Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-met allic systems. We further discuss the case of density matrices for met allic systems at positive electronic temperature. A few other possible applications are also discussed.
A few observations are in order: first, the decay result ) is asymptotic, that is, it implies fast decay at sufficiently large distances @math only. Second, ) is consistent not only with strict exponential decay, but also with decay of the form @math where @math is arbitrary (positive or negative) and @math . Hence, the actual decay could be faster, but also slower, than exponential. Since the result in ) provides only an estimate (rather than an upper bound) for the density matrix in real space, it is not easy to use in actual calculations. To be fair, such practical aspects were not discussed by Kohn until much later (see, e.g., @cite_12 ). Also, later work showed that the asymptotic regime is achieved already for distances of the order of 1-2 lattice constants, and helped clarify the form of the power-law prefactor, as discussed below.
@cite_12
2072565812
A widely applicable nearsightedness'' principle is first discussed as the physical basis for the existence of computational methods scaling linearly with the number of atoms. This principle applies to the one particle density matrix @math but not to individual eigenfunctions. A variational principle for @math is derived in which, by the use of a penalty functional @math , the (difficult) idempotency of @math need not be assured in advance but is automatically achieved. The method applies to both insulators and met als.