The Hubbard dimer: a density functional case study of a many-body problem

Carrascal, Diego; Ferrer, Jaime; Smith, Justin C.; Burke, Kieron

doi:10.1088/0953-8984/27/39/393001

Cited by 123 publications

(200 citation statements)

References 260 publications

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“…There has been recent work on the exact functional in this model from Fuks et al [8,9], Carrascal et al [10], Pastor and co-workers [11,12], Requist et al [13], and in other systems [14][15][16].…”

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confidence: 99%

Landscape of an exact energy functional

Cohen

Mori‐Sánchez

2016

Phys. Rev. A

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One of the great challenges of electronic structure theory is the quest for the exact functional of density functional theory. Its existence is proven, but it is a complicated multivariable functional that is almost impossible to conceptualize. In this paper the asymmetric two-site Hubbard model is studied, which has a two-dimensional universe of density matrices. The exact functional becomes a simple function of two variables whose threedimensional energy landscape can be visualized and explored. A walk on this unique landscape, tilted to an angle defined by the one-electron Hamiltonian, gives a valley whose minimum is the exact total energy. This is contrasted with the landscape of some approximate functionals, explaining their failure for electron transfer in the strongly correlated limit. We show concrete examples of pure-state density matrices that are not v representable due to the underlying nonconvex nature of the energy landscape. The exact functional is calculated for all numbers of electrons, including fractional, allowing the derivative discontinuity to be visualized and understood. The fundamental gap for all possible systems is obtained solely from the derivatives of the exact functional.DOI: 10.1103/PhysRevA.93.042511In 1964 Hohenberg and Kohn [1] established density functional theory (DFT) showing that the electron density ρ is all that is is necessary to determine the exact energy of many-electron systems. However, all the challenge of electronic structure is then moved into an unknown universal functional of the density F [ρ]. For a wave function v that is the ground-state solution of the Schrödinger equation with potential v,simply subtracting off the one-electron term, gives the exactThis procedure can be carried out for many different v, to obtain many points of the exact functional F HK [ρ v ]. A question arises of whether all possible densities are achievable. This is the problem of v representability that is addressed by the constrained search by Levy and Lieb [2,3] following earlier work by Percus [4],This functional is defined for all possible densities coming from a N -electron wave function, including those that are not obtainable as the ground-state solution of a Schrödinger equation (not v representable). Once the exact functional is known, the total energy is obtained by minimization only over densities,The exact functional of the first-order density matrix γ can be derived [2,5]and used similarly, where the kinetic energy term is now a known linear functional of γ ,In this paper, the nature of the exact first-order density matrix functional is revealed by considering the asymmetric two-site Hubbard model. In this universe, the fundamental equations are tractable and the exact functional becomes a visualizable three-dimensional energy landscape in the space of density matrices. We demonstrate how this one universal landscape gives the exact energy of all possible systems, for all numbers of electrons including fractional. This connected view of the functional for all density matri...

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confidence: 99%

Landscape of an exact energy functional

Cohen

Mori‐Sánchez

2016

Phys. Rev. A

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“…Further, we do not allow spin-symmetry breaking.Recently, [40][41][42] established an explicit connection between the xc potentials and the wave functions allowing to directly compute the xc potentials from the many-electron wave functions in a numerically robust manner. Studies of exact ground-state xc-functionals for lattice models include the exact one-to-one map between ground-state densities and potentials computed for a half-filled one-dimensional Hubbard chain in [43] using the Bethe Ansatz, for the one-site and double-site Hubbard models in full Fock space in [44,45] and for the two-electron Hubbard dimer via constraint search in [46], among others. For such lattice models the Hohenberg-Kohn theorem [47] can be generalized by replacing the real-space potentials and densities by on-site potentials and on-site occupations [48,49].…”

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confidence: 99%

“…. The third row of figure 9 shows the exact Hohenberg-Kohn functional ( j = 0) discussed previously in literature [43][44][45][46], the first and second row show the first and second excited-state energy functional ( j=1, 2), respectively. The gradient of all three functionals [ ] d F n jj 00 steepens approaching the limit of strictly localized electrons, just as previously observed for the density-to-potential map in section 4 and the density-to-wavefunction map in section 5.1.…”

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confidence: 99%

Exact maps in density functional theory for lattice models

et al. 2016

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In the present work, we employ exact diagonalization for model systems on a real-space lattice to explicitly construct the exact density-to-potential and graphically illustrate the complete exact density-to-wavefunction map that underly the Hohenberg-Kohn theorem in density functional theory. Having the explicit wavefunction-to-density map at hand, we are able to construct arbitrary observables as functionals of the ground-state density. We analyze the density-to-potential map as the distance between the fragments of a system increases and the correlation in the system grows. We observe a feature that gradually develops in the density-to-potential map as well as in the density-towavefunction map. This feature is inherited by arbitrary expectation values as functional of the ground-state density. We explicitly show the excited-state energies, the excited-state densities, and the correlation entropy as functionals of the ground-state density. All of them show this exact feature that sharpens as the coupling of the fragments decreases and the correlation grows. We denominate this feature as intra-system steepening and discuss how it relates to the well-known inter-system derivative discontinuity. The inter-system derivative discontinuity is an exact concept for coupled subsystems with degenerate ground state. However, the coupling between subsystems as in charge transfer processes can lift the degeneracy. An important conclusion is that for such systems with a neardegenerate ground state, the corresponding cut along the particle number N of the exact density functionals is differentiable with a well-defined gradient near integer particle number. IntroductionOver the last decades ground-state density-functional theory (DFT) has become a mature tool in material science and quantum chemistry [1][2][3][4][5]. Provided that the exact exchange-correlation (xc) functional is known, DFT is a formally exact framework of the quantum many-body problem. In practice, the accuracy of observables in DFT highly depends on the choice of the approximate xc-functional. From the local density approximation (LDA) [6], to the gradient expansions such as the generalized gradient approximations (GGAs), e.g. Perdew-Burke-Enzerhof [7] and the hybrid functionals such as B3LYP [8], to the orbital-functionals such as optimized effective potentials [9] and to the range-separated hybrids such as HSE06 [10], the last decades have seen great efforts and achievements in the development of functionals with more accurate and reliable prediction capability.Nonetheless, available approximate functionals such as the LDA, the GGA's and the hybrid functionals have known shortcomings to model gaps of semiconductors [11], molecular dissociation curves [12], barriers of chemical reactions [13], polarizabilities of molecular chains [14, 15], and charge-transfer excitation energies, particularly between open-shell molecules [16].Practical applications of density functional theory encounter two major problems: (i) while the Hohenberg-Kohn theorem tells us that a...

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“…By definition [34], the KS occupation matches the physical one, i.e., the left hand sides of Eqs. (3) and (6) are identical, for a given set of µ, ε, U, and Γ.…”

Section: Kohn-sham Anderson Junctionmentioning

confidence: 95%

Density functional description of Coulomb blockade: Adiabatic versus dynamic exchange correlation

Liu

Burke

2015

Phys. Rev. B

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Above the Kondo temperature, the Kohn-Sham zero-bias conductance of an Anderson junction has been shown to completely miss the Coulomb blockade. Within a standard model for the spectral function, we deduce a parameterization for both the onsite exchange-correlation potential and the bias drop as a function of the site occupation that applies for all correlation strengths. We use our results to sow doubt on the common interpretation of such corrections as arising from dynamical exchange-correlation contributions.

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The Hubbard dimer: a density functional case study of a many-body problem

Cited by 123 publications

References 260 publications

Landscape of an exact energy functional

Landscape of an exact energy functional

Exact maps in density functional theory for lattice models

Density functional description of Coulomb blockade: Adiabatic versus dynamic exchange correlation

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