What has made DFT popular for determining properties of complex many-body systems is the Kohn—Sham KS construction, 5 where instead of modeling the kinetic energy directly in terms of the density an auxiliary noninteracting quantum system is used that has the same density. The kinetic energy of this computationally cheap auxiliary system is then corrected by so-called Hartree-exchange-correlation Hxc contributions that incorporate the missing interaction and kinetic-energy contributions. Already simple approximations to this unknown expression give reasonably accurate answers.
However, it is hard to systematically increase the accuracy of approximations while still keeping the favorable numerical costs. There are several other approaches for dealing with the quantum many-body problem that also avoid the many-body wave function, while the basic variable used makes it easier to model the desired physical quantities. In one-body RDM 1RDM functional theory, 11 the kinetic energy is an explicit functional of the 1RDM, thus only part of the interaction energy needs to be approximated, while in the two-body case 10 even the interaction is given by an explicit functional.
Although the explicit use of wave functions can be avoided in these cases, it is still necessary for the RDM to be representable by a wave function. However, the so-called N -representability conditions that guarantee an underlying wave function associated with an RDM are anything but trivial. There are now several possible ways to remedy the above-mentioned deficiencies. For 1RDM theory, it is helpful to consider the many-body problem at finite temperature and indefinite numbers of particles. Another possibility is to construct approximate natural orbitals, which are eigenfunctions of single-particle Hamiltonians with a local effective-potential.
This however implies that an additional auxiliary potential, which couples to the kinetic-energy density, has to be introduced. A similar approach has recently appeared in a different context, that is, in thermal DFT, 25 , 26 where the additional auxiliary potential corresponds to a proxy for local temperature variations and couples to the entire energy density, including kinetic and interaction contributions.
The concept of local temperature was also introduced in the local thermodynamic ansatz of DFT. When treated within the generalized KS framework, 30 meta-GGAs lead to a local potential coupling to the kinetic-energy density, which can be interpreted as a position-dependent mass. In this Article, we investigate the possibility to include the kinetic-energy density as a basic functional variable in DFT alongside the density.
The idea is that by doing so one can increase the accuracy of density-functional approximations. We investigate this by constructing the exact density functionals of standard DFT and comparing them to the combined kinetic-energy density and density functionals of this extended approach we call kinetic-energy density-functional theory keDFT.
In this way, we want to assess possible advantages of such an approach when considering strongly correlated systems. The so-called kinetic contribution 32 to the exchange correlation potential is important for the description of such systems.
It has been shown that standard DFT functionals fail to describe the effects of this kinetic contribution such as the band narrowing due to interactions. Further, we want to consider the quality of possible approximation schemes to keDFT based on a kinetic-energy KS keKS construction and test them in practice. As is clear from the extent of the proposed program, this is not possible for real systems. We therefore consider lattice keDFT. In this way, we not only avoid the prohibitively expensive calculation of reference data for realistic interacting many-body systems but also avoid mathematical issues connected to the continuum case, like the nonexistence of ground states and nondifferentiability of the involved functionals 38 , 39 or having to deal with the kinetic-energy operator, which is unbounded.
We also highlight how simple approximations carry over from our model systems to more complex lattice systems and even to the full continuum limit. The results hint at the possibility to treat weakly and strongly correlated systems with the same simple approximation to keDFT. This Article is structured as follows: In section 2 we introduce our lattice model, define the density and kinetic-energy density on the lattice, and highlight for a simple two-site case that the kinetic-energy density is a natural quantity to be reproduced by an extended KS construction.
We then introduce the resulting keKS construction assuming the existence of the underlying maps between densities and fields. Still we can provide a bijective mapping between densities and fields for specific cases. In section 4 we then show how we numerically construct the mappings beyond these specific cases and hence find that keDFT on a lattice can be defined also for more general situations. In section 5 we then use the constructed mappings to determine the exact correlation expressions for the KS and the keKS construction, respectively. In section 6 we then compare the results of self-consistent calculations for similar approximations for the KS and the keKS systems, respectively.
Finally, we conclude in section 7. In the following, we consider quantum systems consisting of N Fermions electrons on a one-dimensional lattice of M discrete sites. We assume that these particles can move from site to site only via nearest-neighbor hopping corresponding to a second-order finite-differencing approximation to the Laplacian and employ zero boundary conditions for definiteness the extension to periodic boundary conditions is straightforward. This leads to a Hamiltonian of the following type:.
The nonlocal first term corresponds to the kinetic energy. Let us point out that usually the hopping amplitude is site-independent. We employ this more general form corresponding to a site-dependent mass to establish the necessary mappings see eq However, when we numerically consider interacting systems, we always employ a site-independent hopping, which corresponds to the standard Hubbard Hamiltonian.
The second term corresponds to a local scalar electrostatic potential v i acting on the charged particles at site i. Because we fix the number of particles, the potential v i is physically equivalent to a potential that differs by only a global constant.
In the following, this arbitrary constant is fixed by requiring. From the lattice-version of DFT, 41 we know that for every fixed set of parameters t , U , there is a bijective mapping between the set of all possible potentials in the above gauge to all possible densities for a fixed number of particles.
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In certain situations, for example, for figures, it is more convenient to use the density and potential differences instead of the density and potential. Because it is bijective, we can invert the mapping and find a potential v s where we follow the usual convention and denote the potential of a noninteracting system with an s for a given density n.
The noninteracting mapping allows one to define v i s [ n ], which in turn leads to. The noninteracting Hamiltonian reproduces the prescribed density n as its ground state by construction. This is not yet the KS construction, because we need to know the target density in advance. Only upon connecting the interacting with the noninteracting system by introducing the Hxc potential:.
This problem has as the unique solution the noninteracting wave function that generates the density of the interacting problem without knowing it in advance. Only the latter provides an iterative scheme to predict the density of an interacting references system. When we later present results for the exact KS construction in section 5 , we refer always to the results at the unique fixed point of the KS construction. In practice, however, we do not have the exact v i KS [ v ; n ] available, and hence we need to devise approximations to the unknown Hxc functional.
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As we will see in section 5 , the major problem in these approximations is that the kinetic-energy density of the KS and the interacting system become dramatically different with an increasing U. Here, the kinetic-energy density T i at site i is defined nonlocally because it involves the hopping with the help of the first off-diagonal of the spin-summed 1RDM in site basis representation:.
By analogy to the continuum case, one can also define the charge current J i as. We note that the current obeys the lattice version of the continuity equation:. It is important to note that it is not only the variational minimum-energy principle that ground states have to fulfill, but there are many more exact relations. While for the case of the ground state the EOM of eq 9 is trivial because both sides are individually zero, there are many other nontrivial exact relations that can be based on EOMs and that provide us with exact relations between the densities, the fields, and other physical quantities.
For instance, the second-time derivative of the density provides us with the local force balance of the equilibrium quantum system, 43 which we will use in section 4. Also, while the most common way to find approximations for the Hxc potentials is by obtaining approximate Hxc energy expressions and then taking a functional derivative, the EOMs provide an alternative way to construct approximate Hxc potentials without the need to perform functional variations.
Clearly, if we could enforce that an auxiliary noninteracting system has the same 1RDM as the interacting one, then also the kinetic-energy densities T of the two systems would coincide. This suggests that one can establish a mapping between the interacting 1RDM and a nonlocal potential, that is, a v i , j that connects any two sites of the lattice and thus couples directly to the full 1RDM. However, in general this is not possible as has been realized early on in 1RDM functional theory.
This is also true in more general lattice situations as has been shown in, for example, ref For the 1RDM, two solutions to this problem are known. One is to include temperature and possibly an indefinite number of particles, which introduces off-diagonals that depend on the temperature and the hopping, that is, the nonlocal potential. The other possibility is to make the system degenerate such that we can reproduce any density matrix. Here, we apply a different strategy. While we cannot force the density matrices to coincide, it is possible to require the kinetic-energy densities to be the same.
The crucial difference is that we include the coupling in the Hamiltonian in the definition of the quantity to be reproduced by the KS system. For example, in the two-site case, we merely need to use an interaction-dependent hopping. Thus, the auxiliary noninteracting system reproduces now the pair n , T of the interacting system. Before we move on, let us note that similarly to the continuum case, one could use 1RDM functional theory at zero temperature also on the lattice if one avoids the use of a noninteracting auxiliary system and merely uses functionals based directly on the interacting 1RDM.
Specifically we can then consider a noninteracting auxiliary problem that generates a prescribed pair n , T :. Whether we can construct such an auxiliary system that reproduces the density and kinetic-energy density of an interacting system is something we do not know a priori. In this Article, we provide numerical evidence as well as proofs for specific situations that suggest that such a construction is possible see section 3.
If we introduce then the corresponding mapping differences similar to eq 4 and denote them by mean-field exchange-correlation Mxc :. Only the latter provides an iterative scheme to predict the physical pair n , T of the interacting reference system. When we in the following present results for the exact keKS construction, we refer always to the results at the unique fixed point of the keKS construction.
This also allows us in the following to only use t i ke and v i ke to highlight the difference between the usual KS and the keKS construction. To make the scheme practical, we now need two approximations: one for the Mxc potential and one for the Mxc hopping. Possible routes on how to construct approximations and how this could help to more accurately capture strongly correlated systems we consider in section 5.
At this point, we want to make a first connection to the continuum by considering the appropriate choice of the kinetic-energy density for that case. There are different possible definitions for a local kinetic-energy density, which will give rise to the same total kinetic energy. The single-particle kinetic-energy operator then becomes accordingly , where m r should be substituted with m s r in the noninteracting case.
Similarly to fixing the constant of the local potential, one needs to fix the gauge of the hopping parameter t i ke where the superscript ke in parentheses is used to denote that we refer both to interacting and to noninteracting keKS systems. One of the first things to note is that by letting t i ke change from site to site, we encounter a large equivalence class for the site-dependent hopping parameters.
However, the wave function and also, for example, the 1RDM change. This leaves the density unchanged, as it is just a sum of the squared absolute values of the single-particle wave functions. Also, the kinetic-energy densities stay the same, because the 1RDM switches signs at the same place as the hopping amplitude. As it follows from the discussion above, the sign of t i ke is just a gauge choice, and we need to fix the gauge to establish the sought-after mapping.
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As one can readily see, the density stays the same in both cases, as a consequence of the sign of t being only a gauge choice. A further complication that one encounters in establishing the necessary mappings is that the usual Hohenberg—Kohn approach does not work in our case. The reason is that the control fields t now become explicitly part of the control object T. A similar problem is encountered in current-density-functional theory, when trying to establish a mapping in terms of the gauge-independent physical charge current. However, for specific situations, we are able to show that the discussed mapping is possible.
The most important one in our context is the case of the two-site Hubbard model see Appendix A for details. So we can simply rescale the auxiliary Hamiltonian and thus prove the existence of the mapping in the noninteracting case by employing the Hohenberg—Kohn results. A further simple case is two noninteracting particles, forming a singlet, in a general M -site lattice. Here, the density fixes the single-particle orbital doubly occupied up to a sign, and thus for a given T i only a unique site-dependent hopping t i is possible.
In this case, the mapping is invertible and a unique up to a sign choice is associated from site to site. Note that in this case the KS system and the keKS system yield the same wave function and. It allows us in a simple yet exact way to connect the auxiliary keKS system to the interacting system.
We will use this later to construct a first approximation to t i Mxc. To show that the keDFT mapping can also be defined for other, more general cases, we construct in the following the mappings numerically. Afterward, we make use of the constructed mappings to investigate the properties of the Mxc potentials and the basic functionals, which for the continuum case would be numerically prohibitively expensive. Because, as discussed above, it is not straightforward to show that the mapping 10 is in general, we investigate this question numerically.
Therefore, we construct sets of densities and kinetic-energy densities n , T by solving the interacting problem specified by the Hamiltonian given in eq 2 with a site-independent hopping, which corresponds to the usual Hubbard Hamiltonian , and for every set we determine the potentials v , t of the noninteracting Hamiltonian specified in eq 14 , which yields the target densities n , T.
To determine these potentials, we set up an inversion scheme by using the EOM for the density and the kinetic-energy density, respectively. These provide not only physical relations that connect the quantities v , t with n , T , but they are also suitable to define correlation potentials, as we will explain in the following. Note that in principle the inversion can be done with other techniques, which are used to find the exact local KS potential for a given interacting target density. For instance, in ref 56 , an iteration scheme is introduced that adopts the potential based on the intuition that where the density is too low the potential is made more attractive and where the density is too high it is made less attractive.
It is not so clear how to transfer this intuitive procedure to the kinetic-energy density T i , which is nonlocal, and the control field is part of the observable itself. In the continuum, one could perform an inversion and define the corresponding auxiliary potentials again by EOMs. In Appendix B , the general expressions for any number of sites can be found.
Time-dependent density-functional theory : concepts and applications
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