dyson

dyson: Dyson equation solvers for Green’s function methods

Dyson equation solvers in dyson are general solvers that accept a variety of inputs to represent self-energies or existing Green’s functions, and solve the Dyson equation in some fashion to obtain either

1. a static spectral representation that can be projected into a static Lehmann representation of the Green’s function or self-energy, or

2. a dynamic Green’s function.

The self-energy and Green’s function are represented in the following ways:

Representation	Description
Spectral	Eigenvalues and eigenvectors of the static self-energy supermatrix, from which the Lehmann representation of the self-energy or Green’s function can be constructed.
Lehmann	The Lehmann representation of the self-energy or Green’s function, consisting of pole energies and their couplings to a physical space.
Dynamic	The dynamic self-energy or Green’s function, represented as a series of arrays at each point on a grid of time or frequency points.

The available static solvers are, along with their expected inputs:

Solver	Inputs
Exact	Supermatrix of the static and dynamic self-energy.
Davidson	Matrix-vector operation and diagonal of the supermatrix of the static and dynamic self-energy.
Downfolded	Static self-energy and function returning the dynamic self-energy at a given frequency.
MBLSE	Static self-energy and moments of the dynamic self-energy.
MBLGF	Moments of the dynamic Green’s function.
AufbauPrinciple	Static self-energy, Lehmann representation of the dynamic self-energy, and the target number of electrons.
AuxiliaryShift	Static self-energy, Lehmann representation of the dynamic self-energy, and the target number of electrons.
DensityRelaxation	Lehmann representation of the dynamic self-energy, function returning the Fock matrix at a given density, and the target number of electrons.

For dynamic solvers, all solvers require the grid parameters, along with:

Solver	Inputs
CorrectionVector	Matrix-vector operation and diagonal of the supermatrix of the static and dynamic self-energy.
CPGF	Chebyshev polynomial moments of the dynamic Green’s function.

For a full accounting of the inputs and their types, please see the documentation for each solver.

A number of classes are provided to represent the expressions needed to construct these inputs at different levels of theory. These expressions are all implemented for RHF references, with other spin symmetries left to the user to implement as needed. The available expressions are:

Expression	Description
HF	Hartree–Fock (mean-field) ground state, exploiting Koopmans’ theorem for the excited states.
CCSD	Coupled cluster singles and doubles ground state, and the respective equation-of-motion method for the excited states.
FCI	Full configuration interaction (exact diagonalisation) ground and excited states.
ADC2	Algebraic diagrammatic construction second order excited states, based on a mean-field ground state.
ADC2x	Algebraic diagrammatic construction extended second order excited states, based on a mean-field ground state.
TDAGW	GW theory with the Tamm–Dancoff approximation for the excited states, based on a mean-field ground state.
Hamiltonian	General Hamiltonian expression, which accepts an array representing the supermatrix of the self-energy, and supports scipy.sparse matrices.

Submodules

dyson.expressions	Expressions for constructing Green's functions and self-energies.
dyson.grids	Grids for Green's functions and self-energies.
dyson.representations	Representations for Green's functions and self-energies.
dyson.solvers	Solvers for solving the Dyson equation.
dyson.util	Utility functions.