# Sigma code overview

Sigma is the second half of the GW code. It should be run after calculating epsmat[.h5]/esp0mat[.h5] with epsilon. It gives the quasiparticle self-energies and dispersion relation for quasielectron and quasihole states. The main result is written to the standard output and to the sigma_hp.log file.

## Summary of input and output files

### Required input files

• sigma.inp: Input parameters.

• epsmat[.h5]: Inverse dielectric matrix ($q\ne0$). Created using epsilon. The file has a h5 extension if the code is compiled with HDF5 support (for specification, see epsmat.h5.spec).

• eps0mat.[h5]: Inverse dielectric matrix ($q\rightarrow0$). Created using epsilon. The file has a .h5 extension if the code is compiled with HDF5 support (for specification, see epsmat.h5.spec).

• WFN_inner: Wavefunctions used to expand the Green's function $G$ in $\Sigma=iGW$, and, whenever WFN_outer is abset, file used to load the states $|n\rangle$ and $|m\rangle$ used to compute the self-energy matrix element $\langle n|\Sigma|m \rangle$. For a given k-point in WFN_inner, all other k-points connected by any q-point in epsmat[.h5] should also be in WFN_inner.

• RHO: Charge density $\rho(G)$ (only needed for generalized plasmon-pole model), from SCF calculation on which WFN_inner is based.

• vxc.dat: Matrix elements of the exchange-correlation potential, from a mean-field calculation or a previous Sigma run. VXC file, which stores the whole exchange-correlation matrix in reciprocal space, $V_{xc}(G)$, may be used instead via keyword dont_use_vxcdat.

• WFN_outer: Outer wavefunctions between which the self-energy operator and exchange-correlation potential are sandwiched, i.e., the file used to load the states $|n\rangle$ and $|m\rangle$ used to compute the self-energy matrix element $\langle n|\Sigma|m \rangle$. If absent, the inner wavefunctions from WFN_inner will be used instead. Note that VXC should be consistent with outer wfn while RHO with inner wfn. If outer wfn is generated using a hybrid functional, VXC contains the local part of exchange-correlation potential. In this case set bare_exchange_fraction in sigma.inp to compensate for the non-local part, or use vxc.dat instead of VXC.

The file below will be read only if eqp_corrections is set in sigma.inp.

• eqp.dat: A list of quasiparticle energy corrections for the bands in WFN_inner. The corrected eigenvalues are used for constructing the self-energy operator.

The file below will be read only if eqp_outer_corrections is set in sigma.inp.

• eqp_outer.dat: A list of quasiparticle energy corrections for the bands in WFN_outer. The corrected eigenvalues determine the energies at which the self-energy operator is calculated.

• vxc2.dat: Matrix elements of the exchange-correlation potential, from a mean-field calculation or a previous Sigma run. This file is analogous to vxc.dat, but used for a one-shot hybrid-functional-like calculation instead of a Sigma calculation. VXC2 file, which stores the whole exchange-correlation matrix in reciprocal space, $V_{xc}(G)$, may be used instead via keyword dont_use_vxc2dat.

### Auxiliary files

• x.dat: Matrix elements of the bare exchange, generated by a previous Sigma run to speed up subsequent calculations. Read if use_xdat keyword is present, written otherwise.

• VXC: Exchange-correlation potential (whose matrix elements are subtracted from DFT eigenvalues). Produced by a mean-field code, from the SCF calculation on which WFN_inner is based (or WFN_outer, if present). Used only if vxc.dat is not present or keyword dont_use_vxcdat is set.

• VXC2: Exchange-correlation potential (whose matrix elements are added to the hybrid-functional-like DFT eigenvalues). Produced by a mean-field code, from the SCF calculation on which WFN_inner is based (or WFN_outer, if present). Used only if vxc2.dat is not present or keyword dont_use_vxc2dat is set.

### Output files

The file below will be read only if eqp_corrections is set in sigma.inp.

• eqp0.dat: Contains the on-shell QP energies, $E^\mathrm{mf} - V_\mathrm{xc} - \Sigma(E_0)$. This is not the recommended quantity to use for QP properties. This is Eqn. 36 from Hybertsen & Louie PRB 34 5390.

• eqp1.dat: Contains the off-shell solution to the linearized Dyson's equation, $E^\mathrm{qp}_0 + (d\Sigma/dE) / (1 - d\Sigma/dE) (E^\mathrm{qp}_0 - E_0)$, or a full linear interpolation if more freq. points where computed. This is the recommended quantity to use for QP properties. If you only evaluate $\Sigma(\omega)$ at two frequencies, this is Eqn. 37 from Hybertsen & Louie PRB 34 5390. If there are more frequencies available, the code uses them to find a better linearized solution. For GW calculations without plasmon-pole models, the code reports how many solutions to Dyson's equation were found.

• sigma_hp.log: The log file containing quasiparticle energy values for desired states. For a full-frequency calculation only the value for Sigma calculated at energy closest to the outer wavefunction eigenvalue is shown. The file spectrum.dat contains the full $\Sigma(\omega)$ spectra.