We developed a novel computational method, called pristine unit-cell matrix projection (PUMP), to calculate excited states in systems with >1000 atoms using first principles, parameter-free GW-Bethe Salpeter equation. This method enabled the calculation of the excitonic spectrum of twisted bilayer WS2 in a record breaking 4000 atom moiré unit-cell. 

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In this approach, we first express the twisted bilayer WS2 moiré superlattice electronic wavefunctions (of the valence and conduction states) as a linear combination of pristine wavefunctions of the individual layers. We denote the bottom pristine layer as ‘a’ and top layer as ‘b’. Using the expansion coefficients for each electronic state, the BSE electron-hole interaction kernel matrix elements of the moiré superlattice can be approximated as a linear combination of pristine unit-cell kernel matrix elements of layer ‘a’ and layer ‘b’. The screened Coulomb interaction, W, is assumed to be identical to that of the pristine unit-cell bilayer stacking. This approach led to a speed up of about six orders of magnitude in our calculations.

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Nature 609, 52–57 (2022). https://doi.org/10.1038/s41586-022-04991-9

Nature Materials (accepted). https://doi.org/10.48550/arXiv.2306.00859