Overview
A method has been presented to adapt the proven exact 2-component (X2C) method for use with an adaptive non-uniform grid basis of the Hilbert space, specifically Multiwavelets. The X2C method is traditionally applied to Gaussian-type fixed Hilbert bases. The core function of the X2C method involves block-diagonalizing the Dirac Hamiltonian. This block-diagonalization occurs within the subspace defined by the basis and is achieved in an energy-independent fashion. The inspiration for this method derives from the established efficacy of the atomic mean-field X2C method when applied to Gaussian bases.
Approach
The proposed method translates the X2C formalism, which typically operates within Gaussian-type fixed Hilbert bases, to an adaptive non-uniform grid basis. This grid basis is specifically Multiwavelets. The X2C method's primary objective, as applied here, is to perform an energy-independent block-diagonalization of the Dirac Hamiltonian. This block-diagonalization is restricted to the subspace delineated by the chosen basis. A key aspect of this method involves representing the coupling operator. This representation is conceptualized as a projector onto the small components of the eigenstates belonging to the constituent atoms of the system. This particular approach to the coupling operator draws inspiration from the success demonstrated by the atomic mean-field X2C method in the context of Gaussian bases.
Why This Matters
The described method extends the applicability of the exact 2-component (X2C) approach beyond its traditional domain of Gaussian-type fixed Hilbert bases to adaptive non-uniform grid bases like Multiwavelets. This adaptation maintains the X2C method's ability to block-diagonalize the Dirac Hamiltonian within the basis subspace in an energy-independent manner, a characteristic fundamental to its utility.