Supporting information for "Parquet theory for molecular systems: Formalism and static kernel parquet approximation"

This document contains the detailed derivations of the equations reported in the associated main manuscript, where missing notations can be found. The first three sections set the stage for the parquet formalism that is discussed in the remaining sections. Section I defines the time and Fourier transform conventions that are used in this work. The electron-hole (eh) and the particle-particle (pp) Bethe-Salpeter equations (BSEs) are presented in detail in Secs. II and III, respectively. In particular, these include their transformation to frequency space, their solution in a basis of spin-orbitals, as well as their spin adaptation. The presentation of parquet formalism starts with the Fourier transform of all its equations performed in Sec. IV. These equations are then projected in a basis of spin orbitals in Sec. V. Finally, Sec. VI reports the spin-adapted parquet equations.