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 Pn (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.