To achieve this, the AO Hamiltonian is first transformed to into the localized orbital (LO) basis through various schemes, including Löwdin, Foster-Boys, Intrinsic Atomic Orbitals plus Projected Atomic Orbitals (IAO+PAO), and others, then a Schmidt Decomposition on the truncated Hartree-Fock density matrix P frag , which has the dimension of n orbs × n frag , where n orbs is the total number of molecular orbitals in the system and n frag is the number of localized orbitals in a fragment. A Singular Value Decomposition (SVD) of this matrix then yields three matrices, U, Σ , and