Ab Initio Many Body Quantum Embedding and Local Correlation in Crystalline Materials using Interpolative Separable Density Fitting

TL;DR

This work tackles the high computational cost of correlated wavefunction methods for extended solids by marrying translational symmetry with interpolative separable density fitting in a periodic Gaussian-orbital framework. The authors implement an FFT-adapted ISDF (FFTISDF) representation to achieve linear scaling with the number of k-points and integrate it with density matrix embedding (DMET) and local natural orbital (LNO) correlation methods, enabling CCSD(T)-level descriptions in crystals in the thermodynamic limit. Benchmark results on diamond, CO$_2$, NiO, and CaCuO$_2$ demonstrate accurate ground-state energies and practical extrapolation to the thermodynamic limit, with notably efficient handling of 1000 k-points. The approach significantly broadens access to high-accuracy, many-body treatments for periodic materials and can be extended to other embedding schemes such as dynamical mean field theory.

Abstract

We present an efficient implementation of ab initio many-body quantum embedding and local correlation methods for infinite periodic systems through translational symmetry adapted interpolative separable density fitting, an approach which reduces the scaling of the calculations to only linear with the number of k-points. Employing this methodology, we compute correlated ground-state coupled cluster energies within density matrix embedding and local natural orbital correlation frameworks for both weakly and strongly correlated solids, using up to 1000 k-points. By extrapolating the local correlation domains and k-point sampling we further obtain estimates of the full coupled cluster with singles, doubles, and perturbative triples ground-state energies in the thermodynamic limit.

Figures (4)

• Figure 1: Per-atom energy errors for diamond obtained with: FFTDF ($E_{\text{cut}} = 60~\text{a.u.}$, green), GDF ($\beta = 2.0$, red), and FFTISDF ($E_{\text{cut}} = 60~\text{a.u.}$, blue). For FFTISDF, results are shown for $c_{\text{IP}} = 10$ (dashed) and $c_{\text{IP}} = 14$ (solid). Errors are computed with respect to a reference FFTDF calculation ($E_{\text{cut}} = 160~\text{a.u.}$). Subfigures correspond to (a) k-HF and (b) k-MP2.
• Figure 2: Comparison of wall-clock times for different density-fitting schemes applied to the diamond system, plotted against the number of k-points ($N_{\text{k}}$). Panels show the timing for: (a) integral evaluation, (b) Coulomb matrix construction, (c) exchange matrix construction, and (d) embedding Hamiltonian construction. Solid lines indicate fitted trends highlighting the computational scaling (linear or quadratic) of each method.
• Figure 3: Extrapolation of CO$_2$ correlation energies. LNO-MP2, LNO-CCSD, and LNO-CCSD(T) correlation energies plotted against $\Delta_{\mathrm{LNO}}$ (the difference between k-SOS-MP2 and LNO-SOS-MP2 correlation energies) for k-point meshes: (a) $1 \times 1 \times 1$ ($\Gamma$-point only), (b) $2 \times 2 \times 2$, and (c) $4 \times 4 \times 4$. Small circles represent LNO correlation energies, triangles show global k-point counterparts, crosses mark extrapolated values, and solid lines are fitted lines.
• Figure 4: Thermodynamic limit extrapolation for diamond. Upper panel: k-HF total energy as a function of $1/N_{\text{k}}$. Lower panel: correlation energies from various methods including k-SOS-MP2, LNO-MP2, k-dRPA, LNO-CCSD, and LNO-CCSD(T). Solid lines represent linear fits used for extrapolation to the thermodynamic limit ($1/N_{\text{k}} \to 0$).