Theor Chem Acc (2007) 117:371–381 DOI 10.1007/s00214-006-0165-6 REGULAR ARTICLE Computing the Fukui function from ab initio quantum chemistry: approaches based on the extended Koopmans’ theorem Paul W. Ayers · Junia Melin Received: 17 March 2006 / Accepted: 25 July 2006 / Published online: 23 September 2006 © Springer-Verlag 2006 Abstract The extended Koopmans’ theorem is related to Fukui function, which measures the change in electron density that accompanies electron attachment and removal. Two approaches are used, one based on the extended Koopmans’ theorem differential equation and the other based directly on the expression of the ionized wave function from the extended Koopmans’ theorem. It is observed that the Fukui function for elec- tron removal can be modeled as the square of the first Dyson orbital, plus corrections. The possibility of useful generalizations to the extended Koopmans’ theorem is considered; some of these extensions give approxima- tions, or even exact expressions, for the Fukui function for electron attachment. Keywords Fukui function · Dyson orbital · Extended Koopmans’ theorem · Conceptual density-functional theory 1 Introduction and background Over the last two decades, density-functional theory (DFT) has emerged as the method of choice for rou- tine calculations in quantum chemistry, especially for larger systems. The primary reason for this emergence is indubitably the fact that more rigorous wave-func- tion based techniques cannot compete with DFT if one P. W.Ayers (B ) Department of Chemistry, McMaster University, Hamilton, ON, Canada L8S 4M1 e-mail: ayers@mcmaster.ca J. Melin Department of Chemistry, Kansas State University, Manhattan, KS 66506-3701, USA measures average accuracy per unit computational cost. Another reason, however, is that the language of DFT lends itself to chemical interpretation [1–3]. The sci- ence of interpreting chemical results with DFT, usually called conceptual density-functional theory, supersedes conventional approaches based on molecular orbitals or resonance because it makes contact with density- functional theory, which is in principle exact. The reactivity indicators of conceptual DFT, then, fully accommodate the effects of orbital relaxation and elec- tron correlation. Studies suggest that these effects are sometimes very important [4–6]. However, the reactivity indicators associated with conceptual DFT are usually computed at a relatively low level of theory, typically Kohn–Sham DFT with approx- imate exchange-correlation functionals. It is well known that these methods give poor predictions of reaction barriers and thus, while they are generally adequate for conceptual purposes, they are not robust [7, 8]. There is no reason, however, not to use more accurate, wave- function based, methods to compute the reactivity indi- cators of Kohn–Sham DFT. Most of the key reactivity indicators are readily computed if the energies and elec- tron densities of the system, its cation, and its anion are known. Specifically, this is enough information to compute the chemical potential, μ [9], and the chemi- cal hardness, η [10], using the quadratic model for the energy [10] μ =- I + A 2 =  ∂ E ∂ N  v(r) (1) η = I - A =  ∂ 2 E ∂ N 2  v(r) (2)