The electron propagator for non-interacting particles: An alternative approach to quasi-degenerate perturbation theory (QDPT)

TL;DR

This paper introduces an alternative formulation of quasi-degenerate perturbation theory (QDPT) using non-interacting particles, simplifying many-body techniques and deriving explicit third-order expansions for effective Hamiltonians.

Contribution

It develops a novel NIP-based formalism for QDPT, providing explicit perturbation expansions and closed-form expressions, applicable to arbitrary quantum systems beyond actual particles.

Findings

Derived third-order QDPT expansions using NIP-ADC

Constructed effective Hamiltonian and EEC matrices explicitly

Demonstrated applicability to general quantum systems

Abstract

This article presents an alternative formulation of quasi-degenerate perturbation theory (QDPT). The development results by simplifying established many-body (MB) techniques to systems of non-interacting particles (NIP). While the physical information of a NIP system is fully equivalent with that of the underlying one-particle system, NIP adaptions of MB schemes can acquire useful new roles in the one-particle context. In particular, the algebraic-diagrammatic construction (ADC) approach to the particle-detachment part of the electron propagator, turns into a new formalism of QDPT for the original one-particle system, establishing a diagrammatic perturbation theory for the matrix elements of the effective hamiltonian and the associated effective eigenvector coefficients (EEC). Using the NIP-ADC procedure explicit QDPT expansions are derived through third order. Alternatively, the intermediate state representation (ISR), underlying the ADC approach, allows one to construct the effective secular quantities directly from successive orders in the PT expansions of the NIP ground state and ground-state energy. Moreover, exact closed-form expressions can be derived for the effective hamiltonian and the EEC matrix in terms of 'target state' eigenvectors and eigenvalues of the one-particle system. These results are fully consistent with corresponding expressions derived within the QDPT context. The NIP approach to QDPT is not confined to actual particles but applies to arbitrary quantum systems. Any quantum system can be seen as formally constituting a "hyper-particle" giving rise to a system of non-interacting hyper-particles. The NIP adaptions of the biorthogonal coupled-cluster (BCC) and unitary coupled-cluster (UCC) methods are briefly discussed as well.