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Register a new userLies My Teacher Told Me About Density Functional Theory: Seeing Through Them with the Hubbard Dimer
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Most realistic calculations of moderately correlated materials begin with a ground-state density functional theory (DFT) calculation. While Kohn-Sham DFT is used in about 40,000 scientific papers each year, the fundamental underpinnings are not widely appreciated. In this chapter, we analyze the inherent characteristics of DFT in their simplest form, using the asymmetric Hubbard dimer as an illustrative model. We begin by working through the core tenets of DFT, explaining what the exact ground-state density functional yields and does not yield. Given the relative simplicity of the system, almost all properties of the exact exchange-correlation functional are readily visualized and plotted. Key concepts include the Kohn-Sham scheme, the behavior of the XC potential as correlations become very strong, the derivative discontinuity and the difference between KS gaps and true charge gaps, and how to extract optical excitations using time-dependent DFT. By the end of this text and accompanying exercises, the reader will improve their ability to both explain and visualize the concepts of DFT, as well as better understand where others may go wrong.
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We present in full detail a newly developed formalism enabling density functional perturbation theory (DFPT) calculations from a DFT+$U$ ground state. The implementation includes ultrasoft pseudopotentials and is valid for both insulating and metallic systems. It aims at fully exploiting the versatility of DFPT combined with the low-cost DFT+$U$ functional. This allows to avoid computationally intensive frozen-phonon calculations when DFT+$U$ is used to eliminate the residual electronic self-interaction from approximate functionals and to capture the localization of valence electrons e.g. on $d$ or $f$ states. In this way, the effects of electronic localization (possibly due to correlations) are consistently taken into account in the calculation of specific phonon modes, Born effective charges, dielectric tensors and in quantities requiring well converged sums over many phonon frequencies, as phonon density of states and free energies. The new computational tool is applied to two representative systems, namely CoO, a prototypical transition metal monoxide and LiCoO$_2$, a material employed for the cathode of Li-ion batteries. The results show the effectiveness of our formalism to capture in a quantitatively reliable way the vibrational properties of systems with localized valence electrons.
We propose a lattice density-functional theory for {it ab initio} quantum chemistry or physics as a route to an efficient approach that approximates the full configuration interaction energy and orbital occupations for molecules with strongly-correlated electrons. We build on lattice density-functional theory for the Hubbard model by deriving Kohn-Sham equations for a reduced then full quantum chemistry Hamiltonian, and demonstrate the method on the potential energy curves for the challenging problem of modelling elongating bonds in a linear chain of six hydrogen atoms. Here the accuracy of the Bethe-ansatz local-density approximation is tested for this quantum chemistry system and we find that, despite this approximate functional being designed for the Hubbard model, the shapes of the potential curves generally agree with the full configuration interaction results. Although there is a discrepancy for very stretched bonds, this is lower than when using standard density-functional theory with the local-density approximation.
Quantum embedding based on the (one-electron reduced) density matrix is revisited by means of the unitary Householder transformation. While being exact and equivalent to (but formally simpler than) density matrix embedding theory (DMET) in the non-interacting case, the resulting Householder transformed density matrix functional embedding theory (Ht-DMFET) preserves, by construction, the single-particle character of the bath when electron correlation is introduced. In Ht-DMFET, the projected impurity+bath clusters Hamiltonian (from which approximate local properties of the interacting lattice can be extracted) becomes an explicit functional of the density matrix. In the spirit of single-impurity DMET, we consider in this work a closed (two-electron) cluster constructed from the full-size non-interacting density matrix. When the (Householder transformed) interaction on the bath site is taken into account, per-site energies obtained for the half-filled one-dimensional Hubbard lattice match almost perfectly the exact Bethe Ansatz results in all correlation regimes. In the strongly correlated regime, the results deteriorate away from half-filling. This can be related to the electron number fluctuations in the (two-site) cluster which are not described neither in Ht-DMFET nor in regular DMET. As expected, the per-site energies dramatically improve when increasing the number of embedded impurities. Formal connections with density/density matrix functional theories have been briefly discussed and should be explored further. Work is currently in progress in this direction.
The solution of complex many-body lattice models can often be found by defining an energy functional of the relevant density of the problem. For instance, in the case of the Hubbard model the spin-resolved site occupation is enough to describe the system total energy. Similarly to standard density functional theory, however, the exact functional is unknown and suitable approximations need to be formulated. By using a deep-learning neural network trained on exact-diagonalization results we demonstrate that one can construct an exact functional for the Hubbard model. In particular, we show that the neural network returns a ground-state energy numerically indistinguishable from that obtained by exact diagonalization and, most importantly, that the functional satisfies the two Hohenberg-Kohn theorems: for a given ground-state density it yields the external potential and it is fully variational in the site occupation.
A curious behavior of electron correlation energy is explored. Namely, the correlation energy is the energy that tends to drive the system toward that of the uniform electron gas. As such, the energy assumes its maximum value when a gradient of density is zero. As the gradient increases, the energy is diminished by a gradient suppressing factor, designed to attenuate the energy from its maximum value similar to the shape of a bell curve. Based on this behavior, we constructed a very simple mathematical formula that predicted the correlation energy of atoms and molecules. Combined with our proposed exchange energy functional, we calculated the correlation energies, the total energies, and the ionization energies of test atoms and molecules; and despite the unique simplicities, the functionals accuracies are in the top tier performance, competitive to the B3LYP, BLYP, PBE, TPSS, and M11. Therefore, we propose that, as guided by the simplicities and supported by the accuracies, the correlation energy is the energy that locally tends to drive the system toward the uniform electron gas.
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