three-electron systems in a self-consistent manner. The method exploits the natural symmetries of the problem without assumptions of any basis functions for expressing the wave functions of the electrons or the commonly employed adiabatic approximation. It is found that the results obtained here for a few of the most tightly bound states of each of the atoms, helium and lithium, are in good agreement with findings elsewhere. In this regard, we report data for two states of lithium that were lacking thus far. It is also seen that the pseudospectral method employed here is considerably more economical, from a computational point of view, than previously employed methods such as a finite-element-based approach. The key enabling advantage of the method described here is the short computational times, which are on the order of seconds for obtaining accurate results for heliumlike systems.

Figure 1

Computational times shown as a function for the number of grid points, for typical calculations for both helium and lithium. The dotted vertical line corresponds to the number of grid points typically required to attain an accuracy of $\epsilon \approx 0.1\%\equiv {10}^{-4}{E}_{Z,\infty }$. The number of cores or processors employed by the parallel code is equal to the number of electrons in the atom. The dependence of the computational time on the number of grid points is roughly $O\left(N^3\right)$, while the dependence on the number of electrons is more or less linear, i.e., $O\left(n_e\right)$.

Figure 2

Convergence of the binding energy with mesh refinement for four different domain sizes for the $1^3{\left(0\right)}^{+}$ state of helium. The level of mesh refinement corresponds to $N$ points in each of the two orthogonal directions. The size of the matrix of the coupled eigenvalue problem for a given level of mesh refinement is given by $n_e{(N-1)}^2\times n_e{(N-1)}^2$, where $n_e$ is the number of electrons, which for helium is 2. The levels of mesh refinement employed correspond to $N=21$, 31, 41, 51, 61, and 71 points in each of the $x$ and $y$ directions. The exponential convergence of the spectral method can readily be seen.

Figure 3

Convergence of the binding energy with mesh refinement for four different domain sizes for the $1^3{(-1)}^{+}$ state of helium. The level of mesh refinement corresponds to $N$ points in each of the two orthogonal directions. The size of the matrix of the coupled eigenvalue problem for a given level of mesh refinement is given by $n_e{(N-1)}^2\times n_e{(N-1)}^2$, where $n_e$ is the number of electrons, which for helium is 2. The levels of mesh refinement employed correspond to $N=21$, 31, 41, 51, 61, 71, and 81 points in each of the $x$ and $y$ directions for helium. The exponential convergence of the spectral method can readily be seen for computations in the different domain sizes.

Figure 4

Pictorial representation of the domain $[-1,1]\otimes [-1,1]$ discretized using $N+1$ points in each direction, with $N=3$. The outer boundaries have Dirichlet conditions imposed.

Figure 5

Pictorial representation of the operator $L$ acting upon the vector $\mathbf{p}$. The number of points in either direction is $N=3$ in the current example.

Figure 6

Pictorial representation of the operator $B_x$ acting upon the vector $\mathbf{p}$. The number of points in either direction is $N=3$.

Figure 7

Pictorial representation of the operator $B_y$ acting upon the vector $\mathbf{p}$. The number of points in either direction is $N=3$.

Figure 8

Pictorial representation the operator $M$ acting upon the vector $\left(\begin{array}{c}\mathbf{p}\\ \mathbf{q}\end{array}\right)$. The number of points in either direction, is $N=3$.