Theory

Green implements many-body methods based on the language of Green’s functions for the simulation of realistic molecules and solids. The following pages introduce the theoretical background behind the methods available in the package.

Introduction

An introduction to the language of Green’s functions, the many-body Hamiltonian Green solves, and pointers to textbooks and review articles for readers new to diagrammatic perturbation theory.

Hartree-Fock and Starting Points

Hartree-Fock theory, the static Hartree and exchange contributions, their relation to $\Sigma^\infty$, and how HF or DFT calculations provide starting density matrices for GF2 and GW.

Matsubara Basis

Compact representations of imaginary-time and Matsubara-frequency functions, including Legendre, spline, Chebyshev, intermediate-representation, sparse-sampling, and DLR grids.

Green’s Functions

Imaginary-time Green’s functions, Matsubara-frequency transforms, spectral functions, self-energies, the grand potential, and energy expressions used by self-consistent many-body methods.

GF2

Self-consistent second-order perturbation theory (GF2), also known as Second Order Born — a conserving diagrammatic approximation that is accurate whenever gaps are large and interactions are weak, though it is known to fail for metals.

GW

The fully self-consistent $GW$ approximation, implementing Hedin’s equations with full frequency dependence and self-consistency on the imaginary axis, making the solution thermodynamically consistent and conserving.

Analytical Continuation

Theory on the analytic continuation methods used to obtain the spectral representation of a Green’s function from imaginary-time or Matsubara-frequency data.

Thermodynamics

Self-consistent finite $\Phi$-derivable theories give direct access to thermodynamic properties such as entropy, free energy, and specific heat.

Convergence

Theory behind the convergence acceleration techniques available in Green, such as direct inversion in the iterative subspace (DIIS), for stabilizing and speeding up self-consistent iterations.