Telescoping local-shell decomposition
= Telescoping local-shell decomposition
Filter a fixed local term using a nested sequence of neighbourhood Hamiltonians $H_s$, and call the results $G_s$. The differences $G_s-G_{s-1}$ give a shell decomposition. When $H_0=h_Z$ and $H_D=H$, normalization gives $G_0=h_Z$ and telescoping gives $g_Z=h_Z+\sum_{s=1}^D(G_s-G_{s-1})$. Individual shell terms need not commute with the full ground projector.