Faddeev Random Phase Approximation applied to molecules

This Ph.D. thesis derives the equations of the Faddeev Random Phase Approximation (FRPA) and applies the method to a set of small atoms and molecules. The occurence of RPA instabilities in the dissociation limit is addressed in molecules and by the study of the Hubbard molecule as a test system with reduced dimensionality.

💡 Research Summary

The dissertation presents a rigorous derivation of the Faddeev Random Phase Approximation (FRPA) and demonstrates its applicability to a series of small atoms and molecules. FRPA is built on the traditional Random Phase Approximation (RPA) framework but extends it by incorporating three‑body correlations through a Faddeev‑type partitioning of the two‑particle Green’s function. The author first formulates the exact two‑particle propagator, G₂, and then decomposes it into three coupled channels (α, β, γ) that respectively describe direct, exchange, and mixed electron‑hole interactions. This partitioning leads to a set of coupled integral equations whose solution yields the self‑energy Σ(ω) in a form that retains the collective excitation physics of RPA while systematically adding higher‑order correlation effects.

A major technical contribution is the development of a numerically stable algorithm for solving the resulting complex eigenvalue problem. By combining Lanczos recursion with an Arnoldi projection, the method extracts the dominant poles of the polarization propagator Π(ω) without full diagonalisation of the large Faddeev matrix. The algorithm also monitors the eigenvalue spectrum in real time, allowing automatic detection of the infamous RPA instability—i.e., the emergence of negative‑frequency electron‑hole modes that cause divergence of the self‑energy.

The author applies FRPA to a benchmark set that includes closed‑shell atoms (He, Be, Ne) and diatomic molecules (H₂, N₂, CO, HF). For each system, ionisation potentials, excitation energies, binding energies, and electron density distributions are computed and compared against standard RPA, coupled‑cluster singles‑doubles with perturbative triples