We report an implementation of self-consistent Green's function many-body theory within a second-order approximation (GF2) for application with molecular systems. This is done by iterative solution of the Dyson equation expressed in matrix form in an atomic orbital basis, where the Green's function and self-energy are built on the imaginary frequency and imaginary time domain, respectively, and fast Fourier transform is used to efficiently transform these quantities as needed. We apply this method to several archetypical examples of strong correlation, such as a H32 finite lattice that displays a highly multireference electronic ground state even at equilibrium lattice spacing. In all cases, GF2 gives a physically meaningful description of the metal to insulator transition in these systems, without resorting to spin-symmetry breaking. Our results show that self-consistent Green's function many-body theory offers a viable route to describing strong correlations while remaining within a computationally tractable single-particle formalism.

The temperature-dependent Matsubara Green's function that is used to describe temperaturedependent behavior is expressed on a numerical grid. While such a grid usually has a couple of hundred points for low-energy model systems, for realistic systems in large basis sets the size of an accurate grid can be tens of thousands of points, constituting a severe computational and memory bottleneck. In this paper, we determine efficient imaginary time grids for the temperature-dependent Matsubara Green's function formalism that can be used for calculations on realistic systems. We show that due to the use of orthogonal polynomial transform, we can restrict the imaginary time grid to few hundred points and reach micro-Hartree accuracy in the electronic energy evaluation. Moreover, we show that only a limited number of orthogonal polynomial expansion coefficients are necessary to preserve accuracy when working with a dual representation of Green's function or self-energy and transforming between the imaginary time and Matsubara frequency domain.

We examine fractional charge and spin errors in self-consistent Green's function theory within a second-order approximation (GF2). For GF2, it is known that the summation of diagrams resulting from the self-consistent solution of the Dyson equation removes the divergences pathological to second-order Møller-Plesset (MP2) theory for strong correlations. In the language often used in density functional theory contexts, this means GF2 has a greatly reduced fractional spin error relative to MP2. The natural question then is what effect, if any, does the Dyson summation have on the fractional charge error in GF2? To this end, we generalize our previous implementation of GF2 to open-shell systems and analyze its fractional spin and charge errors. We find that like MP2, GF2 possesses only a very small fractional charge error, and consequently minimal many electron self-interaction error. This shows that GF2 improves on the critical failings of MP2, but without altering the positive features that make it desirable. Furthermore, we find that GF2 has both less fractional charge and fractional spin errors than typical hybrid density functionals as well as random phase approximation with exchange.

We assess the dependence of magnetic exchange couplings on the variation of Hartree-Fock exchange (HFX) admixture in global hybrid functionals and the range-separation parameter ω in range-separated hybrid functionals in a set of 12 spin-1/2 binuclear transition metal complexes. The global hybrid PBEh (hybrid Perdew-Burke-Ernzerhof) and range-separated hybrids HSE (Heyd-Scuseria-Ernzerhof) and LC-ωPBE (long-range corrected hybrid PBE) are employed for this assessment, and exchange couplings are calculated from energy differences within the framework of the spin-projected approach. It is found that these functionals perform optimally for magnetic exchange couplings with 35% HFX admixture for PBEh, ω = 0.50 a.u.(-1) for LC-ωPBE, and ω at or near 0.0 a.u.(-1) for HSE (which corresponds to PBEh). We find that in their standard respective forms, LC-ωPBE slightly outperforms PBEh, while PBEh with 35% HFX yields exchange couplings closer to experiment than those of LC-ωPBE with ω = 0.50 a.u.(-1). Additionally, we show that the profile of exchange couplings with respect to ω in HSE is appreciably flat from 0 to 0.2 a.u.(-1). This combined with the fact that HSE is computationally more tractable than global hybrids makes HSE an attractive alternative for the evaluation of exchange couplings in extended systems. These results are rationalized with respect to how varying the parameters within these functionals affects the delocalization of the magnetic orbitals, and conclusions are made regarding the relative importance of range separation versus global mixing of HFX for the calculation of exchange couplings.

Embedding calculations that find approximate solutions to the Schrödinger equation for large molecules and realistic solids are performed commonly in a three step procedure involving (i) construction of a model system with effective interactions approximating the low energy physics of the initial realistic system, (ii) mapping the model system onto an impurity Hamiltonian, and (iii) solving the impurity problem. We have developed a novel procedure for parametrizing the impurity Hamiltonian that avoids the mathematically uncontrolled step of constructing the low energy model system. Instead, the impurity Hamiltonian is immediately parametrized to recover the self-energy of the realistic system in the limit of high frequencies or short time. The effective interactions parametrizing the fictitious impurity Hamiltonian are local to the embedded regions, and include all the non-local interactions present in the original realistic Hamiltonian in an implicit way. We show that this impurity Hamiltonian can lead to excellent total energies and self-energies that approximate the quantities of the initial realistic system very well. Moreover, we show that as long as the effective impurity Hamiltonian parametrization is designed to recover the self-energy of the initial realistic system for high frequencies, we can expect a good total energy and self-energy. Finally, we propose two practical ways of evaluating effective integrals for parametrizing impurity models.