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Static correlation beyond the random phase approximation:Dissociating H2 with the Bethe-Salpeter equation and time-dependent GW

Thomas Olsen, Kristian Sommer Thygesen

Published Jan 1, 2014• 35 citations
OpenAlex

Abstract

We investigate various approximations to the correlation energy of a H$_2$ molecule in the dissociation limit, where the ground state is poorly described by a single Slater determinant. The correlation energies are derived from the density response function and it is shown that response functions derived from Hedin's equations (Random Phase Approximation (RPA), Time-dependent Hartree-Fock (TDHF), Bethe-Salpeter equation (BSE), and Time-Dependent GW (TDGW)) all reproduce the correct dissociation limit. We also show that the BSE improves the correlation energies obtained within RPA and TDHF significantly for intermediate binding distances. A Hubbard model for the dimer allow us to obtain exact analytical results for the various approximations, which is readily compared with the exact diagonalization of the model. Moreover, the model is shown to reproduce all the qualitative results from the \textit{ab initio} calculations and confirms that BSE greatly improves the RPA and TDHF results despite the fact that the BSE excitation spectrum breaks down in the dissociation limit. In contrast, Second Order Screened Exchange (SOSEX) gives a poor description of the dissociation limit, which can be attributed to the fact that it cannot be derived from an irreducible response function.

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We investigate various approximations to the correlation energy of a H2_22​ molecule in the dissociation limit, where the ground state is poorly described by a single Slater determinant. The correlation energies are derived from the density response function and it is shown that response functions derived from Hedin's equations (Random Phase Approximation (RPA), Time-dependent Hartree-Fock (TDHF), Bethe-Salpeter equation (BSE), and Time-Dependent GW (TDGW)) all reproduce the correct dissociation limit. We also show that the BSE improves the correlation energies obtained within RPA and TDHF significantly for intermediate binding distances. A Hubbard model for the dimer allow us to obtain exact analytical results for the various approximations, which is readily compared with the exact diagonalization of the model. Moreover, the model is shown to reproduce all the qualitative results from the \textit{ab initio} calculations and confirms that BSE greatly improves the RPA and TDHF results despite the fact that the BSE excitation spectrum breaks down in the dissociation limit. In contrast, Second Order Screened Exchange (SOSEX) gives a poor description of the dissociation limit, which can be attributed to the fact that it cannot be derived from an irreducible response function.
We investigate various approximations to the correlation energy of a H2 molecule in the dissociation limit, where the ground state is poorly described by a single Slater determinant. The correlation energies are derived from the density response function and it is shown that response functions derived from Hedin's equations (Random Phase Approximation (RPA), Time-dependent Hartree-Fock (TDHF), Bethe-Salpeter equation (BSE), and Time-Dependent GW) all reproduce the correct dissociation limit. We also show that the BSE improves the correlation energies obtained within RPA and TDHF significantly for intermediate binding distances. A Hubbard model for the dimer allows us to obtain exact analytical results for the various approximations, which is readily compared with the exact diagonalization of the model. Moreover, the model is shown to reproduce all the qualitative results from the ab initio calculations and confirms that BSE greatly improves the RPA and TDHF results despite the fact that the BSE excitation spectrum breaks down in the dissociation limit. In contrast, second order screened exchange gives a poor description of the dissociation limit, which can be attributed to the fact that it cannot be derived from an irreducible response function.
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function LLL. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential WWW in the BSE for LLL leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
The helium atom is the simplest many-body electronic system provided by nature. The exact solution to the Schrödinger equation is known for helium ground and excited states, and it represents a benchmark for any many-body methodology. Here, we check the ab initio many-body GW approximation and the Bethe-Salpeter equation (BSE) against the exact solution for helium. Starting from the Hartree-Fock method, we show that the GW and the BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving the time-dependent Hartree-Fock method. These findings suggest that the accuracy of the BSE and GW approximations is not significantly limited by self-interaction and self-screening problems even in this few electron limit. We further discuss our results in comparison to those obtained by time-dependent density-functional theory.
We develop a time-dependent Gutzwiller approximation (GA) for the Hubbard model analogous to the time-dependent Hartree-Fock (HF) method. This new formalism incorporates ground state correlations of the random phase approximation (RPA) type beyond the GA. Static quantities like ground state energy and double occupancy are in excellent agreement with exact results in one dimension up to moderate coupling and in two dimensions for all couplings. We find a substantial improvement over traditional GA and HF+RPA treatments. Dynamical correlation functions can be computed and are also substantially better than HF+RPA ones and obey well behaved sum rules.
The random phase approximation (RPA) builds in correlations left out by mean-field theory. In full $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces we calculate the Hartree-Fock (HF)+RPA(\mathrm{HF})+\mathrm{RPA}(HF)+RPA binding energy, and compare it to exact diagonalization. We find that often, but not always, the HF+RPA\mathrm{HF}+\mathrm{RPA}HF+RPA gives a good approximation to the ``exact'' ground state energy. In those cases where the RPA is less satisfactory, however, there is no obvious correlation with properties of the HF state, such as deformation or overlap with the exact ground state wave function. This weakens the reliability of the RPA for computing binding energies.
The ground-state energy and static and dynamic correlation functions are investigated in the inhomogeneous-Hartree-Fock (IHF) plus random-phase-approximation (RPA) approach applied to a one-dimensional spinless-fermion model showing self-trapped doping states at the mean-field level. Results are compared with those obtained using homogeneous HF and exact diagonalization. RPA fluctuations added to the generally inhomogeneous-HF ground state allow the computation of dynamical correlation functions that compare well with exact diagonalization results. The RPA correction to the ground-state energy agrees well with the exact results at strong- and weak-coupling limits. We also compare it with a related quasiboson approach. The instability towards self-trapped behavior is signaled by a RPA mode with frequency approaching zero.