Variational Excitations in Real Solids: Optical Gaps and Insights into Many-Body Perturbation Theory

We present an approach to studying optical band gaps in real solids in which quantum Monte Carlo methods allow for the application of a rigorous variational principle to both ground and excited state wave functions. In tests that include small, medium, and large band gap materials, optical gaps are predicted with a mean absolute deviation of 3.5% against experiment, less than half the equivalent errors for typical many-body perturbation theories. The approach is designed to be insensitive to the choice of density functional, a property we exploit in order to provide insight into how far different functionals are from satisfying the assumptions of many-body perturbation theory. We explore this question most deeply in the challenging case of ZnO, where we show that, although many commonly used functionals have shortcomings, there does exist a one-particle basis in which perturbation theory’s zeroth-order picture is sound. Insights of this nature should be useful in guiding the future application and improvement of these widely used techniques.

Figure 1

VMC-CISD optical gap predictions plotted against experimental results. See Table 1 for more details.

Figure 2

Here we investigate the appropriateness of various one-particle orbital sets for MBPT by plotting the VMC-CISD residual weight fraction, which we define as the sum of squared CI coefficients on all configurations other than the primary $\mathrm{VBM}\to \mathrm{CBM}$ transition when working in a particular orbital basis. In cases where degeneracy in the VBM leads to multiple equal-energy $\mathrm{VBM}\to \mathrm{CBM}$ configurations, the sum excludes all such configurations.

Figure 3

Optical gap and single-particle transition energy data for ZnO. On the left, we compare $G_0{W}_0$ fundamental gaps using one-particle starting points that employ different fractions of exact exchange with our VMC-CISD optical gaps based on the same starting points. For the various $i\to a$ transitions, we plot on the right histograms of the differences ${\mathcal{D}}_{ia}={\mathrm{\Delta }}_{ia}^{\mathrm{DFT}}-{\mathrm{\Delta }}_{ia}^{\mathrm{VMC}}$ between the DFT estimates (i.e., the orbital energy differences ${\mathrm{\Delta }}_{ia}^{\mathrm{DFT}}={\epsilon }_a-{\epsilon }_i$) for the energy cost of promoting an electron from orbital $i$ to orbital $a$ and the analogous quantities ${\mathrm{\Delta }}_{ia}^{\mathrm{VMC}}$, which are the VMC energy differences between the $i\to a$ excited and the ground state Jastrow-modified Slater determinants. $G_0{W}_0$ data from Fuchs et al. [47]. Experimental result from Lauck and co-workers [48].

Figure 4

A cut along ZnO’s ($\overline{1}2\overline{1}0$) plane in which we investigate the lowest energy excitation’s hole density in the vicinity of the Zn atom. For each method, we plot the contour along which the number of holes per ${\AA }^3$ is equal to 1.2.