Self-consistent Green's function embedding for advanced electronic structure methods based on a dynamical mean-field concept

We present an embedding scheme for periodic systems that facilitates the treatment of the physically important part (here a unit cell or a supercell) with advanced electronic structure methods, that are computationally too expensive for periodic systems. The rest of the periodic system is treated with computationally less demanding approaches, e.g., Kohn-Sham density-functional theory, in a self-consistent manner. Our scheme is based on the concept of dynamical mean-field theory formulated in terms of Green's functions. Our real-space dynamical mean-field embedding scheme features two nested Dyson equations, one for the embedded cluster and another for the periodic surrounding. The total energy is computed from the resulting Green's functions. The performance of our scheme is demonstrated by treating the embedded region with hybrid functionals and many-body perturbation theory in the GW approach for simple bulk systems. The total energy and the density of states converge rapidly with respect to the computational parameters and approach their bulk limit with increasing cluster (i.e., computational supercell) size.

Figure 1

Schematic depiction of the embedding concept. The embedded region (red) is treated with a more accurate theory, whereas the surrounding (gray) is calculated with less accurate and thus computationally less expensive theories. The key challenge of our and all previous embedding approaches is the appropriate treatment of the red/gray boundary.

Figure 2

The DMFT embedding concept for a Si unit cell. The atoms in the unit cell (red region) constitute the embedded submanifold. Each unit cell of the periodic system is treated as a localized region, i.e., only local interactions ${\mathrm{\Sigma }}^{\mathrm{loc}}$ are treated. The unit cells are coupled to the rest of the system via the hybridization self-energy $\mathrm{\Delta }\left(i\omega \right)$ (green arrow).

Figure 3

The embedded [Eq. (15)] and the onsite [Eq. (7)] Green's functions define two Dyson equations that form two nested loops. The two loops are iterated until self-consistency is reached.

Figure 4

Typical change in the chemical potential during the self-consistency cycle for a bulk Si calculation. Convergence is reached after five iterations of the outer loop. Already at the second iteration the chemical potential is close to its converged value.

Figure 5

RDMFE(PBE0) total energy (upper panel) and cohesive energy (lower panel) for bulk Si with a two-atom unit cell as function of the basis size. The energy zero is set at the value of the tier 1 basis.

Figure 6

RDMFE(PBE0) DOS comparison at each iteration of the main loop. Convergence was achieved after five main-loop iterations.

Figure 7

Comparison of the RDMFE(PBE0) DOS, the periodic PBE, and the periodic PBE0 DOS for a two-atom unit cell.

Figure 8

Comparison of the RDMFE(PBE0) DOS, with the periodic PBE0 DOS for an 8-atom unit cell (upper panel) and a 16-atom unit cell (lower panel).

Figure 9

Upper panel: the RDMFE(PBE0) band structure for bulk Si compared to the periodic PBE0 one. The local self-energy breaks the translation symmetry and the degeneracy gets shifted at some high-symmetry $\mathbf{k}$ points. Middle panel: the RDMFE(PBE0) unfolded band structure for the 16-atom unit cell. The degeneracy shifting gets reduced compared to the 2-atom case. Lower panel: the RDMFE(PBE0) unfolded band structure for the 32-atom unit cell. Also, here the degeneracy is restored in most of the high-symmetry points.

Figure 10

The band structure of bulk Na determined from the k-resolved RDMFE(GW) spectral function along the $\mathrm{\Gamma }\text{−}N$ direction, compared to the periodic PBE one (blue solid line). Both band structures are aligned at the same chemical potential.

Figure 11

Gaussian broadened (with broadening $\sigma =0.01$ eV) quasiparticle spectrum for the RDMFE(GW) self-energy at first iteration (red curve) and at self-consistency (blue curve). Only occupied states are shown.

Figure 12

Upper panel: the RDMFE(PBE0) cohesive energy for bulk Si for increasing unit-cell size. Lower panel: the RDMFE(GW) total energy for bulk He for increasing unit-cell size, where the blue region indicates a change in the range of $\sim 20$ meV. Both curves are referenced to the PBE total energy.

Figure 13

Upper panel: the RDMFE(HSE) total energy for bulk Si for increasing unit-cell size and different screening parameter $\gamma$. The range-separated total energy converges faster than the PBE0 one (black curve). All the RDMFE(HSE) total energies are shifted to the 2-atom unit-cell value of the embedded PBE0 curve to allow better comparison. Lower panel: the change of the sphere radius and screening parameter $\gamma$ (on a reciprocal scale) with the volume of the sphere surrounding the 2-, 4-, 8-, and 16-atom unit cells.

Figure 14

The RDMFE total energy for bulk Si as a function of the lattice constant: RDMFE(PBE0) is shown in the upper panel, where the values for the 2-atom unit cell (blue curve) and 16-atom unit cell (red curve) are compared to the periodic PBE0 (green curve) and experiment [88] (vertical cyan solid line). The equilibrium lattice constant for the 8- and 16-atom unit cells is indicated with the vertical red solid line. The RDMFE(GW) total energy (red curve) is illustrated in the lower panel. Comparison is made with the periodic PBE (black curve). Lattice constant from periodic scGW from the work of Kutepov et al. [54] is indicated by the vertical green solid line.