Large strongly correlated systems provide a challenge to modern electronic structure methods, because standard density functionals usually fail and traditional quantum chemical approaches are too demanding. The density-matrix renormalization group method, an extremely powerful tool for solving such systems, has recently been extended to handle long-range interactions on real-space grids, but is most efficient in one dimension where it can provide essentially arbitrary accuracy. Such 1d systems therefore provide a theoretical laboratory for studying strong correlation and developing density functional approximations to handle strong correlation, if they mimic three-dimensional reality sufficiently closely. We demonstrate that this is the case, and provide reference data for exact and standard approximate methods, for future use in this area.

The purpose of this short essay is to introduce students and other newcomers to the basic ideas and uses of modern electronic density functional theory, including what kinds of approximations are in current use, and how well they work (or not). The complete newcomer should find it orients them well, while even longtime users and aficionados might find Electronic Structure ProblemFor the present purposes, we define the modern electronic structure problem as finding the ground-state energy of nonrelativistic electrons for arbitrary positions of nuclei within the Born-Oppenheimer approximation.[1] If this can be done sufficiently accurately and rapidly on a modern computer, many properties can be predicted, such as bond energies and bond lengths of molecules, and lattice structures and parameters of solids. Consider a diatomic molecule, whose binding energy curve is illustrated in Figure 1. The binding energy is given bywhere E 0 (R) is the ground-state energy of the electrons with nuclei separated by R, and E A and Z A are the atomic energy and charge of atom A and similarly for B. The minimum tells us the bond length (R 0 ) and the well-depth (D e ), corrected by zero-point energy ( hx=2), gives us the dissociation energy (D 0 ). The Hamiltonian for the N electrons iŝwhere the kinetic energy operator iŝthe electron-electron repulsion operator isV ee ¼ 1 2and the one-body operator iŝFor instance, in a diatomic molecule, v(r) ¼ ÀZ A /r À Z B /|r À R|. We use atomic units unless otherwise stated, setting e 2 ¼ h ¼ m e ¼ 1, so energies are in Hartrees (1 Ha ¼ 27.2 eV or 628 kcal/mol) and distances in Bohr radii (1 a 0 ¼ 0.529 Å ). The ground-state energy satisfies the variational principle:where the minimization is over all antisymmetric N-particle wavefunctions. This E was called E 0 (R) in Eq. (1).* Many traditional approaches to solving this difficult manybody problem begin with the Hartree-Fock (HF) approximation, in which W is approximated by a single Slater determinant (an antisymmetrized product) of orbitals (single-particle wavefunctions)[2] and the energy is minimized.[3] These include configuration interaction, coupled cluster, and Møller-Plesset perturbation theory, and are mostly used for finite systems, such as molecules in the gas phase.

Using the dual Kantorovich formulation, we compute the strictly correlated electrons (SCE) functional (corresponding to the exact strong-interaction limit of density functional theory) for the hydrogen molecule along the dissociation curve. We use an exact relation between the Kantorovich potential and the optimal map to compute the comotion function, exploring corrections based on it. In particular, we analyze how the SCE functional transforms in an exact way the electron-electron distance into a one-body quantity, a feature that can be exploited to build new approximate functionals. We also show that the dual Kantorovich formulation provides in a natural way the constant in the Kohn-Sham potential recently introduced by Levy and Zahariev [Phys. Rev. Lett. 2014, 113, 113002] for finite systems.

We investigate the construction of approximated exchange-correlation functionals by interpolating locally along the adiabatic connection between the weak- and the strong-coupling regimes, focussing on the effect of using approximate functionals for the strong-coupling energy densities. The gauge problem is avoided by dealing with quantities that are all locally defined in the same way. Using exact ingredients at weak coupling we are able to isolate the error coming from the approximations at strong coupling only. We find that the nonlocal radius model, which retains some of the non-locality of the exact strong-coupling regime, yields very satisfactory results. We also use interpolation models and quantities from the weak- and strong-coupling regimes to define a correlation-type indicator and a lower bound to the exact exchange-correlation energy. Open problems, related to the nature of the local and global slope of the adiabatic connection at weak coupling, are also discussed.

We extend the density matrix renormalization group to compute exact ground states of continuum many-electron systems in one dimension with long-range interactions. We find the exact ground state of a chain of 100 strongly correlated artificial hydrogen atoms. The method can be used to simulate 1d cold atom systems and to study density functional theory in an exact setting. To illustrate, we find an interacting, extended system which is an insulator but whose Kohn-Sham system is metallic.PACS numbers: 71.15. Dx, 05.10.Cc, 71.15.Mb, 31.15.EFor electronic structure calculations, these are the best of times and the worst of times. When correlations are weak, density functional theory (DFT) makes it possible to tackle extremely realistic Hamiltonians and large system sizes with reasonable accuracy [1]. For strongly correlated systems, there exist powerful and controllable numerical methods [2] for simulating lattice Hamiltonians, such as the Hubbard model. However, few numerical tools can treat the combination of strongly correlated electronic systems and realistic microscopic Hamiltonians. In the strongly correlated regime, DFT approximations are neither systematic nor controllable, often leading to unrestrained parameter multiplication and empiricism. Model Hamiltonians rely on the arbitrary truncation of terms that may be crucial in tipping the balance between competing phases. Attempts to bridge the gap between realistic Hamiltonians and strong correlation techniques, such as dynamical mean field theory coupled to DFT [3,4], may contain both arbitrary truncations and a less than ideal treatment of correlations.Therefore we would like to study DFT in an exact setting to see how density functional approximations break down and whether new approximations contain the right physics. But very few continuum, three-dimensional, long-range interacting systems can be easily treated exactly. Here, we show that by studying one dimensional (1d) systems instead, we can treat realistic Hamiltonians and strong electron correlations essentially exactly, even for a very large number of atoms. Because they preserve the continuum, our 1d models mimic key features of three-dimensional reality surprisingly well [5].Our approach is based on the density matrix renormalization group (DMRG) [6], the most powerful of the strongly correlated techniques for 1d lattice models. Here we extend DMRG to treat continuum electron systems with long-range interactions. This new approach retains DMRG's exponential convergence and near linear scaling with system size. As an example, we present a near exact calculation of a system with 100 strongly interacting pseudo-hydrogen atoms (Fig. 1).A key motivation for this method is to study DFT in an exact setting, both when correlations are strong and near the thermodynamic limit. Generically, 1d systems have strong quantum fluctuations, making them an especially rigorous test of DFT approximations; they can also be pushed to large size with less effort. As in Fig. 1, we can easily compare various DFT approx...