Density Functional and Density Matrix Method Scaling Linearly with the Number of Atoms

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 n(r,r^') but not to individual eigenfunctions. A variational principle for n(r,r^') is derived in which, by the use of a penalty functional P[n(r,r^')], the (difficult) idempotency of n(r,r^') need not be assured in advance but is automatically achieved. The method applies to both insulators and metals.