# GW¶

## Overview¶

Electron attachment and detachment energies can be accurately described by many-body perturbation theory (MBPT) methods. In particular, the GW approximation (GWA) to the self-energy is a MBPT method that has seen recent interest in its application to molecules due to a promising cost/accuracy ratio.

The GW module implemented in NWChem takes a DFT mean-field approximation to the Green’s function, G0, in order to solve the quasiparticle equation at the one-shot G0W0 or at various levels of the eigenvalue self-consistent GW approach (evGW). Since the mean-field orbitals are kept fixed in all these approaches, the results depend on the actual starting point G0 (hence, they depend on the exchange-correlation functional chosen for the underlying DFT calculation). For example, it has been known that a large fraction of exact exchange is needed for the accurate prediction of core-level binding energies at the one-shot G0W0 level.

For further theoretical insights and details about the actual implementation in NWChem, please refer to the paper by Mejia-Rodriguez et al1.

GW input is provided using the compound directive

GW
...
END


The actual GW calculation will be performed when the input module encounters the TASK directive.

TASK DFT GW


Note that DFT must be specified as the underlying QM theory before GW. The charge, geometry, and DFT options are all specified as normal.

In addition to an atomic orbital basis set, the GW module requires an auxiliary basis set to be provided in order to fit the four-center electron repulsion integrals. The auxliary basis set can have either the cd basis or ri basis names (see also DFT). Three combinations can be obtained:

• If a ri basis is given without a cd basis, the ground-state DFT will be performed without density fitting, and the GW task will use the ri basis to fit the integrals.
• If a cd basis is given without a ri basis, both DFT and GW tasks will be performed using the cd basis to fit the integrals.
• If both cd basis and ri basis are present, the cd basis will be used for the DFT task, while the ri basis will be used for the GW task.

## GW Input directive¶

There are sub-directives which allow for customized GW calculations. The most general GW input block directive will look like:

GW
RPA
CORE
EVGW [<integer eviter default 4>]
EVGW0 [<integer eviter default 4>]
FIRST <integer first_orbital default 1>
METHOD [ [analytic] || [cdgw <integer grid_points default 200>] ]
ETA <real infinitesimal default 0.001>
SOLVER [ [newton <integer maxiter default 10> ] || [graph] ]
STATES [ [ alpha || beta ] [occ <integer number default 1>] [vir <integer default 0>] ]
CONVERGENCE <real threshold default 0.005> [<string units default ev>]
END


The following sections describe these keywords.

### RPA¶

The keyword RPA triggers the computation of the RPA correlation energy. This adds a little overhead to the CD-GW approach.

### CORE and FIRST¶

The CORE keyword forces to start counting the STATES from the FIRST molecular orbital upwards.

The FIRST keyword has no meaning without CORE specified.

### EVGW and EVGW0¶

The EVGW keyword trigger the partial self-consistnet evGW approach, where both the Green’s function G and the screened Coulomb W are updated by using the quasiparticle energies from the previous step in their construction.

Similarly, the EVGW0 triggers the evGW0 approach, where only the Green’s function G is updated with the quasiparticle energies of the previous iterations. W0 is kept fixed.

Both partial self-consistent cycles run for eviter number of cycles.

The use of EVGW or EVGW0 will trigger the use of a scissor-shift operator for all states not updated in the evGW cycle.

### METHOD¶

Two different techniques to obtain the diagonal self-energy matrix elements are implemented in NWChem.

The analytic method builds and diagonalizes the full Casida RPA matrix in order to obtain the screened Coulomb matrix elements. The Casida RPA matrix grows very rapidly in size (Nocc × Nvir) and ultimately yields a N6 scaling due to the diagonalization step. It is therefore recommended to link the ELPA and turn on its use by setting

SET dft:scaleig e


The cdgw method uses the Contour-Deformation technique in order to avoid the N6 diagonalization step. The diagonal self-energy matrix elements Ʃnn are obtained via a numerical integration on the imaginary axis and the integrals over closed contours on the first and third quadrants of the complex plane. The grid_points value controls the density of the modified Gauss-Legendre grid used in the numerical integration over the imaginary axis.

Both analytic and cdgw methods are suitable for core and valence calculations.

### ETA¶

The magnitude of the imaginary infinitesimal can be controlled using the keyword ETA. The default value of 0.001 should work rather well for valence calculations, but CORE calculations might need a larger value, sometimes between 0.005 or even 0.01.

### SOLVER¶

Two methods to solver the quasiparticle equations are implemented in NWChem.

The newton method uses a modified Newton approach to find the fixed-point of the quasiparticle equations. The Newton method tries to bracket the solution and switches to a golden section method whenever the Newton step goes beyond the bracketing values.

The graph method uses a frequency grid in order to bracket the solution between two consecutive grid points. The number of grid points is controlled heuristically depending on the METHOD and on the presence, or not, of nearby states in a cluster of energy (see below).

Regardless of the solver, the energies of the states are always classified in clusters with a maximum extension of 1.5 eV. For a given cluster of energies, the newton method will start with the state closer to the Fermi level and use its solution as guess for the rest of the states in the cluster. The graph method will look for the solution of all the states in a given cluster at once with a frequency grid with range large enough to encompass all the cluster ± 0.2 eV.

### STATES¶

The keyword STATES controls for which particular state the GW quasiparticle equations are to be solved. The keyword might appear twice, one for the alpha spin channel and one for the beta channel. The beta channel keyword is meaningless for restricted closed-shell DFT calculations (MULT 1 without ODFT in the DFT input block).

The number of occupied states will be counted starting from the state closest to the Fermi level (HOMO) unless the keyword CORE is present. The virtual states will always be counted from the state closest to the Fermi level upwards.

A -1 following either occ or vir stands for all states in the respective space.

### CONVERGENCE¶

The converegnce threshold of the quasiparticle equations can be controlled with the keyword CONVERGENCE and might be given either in eV or Hartree au.

## Sample Input File¶

• A GW calculation requesting the core-level binding energies of all 1s states (6 Fluorines and 6 Carbons) using the CD-GW method.
title "CDGW C6F6 core"
start
echo

memory 2000 mb

geometry
C     -0.21589696     1.38358991     0.00000000
C     -1.30618181     0.50480033     0.00000000
C     -1.09023026    -0.87871037     0.00000000
C      0.21590562    -1.38360671     0.00000000
C      1.30610372    -0.50476737     0.00000000
C      1.09020243     0.87883094     0.00000000
F     -0.42025331     2.69273557     0.00000000
F     -2.54211642     0.98238922     0.00000000
F     -2.12174279    -1.71033945     0.00000000
F      0.42026196    -2.69275237     0.00000000
F      2.54203111    -0.98237286     0.00000000
F      2.12188428     1.71024875     0.00000000
end

basis "ao basis" spherical
* library cc-pvdz
end

basis "cd basis" spherical
* library cc-pvdz-ri
end

dft
xc xpbe96 0.55 hfexch 0.45 cpbe96 1.0
direct
end

gw
core
eta 0.01
method cdgw
solver newton 15
states alpha occ 12
end


• A valence GW calculation to obtain the vertical ionization potential and the vertical electron affinity of the water molecule using the analytic method.
start

geometry
O     -0.000545       1.517541       0.000000
H      0.094538       0.553640       0.000000
H      0.901237       1.847958       0.000000
end

basis "ao basis" spherical
h library def2-svp
o library def2-svp
end

basis "cd basis" spherical
h library def2-universal-jkfit
o library def2-universal-jkfit
end

dft
mult 1
xc pbe96
grid fine
direct
end

gw
states alpha occ 1 vir 1
end


• An evGW0 calculation with 10 iterations using the analytic method. All occupied energies, but only 10 virtual ones, are updated using GW. The rest of the virtual states are shifted using the so-called scissor operator.
start

geometry
O     -0.000545       1.517541       0.000000
H      0.094538       0.553640       0.000000
H      0.901237       1.847958       0.000000
end

basis "ao basis" spherical
h library def2-svp
o library def2-svp
end

basis "cd basis" spherical
h library def2-universal-jkfit
o library def2-universal-jkfit
end

dft
mult 1
xc pbe96
grid fine
direct
end

gw
evgw0 10
states alpha occ -1 vir 10
end