Density functional theory in the canonical ensemble: I. General formalism

Abstract

Abstract. Density functional theory stems from the Hohenberg-Kohn-Sham-Mermin (HKSM) theorem in the grand canonical ensemble (GCE). However, as recent work shows, although its extension to the canonical ensemble (CE) is not straightforward, work in nanopore systems could certainly benefit from a mesoscopic DFT in the CE. The stumbling block is the fixed N constraint which is responsible for the failure in proving the interchangeability of density profiles and external potentials as independent variables. Here we prove that, if in the CE the correlation functions are stripped off of their asymptotic behaviour (which is not in the form of a properly irreducible nbody function), the HKSM theorem can be extended to the CE. In proving that, we generate a new hierarchy of N -modified distribution and correlation functions which have the same formal structure that the more conventional ones have (but with the proper irreducible n-body behaviour) and show that, if they are employed, either a modified external field or the density profiles can indistinctly be used as independent variables. We also write down the N -modified free energy functional and prove that the thermodynamic potential is minimized by the equilibrium values of the new hierarchy.

Variational principle of classical density functional theory via Levy’s constrained search method

“…Hence, whether the definition (22) helps to construct approximations for grand-canonical freeenergy functionals remains an open question. For the case of the canonical ensemble we point the reader to the very significant body of work that has been carried out to formulate a computational scheme that permits to capture the effects that arise due to the constraint of fixed number of particles [21][22][23][24][25][26]. While the generalization to equilibrium mixtures is straightforward, we expect the application of Levy's method to DFT for quenched-annealed mixtures [28][29][30][31] to constitute an interesting topic for future work.…”

“…The notation f → ρ in (22) indicates the relationship (23). Note that (i) in general there will be many different forms of f that yield the same ρ, and (ii) no further conditions on f are imposed, apart from its normalization.…”

“…where the inner minimization is a search under the contraint the f generates ρ [via relationship (23)]. For Hamiltonians of the form (12) the above can be written as…”

“…There is considerable current interest in the theoretical description of the behavior of small systems, where the grand and the canonical ensembles are inequivalent in general, and the latter might model certain (experimental) realizations of strongly confined systems more closely. Several relatively recent contributions address the problem of formulating DFT in the canonical ensemble [21][22][23][24][25][26]. The authors of these papers consider the important problem of how to obtain DFT approximations that make computations in the canonical ensemble feasible.…”

Theory of Solutions in Energy Representation in NPT-ensemble: Derivation Details

“…This implies that there should be only one u e sw,λ (ǫ) to which a given ρ e sw,λ (ǫ) corresponds. Both in coordinate and energy representation it is not the case if we consider ensembles where the number of particles is fixed [3,10,11,12]. This can be easily seen from the definition of ρ e sw,λ (ǫ) (Eq.…”