GW¶
Overview¶
Electron attachment and detachment energies can be accurately described by manybody perturbation theory (MBPT) methods. In particular, the GW approximation (GWA) to the selfenergy 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 meanfield approximation to the Green’s function, G_{0}, in order to solve the quasiparticle equation at the oneshot G_{0}W_{0} or at various levels of the eigenvalue selfconsistent GW approach (evGW). Since the meanfield orbitals are kept fixed in all these approaches, the results depend on the actual starting point G_{0} (hence, they depend on the exchangecorrelation 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 corelevel binding energies at the oneshot G_{0}W_{0} level.
For further theoretical insights and details about the actual implementation in NWChem, please refer to the paper by MejiaRodriguez et al^{1}.
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 fourcenter
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 acd basis
, the groundstate DFT will be performed without density fitting, and the GW task will use theri basis
to fit the integrals.  If a
cd basis
is given without ari basis
, both DFT and GW tasks will be performed using thecd basis
to fit the integrals.  If both
cd basis
andri basis
are present, thecd basis
will be used for the DFT task, while theri basis
will be used for the GW task.
GW Input directive¶
There are subdirectives 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 CDGW 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 selfconsistnet 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 evGW_{0} approach, where
only the Green’s function G is updated with the quasiparticle energies
of the previous iterations. W_{0} is kept fixed.
Both partial selfconsistent cycles run for eviter
number of cycles.
The use of EVGW
or EVGW0
will trigger the use of a scissorshift
operator for all states not updated in the evGW cycle.
METHOD¶
Two different techniques to obtain the diagonal selfenergy 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 (N_{occ} × N_{vir})
and ultimately yields a N^{6} 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 ContourDeformation technique in order to
avoid the N^{6} diagonalization step. The diagonal selfenergy
matrix elements Ʃ_{nn} 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 GaussLegendre 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 fixedpoint
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 closedshell 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 corelevel binding energies of all 1s states (6 Fluorines and 6 Carbons) using the CDGW 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 ccpvdz
end
basis "cd basis" spherical
* library ccpvdzri
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
task dft gw
 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 def2svp
o library def2svp
end
basis "cd basis" spherical
h library def2universaljkfit
o library def2universaljkfit
end
dft
mult 1
xc pbe96
grid fine
direct
end
gw
states alpha occ 1 vir 1
end
task dft gw
 An evGW_{0} 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 socalled 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 def2svp
o library def2svp
end
basis "cd basis" spherical
h library def2universaljkfit
o library def2universaljkfit
end
dft
mult 1
xc pbe96
grid fine
direct
end
gw
evgw0 10
states alpha occ 1 vir 10
end
task dft gw
References¶

MejiaRodriguez, D.; Kunitsa, A.; Aprà, E.; Govind, N. Scalable Molecular GW Calculations: Valence and Core Spectra. Journal of Chemical Theory and Computation 2021, 17 (12), 7504–7517. https://doi.org/10.1021/acs.jctc.1c00738. ↩