A fermion ground state energy functional is set up in terms of particle density, relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact sum rules. Prototypical application is made to several one-dimensional spinless noninteracting models. The effectiveness of atomic'' constraints on model molecules'' is observed, as well as the structure of systems with only finitely many bound states.
A method is presented for relating a ground-state density distribution to the effective single-particle potential which has this density for its ground state for a noninteracting system of particles. The method is useful for systems consisting of a small number of spinless fermions and in the case of two particles a second-order, nonlinear differential equation relates the density and the effective potential. The question of
v representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form ${e}^{\ensuremath{-}ar}$ for two-particle systems are not ground-state
v representable. The added flexibility of three dimensions over one allows an effective potential which does not necessarily have the chosen density as its ground state, but as its lowest nondegenerate state.
It is shown that interacting many-particle quantum systems can be described in terms of fully correlated
n-particle densities, which determine uniquely the potential acting on the system and satisfy a minimum principle with respect to the ground-state energy. This leads to a generalization of ordinary density functional theory in terms of
n-particle densities which allows the direct and self-consistent treatment of correlation effects within electronic structure methods for atoms, molecules, and solids.
In pair density-functional theory, the only unknown piece of the energy is the kinetic energy
T as a functional of the pair density
P(x1,x2). Although
T[P] has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy
T2[P] is shown to be exact for two-electron systems and a lower bound to
T[P] for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning
T[P]=T2[P]+Teff[P], where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of
Teff[P]. This self-consistent scheme to minimize the energy with respect to
P is more efficient than previous procedures. Further information on the functional derivative of
Teff[P] is derived from a contracted Schr\"odinger equation. Since
Teff[P] is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.
The kinetic energy, or thermodynamic potential, of a non-interacting Fermion system, as a functional of the non-uniform density pattern evoked by an external potential, serves as a crucial tool in the analysis of interacting system ground states. Examples from classical lattice gas and continuum fluid theory are cited to show that analytic tractability can be dramatically enhanced by the introduction of auxiliary density fields with respect to which a suitable thermodynamic potential is stationary. This strategy is developed for the prototype of one-dimensional spinless Fermions, resulting in an exact highly overcomplete representation. A semi-classical variational ansatz recovers the familiar Thomas-Fermi plus renormalized Weiszacker expression, as well as leading corrections, and a simple extension creates a simple modification of the former. The corresponding 3-dimensional format is set up, and evaluated by extrapolation from the 1-dimensional case.
Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites
Na, band filling
Ne, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on
Na, Ne, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive
(U>0) and attractive
(U<0) interactions are contrasted.