Ab Initio Calculations in a Uniform Magnetic Field Using Periodic Supercells

We present a formulation of ab initio electronic structure calculations in a finite magnetic field, which retains the simplicity and efficiency of techniques widely used in first principles molecular dynamics simulations, based on plane-wave basis sets and Fourier transforms. In addition we discuss results obtained with this method for the energy spectrum of interacting electrons in quantum wells, and for the electronic properties of dense fluid deuterium in a uniform magnetic field.

Figure 1

The real-space wave function $\psi (x,y)$ can be Fourier transformed into reciprocal space $c(k_x,k_y)$ in two steps, via an intermediate-space wave function $f(x,k_y)$ (see text). (a) At $B=0$, $f(x,k_y)$ can be regarded as a set of one-dimensional periodic functions, or rings. (b) At $B=h/(eab)$, MPBC requires $f(x,k_y)$ to be a long spiral. The resulting wave function in intermediate and reciprocal space is effectively one dimensional.

Figure 2

(a) Energy spectrum of single electron in quantum dot as a function of magnetic field. The inset shows the geometry of quantum dot (shaded area) and simulation cell (outer square). (b) Energy spectrum of two interacting electrons in quantum dot as a function of magnetic field. Filled and open circles indicate spin singlet and triplet states from this work. Thick and thin lines are for singlet and triplet states from Ref. 12.

Figure 3

(a) Lowest levels of a hydrogen atom as a function of magnetic field: ○ for this work, solid line for previous results [5]. (b) Change of binding energy of an ${\mathrm{H}}_{\mathrm{2}}$ molecule in the ${}^1\Sigma _g$ state (two spins antiparallel) as a function of magnetic field: ○ for our result using Hartree-Fock approximation with inter-proton distance fixed at $0.74\AA$, ● and solid line from Ref. 13.

Figure 4

(a) Total charge density of a dense deuterium fluid (see text), which remains essentially the same as $B$ goes from $0$ to ${10}^4\mathrm{T}$. (b) The charge densities of the HOMO state for $B=0$ (blue) and $B={10}^4\mathrm{T}$ (red) are distributed on different atoms.