# Recent Progress in Lattice Density Functional Theory

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts of Lattice Density Functional Theory

## 3. The Anderson Model

#### 3.1. Two-Level Approximation

#### 3.2. Applications to Anderson Rings

## 4. The Hubbard Model

#### 4.1. Local Perspective to the Interaction-Energy Functional

#### 4.2. Applications of the Scaled Dimer Approximation

#### 4.3. Reciprocal-Space Perspective

#### 4.4. Application to the Half-Filled Hubbard Model

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Parr, R.G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: Oxford, UK, 1989. [Google Scholar]
- Dreizler, R.M.; Gross, E.K.U. Density Functional Theory: An Approach to the Quantum Many-Body Problem; Springer: New York, NY, USA, 1990. [Google Scholar]
- Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev.
**1964**, 136, B864–B871. [Google Scholar] [CrossRef] - Kohn, W. Nobel Lecture: Electronic Structure of Matter—Wave Functions and Density Functionals. Rev. Mod. Phys.
**1999**, 71, 1253–1266. [Google Scholar] [CrossRef] - Kohn, W.; Sham, L.J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev.
**1965**, 140, A1133–A1138. [Google Scholar] [CrossRef] - Von Barth, U.; Hedin, L. A local exchange-correlation potential for the spin polarized case: I. J. Phys. C
**1972**, 5, 1629. [Google Scholar] [CrossRef] - Langreth, D.C.; Mehl, M.J. Beyond the local-density approximation in calculations of ground-state electronic properties. Phys. Rev. B
**1983**, 28, 1809–1834. [Google Scholar] [CrossRef] - Becke, A.D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A
**1988**, 38, 3098–3100. [Google Scholar] [CrossRef] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] - Becke, A.D. Density-Functional Thermochemistry. II. The Effect of the Perdew-Wang Generalized-Gradient Correlation Correction. J. Chem. Phys.
**1992**, 97, 9173–9177. [Google Scholar] [CrossRef] - Heitler, W.; London, F. Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Z. Phys.
**1927**, 44, 455–472. [Google Scholar] [CrossRef] - Hewson, A.C. The Kondo Problem to Heavy Fermions; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Mahan, G.D. Many-Particle Physics, 2nd ed.; Physics of Solids and Liquids; Plenum Press: New York, NY, USA, 1990. [Google Scholar]
- Dagotto, E. Correlated electrons in high-temperature superconductors. Rev. Mod. Phys.
**1994**, 66, 763–840. [Google Scholar] [CrossRef] - Georges, A.; Kotliar, G.; Krauth, W.; Rozenberg, M.J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys.
**1996**, 68, 13–125. [Google Scholar] [CrossRef] - Imada, M.; Fujimori, A.; Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys.
**1998**, 70, 1039–1263. [Google Scholar] [CrossRef] - Toulouse, J.; Colonna, F.; Savin, A. Long-Range–Short-Range Separation of the Electron-Electron Interaction in Density-Functional Theory. Phys. Rev. A
**2004**, 70, 062505. [Google Scholar] [CrossRef] - Toulouse, J.; Gerber, I.C.; Jansen, G.; Savin, A.; Ángyán, J.G. Adiabatic-Connection Fluctuation-Dissipation Density-Functional Theory Based on Range Separation. Phys. Rev. Lett.
**2009**, 102, 096404. [Google Scholar] [CrossRef] - Toulouse, J.; Zhu, W.; Ángyán, J.G.; Savin, A. Range-separated density-functional theory with the random-phase approximation: Detailed formalism and illustrative applications. Phys. Rev. A
**2010**, 82. [Google Scholar] [CrossRef] - Zhu, W.; Toulouse, J.; Savin, A.; Ángyán, J.G. Range-separated density-functional theory with random phase approximation applied to noncovalent intermolecular interactions. J. Chem. Phys.
**2010**, 132, 244108. [Google Scholar] [CrossRef] - Janesko, B.G.; Henderson, T.M.; Scuseria, G.E. Long-range-corrected hybrids including random phase approximation correlation. J. Chem. Phys.
**2009**, 130. [Google Scholar] [CrossRef] - Irelan, R.M.; Henderson, T.M.; Scuseria, G.E. Long-range-corrected hybrids using a range-separated Perdew-Burke-Ernzerhof functional and random phase approximation correlation. J. Chem. Phys.
**2011**, 135. [Google Scholar] [CrossRef] - Becke, A.D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys.
**1993**, 98, 5648–5652. [Google Scholar] [CrossRef] - Perdew, J.P.; Ernzerhof, M.; Burke, K. Rationale for mixing exact exchange with density functional approximations. J. Chem. Phys.
**1996**, 105, 9982–9985. [Google Scholar] [CrossRef] - Heyd, J.; Scuseria, G.E.; Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys.
**2003**, 118, 8207–8215. [Google Scholar] [CrossRef] - Kudin, K.N.; Scuseria, G.E.; Martin, R.L. Hybrid Density-Functional Theory and the Insulating Gap of UO
_{2}. Phys. Rev. Lett.**2002**, 89, 266402. [Google Scholar] [CrossRef] - Prodan, I.D.; Scuseria, G.E.; Martin, R.L. Covalency in the actinide dioxides: Systematic study of the electronic properties using screened hybrid density functional theory. Phys. Rev. B
**2007**, 76. [Google Scholar] [CrossRef] - Wen, X.D.; Martin, R.L.; Roy, L.E.; Scuseria, G.E.; Rudin, S.P.; Batista, E.R.; McCleskey, T.M.; Scott, B.L.; Bauer, E.; Joyce, J.J.; et al. Effect of spin-orbit coupling on the actinide dioxides AnO
_{2}(An=Th, Pa, U, Np, Pu, and Am): A screened hybrid density functional study. J. Chem. Phys.**2012**, 137, 154707. [Google Scholar] [CrossRef] [PubMed] - Eyert, V. VO
_{2}: A Novel View from Band Theory. Phys. Rev. Lett.**2011**, 107, 016401. [Google Scholar] [CrossRef] [PubMed] - Iori, F.; Gatti, M.; Rubio, A. Role of nonlocal exchange in the electronic structure of correlated oxides. Phys. Rev. B
**2012**, 85, 115129. [Google Scholar] [CrossRef] - Sharma, S.; Dewhurst, J.K.; Lathiotakis, N.N.; Gross, E.K.U. Reduced Density Matrix Functional for Many-Electron Systems. Phys. Rev. B
**2008**, 78. [Google Scholar] [CrossRef] - Lathiotakis, N.N.; Sharma, S.; Helbig, N.; Dewhurst, J.K.; Marques, M.A.L.; Eich, F.; Baldsiefen, T.; Zacarias, A.; Gross, E.K.U. Discontinuities of the Chemical Potential in Reduced Density Matrix Functional Theory. Z. Phys. Chem.
**2010**, 224, 467–480. [Google Scholar] [CrossRef] - Gilbert, T.L. Hohenberg-Kohn theorem for nonlocal external potentials. Phys. Rev. B
**1975**, 12, 2111–2120. [Google Scholar] [CrossRef] - Müller, A.M.K. Explicit approximate relation between reduced two- and one-particle density matrices. Phys. Rev. A
**1984**, 105, 446–452. [Google Scholar] [CrossRef] - Goedecker, S.; Umrigar, C.J. Natural Orbital Functional for the Many-Electron Problem. Phys. Rev. Lett.
**1998**, 81, 866–869. [Google Scholar] [CrossRef][Green Version] - Gritsenko, O.; Pernal, K.; Baerends, E.J. An improved density matrix functional by physically motivated repulsive corrections. J. Chem. Phys.
**2005**, 122, 204102. [Google Scholar] [CrossRef] [PubMed] - Buijse, M.A.; Baerends, E.J. An approximate exchange-correlation hole density as a functional of the natural orbitals. Mol. Phys.
**2002**, 100, 401–421. [Google Scholar] [CrossRef] - Rohr, D.R.; Pernal, K.; Gritsenko, O.V.; Baerends, E.J. A density matrix functional with occupation number driven treatment of dynamical and nondynamical correlation. J. Chem. Phys.
**2008**, 129, 164105. [Google Scholar] [CrossRef][Green Version] - Piris, M. A new approach for the two-electron cumulant in natural orbital functional theory. Int. J. Quantum Chem.
**2006**, 106, 1093–1104. [Google Scholar] [CrossRef] - Piris, M.; Lopez, X.; Ruipérez, F.; Matxain, J.M.; Ugalde, J.M. A natural orbital functional for multiconfigurational states. J. Chem. Phys.
**2011**, 134, 164102. [Google Scholar] [CrossRef] [PubMed] - Marques, M.A.L.; Lathiotakis, N.N. Empirical functionals for reduced-density-matrix-functional theory. Phys. Rev. A
**2008**, 77. [Google Scholar] [CrossRef][Green Version] - Lathiotakis, N.N.; Marques, M.A.L. Benchmark calculations for reduced density-matrix functional theory. J. Chem. Phys.
**2008**, 128, 184103. [Google Scholar] [CrossRef][Green Version] - Lathiotakis, N.N.; Sharma, S.; Dewhurst, J.K.; Eich, F.G.; Marques, M.A.L.; Gross, E.K.U. Density-matrix-power functional: Performance for finite systems and the homogeneous electron gas. Phys. Rev. A
**2009**, 79, 040501. [Google Scholar] [CrossRef][Green Version] - Rohr, D.R.; Toulouse, J.; Pernal, K. Combining density-functional theory and density-matrix-functional theory. Phys. Rev. A
**2010**, 82. [Google Scholar] [CrossRef][Green Version] - Pariser, R.; Parr, R.G. A Semi-Empirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. I. J. Chem. Phys.
**1953**, 21, 466–471. [Google Scholar] [CrossRef] - Pople, J.A. Electron interaction in unsaturated hydrocarbons. Trans. Faraday Soc.
**1953**, 49, 1375–1385. [Google Scholar] [CrossRef] - Anderson, P.W. Localized Magnetic States in Metals. Phys. Rev.
**1961**, 124, 41–53. [Google Scholar] [CrossRef] - Hubbard, J. Electron Correlations in Narrow Energy Bands. Proc. R. Soc. Lond. A
**1963**, 276, 238–257. [Google Scholar] - Kanamori, J. Electron Correlation and Ferromagnetism of Transition Metals. Prog. Theor. Phys.
**1963**, 30, 275–289. [Google Scholar] [CrossRef][Green Version] - Gutzwiller, M.C. Effect of Correlation on the Ferromagnetism of Transition Metals. Phys. Rev. Lett.
**1963**, 10, 159–162. [Google Scholar] [CrossRef] - Parks, R.D. Superconductivity; Marcel Dekker: New York, NY, USA, 1969; Volumes 1 and 2. [Google Scholar]
- Fulde, P. Electron Correlations in Molecules and Solids, 3rd ed.; Number 100 in Springer series in solid-state sciences; Springer: Berlin/Heidelberg, Germany, 1995. [Google Scholar]
- Gunnarsson, O.; Schönhammer, K. Density-Functional Treatment of an Exactly Solvable Semiconductor Model. Phys. Rev. Lett.
**1986**, 56, 1968–1971. [Google Scholar] [CrossRef] - Svane, A.; Gunnarsson, O. Localization in the self-interaction-corrected density-functional formalism. Phys. Rev. B
**1988**, 37, 9919–9922. [Google Scholar] [CrossRef] - Schindlmayr, A.; Godby, R.W. Density-functional theory and the v-representability problem for model strongly correlated electron systems. Phys. Rev. B
**1995**, 51, 10427–10435. [Google Scholar] [CrossRef][Green Version] - Schönhammer, K.; Gunnarsson, O.; Noack, R.M. Density-functional theory on a lattice: Comparison with exact numerical results for a model with strongly correlated electrons. Phys. Rev. B
**1995**, 52, 2504–2510. [Google Scholar] [CrossRef] - Lima, N.A.; Silva, M.F.; Oliveira, L.N.; Capelle, K. Density Functionals Not Based on the Electron Gas: Local-Density Approximation for a Luttinger Liquid. Phys. Rev. Lett.
**2003**, 90, 146402. [Google Scholar] [CrossRef] [PubMed][Green Version] - Capelle, K.; Campo, V.L., Jr. Density Functionals and Model Hamiltonians: Pillars of Many-Particle Physics. Phys. Rep.
**2013**, 528, 91–159. [Google Scholar] [CrossRef] - Verdozzi, C. Time-Dependent Density-Functional Theory and Strongly Correlated Systems. Phys. Rev. Lett.
**2008**, 101, 166401. [Google Scholar] [CrossRef] [PubMed] - Stefanucci, G.; Kurth, S. Towards a Description of the Kondo Effect Using Time-Dependent Density-Functional Theory. Phys. Rev. Lett.
**2011**, 107, 216401. [Google Scholar] [CrossRef][Green Version] - Bergfield, J.P.; Liu, Z.F.; Burke, K.; Stafford, C.A. Bethe Ansatz Approach to the Kondo Effect within Density-Functional Theory. Phys. Rev. Lett.
**2012**, 108, 066801. [Google Scholar] [CrossRef][Green Version] - Brosco, V.; Ying, Z.J.; Lorenzana, J. Exact Exchange-Correlation Potential of an Ionic Hubbard Model with a Free Surface. Sci. Rep.
**2013**, 3, 2172. [Google Scholar] [CrossRef][Green Version] - Carlsson, A.E. Exchange-correlation functional based on the density matrix. Phys. Rev. B
**1997**, 56, 12058–12061. [Google Scholar] [CrossRef] - Hennig, R.G.; Carlsson, A.E. Density-matrix functional method for electronic properties of impurities. Phys. Rev. B
**2001**, 63, 115116. [Google Scholar] [CrossRef] - López-Sandoval, R.; Pastor, G.M. Density-Matrix Functional Theory of the Hubbard Model: An Exact Numerical Study. Phys. Rev. B
**2000**, 61, 1764–1772. [Google Scholar] [CrossRef][Green Version] - López-Sandoval, R.; Pastor, G.M. Density-Matrix Functional Theory of Strongly Correlated Lattice Fermions. Phys. Rev. B
**2002**, 66, 155118. [Google Scholar] [CrossRef][Green Version] - López-Sandoval, R.; Pastor, G.M. Electronic Properties of the Dimerized One-Dimensional Hubbard Model Using Lattice Density-Functional Theory. Phys. Rev. B
**2003**, 67. [Google Scholar] [CrossRef][Green Version] - López-Sandoval, R.; Pastor, G.M. Interaction-Energy Functional for Lattice Density Functional Theory: Applications to One-, Two-, and Three-Dimensional Hubbard Models. Phys. Rev. B
**2004**, 69. [Google Scholar] [CrossRef][Green Version] - Saubanère, M.; Pastor, G.M. Scaling and Transferability of the Interaction-Energy Functional of the Inhomogeneous Hubbard Model. Phys. Rev. B
**2009**, 79, 235101. [Google Scholar] [CrossRef] - Saubanère, M.; Pastor, G.M. Density-Matrix Functional Study of the Hubbard Model on One- and Two-Dimensional Bipartite Lattices. Phys. Rev. B
**2011**, 84. [Google Scholar] [CrossRef] - Töws, W.; Pastor, G.M. Lattice density functional theory of the single-impurity Anderson model: Development and applications. Phys. Rev. B
**2011**, 83, 235101. [Google Scholar] [CrossRef] - Müller, T.S.; Töws, W.; Pastor, G.M. Exploiting the Links between Ground-State Correlations and Independent-Fermion Entropy in the Hubbard Model. Phys. Rev. B
**2018**, 98, 045135. [Google Scholar] [CrossRef] - Levy, M. Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proc. Natl. Acad. Sci. USA
**1979**, 76, 6062–6065. [Google Scholar] [CrossRef][Green Version] - Lieb, E.H. Density Functionals for Coulomb Systems. Int. J. Quantum Chem.
**1983**, 24, 243–277. [Google Scholar] [CrossRef] - Valone, S.M. Consequences of Extending 1-matrix Energy Functionals from Pure–State Representable to All Ensemble Representable 1 Matrices. J. Chem. Phys.
**1980**, 73, 1344–1349. [Google Scholar] [CrossRef] - Chayes, J.T.; Chayes, L.; Ruskai, M.B. Density Functional Approach to Quantum Lattice Systems. J. Stat. Phys.
**1985**, 38, 497–518. [Google Scholar] [CrossRef] - Töws, W.; Pastor, G.M. Spin-polarized density-matrix functional theory of the single-impurity Anderson model. Phys. Rev. B
**2012**, 86, 245123. [Google Scholar] [CrossRef] - Varma, C.M.; Yafet, Y. Magnetic Susceptibility of Mixed-Valence Rare-Earth Compounds. Phys. Rev. B
**1976**, 13, 2950–2954. [Google Scholar] [CrossRef] - Parlett, B.N. The Symmetric Eigenvalue Problem; SIAM: Philadelphia, PA, USA, 1998. [Google Scholar] [CrossRef]
- Hubbard, J.; Flowers, B.H. Electron Correlations in Narrow Energy Bands III. An Improved Solution. Proc. R. Soc. Lond. A
**1964**, 281, 401–419. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Statistical Physics, 3rd ed.; Course of Theoretical Physics; Pergamon Press Inc.: New York, NY, USA, 1980; Volume 5, p. 160. [Google Scholar]
- Essler, F.H.L.; Frahm, H.; Göhmann, F.; Klümper, A.; Korepin, V.E. The One-Dimensional Hubbard Model; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar] [CrossRef]
- Schrieffer, J.R.; Wolff, P.A. Relation between the Anderson and Kondo Hamiltonians. Phys. Rev.
**1966**, 149, 491–492. [Google Scholar] [CrossRef] - Lieb, E.H.; Wu, F.Y. Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension. Phys. Rev. Lett.
**1968**, 20, 1445–1448. [Google Scholar] [CrossRef] - Hirsch, J.E. Monte Carlo Study of the Two-Dimensional Hubbard Model. Phys. Rev. Lett.
**1983**, 51, 1900–1903. [Google Scholar] [CrossRef] - Moreo, A.; Scalapino, D.J.; Sugar, R.L.; White, S.R.; Bickers, N.E. Numerical Study of the Two-Dimensional Hubbard Model for Various Band Fillings. Phys. Rev. B
**1990**, 41, 2313–2320. [Google Scholar] [CrossRef] - Varney, C.N.; Lee, C.R.; Bai, Z.J.; Chiesa, S.; Jarrell, M.; Scalettar, R.T. Quantum Monte Carlo Study of the Two-Dimensional Fermion Hubbard Model. Phys. Rev. B
**2009**, 80, 075116. [Google Scholar] [CrossRef][Green Version] - Calandra-Buonaura, M.; Sorella, S. Numerical study of the two-dimensional Heisenberg model using a Green function Monte Carlo technique with a fixed number of walkers. Phys. Rev. B
**1998**, 57, 11446–11456. [Google Scholar] [CrossRef] - Hulthén, L. Über das Austauschproblem eines Kristalls. Ark. Mat. Astron. Fys.
**1938**, 26A, 1. [Google Scholar] - Mattis, D.C.; Pan, C.Y. Ground-State Energy of Heisenberg Antiferromagnet for Spins s = $\frac{1}{2}$ and s = 1 in d = 1 and 2 Dimensions. Phys. Rev. Lett.
**1988**, 61, 463–466. [Google Scholar] [CrossRef] [PubMed] - White, S.R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett.
**1992**, 69, 2863–2866. [Google Scholar] [CrossRef] [PubMed] - White, S.R. Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B
**1993**, 48, 10345–10356. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Two-level double-occupation functional ${D}_{\mathrm{SR}}^{2\mathrm{L}}$ of the spin-restricted half-filled Anderson model (${\gamma}_{\uparrow}={\gamma}_{\downarrow}$ and $N=2$) with local impurity occupation ${\gamma}_{ff}=1$ as a function of the degree of charge fluctuations X. The values ${D}_{\mathrm{SR}}^{\infty}$ and ${X}_{\mathrm{SR}}^{\infty}$ (${D}_{\mathrm{SR}}^{0}$ and ${X}_{\mathrm{SR}}^{0}$) refer to the strongly correlated (uncorrelated) limit. The inset shows the corresponding domain of ground-state representability of $\gamma $ (grey area) in terms of ${\gamma}_{ff}$ and X. Adapted with permission from Reference [71]. ©American Physical Society.

**Figure 2.**Ground-state and triplet-state properties of Anderson rings having ${N}_{o}=12$ orbitals and $N=12$ electrons as a function of the Coulomb-repulsion strength $U/t$ at the impurity orbital: (

**a**) interaction energy W; and (

**b**) impurity occupation ${\gamma}_{ff}$. The insets in (

**b**) show the impurity spin polarization $\langle {\widehat{S}}_{fz}\rangle $ and the degree of charge fluctuations X. Lattice density-functional theory (LDFT) results obtained within the two-level approximation (TLA) ${W}^{2\mathrm{L}}$ (solid and dashed curves) are compared with exact Lanczos diagonalizations (symbols) for ${\epsilon}_{f}=0$ and ${V}_{sf}/t=0.4$. In the inset of (

**b**) the unrestricted Hartree–Fock results for $\langle {\widehat{S}}_{fz}\rangle $ are given by the dashed blue curve for the sake of comparison. Reproduced with permission from References [71,77]. ©American Physical Society.

**Figure 3.**Singlet-triplet gap $\Delta E$ and ground-state degree of spin fluctuations ${\sigma}_{\mathrm{sf}}$ in a half-filled Anderson ring having ${N}_{o}=12$ orbitals, ${\epsilon}_{f}=0$ and ${V}_{sf}/t=0.4$. The results obtained with the TLA (curves) are compared with exact Lanczos diagonalizations (symbols) as a function of the Coulomb-repulsion strength $U/t$ at the impurity orbital. Reproduced with permission from Reference [77]. ©American Physical Society.

**Figure 4.**Ground-state properties of bipartite Hubbard rings having ${N}_{a}=14$ sites at half-band filling ($N=14$) as function of the Coulomb repulsion strength $U/t$: (

**a**) average number of double occupations per site $W/(U{N}_{a})$; (

**b**) NN bond order ${\gamma}_{12}$; and (

**c**) charge transfer $\Delta n={\gamma}_{22}-{\gamma}_{11}$. Representative values of the energy-level splitting $\epsilon $ between the sublattices are considered, which are indicated by numbers in (

**a**). The solid curves are the results obtained with LDFT in conjunction with the scaling approximation ${W}_{\mathrm{sc}}$, while the symbols correspond to exact Lanczos diagonalizations. Reproduced with permission from Reference [70]. ©American Physical Society.

**Figure 5.**Relation between the interaction energy $W\left[{\eta}_{\mathit{k}\sigma}\right]$ and the independent Fermion entropy $S\left[{\eta}_{\mathit{k}\sigma}\right]$ in the ground-state of the half-filled Hubbard model for various finite lattices with periodic boundary conditions. The results were obtained by exact Lanczos diagonalizations on finite 1D rings having ${N}_{a}=6$ (plus signs), ${N}_{a}=10$ (crosses) and ${N}_{a}=14$ sites (squares), as well as for 2D square-lattices having ${N}_{a}=2\times 4$ (circles) and ${N}_{a}=3\times 4$ sites (triangles). Reproduced with permission from Reference [72]. ©American Physical Society.

**Figure 6.**Ground-state properties of the periodic 2D Hubbard model on a 4 × 4 square lattice with ${N}_{\uparrow}={N}_{\downarrow}=8$ electrons and periodic boundary conditions. Exact numerical Lanczos diagonalizations (green crosses) are compared, as a function of the Coulomb repulsion strength $U/t$ with the linear independent-Fermion entropy (IFE) ansatz (red curves): (

**a**) ground-state energy ${E}_{0}$; and (

**b**) natural-orbital occupation numbers ${\eta}_{\mathit{k}\uparrow}={\eta}_{\mathit{k}\downarrow}$. The average number of double occupations D and kinetic energy T are shown in the inset of (

**a**). Reproduced with permission from Reference [72]. ©American Physical Society.

**Figure 7.**Ground-state energy of the half-filled Hubbard model on periodic hypercubic d-dimensional lattices as function of the Coulomb-repulsion strength $U/t$. All curves were obtained by means of the linear independent-Fermion entropy (IFE) approximation to LDFT. The exact ground-state energy of the 1D lattice is given by the blue crosses [84]. The green triangles and circles correspond to quantum Monte Carlo simulations for the 2D square lattice [85,86]. For each dimension d, the NN hopping integral ${t}_{d}$ is scaled as ${t}_{d}=t/\sqrt{d}$ in order that the second moment ${w}_{2}=2d{t}_{d}^{2}$ of the local density of states is the same for all d. Reproduced with permission from Reference [72]. ©American Physical Society.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Müller, T.S.; Töws, W.; Pastor, G.M. Recent Progress in Lattice Density Functional Theory. *Computation* **2019**, *7*, 66.
https://doi.org/10.3390/computation7040066

**AMA Style**

Müller TS, Töws W, Pastor GM. Recent Progress in Lattice Density Functional Theory. *Computation*. 2019; 7(4):66.
https://doi.org/10.3390/computation7040066

**Chicago/Turabian Style**

Müller, T. S., W. Töws, and G. M. Pastor. 2019. "Recent Progress in Lattice Density Functional Theory" *Computation* 7, no. 4: 66.
https://doi.org/10.3390/computation7040066