Partition density functional theory

Kohn-Sham density functional theory (KS-DFT)[1, 2] is an efficient and usefully accurate electronic structure method, because it replaces the interacting Schrödinger equation with a set of single-particle orbital equations. Calculations with several hundred atoms are now routine, but there is always interest in much larger systems. Many such systems are treated by a lower-level method, such as molecular mechanics, but a fragment in which a chemical reaction occurs must still be treated quantum mechanically. A plethora of such QM/MM approaches have been tried and tested, with varying degrees of success[3]. These are often combined with attempts at orbital-free DFT, which avoids the KS equations, but at the cost of higher error and unreliability. On the other hand, partition theory (PT) [4, 5] combines the simplicity of functional minimization with a density optimization to define fragments (such as atoms) within molecules, overcoming limitations of earlier approaches to reactivity theory[6, 7]. While there are now many definitions of, e.g., charges on atoms, none have the generality of PT and the associated promise of unifying disparate chemical concepts. However, previous work on PT has been either formal[4, 5] or for two atom systems[8, 9]. In this paper, we unite KS-DFT with PT to produce an algorithm that allows a KS calculation for a molecule to be performed via a self-consistent loop over isolated fragments. Such a fragment calculation exactly reproduces the result of a standard KS calculation of the entire molecule. We demonstrate its convergence on a 12-atom example. This also shows that fragments can be calculated ’on the fly’, as part of solving any KS molecular problem. Thus we present a formally exact framework within which existing practical approximations can be analyzed and, for smaller systems, compared with exact quantities. In practical terms, our method suggests new approximations that can, by construction, scale linearly[10] with the number of fragments (so-called O(N)), and allow embedding of KS calculations within cruder force-field calculations (QM/MM). It also suggests ways to improve XC approximations so as to produce correct dissociation of molecules [11]. To understand the relation between DFT and PT, recall that the Hohenberg-Kohn theorem proves that for a given electron-electron interaction and statistics, the external (one-body) potential v(r) is a unique functional of the density n(r). The total energy can be written as: