Accurate calculation of Wannier centers, position matrix, and composite operators using translationally equivariant and higher-order finite differences

Abstract

The momentum-space derivatives of Bloch wavefunctions are essential for studying quantum geometry and the equilibrium and response properties of solids. In practical first-principles calculations, these derivatives are obtained via Wannier interpolation of position and related composite matrices. These matrices are initially evaluated on a coarse k-point grid using finite-difference approximations and then interpolated to a dense grid. The accuracy of the finite-difference approximation directly impacts the convergence and reliability of the result. In this work, we present two key improvements to the finite-difference calculation of position and composite operators for Wannier interpolation. First, we formulate a translationally equivariant scheme that preserves the underlying symmetries of the system and significantly reduces finite-difference errors. Second, we introduce a higher-order finite-difference approach that yields a more accurate approximation of the k-space derivatives by systematically increasing the convergence rate. From a real-space perspective, these improvements correspond to better approximations of the position operator at the locations of the Wannier functions. We also present a generalization of the finite-difference scheme, which may reduce the number of finite-difference points while maintaining accuracy. We demonstrate the effectiveness of our methods by applying them to the calculation of Wannier centers and spreads, electric polarization, off-diagonal position matrix elements, orbital magnetization, and spin Hall conductivity. Our results demonstrate significant reductions in finite-difference errors, elimination of symmetry-violating errors, and improved convergence with respect to k-point sampling. These methods have been implemented in the open-source packages and can be readily adopted in other Wannier-based codes with minimal computational overhead.