Have a look at http://www.wien2k.at/reg_user/textbooks/ , DFT and the family of ..., this will answer most of your questions > What are the "LOCAL ORBITAL" referring to? see chapters 4 & 5 > And why 2 energies for the same l (angular momentum) if APW needs only 1 > (LAPW needs 2 energies)? see examples in chapter 6 Stefaan