Electric Polarization from a Many-Body Neural Network Ansatz

Ab initio calculation of dielectric response with high-accuracy electronic structure methods is a long-standing problem, for which mean-field approaches are widely used and electron correlations are mostly treated via approximated functionals. Here we employ a neural network wave function ansatz combined with quantum Monte Carlo method to incorporate correlations into polarization calculations. On a variety of systems, including isolated atoms, one-dimensional chains, two-dimensional slabs, and three-dimensional cubes, the calculated results outperform conventional density functional theory and are consistent with the most accurate calculations and experimental data. Furthermore, we have studied the out-of-plane dielectric constant of bilayer graphene using our method and reestablished its thickness dependence. Overall, this approach provides a powerful tool to accurately describe electron correlation in the modern theory of polarization.

Figure 1

A brief illustration of the computational cost and accuracy of different electronic structure methods in polarization calculations. $N$ denotes the number of electrons in the system. High-level correlated wave function methods are not shown in the PBC panel because they have not been applied to PBC polarization calculations so far. MP2, second-order Møller-Plesset perturbation theory; CCSD, coupled cluster with single and double excitations; FCI, full configuration interaction.

Figure 2

Calculations of chains and slabs. (a) Illustrations of hydrogen chain, polyyne, hydrogen slab, and the applied electric field. (b) Hydrogen chain, (c) Polyyne, (d) Hydrogen slab susceptibilities $\chi$. ${\mathrm{\Omega }}_p$ denotes the volume of the primitive cell. Parameters and training curves of DeepSolid are presented in Supplemental Material, Secs. VI–VIII [37]. For the hydrogen system, LDA and HF results were calculated with the 3–21G basis set under PBC [5]. CCSD(T) calculations were performed with the $6–311{\mathrm{G}}^{**}$ basis set under OBC [9]. Intra- and interpair distances are set to 2 and 3 bohr, respectively, for the hydrogen chain. The interchain distance is set to 4.724 bohr for the slab. For polyyne, the alternating distance between carbon atoms is set to 1.18 and 1.4 Å. LDA, HF, and RPA results were calculated with 6–31G, 3–21G, and 4–31G basis sets, respectively, under OBC [49, 50, 51].

Figure 3

Calculated high-frequency dielectric constant ${\epsilon }_{\infty }$ of alkali metal hydrides XH. Statistical errors are negligible for the presented data. LDA and PBE results in PBC are taken from Ref. [55]. HF results are obtained using the vasp code. Experimental data are taken from Refs. [56, 57, 58]. Exp1 is derived from the refractive indexes $n_{\mathrm{D}}$ for sodium doublet (589.29 nm) via Eq. (6), and Exp2 is the extrapolated result toward infinite wave length. RbH experiments are absent.

Figure 4

Calculated effective two-dimensional dielectric constant ${\epsilon }_{\infty }^{\perp }$ of bilayer graphene. Statistical errors are negligible for the presented data. (a) Bilayer graphene under electric field. (b) Calculated ${\epsilon }_{\infty }^{\perp }$ as a function of graphene layer distance $d$. The equilibrium separation of BLG is set to 3.347 Å.