Exact maps in density functional theory for lattice models

Abstract

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...

Exact functionals for correlated electron–photon systems

“…We find a broad smearing of the density-to-potential map. Figure 2 we find a steepening of the gradient in the density-to-potential plot that we have earlier introduced as intrasystem steepening [20]. In figure 4, we show the same mapping for the photon displacement variable q as function of the external variables ( ) v j , ext ext .…”

“…diagonalization [20,39] given in equation (6) and obtain the corresponding ground-state wave function of the system, in the following denoted by Y ( ) v j , 0 ext ext . Using the exact wave function, we have access to the conjugated set of variables, i.e.…”

“…If, however, common approximations for the electron-electron many-body effects are used, then clearly QEDFT will face the same challenges as standard DFT when systems with strong electron-electron correlations are considered. To better understand such situations in DFT, the impact of static correlation and localization for different exact density maps has been analyzed in a recent work [20,21]. By investigating specific integrated quantities of these maps, e.g.…”

Orbital nodal surfaces: Topological challenges for density functionals

“…This success is based on the favorable ratio of accuracy to computational cost that DFT offers, especially with semilocal approximations for the exchange-correlation (xc) energy E xc [n(r)]. However, while the low computational cost of semilocal functionals has very much contributed to the success of DFT because it enables access to large systems of practical relevance, the functional derivatives of typical semilocal functionals, i.e., their corresponding xc potentials, miss important features of the exact xc potential, in particular discontinuities [3,4] and step structures [5][6][7][8][9] that are relevant, e.g., in charge-transfer situations [10][11][12] and ionization processes [5,[13][14][15][16]. Many attempts have been made to incorporate some of the missing features into semilocal DFT [17][18][19][20][21][22][23][24][25][26][27].…”

Challenges for semilocal density functionals with asymptotically nonvanishing potentials

“…Yet their functional derivatives, i.e., the corresponding xc potentials, typically completely miss important features of the exact xc potential [6][7][8][9][10][11][12][13][14][15]. Among them are, e.g., the particle-number discontinuity [16,17] and step structures or steepening effects [8,[18][19][20][21][22][23] that enforce [24], e.g., the principle of integer preference. Particle-number discontinuities and potential step structures and steepenings are mathematically different properties, but they are closely related to each other [8,17].…”