Numerical results for ground state and excited state properties (energies, double occupancies, and Matsubara-axis self energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods. arXiv:1505.02290v2 [cond-mat.str-el] 15

Density functional theory incorporating hybrid exchange–correlation functionals has been extraordinarily successful in providing accurate, computationally tractable treatments of molecular properties. However, conventional hybrid functionals can be problematic for solids. Their nonlocal, Hartree–Fock-like exchange term decays slowly and incorporates unphysical features in metals and narrow-bandgap semiconductors. This article provides an overview of our group’s work on designing hybrid functionals for solids. We focus on the Heyd–Scuseria–Ernzerhof screened hybrid functional [J. Chem. Phys. 2003, 118, 8207], its applications to the chemistry and physics of solids and surfaces, and our efforts to build upon its successes.

Projected Hartree-Fock (PHF) theory has a long history in quantum chemistry. PHF is here understood as the variational determination of an N-electron broken symmetry Slater determinant that minimizes the energy of a projected state with the correct quantum numbers. The method was actively pursued for several decades but seems to have been abandoned. We here derive and implement a "variation after projection" PHF theory using techniques different from those previously employed in quantum chemistry. Our PHF methodology has modest mean-field computational cost, yields relatively simple expressions, can be applied to both collinear and non-collinear spin cases, and can be used in conjunction with deliberate symmetry breaking and restoration of other molecular symmetries like complex conjugation and point group. We present several benchmark applications to dissociation curves and singlet-triplet energy splittings, showing that the resulting PHF wavefunctions are of high quality multireference character. We also provide numerical evidence that in the thermodynamic limit, the energy in PHF is not lower than that of broken-symmetry HF, a simple consequence of the lack of size consistency and extensivity of PHF.

We present an analytic proof demonstrating the equivalence between the Random Phase Approximation (RPA) to the ground state correlation energy and a ring-diagram simplification of the Coupled Cluster Doubles (CCD) equations. In the CCD framework, the RPA equations can be solved in O(N 4 ) computational effort, where N is proportional to the number of basis functions.There has recently been a revival of interest in RPA in the quantum chemistry community. The RPA is popular for calculations of excitation energies both in finite systems 1,2 and in solids, 3,4 and is related to time-dependent density functional theory. 5,6,7 As a technique for describing electronic correlations, RPA has significant advantages, particularly for those interested in density functional theory. It describes dispersion and van der Waals interactions correctly, 8,9 and is exact for long-range correlations. 10 Left-right static correlations seem to be properly described by RPA, 11 and RPA fixes the pathologies of nonlocal Hartree-Fock-type exchange in metallic systems. Readers interested in details about RPA for ground state correlation can refer to the recent paper by Furche 12 where he discusses an interesting simplification to reduce the computational cost of RPA correlation and provides ample background information about RPA. Note that his work focuses on direct RPA, in which the exchange terms are neglected; as discussed later in this communication, this is the form of RPA most useful in the context of density functional theory.A connection between the RPA correlation energy and a ring diagram approximation to CCD was first mentioned by Freeman in his 1977 paper. 13 Very recently, A. Grüneis and G. Kresse reproduced this evidence and found numerical proof of the equivalence between these two approaches. 14 Here, we offer an analytic proof that these two problems yield identical correlation energies. To the best of our knowledge, no such formal proof has been given before.As a method for calculating electronic excitation spectra, RPA requires the solution ofThe matrices A, B, X, and Y are all ov × ov, where o and v are respectively the number of occupied and unoccupied spin-orbitals. The eigenvalue problem above can be completed by noting that if Xi Yi is an eigenvector with eigenvalue ω i , then Yi Xi is also an eigenvector, with eigenvalue −ω i . In the (real) canonical spin-orbital basis we use throughout this letter, we haveHere, ǫ p is a diagonal element of the Fock operator. Indices i, j, k, and l indicate occupied spin-orbitals, while a, b, c, d indicate unoccupied spin-orbitals. For arbitrary spin-orbitals p, q, r, and s, the two-electron integral pq rs is defined bywhere x is a combined space and spin electron coordinate. The RPA correlation energy can be obtained by considering two harmonic excitation energy problems: 12,15 RPA and the Tamm-Dancoff approximation (TDA) thereto, which sets B = 0 and thus solvesIn the quantum chemistry community, TDA is also known as configuration interaction singles (CIS). While TDA includes onl...

The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. However, the gap in the band structure of the exact multiplicative Kohn-Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS densityfunctional theory. Here, we give a simple proof of a theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approximate functional if the GKS potential operator is continuous and the density change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from metageneralized gradient approximations (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen molecules, solid aluminum arsenide, and solid argon provide numerical illustrations. T he most basic property of a periodic solid is its fundamental energy gap G, which vanishes for a metal but is positive for semiconductors and other insulators. G dominates many properties. As the unbound limit of an exciton series, G is an excitation energy of the neutral solid, but it is defined here as a difference of ground-state energies: If EðMÞ is the ground-state energy for a solid with a fixed number of nuclei and M electrons, and if M = N for electrical neutrality, thenis the difference between the first ionization energy IðNÞ and the first electron affinity AðNÞ of the neutral solid. Whereas I and A can be measured for a macroscopic solid, they can be computed directly (as ground-state energy differences) either by starting from finite clusters and extrapolating to infinite cluster size or (for I-A) by starting from a finite number of primitive unit cells, with periodic boundary condition on the surface of this finite collection, and extrapolating to an infinite number. Here we shall follow both approaches, which have been discussed in a recent study (1). (The energy to remove an electron to infinite separation cannot depend upon the crystal face through which it is removed, although the energy to remove an electron to a macroscopic separation, but much smaller than the dimensions of that face, may so depend. The gap is of course a bulk property.) Band-Gap Problem in Kohn-Sham Density-Functional TheoryKohn-Sham density-functional theory (2, 3) is a formally exact way to compute the ground-state energy and electron density of M interacting electrons in a multiplicative external potential. This theory sets up a fictitious system of noninteracting electrons with the same ground-state density as the real interacting system, found by solving self-consistent one-electron Schrödinger equations. These electrons move in a multiplicative effective Kohn-Sham (KS) potential, the sum of the external and Hartree potentials and the derivative of the density functional for...