Efficient Bethe-Salpeter Equation Calculations Based on Numerical Atomic Orbitals and Norm-Conserving Pseudopotentials: Dual-${\boldsymbol k}$-Mesh Strategy

We present an efficient implementation of the Bethe--Salpeter equation (BSE) based on numerical atomic orbitals (NAOs) and norm-conserving pseudopotentials within the ABACUS+LibRPA framework. By exploiting the localized resolution-of-identity (LRI) technique, the screened Coulomb interaction is cast into a real-space, unit-cell-indexed form $W_{\mu\nu}(\boldsymbol R)$ that is inherently short-ranged and well localized. This spatial locality enables an efficient Fourier interpolation of the BSE kernel from the coarse $\boldsymbol k$-mesh used in the preceding $GW$ calculation to an arbitrarily dense $\boldsymbol k$-mesh on which the BSE Hamiltonian is assembled and diagonalized, thereby giving rise naturally to a dual-$\boldsymbol k$-mesh workflow. Building on this scheme, we systematically examine the convergence of the absorption spectra with respect to the NAO basis set, the auxiliary basis set, and the $\boldsymbol k$-point sampling. Benchmark calculations for both molecular and periodic systems collectively validate the accuracy of the present implementation and establish the dual-$\boldsymbol k$-mesh strategy as a practical and reliable approach for $GW$+BSE calculations.