On the Construction of Size-Extensive Effective Hamiltonians for Time-Independent and Time-Dependent Quasi-Degenerate Systems

Abstract

The paramount importance of maintaining size-extensivity, i.e. of ensuring the correct scaling behaviour of all computed extensive properties (such as energy) with number of particles, for many-body systems is now widely appreciated [1–3]. The earliest many-body perturbative formulation of Bruckner, Goldstone and Hubbard [4] took account of the size-extensivity for the energy of closed shell many-fermion systems by proving the now famous Linked Cluster Theorem. Subsequently, such a theorem was formulated within a non-perturbative framework also [5]. The generalization encompassing open-shell systems followed later, both in the perturbative [6] and the non-perturbative [7] contexts. It became increasingly clear that size-extensivity is essential not only for computing total energies per se for open-shell systems (as is encountered in generating potential surfaces) but also for energy differences (such as ionization potential, electron affinity, excitation energy or Auger energies). What seems not to have been realized that widely yet is the fact that size-extensivity is equally important for treating time-dependent problems involving many-body systems. Size-extensive formulations would bring out the essentially localized nature of interactions and would aid in the proper description of fragmentation and combination of the reacting species as they evolve in time.