The notation for the ALMO-EDA derivation and terms found below follow the notation used in previous works on ALMO-EDA (10.1039/C8SC02990C). The molecular fragments are represented using the indices x and y. The atomic orbital (AO) basis functions are denoted as |f i with indices µ and n. Molecular orbitals (MOs) with the indices i and j representing occupied MOs, the indices a and b representing virtual MOs, and the indices p and q representing general MOs (occupied or virtual). Since the orbitals isolated on fragments x and y are nonorthogonal, we are required to use biorthogonal orbitals, which necessitates the use of covariant and contravariant AO/MO basis vectors. Similar to the overlap matrix in a non-biorthogonal basis, the inner product of two covariant AOs on fragmens x and y defines the covariant overlap metric