Bond breaking and bond formation: how electron correlation is captured in many-body perturbation theory and density-functional theory.

For the paradigmatic case of H(2) dissociation, we compare state-of-the-art many-body perturbation theory in the GW approximation and density-functional theory in the exact-exchange plus random-phase approximation (RPA) for the correlation energy. For an unbiased comparison and to prevent spurious starting point effects, both approaches are iterated to full self-consistency (i.e., sc-RPA and sc-GW). The exchange-correlation diagrams in both approaches are topologically identical, but in sc-RPA they are evaluated with noninteracting and in sc-GW with interacting Green functions. This has a profound consequence for the dissociation region, where sc-RPA is superior to sc-GW. We argue that for a given diagrammatic expansion, sc-RPA outperforms sc-GW when it comes to bond breaking. We attribute this to the difference in the correlation energy rather than the treatment of the kinetic energy.

Correlation potentials for molecular bond dissociation within the self-consistent random phase approximation.

Self-consistent correlation potentials for H(2) and LiH for various inter-atomic separations are obtained within the random phase approximation (RPA) of density functional theory. The RPA correlation potential shows a peak at the bond midpoint, which is an exact feature of the true correlation potential, but lacks another exact feature: the step important to preserve integer charge on the atomic fragments in the dissociation limit. An analysis of the RPA energy functional in terms of fractional charge is given which confirms these observations. We find that the RPA misses the derivative discontinuity at odd integer particle numbers but explicitly eliminates the fractional spin error in the exact-exchange functional. The latter finding explains the improved total energy in the dissociation limit.

Open-shell reduced density matrix functional theory.

Open-shell reduced density matrix functional theory is established by investigating the domain of the exact functional. For spin states that are the ground state, a particularly simple set is found to be the domain. It cannot be generalized to other spin states. A number of conditions satisfied by the exact density matrix functional is formulated and tested for approximate functionals. The exact functional does not suffer from fractional spin error, which is the source of the static correlation error in dissociated molecules. We prove that a simple approximation (called the Buijse-Baerends functional, Müller or square root functional) has a non-positive fractional spin error. In the case of the H atom the error is zero. Numerical results for a few atoms are given for approximate density and density matrix functionals as well as a recently developed range-separated combination of both.

A new scheme to calculate isotope effects.

We present a new scheme to calculate isotope effects. Only selected frequencies at the target level of theory are calculated. The frequencies are selected by an analysis of the Hessian from a lower level of theory. We obtain accurate isotope effects without calculating the full Hessian at the target level of theory. The calculated frequencies are very accurate. The scheme converges to the correct isotope effect.

A density matrix functional with occupation number driven treatment of dynamical and nondynamical correlation.

A recently proposed series of corrections to the earliest JK-only functionals has considerably improved the prospects of density matrix functional theory (DMFT). Still, the most advanced of these functionals (correction C3) requires a preselection of the terms in the pair density Gamma(r(1),r(2)) involving the bonding and antibonding natural orbitals (NOs) belonging to an electron pair bond. Ideally, a DMFT functional should only depend on the NOs and their occupation numbers, and we propose a functional with an occupation number driven weighing of terms in the pair density. These are formulated as "damping" for certain ranges of occupation numbers of the two-electron cumulant that arises in the expansion of the two-particle density matrix of the paradigmatic two-electron system. This automatic version of C3, which we denote AC3, provides the correct dissociation limit for electron pair bonds and it excellently reproduces the potential energy curves of the multireference configuration interaction (MRCI) method for the dissociation of the electron pair bond in the series of the ten-electron hydrides CH(4), NH(3), H(2)O, and HF. AC3 reproduces closely the experimental equilibrium distances and at R(e) it yields correlation energies of the ten-electron systems with an average error in the absolute values of only 3.3% compared to the MRCI values. We stress the importance of treatment of strong correlation cases (NO occupation numbers differing significantly from 2.0 and 0.0) by appropriate terms in the cumulant.