Faddeev Random Phase Approximation Applied to Molecules

In this work we have introduced a Green's function technique for the calculation of ground-state energies and ionization energies in quantum many-body systems. The elementary building blocks of the theory are the RPA (random phase approximation) excitations. These are used to construct an approximation for the self-energy in a specific manner, called the FRPA (Faddeev RPA). The FRPA holds the promise to be a Green's function method with a wide applicability, from finite systems like atoms and molecules to extended systems like the uniform electron gas and nuclear matter. The fully self-consistent FRPA is conserving in the Baym-Kadanoff sense and consequently obeys important conservation laws. In Chapter 2 we will present all the theoretical tools that are needed to construct the FRPA. The definition of the single-particle Green's function. The Dyson equation will be given in the form that requires the irreducible 2p1h/2h1p propagator. After this the definition of the polarization propagator and two-particle Green's function will be given. These propagators will be used in a suitable approximation called the RPA. Chapter 3 will deal with the derivation of the FRPA mechanism. For this the six-point vertex function has to be reduced from a six-time object to a two-time object. The reduction requires that the propagators are two-time quantities and that the class of diagrams is restricted. The Faddeev procedure increases the matrix dimensions by a factor of three. However, by elimination of the spurious solutions, the matrix dimension brought back to the original dimension through a simple projection. The ADC(3) (Algebraic Diagrammatic Construction method of third order) is encompassed by the FRPA as a limit case which can easily be derived. The results obtained with the previously derived method will be given in Chapter 4. We have applied the FRPA to a series of closed-shell atoms, a set of simple diatomic molecules and a schematic model. Through this schematic model we try to give an understanding of the problems that occur in the dissociation limit with the FRPA. In Chapter 5 we will present the conclusions of this work, together with an outlook on possibilities for future research