Density functional embedding for periodic and nonperiodic diffusion Monte Carlo calculations

We present density functional embedding for diffusion Monte Carlo calculations (DMC). Using two test systems, an H chain and a Be slab, we demonstrate the feasibility and show that the approach can give high-quality results. DMC embedded in potentials from density functional embedding theory thus presents an alternative to conventional embedded quantum chemical methods, such as embedded complete active space calculations. This is especially true for cases in which the cluster size becomes too large for accurate wave-function theory based methods or for cases in which basis-set superposition errors become dominant. Furthermore, we show that embedded DMC can easily be used for systems in which the cluster of interest is a one-dimensional or two-dimensional periodic object and cannot be described as isolated impurity. This extends the applicability of the embedding approach to new fields.

Figure 1

(a) Sketch of the ${\mathrm{H}}_4$ chain. The light gray atom is replaced by an embedding potential in the embedded cluster calculations. (b) Change in binding energy for different values of $\mathrm{\Delta }$, where $\mathrm{\Delta }$ is defined in subplot (b). The error bars of the DMC results are not shown, but they are all smaller than $10\mathrm{meV}$ and are thus negligible.

Figure 2

Cut of the density differences $\mathrm{\Delta }\rho$ between the sum of the cluster density and the environment density, and the density of the full calculation along the chain of H atoms for different values of $E_{\text{cut}}$ (left $y$ axis). The total density is shown for reference (right $y$ axis). The positions of the H atoms in the box are indicated by black arrows. The leftmost atom is the one that is replaced by an embedding potential.

Figure 3

Energy change associated with the bond stretch in the ${\mathrm{H}}_4$ chain: (a) DMC results for the full system compared to embedded DMC (DMC in LDA) results and DMC results for the cluster without embedding potential (see text for details). (b) Comparison of embedded DMC results with embedded DFT results. The error bars of the DMC results are not shown, but they are all smaller than $10\mathrm{meV}$ and are thus negligible.

Figure 4

Sketch of the geometries used to determine the cohesive energy [see Eq. (15)] of Be. The slabs are periodic in the $x-y$ plane.

Figure 5

Absolute values of the maximum density differences between the full and the embedded system obtained for a 6 layer Be slab at different values of $\lambda$.

Figure 6

2D cut through the optimized embedding potential for a six-layer slab of beryllium, where the two outermost layers on the top and the two outermost layers on the bottom are described by the environment and the two central layers are cluster atoms. (a) Sketch of the geometry. The direction of the 2D cut used in (b) and (c) is indicated by a bold dashed line. Dark green balls indicate first-layer atoms, light green balls second-layer atoms. The thin dotted line shows the unit cell. (b) Embedding potential obtained using $\lambda =1\times {10}^{-6}$; black arrows indicate the region in the embedding potential expected to be responsible to replicate the binding between the cluster and the environment. (c) Same as (b) but using an $\mathbf{r}$-dependent $\lambda$ function with ${\lambda }_i=1\times {10}^{-7},{\lambda }_o=1\times {10}^{-5}$, and $r_{\text{cut}}=1.0\AA$. The blue and green circles represent the positions of the environment and cluster atoms in the cut plane, respectively.

Figure 7

Maximum and integrated density difference for a three- and a six-layer slab of beryllium for different values of $\lambda$. Since the density differences at high and at low plane-wave cutoff should both be low, we only report the larger of the differences obtained for $E_{\text{cut}}=600\mathrm{eV}$ or for an embedding potential computed at $E_{\text{cut}}=600\mathrm{eV}$ and used at $E_{\text{cut}}=1720\mathrm{eV}$, unless noted otherwise. Horizontal lines: values obtained for constant values of $\lambda$ (${\lambda }_o={\lambda }_i$); dotted horizontal lines: result obtained at $E_{\text{cut}}=600\mathrm{eV}$ (shown for reference at $\lambda =1\times {10}^{-8}$ and $\lambda =1\times {10}^{-6}$ only); thick black line: results obtained for different combinations of ${\lambda }_i,{\lambda }_o$, and $r_{\text{cut}}$ (Bohr).