Exact Maps in Density Functional Theory for Lattice Models

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 for the first time the 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-to-wavefunction 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. We show that for fully decoupled subsystems the intra-system steepening transforms into the well-known inter-system derivative discontinuity. An important conclusion is that for e.g. charge transfer processes between localized fragments within the same system it is not the usual inter-system derivative discontinuity that is missing in common ground-state functionals, but rather the differentiable intra-system steepening that we illustrate in the present work.


Introduction
Over 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 arbitrary ground-state observables are functionals of the ground-state density, it does study lattice models with soft-Coulomb interaction between particles. Reference calculations of [53] show that one-dimensional model systems capture the essence of three-dimensional systems when studying strongcorrelation in DFT.
In this work, we study the exact density-to-potential and density-to-wavefunction map of a onedimensional lattice model with a system size that still allows to exactly diagonalize the Hamiltonian in full Fock space. The wavefunction-to-density map has been fully realized for fixed particle number, i.e. for the twoelectron singlet case by [54,55] and for the more-electron case by [56,57]. In principle, a small system size of the model used in this work allows to graphically illustrate the complete density-to-wavefunction map in the full Fock space of the system, and can be used to show how features such as the intra-system steepening and the inter-system derivative discontinuity of the density-to-potential map appear in the density-to-wavefunction map. As example, we explicitly show the complete density dependence of the two-electron singlet ground-state wavefunction. For different values of the external potential in the Hamiltonian we perform exact diagonalization of the Hamiltonian. Each diagonalization gives us all eigenfunctions and eigenenergies of the system, where the eigenstate with lowest eigenenergy corresponds to the ground state. We use the ground-state of each exact diagonalization corresponding to a fixed external and fixed chemical potential to construct both one-to-one maps, i.e.the map between on-site potentials and ground-state on-site occupations (ground-state densities), and the map between ground-state densities and ground-state wave-functions. To illustrate the latter, we numerically construct the configuration-interaction (CI) coefficients of the wave-function expansion as functionals of the ground-state density. We study the exact features of these maps for systems with different ratio of discrete values of the kinetic hopping probability l t to the electron-electron interaction strength l w . This allows us to study the exact maps from the non-interacting to the strictly localized electron limit while we gradually change the correlation of the system. We illustrate how the distinctive features of the exact density-topotential map transmit into the wavefunction-to-density map, and further into expectation values and transition matrix elements of arbitrary operators as functionals of the ground-state density.
We show that in approaching the limit of strongly correlated electrons, i.e.  l l 0 t w , the gradient of the exact density-to-potential map of our model system steepens. We denote this feature as intra-system steepening which gradually builds up within the system as the hopping probability favoring the delocalization of electrons decreases and the electron-electron interaction favoring the localization increases. In the strictly localized electron limit, where l = 0 t , we see that the ground-state becomes degenerate and hence, the intrasystem steepening transforms into the step-like inter-system derivative discontinuity.
We find that qualitative features such as the intra-system steepening and the inter-system derivative discontinuity of the density-to-potential map are already captured by a two-site lattice model. In the case of a two-site model, each site can be regarded as a subsystem. With increasing distance between the subsystems, the hopping probability decreases and the localization of the electrons on each site increases. If the sites are infinitely apart and the subsystems are truly separated, the electrons are strictly localized on each site. We simulate the infinite separation in the two-site model by setting the hopping parameter in the kinetic operator l t strictly to zero. Since the kinetic energy is strictly zero, this limit is a classical limit. However, setting l t equal to zero allows us to imitate the infinite bond-stretching of the molecular model, where the distance of the molecular wells d goes to ¥. In this limit, l = 0 t implying  ¥ d , the intra-system steepening of the density-to-potential map becomes the standard step-like inter-system derivative discontinuity.
Arbitrary observables and transition-matrix elements are affected by the presence of the intra-system steepening and the inter-system derivative discontinuity, and in particular by the lack of it in approximate functionals. We illustrate how both features are transmitted to the ground-and excited-state energy, the excitedand transition-state density and to the correlation entropy functionals.
The paper is organized as follows. In section 2 we present the exact maps between local potentials, groundstate wavefunctions and ground-state densities. In section 3 we introduce the lattice model and the methodology that we employ in the present work. Section 4 is dedicated to the study of the intra-system steepening of the exact one-to-one density-to-potential map when approaching the strictly localized limit (i.e. the strongly correlated limit) and its transition into a step function with only allowed discrete integer density values for truly separated subsystems. In section 4.1, we discuss how theses discrete integer density values of the decoupled systems connect if a weak coupling between the systems is introduced. In section 5 we use the potential-to-density map to construct the CI coefficients of the ground-and excited-state wavefunction expansions as explicit functionals of the ground-state density. Ground-state degeneracies leave topological scars in the electron density [58]. We illustrate how these degeneracies and furthermore also near-degeneracies of the eigenenergies of the system affect the ground-and excited-state expectation values and transition matrix elements of relevant operators as functionals of the ground-state density. Finally, in section 6, we summarize our findings and give an outlook for future work.

Exact mappings
To understand which features approximate functionals are missing, it is instructive to explicitly construct and to analyze the exact maps between the ground-state wavefunction Y 0 , the local potential V, and the ground-state density n 00 , sketched in figure 1. For fixed electron-electron interactionŴ , the many-body Schrödinger equation defines a unique map between the set of local potentials V and the set of energy eigenstates Y k , depicted as map  in figure 1. The ground-state density n 00 can be computed as usual according to which establishes a unique map from the set of N-electron ground-state wavefunctions Y 0 to the set of Nelectron ground-state densities n 00 . Hohenberg and Kohn [47] proved that the map  in figure 1 between n 00 and V is one-to-one and unique if V-representability is fulfilled [59][60][61][62][63]. Assuming existence of this one-to-one density-to-potential map allows in principle to construct any ground-state observable as a unique functional of the ground-state density n 00 , 00 00 0 00 0 00 Note that in addition to the one-to-one V-to-n 00 map, the Schrödinger equation establishes a map between n 00 and the excited-state wavefunctions Y ¹ k , 0 k . As a consequence, excited-state expectation values with = > k l 0, and transition matrix elements with ¹ k l, can be computed as functionals of the ground-state density using The ground-state energy E 0 and the ground-state density n 00 can be accessed using the variational principle In the following, we illustrate the features of the exact density-to-potential and density-to-wavefunction maps explicitly for our model systems.

Lattice model
In the present work, we restrict ourselves to one-dimensional lattice systems for which the construction of exact functionals via exact diagonalization is numerically feasible. On a lattice, the potential becomes an on-site Figure 1. Schematic illustration of the exact mapping between N-electron wavefunctions Y k , local potentials V, and ground-state electron densities n 00 . The maps are depicted as red arrows, where  maps V onto Y k ,  maps Y k onto n 00 and  maps n 00 onto V. Black arrows indicate the bijectivity of each of these one-to-one maps. Note that every element in V has an exact one-to-one equivalent in n 00 and Y 0 .
the density a site-occupation, ( ) ( )  n x n x i , and the integral becomes a sum over sites [48]. Furthermore, the kinetic energy operator becomes a nearest-neighbor hopping term.

Lattice Hamiltonian
For N interacting electrons in one spatial dimension we consider Hamiltonians of the formˆ( where the parameter μ, connected to the particle number operatorN , acts as a Lagrange multiplier shifting the state with lowest energy to blocks with different particle number N in Fock space. To switch between different coupling limits, we introduce the amplitude of the kinetic hopping ( ) l j = r cos t and the strength of the electron-electron interaction ( ) l j = r sin w as parameters in polar representation with radius r and angle j, see   and atomic separation d=5 a.u. and the corresponding exact exchange-correlation potential and ground-state density in real-space. We denote the blue-colored left halfspace as system S 1 and the red-colored right halfspace as system S 2 . With increasing distance d of the molecular wells the coupling of both systems decreases. Figure on the righthand: exact effective density-to-potential map, where we vary the potential difference d = v Z Z 1 2 from −5 to 5. The density difference dn 00 corresponds to the electronic density summed over the left half-space minus the density summed over the right halfspace as defined in equation (20). The point at ( in this effective map corresponds to the three panels in real-space on the left-hand side. The red-framed zoom at the top corner shows that the density-to-potential map has a smooth behavior on the scale of the coupling of both systems. whereˆ † s c l, andˆs c l, denote creation and annihilation operators of an electron placed on site l with spin projection onto the z-axis σ. Usually, the hopping t 0 changes with the lattice spacing dx. However, we choose to leave dx fixed and use the parameter l t and l w instead. The last term in equation (8) corresponds to on-site hopping. For model systems this term is usually not taken into account. Here we keep the term to allow for a consistent first and second quantized treatment of the Hamiltonian. We study the non-local soft-Coulomb electron-electron interactionˆˆˆˆ( introduces a potential difference between the sites in the lattice, which depending on its strength, shifts the electron density among the sites in the lattice. We restrict ourselves to two different scenarios for which exact diagonalization is still possible, similar to [66,67]. In case (i) we consider two spin-singlet electrons on M=206 sites. The particles are confined in a box from = - . To mimic the bond-stretching in molecular systems, we consider an external potential with two atomic wells separated by distances ranging from d=2 to d=8 a.u.. The depth of the wells is given by the nuclear charges Z 1 and Z 2 which we modulate with the parameter [ ] a Î 0, 2 . We will see that the essence of such a system is already captured by a two-site model. As case (ii), we consider M=2 sites in the lattice with a distance = x d 1 2 , where we vary the parameters l t and l w . In this case, the system size allows to perform exact diagonalization in the full Fock space of the model.

Methodology
To explicitly construct the one-to-one map between external potentials and ground-state densities, we diagonalize the Hamiltonian introduced in equation (7) for different external potentials v m , but fixed j. The external potential takes values = D v m m v , where D v is the numerical step size, and m is the step number. Although not shown in the present work, similarly the Hamiltonian can be diagonalized for different chemical potentials m k with the chemical potential values m = D m k k , the numerical step size D m and the step number k. In this way all functionals are constructible as functions of the particle number N and can be studied in complete Fock-space. Here we fix the chemical potential and select a discrete and uniformly distributed set of potentials from the continuous set V of possible external potentials in figure 1. In the next step we use exact diagonalization to compute the ground-state wavefunction Y j 0 and energy j E 0 corresponding to each value of v i and m i (but fixed j). For each ground-state wavefunction, we compute the corresponding on-site ground-state density where j is the site subindex, and the spin-summed density operator readŝ In addition to the ground-state wavefunction, the exact diagonalization gives us access to the excited-state wavefunctions Y j ¹ k 0 , which allows us to compute excited-state observables and transition matrix elements of operators as functionals of the ground-state density according to equation (4). On a lattice with M sites, the continuous one-dimensional ground-state density ( ) n x 00 becomes a vector ( ( ) ( ) ( )) n x n x n x , ,..., M 00 1 00 2 00 . Hence, expectation values and transition matrix elements as functionals of the density become rank M tensors In case (ii) where two sites in the lattice are considered, all functionals depend on the on-site densities ( ) n x Instead of expressing all functional dependencies in terms of the variables ( ) n x 00 1 and ( ) n x 00 2 , we rotate the coordinate system to the total particle number ( ) ( ) = + N n x n x 00 1 00 2 and the occupation difference ( ) ( ) d = n n x n x 00 00 1 00 2 between the sites [46]. To illustrate the wavefunction-to-density map, we expand the ground ( ) = k 0 and the excited ( , 00 where we have chosen |F ñ q to be the eigenstates of the kinetic operatorT . This gives rise to the CI coefficients as explicit functionals of dn 00 , we gain access to all ground-and excited-state expectation values or transition matrix elements of any operator, i.e. A prime example is the Hohenberg-Kohn energy functional defined in equation (6), which is the ground-state expectation value of the Hamiltonianˆ( )ˆ( )l on the angle j, since the Hohenberg-Kohn proof can only be established for fixed and given kinetic energy and particle-particle interaction. By changing the angle j, we construct the exact energy functional for different electron-electron interactions and kinetic terms, where j = 0 is the non-interacting and j = p 2 the infinitely correlated limit. Our approach allows to construct the exact density functionals for any observable of interest. We illustrate this for a few selected examples in the following sections. Also we emphasize that throughout this work all functionals are constructed in the zero-temperature limit.

Features of the exact density-to-potential map
We start our analysis for the Hamiltonian of case (i), where we consider a diatomic molecule with different interatomic separations. While the full density-to-potential map is a high-dimensional function for M=206 sites and impractical to visualize, the essence of the bond stretching can be captured by the integrated densities of fragments of the system. A natural choice to partition the system into its fragments, is to divide the total molecular charge distribution at its minima into different Bader basins [68,69]. By integrating the density over each of these Bader basins, the high dimensionality of the density in real-space reduces drastically. For our diatomic model the partitioning reduces the dimensionality from 206 to two, by mapping the sites in the grid onto the basins. We can then refer to each basin as a effective site in real space and regard the density difference between the basins as density difference between the two sites. For the simple diatomic molecule in onedimension, we simply divide the system in two equal half-spaces, and construct the density-difference according to To obtain the potential difference between the two basins, we take the difference between the maximum depth of the molecular potential wells of each basin and define the potential difference as . Note, the potential difference can be tuned by changing the nuclear charge of the two atoms with the parameter α of equation (12). The resulting effective density-to-potential map for our diatomic model is shown in figure 4. Starting from left to right we increase the distance between the molecular wells. The effective density-topotential map starts out with a smooth monotonic shape. When the distance of the atoms is increased the gradient of the effective density-to-potential map steepens, leading ultimately to steps in the density values for the infinitely separated limit.
The very same qualitative behavior can be found for a simple two-site lattice system. As a second example we consider therefore the Hamiltonian of case (ii). In this case we construct the exact density-to-potential map for the two-particle singlet states of the two-site lattice model. The results are shown in figures 5 and 6. Figure 5 shows the Hartree exchange-correlation part of the exact potential as functional of the ground-state density, whereas the first row of figure 6 shows the dependence of the exact density as function of the external potential difference between the sites. Note the density-to-potential map in figure 6 is one-to-one, and therefore invertible, the domain and co-domain of the selected independent variable and the corresponding function values can be swapped. In addition to the ground-state density-to-potential map in the first row of figure 6, the second and the third row show the first and the second excited-state density-to-potential map, and the fourth row shows the eigenenergies E 0 , E 1 and E 2 as function of the external potential difference between the two sites in lattice. From left to right, j increases, i.e. the electron-electron interaction favoring the localization of the electrons increases, whereas the kinetic energy favoring the delocalization decreases, i.e.
For the potential difference we select values from −5 to 5, shifting the electron density from one site in the lattice to the other. Setting j = 0 in equation (7) corresponds to non-interacting electrons, where the eigenfunctions are single-particle Slater-determinants. In this limit the density-to-potential map for our model can be found analytically The map behaves smoothly as can be seen in the leftmost figure in the first row of figure 6. Approaching the strictly localized limit, i.e. j  p 2 , the slope of the exact density-to-potential map sharpens until the map develops a characteristic feature, which we denote as intra-system steepening. The intra-system steepening of the gradient of the density-to-potential map corresponds to the localization of the electrons in the respective subsystems. Near the strictly localized limit, e.g. j = -p 2 1 100 , the electrons are highly localized on the sites.
In the strictly localized electron limit j = p 2 the hopping parameter is equal to zero. In this limit the system 'breaks' into two kinetically decoupled sites and the Hamiltonian reduces toˆˆ= +  Figure 4. Exact density-to-potential map for a one-dimensional diatomic molecule with nuclear charges Z 1 and Z 2 , where we vary the potential difference d = v Z Z 1 2 from −5 to 5 for different atomic separations =d 2 8 a.u. similar to [66,67]. The density difference dn 00 corresponds to the electronic density summed over the left half-space minus the density summed over the right halfspace as defined in equation (20) and illustrated in figure 3. The graph illustrates the influence of electron localization on the groundstate density-to-potential map. From left to right the distance of the molecular wells increases while the gradient of the density-topotential steepens with increasing distance d and hence, decreasing coupling of the fragments of the system. We denote this feature of the density-to-potential map as intra-system steepening (see text for details). . Depending on the ratio between the external potential difference dv and the electron-electron repulsion strength l w , (except in the points of degeneracy) one of these three eigenstates is energetically more favorable and becomes the ground-state of the electronic system, see lower panel of figure 6. In the strictly localized limit, the density difference dn 00 transitions from a continuous variable to a discrete set of integer values. Namely, the only possible values for the ground-state density differences are the integer values in the strictly localized limit. Inset at the bottom at the right-hand side: avoided and real crossings of eigenenergies. From left to right the angle j increases the correlation in the system going from the non-interacting (j = 0) to the strictly site-localized electron limit ( ) j = p 2 . In the molecular model system of figure 4 this corresponds to an increasing distance d of the molecular wells. The gradient of all three densities steepens whenever the corresponding eigenstate as functional of the external potential comes close to an avoided crossing. We denote this exact feature of the density-to-potential map as intra-system steepening.
In the strictly localized limit ( ) j = p 2 the intra-system steepening becomes the inter-system derivative discontinuity while the avoided crossing turns into a real-crossing with degenerate eigenenergies. For fully decoupled systems the different values of the external potential lead to the same density difference dn 00 as can be seen in the map for j = p 2 in figure 6. Therefore, the one-to-one map between dn 00 and dv breaks down. The Hohenberg-Kohn theorem is applicable to each disjoint system but not to the combined system. For j = p 2 , the two sites kinetically decouple, however both sites are still coupled via the long-range soft-Coulomb interaction. For zero kinetic coupling, the ground-state of this combined system becomes degenerate in two points near d = v 0. Due to the points of degeneracy of the ground-state of the system, the distributional points of the density-to-potential map can be connected via straight-lines and the intra-system steepening transitions into the inter-system derivative discontinuity. Consequently, functionals along the cut in the variable dn are a linear combination of the functionals of the degenerate densities as has been shown for the ground-state energy functional as functional of the particle number [25,70]. Therefore, we connect the distributional points along the variable dn for all functionals via straight lines, i.e.
( ) d w =  n 2 1 00 and   w 0 1. In a physical picture each one of the kinetically disconnected sites can be seen as a system infinitesimally weakly connected to a grandcanonical particle reservoir.
Contrary to the widely discussed inter-system derivative discontinuity, which describes the piece-wise linear behavior of the energy as a function of the particle number [ ] E N , the intra-system steepening describes the smooth behavior of the energy as functional of the density difference between fragments within the system [ ] d E n 00 . Both features already show up in the density-to-potential map and transmit to all observables. The Hohenberg-Kohn energy functional is therefore only one specific example for the appearance of the intersystem derivative discontinuity and the intra-system steepening. The smooth behavior of the intra-system steepening is a consequence of the mixing of different quantum eigenstates around avoided crossings, and the steps related to the inter-system derivative discontinuity directly result from intersections of eigenenergies, thus Figure 7. Differences between the strictly localized limit (a), the intra-system steepening (b), (c) and the inter-system derivative discontinuity (c), see discussion in section 4.1. Note, all three cases are mathematically well-defined limits. However, limit (a) and in general also the inter-system derivative discontinuity of limit (c) (except for an accidental degeneracy of the ground-state) are purely mathematically constructions, whereas processes in nature happen in the limit of (b) since neither the kinetic coupling nor the particle interaction can be fully switched off. We consider two systems S 1 with particle number N 1 and S 2 with particle number N 2 . In figure (a) both systems are decoupled, and the ground-state of each isolated system is a pure state with fixed, integer particle number leading to a distributional density-to-potential map. The Hohenberg-Kohn theorem is valid in the disjoint Hilbert space of system S 1 and S 2 , but not in the combined Hilbert space of both systems. Figure (b) shows the time-evolution of the system S 1 with initially -N 1 1 particles to + N 1 1 . The particle number of system S 1 can only change if the system is coupled to a bath, here system S 2 with initially + N 1 2 particles. Figure (c) shows the eigenenergies of the two-particle singlet states as in figure 6 and illustrates the difference between the intra-system steepening and the inter-system derivative discontinuity. The question is how the functionals of the densities of system S 1 and S 2 in figure (a) connect if a weak coupling between the systems is introduced. Note, although in the two-site model we kinetically decouple both sites, the sites are still coupled via the long-range soft-Coulomb interaction. For an accidental degeneracy of the ground-state, functionals are constructible as linear combination of the degenerate subspace. This construction then leads to the wellknown straight-line behavior along the cut in the variable N of ground-state density functionals which are not differentiable at integer particle number  = + N N 1 2 . In general a weak coupling lifts the degeneracy of the ground-state of the system as indicated on the bottom of the figure and leads to a smooth density-to-potential map. The gradient of this map steepens with decreasing coupling of the systems. The intra-system steepening is valid from strong to weak coupling. real crossings, see lower panel and inset of figure 6. The inter-system derivative discontinuity appears when electrons are strictly localized in states with different particle number but an accidental degeneracy of the ground-state allows the statistical mixture of the degenerate ground-states. Note that the steepening of the gradient for dn 00 as well as for dn 11 and dn 22 arises whenever the eigenvalues of the Hamiltonian in equation (7) as function of the external potential become nearly degenerate. The connection between the avoided crossing and the steepening of the gradients functional is closely related to the finding of [36], i.e. that the step feature of the exact xc-potential in space arises in the vicinity of the avoided crossing, when the bonding and antibonding orbitals become nearly degenerate. Without this step feature (and the peaks) of the exact xc-potential, the noninteracting electron density would artificially smear out over both basins and lack the intra-system steepening of the exact electron density-to-potential map. For j = 0 all eigenvalues are non-degenerate, hence the densityto-potential map of all eigenstates behaves smoothly. When we approach the strongly correlated limit at j  (see inset figure 6). Because of this degeneracy of the first and second excited-state, the distributional points +2 and −2 of the first and second excited-state density as functional of the ground-state density can be connected via straight lines along the cut in the variable dn, see also figure 10. Caused by a real crossing of the eigenenergies in the strictly localized limit, the one-to-one correspondence with an external potential breaks down for all densities, i.e. the ground-state and the excited-state densities. The density-to-potential map becomes a distribution in this limit and the Hohenberg-Kohn theorem does not apply.

Intra-system steepening versus intersystem derivative discontinuity
In DFT, the ground-state wavefunction and all ground-state observables are unique functionals of the groundstate density. In this work, we also show the functional behavior of excited-state observables and transition matrix elements, for which in general no one-to-one correspondence exist. Nevertheless, as we will see in the following sections, features of the exact density-to-potential map directly translate to both types of functionals, i.e. the unique ground-state functionals and the generally non-unique functionals of the excited-state observables and the transition matrix elements. Therefore, it is of fundamental importance to identify and classify these features of the exact density-to-potential map. In the following, we distinguish between the three different cases displayed in figure 7. For all cases let us assume, that we have two systems S 1 and S 2 . In the two-site lattice model one site corresponds to S 1 and one to S 2 , whereas in the many-site model one basin corresponds to S 1 and the other one to S 2 . In the first case as in figure 7(a) the kinetic coupling as well as the coupling via electron-electron interaction between the systems are strictly zero. This is a purely mathematical limit, since each system S 1 and S 2 is completely isolated from its environment and thus, the particle number in the system can not change. The Hilbert space of both systems is disjoint. The ground-state of each system corresponds to a pure state with fixed particle number. The Hohenberg-Kohn theorem is valid in the Hilbert space of each system, but not in the combined Hilbert space of both decoupled systems S 1 and S 2 . The cut along dn 00 of the density-to-potential map of the total system is a step function, where the values of dn 00 are the particle number of system S 1 and S 2 , i.e. N 1 and N 2 . For fully decoupled systems S 1 and S 2 , the functionals of this step-like densityto-potential map can only take values at the integer particle number of system S 1 and S 2 , i.e. F [N 1 and N 2 ], where the values of the functional at N 1 and N 2 are disconnected. In the second case displayed in figure 7(b) the coupling of the systems ranges from strong to infinitesimally weak. Without loss of generality, we refer to S 1 as system and to S 2 as bath. This case is the underlying concept of charge transfer processes or in the description of band gaps of a system, where the particle number is changed by adding or removing a particle. These processes are not instantaneous processes, but should be regarded as processes in which a system with initially N particles evolves in real-time to a system of -N 1 or + N 1 particles. The particle number of a system S 1 can only change if it is coupled to a bath S 2 . The particle number of the total system-and-bath, i.e.  = + N N 1 2 , is fixed and integer, whereas the particle number N 1 and N 2 can take fractional values. In general, an infinitesimal small kinetic coupling term of a system to a bath lifts the degeneracy of the ground-state of the combined system + S S 1 2 as we have exemplified for the eigenenergies of the two-site model in figure 6, ultimately leading to a smooth density-to-potential map. Functionals of this smooth map are also smooth, as we exemplify in the following sections for the wavefunction and selected observables as functional of the ground-state density. We will see that in the weakly coupled limit, i.e. the limit j = -p 1 100 2 , the straight-line behavior is an excellent approximation to the exact functional behavior along the cut in the variable dn but that the exact functional shows a smooth behavior since the density-to-potential map is also smooth. The deviation of the straight-line behavior depends on the energy-scale of the coupling term between the system and the bath. To see how the functionals deviate from the straight-line behavior we included a zoom of the functionals in the respective limit whenever it improved the clarity. The third case in figure 7(c) is valid for an infinitesimally small coupling of a system to a bath for which the ground-state of the system shows an accidental degeneracy. Here the coupling of both sites is realized via the long-range soft Coulomb interaction in the Hamiltonian. Functionals of the distributional points of the density-to-potential map are then linear combinations of the densities of the degenerate subspace, and can be connected via straight-lines [25,70]. For such weakly coupled processes with a degenerate ground-state, the straight-line behavior and hence the derivative discontinuity at integer particle number N introduced in the original work of Perdew is exact.
To give a mathematical more rigorous definition of the concept of the steepening, we introduce the maximum value of the gradient of the effective density-to-potential map, i.e.

( )
. To obtain the effective density-to-potential map, it is necessary to first partition the density of the system, e.g. using a Bader partitioning scheme. Second, the integrated density over each basin corresponds to a subsystem, e.g. for three subsystems we obtain the densities n n , 1 2 and n 3 . To determine the localization and hence, the correlation between two of the three subsystems, we construct the density difference d =n n n 12 1 2 and d =n n n 2 3 2 3 as functional of the potential difference, i.e. 12 12 23 and 23 12 23 , where e.g. d = v Z Z kl k l and Z k is the maximal depth of the external potential present in basin k. The gradient f ijkl is a measure how the localization over subsystems changes subject to small changes in the external effective potential and therefore, can be used to illustrate the intra-system steepening while e.g. stretching the one-dimensional molecule. The size of the tensor f ijkl depends on the number of subsystems present in the system. The steepening presents a slope that gets steeper as the correlation between subsystems in e.g. our bipartite system increases. The relation between dn ij and dv kl remains locally one-to-one and therefore invertible, the domain and co-domain of the selected independent variable and the corresponding functional values can be swapped as long as the corresponding value of f ijkl is bounded. In the limit of the inter-system derivative discontinuity the functional value of f ijkl for given ijkl diverges, indicating that an infinitesimally small change in the effective potential changes the electron localization in the system dramatically. For our specific model system, we illustrate that the transition of the steepening to the non-differentiable derivative discontinuity corresponds to a maximum in the von-Neumann correlation functional which provides a good measure for the correlation present in the system. For systems with degenerate ground-state (each of which corresponds to a Hilbert space with fixed particle number), the ground-state density can be constructed by a linear combination in the degenerate subspace resulting in a straight-line behavior whose slope changes at integer d = n N. Functionals in these points (at integer N) are not differentiable. Whenever an intra-system steepening in the effective density-to-potential map appears as a consequence of high correlation, we expect such a feature to be present in the xc functional as a functional of the ground-state density.

Features of the exact density-to-wavefunction map
The inter-system derivative discontinuity and the intra-system steepening discussed in the previous section are exact properties of the density-to-potential map. As a consequence, also the exact wavefunction and hence, all exact observables as function of the exact density inherit the intra-system steepening and the inter-system derivative discontinuity. In the following sections we illustrate this fact. In particular, we show how these features show up in the CI-coefficients, and consequently in the energy, the excited-densities and in the correlation entropy functional.

Exact configuration interaction coefficients as functionals of the ground-state density
To construct the density-to-wavefunction map, we expand the correlated ground-and excited-state wavefunctions from the exact diagonalization of the Hamiltonian in a complete set of Slater determinants |F ñ q . This gives rise to CI coefficients as functionals of the ground-state density as defined in equation (17). Clearly, each choice for the set of Slater determinants |F ñ q induces a different set of CI functionals. Here we choose as basis set the determinants which are eigenfunctions of the free kinetic energy operator. More specifically, we project the two-particle singlet ground-state wavefunction of the Hamiltonian in equation (7) onto the three two-particle singlet eigenstates of the kinetic operatorT to construct one of these sets for each different j. The CI coefficients of the non-interacting electrons are shown in the leftmost column of figure 8, where j = 0. Approaching the strictly localized electron limit, i.e. from left to right in figure 8, the gradient of the CI coefficients sharpens. This sharpening corresponds to the intra-system steepening of the dn 00 -to-dv map introduced in section 4 and is inherited by the CI coefficients. Furthermore, the inter-system derivative discontinuity shows up in the CI coefficients for j =

Exact ground-state and excited-state energy functionals
Since the CI coefficients a j q of the wavefunction inherit the intra-system steepening and the inter-system derivative discontinuity, arbitrary ground-state expectation values, defined in equation (18), also inherit the intra-system steepening and the inter-system derivative discontinuity. Note, the excited-state CI coefficients also show the same exact features, which are then inherited by excited-state functionals in the respective limit. As particular examples for this inheritance, we illustrate in figure 9 the intra-system steepening and the intersystem derivative discontinuity for the exact Hohenberg-Kohn functional ( j=0) and the excited-state energy Figure 8. CI coefficients of the two-particle ground-state wavefunction in the kinetic operator basis. From left-to-right we approach the strictly localized limit (j = p 2 ) and the gradient of all three CI coefficients steepens. For j = p 2 the CI coefficients take only discrete values which can be interpolated linearly (dashed lines) due to the degeneracy of the eigenstates in the strictly localized limit.
j of the ground-, the firstand second-excited state for different strengths of the electron localization j. First row: second excited-state energy F 22 as functional of the ground-state density. Second row: first excited-state energy F 11 as functional of the ground-state density. Third row: ground-state energy F 00 as functional of the ground-state density, i.e. the Hohenberg-Kohn functional. From the non-interacting limit (left) to the strictly localized limit (right), the gradient of all energy functionals steepens. In the highly localized limit, where j = -p 2 1 100 , all energy functionals are continuous. In particular, the ground-state energy functional shows a convex behavior as can be seen in the detailed view of the intra-system steepening of highly localized electrons and the inter-system derivative discontinuity of strictly localized electrons at the bottom of the figure. Note, that here the x-axis has been scaled by one order of magnitude. In the strictly localized limit, for all energy functionals only the three distributional points d =  n 2 00 and d = n 0 00 exist. Due to the degeneracy of the eigenstates in the strictly localized limit which is shown in the lower panel of figure 6, these three distributional points connect via straight lines indicated by a blackdashed line.
as functionals of the ground-state density dn 00 . The excited-state density functionals are shown in the second and third row of figure 10 respectively. For completeness, also the trivial linear behavior of the ground-state density as functional of the ground-state density is shown in the first row of the figure. From the non-interacting (left) to the strictly localized limit (right), the gradient of the excited-state density functionals steepens up to the 100 and the inter-system derivative discontinuity is given on the right. . From the non-interacting (left) to the strictly localized limit (right), approaching the strictly localized limit the gradient of both transition density functionals steepens. In the strictly localized limit, the sites are disconnected. Therefore, there are no transitions between the three two-particle singlet states, and the transition densities are zero. strictly localized limit where the excited-state density functionals obey the straight-line condition due to the degeneracy of the ground-state. To highlight the difference of the intra-system steepening and inter-system derivative discontinuity of the excited-state density functionals a detailed view of the critical region can be found on the right-hand side of figure 10.
Transition densities are an important ingredient for linear response calculations in time-dependent DFT. For our model system, we show the exact transition densities as functionals of the ground-state density. In contrast to the excited-state density functionals, the transition density functionals are phase-dependent. Figure 11 shows the absolute value of the transition density as functional of the ground-state density. The first and second row of figure 11 show the absolute value of the transition density from the first to the second and from the ground-to the second excited state, respectively. Approaching the strictly localized limit, both transition-state densities show clearly the intra-system steepening. In the strictly localized limit, there is no transition between the eigenstates of the system and the transition-state densities are zero, see j = p 2 in panel one and two.

Exact correlation entropy functional
As final example we illustrate the functional behavior of the correlation entropy. The correlation entropy , discussed in detail in [38] measures the correlation and entanglement present in a many-body system. It can be understood as well as a measure of the Slater rank [38,39] as can be seen if we compare the correlation entropy plotted in figure 12 with the mixing of the eigenstates in lower panel and inset of figure 6 for the different values of the parameter j. In the two-site model, where we have access to all eigenvectors and eigenvalues, we can compute the correlation entropy of the system where n j are the eigenvalues of the reduced one-body density matrix j j 00 0 0 The correlation entropy is zero for pure states, and has its maximum for maximally mixed states [38,39,71]. In figure 12 we see that the correlation entropy increases with increasing correlation while the gradient of the correlation entropy functional obeys the intra-system steepening and transitions into the inter-system derivative discontinuity due to the degeneracy of the ground-state for j = p 2 . In the limit of non-interacting electrons, where there is no correlation, the correlation entropy vanishes. The maximum value of the correlation entropy is reached in the strictly localized limit for d = n 0 00 where all three eigenenergies are degenerate.

Summary
In the present work we have illustrated for a simple but general interacting lattice model how the intra-system steepening, an exact feature of the ground-state density-to-potential map, develops gradually with increasing decoupling between fragments of a system and transforms into the well-known inter-system derivative discontinuity for an accidental degeneracy of the ground-state of weakly coupled systems. As a consequence of the Hohenberg-Kohn theorem, the wavefunction-to-density map inherits the exact features of the density-topotential map as well as the ground-and excited-state observables and transition-matrix elements. Figure 12. Correlation entropy as functional of the ground-state density indicating the correlation within the system. For noninteracting electrons the correlation entropy is zero. From left to right, approaching the strictly localized limit, the correlation and the mixing of the eigenstates and hence the correlation entropy increases. Furthermore, the gradient of the functional obeys the intrasystem steepening and the inter-system derivative discontinuity for j = p 2 .
Although both exact features are linked to the localization of the electrons, we carved out that the intrasystem steepening and the inter-system derivative discontinuity are conceptually different features within density functional theory. The inter-system derivative discontinuity corresponds to the electron localization in weakly coupled subsystems with degenerate ground-state. The inter-system derivative discontinuity coincides with a real crossing of the eigenenergies of the system as function of the external potential. In this limit, the cut along the variable dn of the generally very high-dimensional ground-state density functionals have straight lines between different values for the particle number N due to the mixture of the states in degenerate subspaces, ( ) w w = -+ + F F F 1 N N 1 with the mixing parameter   w 0 1. The intra-system steepening instead corresponds to the electron localization in weakly to strong coupled fragments of a system. The intra-system steepening coincides with an avoided crossing of the eigenenergies as function of the external potential and sharpens when approaching the real crossing. Ground-state density functionals result directly from the one-toone correspondence of the Hohenberg-Kohn theorem, such as the convex ground-state energy as function of the density difference between the fragments of the system. The inter-system derivative discontinuity plays a crucial role whenever the particle number of the total system changes which is the case for observables such as the electron of a system. The intra-system steepening is linked to processes where particles are transferred from one fragment to another within a system of fixed particle number such as stretched molecules, charge-transfer processes and any problem involving highly localized electrons. We expect approximate functionals may fail to describe such problems because of the lack of the intrasystem steepening feature. Given the relevance of the above mentioned problems it is crucial to develop improved density functionals that capture this exact condition of the exact density-to-potential and density-towavefunction maps. In the highly localized electron limit the exact xc-functional does not present a straight line behavior but rather a sharp but differentiable one. Our work illustrates those fundamental concepts of density functional theory. To improve the accuracy of DFT observables, approximate functionals should capture both, the inter-system derivative discontinuity and the intra-system steepening respectively. Work about how to generalize the present results from lattice Hamiltonians to real-space systems is currently in progress. Our results also allow to get insight about spin DFT functionals as the magnetization of the N electron system can be written in terms of the ground-state density (as all other observables we discussed in this paper). This is a way to solve the known problems of spin DFT [72,73] (however it would require going beyond present adiabatic functionals, work along those lines is in progress).