By introducing a phase field and solving the eigen-functional equation of particles, we obtain the exact expressions of the ground state energy as a functional of the particle density for interacting electron/boson systems, and a two-dimensional electron gas under an external magnetic field, respectively. With the eigen-functionals of the particles, we can construct the ground state wave-function of the systems. Moreover, with the expressions of the ground state energy, we can exactly determine the ground state energy and the ground state particle density of the systems by taking $% δE_g[ρ]/δρ(x)=0$ .By introducing a phase field and solving the eigen-functional equation of particles, we obtain the exact expressions of the ground state energy as a functional of the particle density for interacting electron/boson systems, and a two-dimensional electron gas under an external magnetic field, respectively. With the eigen-functionals of the particles, we can construct the ground state wave-function of the systems. Moreover, with the expressions of the ground state energy, we can exactly determine the ground state energy and the ground state particle density of the systems by taking $% δE_g[ρ]/δρ(x)=0$.

arXiv:cond-mat/0101428Jan 29, 20011 citation

Exact density functionals for 1d systems

Lucas Wagner, Edwin Miles Stoudenmire, Kieron Burke +1 more

Bulletin of the American Physical SocietyFeb 27, 20120 citations

Symmetry breaking with a proper approximate density functional: Exact ground-state energy as a close lower bound on the computed energy

John P. Perdew

In the widely-used Kohn-Sham density functional theory for the manyelectron ground state, only the density functional for the exchange-correlation energy is approximated. While the exact density functional minimizes at the exact ground-state energy for any system, widely used approximations can at best describe normally-correlated ground states. A “proper” approximation for a given problem is defined as one that well describes relevant normally-correlated states. Strong correlation, present in many quantum materials, can arise due to abnormally high degeneracy of the Kohn-Sham occupied and unoccupied one-electron energies near the Fermi level. In such situations, the approximate functional may lower its energy by spontaneously breaking symmetries that the ground-state density shares with the many-electron Hamiltonian, reducing the degeneracies that lead to strong correlation. Here the Levy constrained search is invoked to argue that the total energy of a “proper” functional with maximal spontaneous symmetry breaking is bounded from below by the exact total energy, and that the bound may be close to the extent that the abnormal degeneracies are removed. This is a consequence of a variational principle, and it may not hold in non-variational but more accurate approaches to normal correlation such as coupled cluster theory. Since the correlation energy of any one-electron state is zero, a widely “proper” approximate functional should be nearly or exactly self-interaction free, and progress toward a proper self-interaction correction is reviewed here. The time-centered physical interpretation of symmetry breaking is also discussed.In the widely-used Kohn-Sham density functional theory for the manyelectron ground state, only the density functional for the exchange-correlation energy is approximated. While the exact density functional minimizes at the exact ground-state energy for any system, widely used approximations can at best describe normally-correlated ground states. A “proper” approximation for a given problem is defined as one that well describes relevant normally-correlated states. Strong correlation, present in many quantum materials, can arise due to abnormally high degeneracy of the Kohn-Sham occupied and unoccupied one-electron energies near the Fermi level. In such situations, the approximate functional may lower its energy by spontaneously breaking symmetries that the ground-state density shares with the many-electron Hamiltonian, reducing the degeneracies that lead to strong correlation. Here the Levy constrained search is invoked to argue that the total energy of a “proper” functional with maximal spontaneous symmetry breaking is bounded from below by the exact total energy, and that the bound may be close to the extent that the abnormal degeneracies are removed. This is a consequence of a variational principle, and it may not hold in non-variational but more accurate approaches to normal correlation such as coupled cluster theory. Since the correlation energy of any one-electron state is zero, a widely “proper” approximate functional should be nearly or exactly self-interaction free, and progress toward a proper self-interaction correction is reviewed here. The time-centered physical interpretation of symmetry breaking is also discussed.

Low Temperature PhysicsSep 1, 20260 citations

Single-particle-exact density functional theory

Martin-Isbjörn Trappe, Jun Hao Hue, Jonah Zi Chao Huang +4 more

arXiv:2305.03233Annals of PhysicsOct 21, 20231 citation

Degenerate Ground States and a Fractional Number of Electrons in Density and Reduced Density Matrix Functional Theory

Weitao Yang, Yingkai Zhang, Paul W. Ayers

For a linear combination of electron densities of degenerate ground states, it is shown that the value of any energy functional is the ground state energy, if the energy functional is exact for ground state densities, size consistent, and translational invariant. The corresponding functional of kinetic and interaction energy is the linear combination of the functionals of the degenerate densities. Without invoking ensembles, it is shown that the energy functional of fractional number electrons is a series of straight lines interpolating its values at integers. These results underscore the importance of grand canonical ensemble formulation in density functional theory.For a linear combination of electron densities of degenerate ground states, it is shown that the value of any energy functional is the ground state energy, if the energy functional is exact for ground state densities, size consistent, and translational invariant. The corresponding functional of kinetic and interaction energy is the linear combination of the functionals of the degenerate densities. Without invoking ensembles, it is shown that the energy functional of fractional number electrons is a series of straight lines interpolating its values at integers. These results underscore the importance of grand canonical ensemble formulation in density functional theory.

Physical Review LettersMay 29, 2000447 citations

Exact and Approximate Ensemble Density Functionals for a Simple model

Francisca Sagredo, Kieron Burke

Bulletin of the American Physical SocietyMar 8, 20180 citations

% δE_g[ρ]/δρ(x)=0