Pseudopotential for the electron-electron interaction

We propose a pseudopotential for the electron-electron Coulomb interaction to improve the efficiency of many-body electronic structure calculations. The pseudopotential accurately replicates the scattering properties of the Coulomb interaction, and recovers the analytical solution for two electrons in a parabolic trap. A case study for the homogeneous electron gas using the diffusion Monte Carlo and configuration interaction methods recovers highly accurate values for the ground state energy, and the smoother potential reduces the computational cost by a factor of $\sim 30$. Finally, we demonstrate the use of the pseudopotential to study isolated lithium and beryllium atoms.

Figure 1

(a) Interaction potentials: the Coulomb potential is shown in red, the pseudopotential with cutoff radius $c=r_0$ is in cyan, and the pseudopotential with cutoff radius $c=2r_0$ is in blue. (b) The error in the logarithmic derivative of the scattering wave function with cutoff radius for an electron gas with $r_{\text{s}}=2$. $\delta$ shows the error summed over angular momentum channels, ${\delta }_0$ is the contribution from the $\ell =0$ channel, and ${\delta }_1$ from the $\ell =1$ channel. (c) The pseudopotential error for a range of $r_{\text{s}}$ values. (d) The error in the logarithmic derivative of the scattering wave function with incident wave vector for the $\ell =0$ and $\ell =1$ scattering channels. The filled blue curve plotted on an arbitrary linear scale on the secondary $y$ axis shows the weighting factor ${\rho }_{\ell }\left(k\right)$ used in evaluating the overall error in the logarithmic derivative.

Figure 2

(a) Wave function of the relative motion of two electrons in a harmonic trap in the $\ell =0$ angular momentum state. The green curve shows the wave function for noninteracting electrons, the red shows electrons interacting via the Coulomb interaction, and blue shows interactions via the pseudopotential. (b) The error per electron in the estimate of the ground state energy of two opposite-spin ($\ell =0$) and same-spin ($\ell =1$) electrons in a parabolic trap as a function of cutoff radius. The vertical blue dotted line shows the typical separation of the electrons in the harmonic trap.

Figure 3

(a) Upper: the error in the energy of the HEG with DMC time step. The $y$-axis origin is set by the extrapolation to zero time step of the energy obtained with the Coulomb interaction. The red error bars show the Coulomb interaction and the blue error bars the pseudopotential; the solid lines show linear extrapolation to zero time step. Lower: the uncertainty in energy predictions, with the lines showing the ${\tau }^{-1/2}$ fit. (b) The error in the energy with cutoff radius for different time steps $\tau \in \{0,\cdots ,1\}$. Each curve has a minimum with cutoff radius, the locus of these minima with varying time step is tracked by the green dashed line. (c) The relative statistical uncertainty with cutoff radius in DMC. (d) The spin-resolved pair correlation functions for same-spin ($g_{\uparrow \uparrow }$) and opposite-spin ($g_{\uparrow \downarrow }$) electrons for the Coulomb interaction are shown in red and those for the pseudopotential are in blue. The dotted blue curve shows the analytic correction ($g_{\uparrow \downarrow }^{\text{2body}}$) applied to the cyan $g_{\uparrow \downarrow \text{PP}}$ obtained directly from the pseudopotential. (e) The error in the energy and speedup obtained with density.

Figure 4

(a),(b) CID wave function for opposite-spin electrons passing through coalescence for the HEG at $r_{\text{s}}=2$ with increasing plane-wave orbital basis sets of size $M$. The exact solution is shown in red (blue) for the Coulomb interaction potential and the pseudopotential. (c) The average relative error in the wave function, and (d) the percentage error in the energy with basis set size for the Coulomb potential (red) and pseudopotential (blue) with dotted trend lines.

Figure 5

(a) Error $\mathrm{\Delta }E$ in the total energy with cutoff radius. The red points are for the isolated Li atom and blue are for the Be atom. The gray shading denotes where the results attain chemical accuracy. (b) The error in the total energy of the Li (red) and Be (blue) ions. (c) The error in the ionization energy of a Li (red) and Be (blue) atom. (d) The speedup of the DMC calculation. The red points show the Li atom and blue the Be atom.