A Transcorrelated Wave-Function Framework for Solids: An Application to Bulk and Defected Silicon

TL;DR

This work develops a periodic transcorrelated embedding framework (periodic xTC-PP) that transforms the Hamiltonian with a Jastrow factor to dramatically improve basis-set convergence for solids, and couples this with fragment-based CC solvers to treat defects. Benchmarking on bulk silicon shows TZ-level xTC-PP-CCSD(T) energies approach FCIQMC and CCSDT results, effectively reaching CBS accuracy with reduced computational cost. Applying the embedding to silicon self-interstitial defects yields formation energies that converge with fragment size and basis level, aligning well with experimental and reference values. Overall, transcorrelation combined with embedding provides a practical route to quantitatively reliable wave-function studies of crystals and defects, significantly reducing the basis-set bottleneck.

Abstract

Accurate wave-function descriptions of pristine and defected solids remain challenging due to the simultaneous presence of finite-size, basis-set, and correlation errors. While embedding techniques alleviate finite-size effects and correlated wave-function approaches systematically improve correlation, basis-set incompleteness continues to limit practical accuracy. Here we present a study of transcorrelated (TC) many-body wave-function methods on properties of solid state systems. We augment the existing xTC theory to periodic systems, and establish an unified transcorrelated embedding framework that integrates periodic TC theory with fragment-based correlated solvers. Using silicon as a test case, we validate the method against coupled-cluster, FCIQMC, and diffusion Monte Carlo benchmarks for bulk. Then we apply TC embedding to calculation of formation energies of two silicon self-interstitials. The TC Hamiltonian yields rapid basis convergence and quantitatively reliable defect formation energies at the triple-$ζ$ level, substantially reducing the basis-set bottleneck for wave-function treatments of crystalline defects.

Figures (7)

• Figure 1: The periodic simulation cells of the relaxed hexagonal (H,left) and split (X, right) silicon self-interstitial defects used in the formation energy calculations, with the highest occupied fragment orbitals plotted.
• Figure 2: Computational workflows used in this work. Boxes represent the calculation stages and the codes employed. Directed arrows indicate the flow of data between stages, and their labels specify the quantities transferred (see Sec. \ref{['sec:Theory']}). The orbitals $\phi_p$ denote periodic HF orbitals, $\phi_i^\mathrm{loc}$ their localized counterparts, and $\phi_{p_f}$ and $\phi_{i_e}$ the $\Gamma$-point Bloch sums of the Wannier functions $i_e$ and fragment orbitals $p_f$, $q_f$, respectively. $E_0$ is the nuclear repulsion energy. Green indicates steps specific to the embedding workflow, blue those specific to fully periodic calculations, and lavender the stages shared by both workflows. For clarity, we show the xTC data passed to CC and FCIQMC only once.
• Figure 3: Embedding fragments used in this work. Each letter denotes a bulk–defect fragment pair: for the H-interstitial series (a–e) correspond to pairs with (7), (9), (15), (21), and (27) atoms in the defect fragments; for the X-interstitial series (a–d) correspond to pairs with (8), (12), (16), and (22) atoms. In bulk fragments corresponding to a certain defect fragment there is always one atom less. Fragment Si atoms are shown in orange; environment Si atoms are shown as smaller midnight-blue balls. For clarity, we only show a subset of the atoms in the periodic simulation cells of Fig. \ref{['fig: defect illustration']}.
• Figure 4: Total electronic energies per primitive unit cell for bulk silicon obtained from eight-atom simulation-cell calculations. Bars show CCSD, DCSD, CCSD(T), and CCSDT results, together with the highest-population FCIQMC (init 10) energies, evaluated using Gaussian DZ and TZ basis sets for the xTC-PP Jastrow cutoff combinations 4 1 1 and 5 3 3, as well as non-TC cases. Horizontal lines indicate fixed-node diffusion Monte Carlo benchmarks with single-determinant (SJ, black dashed) and backflow-corrected (SJB, black solid) trial wave functions; stochastic DMC uncertainty is not visible because it is small. We have also drawn a black solid line accross the bars in the histogram groups to denote the reference energy and the amount of captured correlation energy below the lines. We also show the extrapolated CSB CCSD(T) estimate for non-TC results as horizontal green dash-dotted line.
• Figure 5: Formation energies of H- and X-interstitials obtained from correlated wave function methods at different basis levels. The upper panels show non-transcorrelated (Non-TC) results, and the lower panels their transcorrelated (xTC–PP) counterparts. Horizontal reference lines indicate periodic benchmark values. Experimental references fall within the grey shaded region for H-interstitial. The x-axis labels (a–e for H, a–d for X) correspond to the fragment pairs defined in Fig. \ref{['figure: fragments']}. The Horizontal reference line references are: CCSDhf-cc-hse, CCSD(T)hf-cc-hse, QMCqmc (stochastic errorbar of QMC is omitted for clarity), RPArpa-pbe, HSEhf-cc-hse, and PBErpa-pbe.The experimental range of results from different sourcesexp1exp2exp3exp4exp5 is shown by grey shaded region.