Wannier states in order-N electronic-structure method

The ‘order-N’ or ‘linear-systemsize-scaling’ methods 1) are electronic-structure calculations in which the computational cost is ‘O(N)’, or linearly proportional to the system-size N . In conventional schemes, the computational cost should be O(N), because of the matrix-diagonalization procedures or the orthogonalization constraints of the wavefunctions. The aim of order-N methods is to calculate large systems, for instance, systems containing some thousands of atoms or more, without such operations. So far many order-N methods have been proposed, and one of them is called ‘localized-orbital order-N’ (LO-O(N)) method, which is based on the locality of Wannier states. Here Wannier states are not the conventional ones, defined by Wannier, but are generalized concepts, applicable to non-periodic and/or composite-band cases.