The analog of Koopmans’ theorem in spin-density functional theory

Gritsenko, O. V.; Baerends, E. J.
November 2002
Journal of Chemical Physics;11/22/2002, Vol. 117 Issue 20, p9154
Academic Journal
For spin-unrestricted Kohn-Sham (KS) calculations on systems with an open shell ground state with total spin quantum number S, we offer the analog of the Koopmans'-type relation between orbital energies and ionization energies familiar from the Hartree-Fock model. When (case I) the lowest ion state has spin S- 1/2 (typically when the neutral molecule has a (less than) half filled open shell), the orbital energy of the highest occupied orbital (ø[SUBH]), belonging to the open shell with majority spin (α) electrons, is equal to the ionization energy to this lowest ion state with spin S - 1/2: ε[SUPα][SUBH]-I[SUPS-1/2](ø[SUP-1][SUBH]). For lower (doubly occupied) orbitals the ionization ø[SUP-1][SUBH] leaves an unpaired electron that can couple to the open shell to S± 1/2 states: ε[SUPβ][SUBi]= -I[SUPS+l/2](ø[SUP-1][SUBi]) (exact identity for i = H- 1 ), ε[SUPα][SUBi] = - {[2 S/(2S + 1)]I[SUPS-1/2] (ø[SUP-1][SUBi]) + [ 1/(2S + 1 )]I[SUPS+1/2] (ø[SUP-1][SUBi])}, reducing to a simple average in the case of a doublet ground state (single electron outside closed shells). When the lowest ion state has spin S+ 1/2 (case II; typically for more than half filled open shells): ε[SUPα][SUBH] = ε[SUPβ][SUBH]=- I[SUPS- 1/2](ø[SUP-1][SUBH]; for i


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