Generalized average local ionization energy and its representations in terms of Dyson and energy orbitals

Ryabinkin and Staroverov [J. Chem. Phys. 141, 084107 (2014)] extended the concept of average local ionization energy (ALIE) to correlated wavefunctions by defining the generalized ALIE as Ī(r)=-∑jλj|fj(r)|(2)/ρ(r), where λj are the eigenvalues of the generalized Fock operator and fj(r) are the corresponding eigenfunctions (energy orbitals). Here we show that one can equivalently express the generalized ALIE as Ī(r)=∑kIk|dk(r)|(2)/ρ(r), where Ik are single-electron removal energies and dk(r) are the corresponding Dyson orbitals. The two expressions for Ī(r) emphasize different physical interpretations of this quantity; their equivalence enables one to calculate the ALIE at any level of ab initio theory without generating the computationally expensive Dyson orbitals.