Density functional theory for ensembles

Gross, E. K. U., Oliveira, L. N., Kohn, W., 1988b, Density-Functional Theory for Ensembles of Fractionally Occupied States. I. Basic Formalism , Phys. Rev. A, 37, 2809. 

The density functional theory for ensembles is based on the generalized Rayleigh-Ritz variational principle [7]. The eigenvalue problem of the Hamiltonian H is given by... 

Density Functional Theory for Excited States to good approximation for ensembles of multiplets. 

To complete the definition of the functional derivatives and of the chemical potential /x, we extend the constrained search from wavefunctions to ensembles [49,50]. An ensemble or mixed state is a set of wavefunctions or pure states and their respective probabilities. By including wavefunctions with different electron numbers in the same ensemble, we can develop a density functional theory for non-integer particle number. Fractional particle mun-bers can arise in an open system that shares electrons with its environment, and in which the electron number fluctuates between integers. 

Are there any remedies in sight within approximate Kohn-Sham density functional theory to get correct energies connected with physically reasonable densities, i. e., without having to use wrong, that is symmetry broken, densities In many cases the answer is indeed yes. But before we consider the answer further, we should point out that the question only needs to be asked in the context of the approximate functionals for degenerate states and related problems outlined above, an exact density functional in principle also exists. The real-life solution is to employ the non-interacting ensemble-Vs representable densities p intro-... 

The densities Pi are obtained from a set of degenerate KS wave functions and the w, are the corresponding weights. Without going into details we note that regular density functional theory can be extended to such ensembles. For our problems at hand, we can write down the energy expression as... 

More insight into these processes is obtained by studying the particle number dependent properties of density functionals. This of course requires a suitable definition of these density functionals for fractional particle number. The most natural one is to consider an ensemble of states with different particle number (such an ensemble is for instance obtained by taking a zero temperature limit of temperature dependent density functional theory [84]). We consider a system of N + co electrons where N is an integer and 0 < m < 1. For the corresponding electron density we then have... 

Instead of supposing there to be a single Kohn-Sham potential, one can think of it as a vector in Fock space. For each sheet ft = N of the latter, there is a component vKS(r,N) and a corresponding set of Kohn-Sham equations. Density functional theory and Kohn-Sham theory hold separately on each sheet. Ensemble-average properties are then composed of weighted contributions from each sheet, computable sheet by sheet via the techniques of DFT and the KS equations. Nevertheless, though completely valid, this procedure would yield for the reactivity indices f(r), s(r), and S the results already obtained directly from Eqs. (28). We are left without proper definitions of chemical-reactivity indices for systems with discrete spectra at T = 0 [43]. 

The electron-transfer reactivities are defined as derivatives of the electron-density p(r) with respect to total electron number Jf, Ufr), or chemical potential p, s r). The treatment of JT as a continuous variable [8-12] is justified by reference to the ensemble formulation of density-functional theory [8,18] and, in consequence, of the Kohn-Sham theory. We show in Sect. 4, in previously unpublished work [42], that this ensemble formulation yields either vanishing or infinite local and global softnesses for localized systems with... 

Canonical ensemble density functional theory (CEDFT) has been employed for predicting hysteretic adsorption/desorption isotherms in nanopores of different geometries in the wide range of pore sizes (1 - 12 nm). It is shown that the CEDFT model qualitatively describes equilibrium and spinodal transitions and is in a reasonable quantitative agreement with experiments on well-characterized MCM-41 samples. A DFT-based method for calculating pore size distributions from the adsorption and desorption branches of nitrogen adsorption isotherms has been elaborated and tested against literature data on capillary condensation in MCM-41 samples with pores from 5 to 10 nm. 

I ic. 11. Relationship between the pore filling pressure and the pore width predicted by the modified Kelvin equation (MK). the Horvath-Kawazoe method (HK), density functional theory (DFT). and Gibbs ensemble Monte Carlo simulation (points) for nitrogen adsorption in carbon slit pores at 77 K [11]. 

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