GW
**


General
=======

Warning:

  In AMS2023.102 (or bug-fixed AMS2023 binary with r112249 or higher)
  a bug was fixed in ADF regarding GW. An incorrect Fermi energy was
  used. This is especially important in case the Fermi energy is not
  close to zero.

This page describes the basic procedure, usage, and scope of a GW
calculation. Technical details of the algorithm can be tweaked in the
MBPT input block. In AMS2020, G_0W_0 is implemented. In AMS2021, evGW
and evGW_0 are implemented. In AMS2022, qsGW, qsGW_0, and G3W2 are
implemented. In AMS2023, GW (G_0W_0, evGW, evGW_0, qsGW, qsGW_0, and
G3W2) with spin-orbit coupling is implemented.

See also:

  Tutorial: Accurate Ionization Potential and Electron Affinity with
  GW

  Examples GW

The GW method is a relatively accurate method to obtain information
about so-called charged excitations, or single-particle excitations.
We refer to them as quasiparticle energies. These are especially
important to interpret and predict the outcome of direct and inverse
photo-emission spectroscopy and can be used to obtain, for example,
very accurate ionization potentials and electron affinities which give
access to the so-called fundamental gap (not to be confused with the
optical gap).

GW can be used with scalar and spin-orbit coupled relativistic effects
within the "ZORA", "X2C", or "RA-X2C" formalism.

In practice, fully self-consistent GW is rarely used for molecular
systems. Instead, perturbative approximations, so-called quasiparticle
GW methods are used since they are cheaper and also more accurate than
fully self-consistent GW. The most popular approach is G_0W_0 [1], in
which quasiparticle energies are obtained as a one-shot perturbative
correction to KS eigenvalues. A downside of this approach is the
rather pronounced starting-point dependence. This can be overcome to a
large extent in eigenvalue-only self-consistent GW (evGW), in which
the quasiparticle energies (but not the density) are iteratively
updated until self-consistency is reached. In quasiparticle self-
consistent GW, also the density is updated in each iteration, so that
the results become completely starting-point independent. The
partially self-consistent GW variants usually converge within 6-12
iterations, which makes these approaches a factor of 6-12 more
expensive than G_0W_0.

A GW calculation as implemented in ADF proceeds in five steps as has
been described in detail in a series of papers. [1] [5] [6] The
technique is also known as the GW space-time method. Our
implementation is closely related to the scheme outlined in [4]. A GW
implementation in ADF for closed-shell molecules that includes spin-
orbit coupling is described in Ref. [14].

* A DFT single-point calculation is performed. This can be an LDA,
  GGA, or hybrid calculation. At the moment, ADF does not support the
  use of XCfun, but LibXC is supported. Valid choices could be LDA,
  PBE, BLYP, PBE0, BHandHLYP, and many more. The default is LDA. As
  usual, the functional to be used during the SCF is requested in the
  XC input block block. The following requests a G_0W_0 calculation
  with default settings using a PBE reference:

     XC
        GGA PBE
     END

     GW
     End

* From the KS orbitals and orbital energies, a Green's function (G) is
  evaluated and from the Green's function, the so-called independent-
  particle polarizability is calculated. This is done in imaginary
  time.

* The polarizability is Fourier transformed to the imaginary frequency
  axis where the screened Coulomb potential (W) is evaluated using the
  Coulomb potential and the polarizability.

* The screened Coulomb potential is Fourier transformed to imaginary
  time. Here, the self-energy is calculated using G and W (that's why
  it is called GW), which gives access to spectroscopic properties.

* The self-energy is transformed to the molecular orbital basis from
  where it is Fourier transformed to the imaginary frequency axis from
  where it is analytically continued to the complex plane. Along the
  real frequency axis, the quasiparticle energies are evaluated.

* In case of evGW (evGW_0), the input KS eigenvalues are replaced by
  the quasiparticle energies from the previous iteration and the
  scheme is iterated until self-consistency in the quasiparticle
  energies is reached.

* In case of qsGW (qsGW_0), a non-local, Hermitian, and static
  exchange-correlation potential is constructed from the self-energy.
  This exchange-correlation potential replaces the KS exchange-
  correlation potential. Diagonalization gives a new set of single-
  particle orbitals and quasiparticle energies. The procedure is
  repeated until self-consistency is reached.

* By default, the DIIS algorithm is used to accelerate and stabilize
  convergence of the self-consistent GW schemes. A linear-mixing
  scheme can also be used.

The GW space-time method has the distinct advantage that it can be
very fast, while a full frequency, conventional GW calculation scales
to the 6th power of the system size and is prohibitive for systems
larger than a few tens of atoms. ADF uses advanced density fitting
options to accelerate the space-time method further and in practice
nearly quadratic scaling can be reached. This enables the routine
application of the method to systems of several hundreds of atoms. A
G_0W_0 calculation (without the preceding SCF) is usually not much
slower than a hybrid calculation. The downside of the approach is that
the analytical continuation technique produces large errors (up to
several eV) for core states which are, for example, important in X-ray
spectroscopy. Thus, the GW implementation in ADF should only be used
to predict quasiparticle energies for states in the valence region. In
fact, we have only tested it for HOMO and LUMO states (which are
arguably most important).

The states of interest can be requested in the GW block:

   GW
     nStates 5
   End

This is the default and calculates five occupied and five unoccupied
states.

   GW
     nStates 1
   End

calculates the HOMO and LUMO quasiparticle energy only.


Levels of self-consistency
==========================


Eigenvalue-only self-consistent GW
----------------------------------

An evGW calculation is requested by

   GW
     SelfConsistency evGW
   End

One can also only iterate G by keeping W fixed, which reduces the cost
of each iteration by roughly 50 %. This is requested by

   GW
     SelfConsistency evGW0
   End

On the other hand, much of the starting-point dependence of the G_0W_0
method remains in the evGW_0 method and it is generally not
recommended.


Quasiparticle self-consistent GW
--------------------------------

A qsGW calculation is requested by

   GW
     SelfConsistency qsGW
   End

One can also only iterate G by keeping W fixed, which reduces the cost
of each iteration by roughly 50 %. This is requested by

   GW
     SelfConsistency qsGW0
   End

On the other hand, much of the starting-point dependence of the G_0W_0
method remains in the qsGW_0 method and it is generally not
recommended.

More options:

"QPHamiltonian"
   In quasiparticle self-consistent GW, the frequency-dependent self-
   energy is mapped to a static exchange-correlation potential in each
   iteration. The mapping is not unique and different schemes have
   been suggested. They differ in the way the frequency dependence of
   the self-energy is treated. In ADF, two variants can be used: KSF1
   and KSF2 are both from the paper by Kotani et al. [7]. KSF1 refers
   to Eq. 10 therein and is most commonly implemented, KSF2 refers to
   Eq. 11 therein. KSF2 is the default in ADF since it is numerically
   more stable when qsGW is implemented together with the analytical
   continuation algorithm. In case the analytical integration
   algorithm is used, only KSF1 is currently supported. ADF also
   supports a variant denoted as LQSGW, which has been suggested by
   Kutepov et al. [8]. It uses a first-order expansion around the
   chemical potential. This is also similar to the approach
   implemented in VASP.

"FixedGrids"
   By default, the imaginary frequency and imaginary time grids used
   in the GW calculation are updated in each iteration in a qsGW
   calculation since this is necessary to make the results strictly
   starting-point independent. This can also be turned off, and fixed
   grids are used throughout. This might be helpful in case of
   convergence problems.


Convergence
-----------

The self-consistency can be controlled by a few parameters: For
example

   GW
     Converge HOMO=5e-3
   End

requests that the evGW(0) calculation is considered converged if the
difference between the HOMO quasiparticle energies of 2 consecutive
iterations does not change by more than 5 meV. The default is 1 meV
which is in practice usually a little too tight. We recommend
adjusting this parameter according to your requirements, for example
the experimental resolution you would like to match.

For qsGW (qsGW_0), the change in the norm of the density matrix is
used as an additional criterion to control convergence. In evGW
(evGW_0) it is ignored.

   GW
     Converge Density=1e-07
   End

is the default. For very large systems and when QZ basis sets are
used, it is recommended to decrease that value, for example to

   GW
     Converge Density=1e-08
   End

   GW
     linearmixing 0.2
   End

turns off DIIS and requests to use linear mixing instead with a mixing
parameter of 0.2. This can be useful if, for some reason, convergence
using DIIS cannot be achieved. However, it is usually better to adjust
the number of vectors in the DIIS expansion. This is achieved by (for
example)

   GW
     DIIS 5
   End

The default is 10 expansion vectors.


Vertex corrections beyond GW
============================

Starting from ADF2022 and with many additions in ADF2025, it is
possible to go beyond the GW approximation for the self-energy using
vertex corrections. This is possible in several ways, but not all are
recommended. The algorithms and methods implemented in AMS (all of
these methods can be used with both the ADF and the BAND engine) have
been described in a series of publications. [10] [11] [12] [13]

The easiest way to go beyond the GW is to also calculate the next term
in the self-energy, often called G3W2. For the full G3W2 term, two
nested and slowly converging frequency integrals have to be performed.
[11] However, there are possible approximations to it, for instance,
the SOSEX and 2SOSEX approximations which only require a single
frequency integration. [11] It is also possible to apply a static
approximation to this term which avoids frequency integrations
completely. [10]

None of these variants is recommended. They are expensive, and it can
be shown that they always perform worse than a reasonable GW
calculation. [11] The reason for this is that additional terms in the
self-energy beyond GW should always be combined with additional terms
in the polarizability. Additional self-energy terms always lower the
HOMO, increase the LUMO, and therefore open the fundamental gap.
Additional terms in the polarizability do the opposite. It can be
shown that they largely cancel, [12] but the remaining effect improves
results. This has been shown for a Hartree-Fock starting point (not
recommended for G0W0, but with additional vertex corrections the
performance is very good) [12] and for qsGW. [13]

ADF/BAND implements four possibilities to go beyond the simple RPA
polarizability:

   GW
     Polarizability {RPA|G4V1|G4W1|TDHF|BSE}
   End

* **RPA**        : Standard (direct) RPA without exchange. This is the
  default.

* **G4V1**       : This evaluates the RPA polarizability and adds the
  G4V1 term. This is the next-to-leading order term beyond RPA in the
  bare Coulomb interaction V. [12]

* **G4W1**       : This evaluates the RPA polarizability and adds the
  G4W1 term. This is the next-to-leading order term beyond RPA in the
  statically screened Coulomb interaction W. [12]

* **TDHF**       : Here, instead of improving the RPA polarizability
  order by order, one adds an infinite order correction. TDHF solves
  the usual TDHF linear response equations to calculate the reducible
  polarizability which is then used to calculate the screening. The
  G4V1 term is the first-order approximation to TDHF. [12]

* **BSE**       : This is the recommended option. BSE solves the usual
  BSE linear response equations with the statically screened exchange
  kernel to calculate the reducible polarizability which is then used
  to calculate the screening. This is also the BSE which is usually
  solved in the GW-BSE method. The G4W1 term is the first-order
  approximation to TDHF. [12]

* Both **BSE** and **TDHF** have to be solved in the MO basis using
  analytical integration. This does not have to be specified in the
  input since ADF selects this automatically.

* Both **G4V1** and **G4W1** have to be solved in the MO basis using
  analytical continuation. This does not have to be specified in the
  input since ADF selects this automatically.

These polarizability terms should always be combined with the
compatible additional terms in the self-energy. [12] Compatible means
that it makes sense physically. Most of these options can be combined
at will, and you will get a result. It just might not make sense.

   GW
     SelfEnergy {HF|G3W2dynamic|G3W2|GW|SOSEX|GWGamma|GWGammaInf|GWGammaInfDyn}
   End

* **HF**          : This uses the GW code to perform a HF calculation.
  Not recommended.

* **G3W2dynamic** : This is the fully G3W2 dynamical self-energy. Very
  expensive and bad. **Not** recommended. [11]

* **G3W2**        : This is the fully statical G3W2 self-energy.
  **Not** recommended. [10]

* **GW**          : This is the default.

* **SOSEX**       : This is the SOSEX approximation to G3W2. Should be
  combined either with the **G4V1** or the **TDHF** polarizabilities.
  [11]

* **GWGamma**     : This is the GWGamma approximation to G3W2. In case
  the **G4V1** or the **TDHF**  polarizability is used, it defaults to
  SOSEX. In case the **G4W1** or the **BSE** polarizabilities are
  used, it does something that looks like SOSEX, but the bare V in
  SOSEX is replaced by the statically screened Coulomb interaction W.
  This makes this option compatible with all options for the
  polarizability (except for the default, RPA). [12]

* **GWGammaInf**  : Similar to TDHF or BSE, it resums the additional
  self-energy terms beyond GW to infinite orders. This is the
  **recommended** choice whenever the TDHF or BSE polarizability is
  used. [12]

* **GWGammaInfDyn**  : Finally, we have an option which first performs
  a **GWGammaInf** calculation, and adds a first-order dynamical
  correction. This is not recommended at the moment but could be
  combined with a first-order dynamical correction to the
  polarizability. However, this is not yet implemented.

* In summary, we do recommend the following combinations: [12] [13]

   GW
     Polarizability {BSE|TDHF}
     SelfEnergy GWGammaInf
   End

* Both options give similar results. [12] In principle, BSE should be
  better, and only slightly more expensive. Both options are
  relatively expensive anyway since a full linear response problem has
  to be solved completely.

* This should either be used with a Hartree-Fock starting point, or
  combined with a qsGW calculation. The latter will give very accurate
  results (possibly the best you can get), but it is expensive. You
  might be able to apply it to systems < 20 atoms in a TZ basis set on
  a standard node. Larger systems are potentially possible if run on a
  high-memory node. Other KS starting points can be used, and this
  could potentially work well. But this has not been tested yet.


Embedding
=========

The GW method can be combined with both the COSMO , DIM , and FQ
embedding methods [16] This coupling is only available for the G_0W_0
level of approximation and is highly experimental, therefore users
should exercise caution in interpreting the results. The
implementation is based on the work by Duchemin et al. [15] on the
combination of GW with the polarizable continuum model, which we
further extended to the DIM method.

To add embedding effects in a GW calculation, simply include the
corresponding keywords in the input file as is done for regular DFT
calculations.


Recommendations
===============


Basis sets
----------

The recommended numerical settings depend strongly on the basis set.
The recommended basis set depends on system size and the property of
interest. The following are recommendations which should be seen as
guidelines and not as definite. We recommend to always verify the
basis set convergence for the property of interest. Details about
basis set convergence can be found in this paper [13].

**Basis sets**

   **All-electron:**  All-electron calculations have to be used.
   **Basis set size:** Larger basis sets are needed to achieve the same accuracy as in a DFT calculation.
   **Augmented basis sets:** Should only be used if absolutely necessary. However, they are often necessary for an accurate description of the electron affinity for molecules with LUMO above the vacuum level.

   **Ionization Potentials:** QZ4P or larger
   **Electron affinities, bound LUMO:** Corr/TZ3P or larger
   **Electron affinities, unbound LUMO:** AUG/ATZ2P or larger
   **Fundamental Gaps:** Triple-zeta quality basis sets are typically sufficient.

The GW-BSE excitation energies and the screened Coulomb interaction
only depend on the QP gaps and not on their absolute values.
Therefore, triple-zeta quality basis sets are typically sufficient.

For **highly accurate results** we recommend performing an
extrapolation to the complete basis set limit: for this, perform two
calculations using the Corr/TZ3P and Corr/QZ6P. The basis set limit is
then calculated according to

   \epsilon_n^{CBS} = \epsilon_n^{QZ} -  \frac{1}{N^{QZ}_{bas}
   }\frac{\epsilon_n^{QZ} - \epsilon_n^{TZ}}{\frac{1}{N^{QZ}_{bas}} -
   \frac{1}{N^{TZ}_{bas}}}

The values for N^{TZ}_{bas} and N^{QZ}_{bas} can be found in the
adf.rkf file in the "GW" section under the entry "nBas". Additional
explanations can be found in the references [3] and [13].
Alternatively, this basis set extrapolation can be performed with the
BAND code and the Dunning basis sets cc-pVTZ and cc-pVQZ.


Numerical aspects
-----------------

* According to the choice of basis set, the recommended numerical
  settings can differ. For a discussion, see the MBPT page.

* qsGW has higher requirements on the numerical parameters than evGW
  and G_0W_0. Usually, it is necessary to use a larger value in the
  "Dependency" key; see the MBPT page. Note that starting from AMS2022
  ADF will use a "Dependency bas=5e-3" and "RIHartreeFock
  DependencyThreshold=5e-3" in case of (any variant of) GW. One can
  override these values in the input.

* The implemented GW algorithm is very sensitive to numerical noise
  and depending on the numerical settings and/or the underlying
  exchange-correlation functional, (occupied) quasiparticle energies
  from partially self-consistent GW calculations performed on
  different hardware can differ by a few meV. The discrepancies are
  generally more pronounced for core states for which the analytical
  continuation technique is not applicable. Here, the analytical
  integration algorithm should be used.

* In our experience, these differences only occur with large basis
  sets (QZ4P or larger) and when Minnesota functionals (we tested M06,
  M06-2X and M06-HF) are used to calculate the KS reference. When the
  DIIS algorithm is used to converge the quasiparticle energies, the
  number of iterations needed for convergence can then differ as well.

* The analytical integration solution of GW is numerically more stable
  since there are no problems due to frequency integration, analytical
  continuation, or time-to-frequency Fourier transforms.

* If the RI formalism is used, one avoids time-to-frequency Fourier
  transforms. This can sometimes help in case of unexpected results
  with the analytical continuation based algorithm.


Choosing the KS reference
-------------------------

* As for ground-state properties, it is far from trivial to recommend
  a universal functional. However, for G_0W_0 calculations, a few
  general guidelines can be offered. A good discussion is found in
  [2]. It should be noted that GGA functionals should not be used,
  even though a PBE starting point is a popular choice. We rather
  recommend using a hybrid functional. The relevant parameter in the
  choice of the hybrid is usually the fraction of exact exchange. In
  our experience, PBE0 with 40-50 % exact exchange is a good choice.
  You might also use a range-separated hybrid via LibXC, for example
  LRC-wPBEH.

* If in doubt, one of the partially self-consistent schemes should be
  used. evGW is almost starting-point independent and qsGW is
  completely starting-point independent. Therefore, for qsGW, it is in
  principle irrelevant what starting point is chosen. Convergence
  properties can in principle be affected, although the number of
  iterations until convergence is reached is also more or less
  independent of the starting point [6]. Also for evGW, starting from
  a hybrid functional or a range-separated functional usually results
  in the highest accuracy.

* A Hartree-Fock starting point can be used in G_0W_0, but in that
  case a vertex correction should be applied. See above.


GW key
======

   GW
      nStates integer
      Polarizability [RPA | BSE | G4W1 | G4V1 | TDHF]
      SelfEnergy [HF | GW | G3W2 | SOSEX | GWGamma | G3W2dynamic | GWGammaInf | GWGammaInfDyn]
      SelfConsistency [G0W0 | EVGW0 | EVGW | QSGW0 | QSGW]
      QPHamiltonian [KSF1 | KSF2 | SRG | LQSGW]
      AnalyticalIntegration
         Enabled Yes/No
         SpectralFunctionResolution integer
         TDA Yes/No
         eta float
      End
      nIterations integer_list
      Converge
         Density float_list
         HOMO float
      End
      LinearMixing float_list
      AdaptiveMixing float_list
      DIIS integer
      LinearizeQPequations Yes/No
      ScissorShift Yes/No
   End

"GW"
   Type:
      Block

   Description:
      Perform a GW calculation. G0W0 is the default for
      GW%SelfConsistency.

   "nStates"
      Type:
         Integer

      Default value:
         5

      GUI name:
         N states

      Description:
         Number of Quasiparticle States to be printed to output.
         The default is 5 states which in this case means that min(5,
         Number of particle states) occupied and min(5, Number of hole
         states) hole states are printed. The whole list of states can
         be printed by setting this parameter to -1'

   "Polarizability"
      Type:
         Multiple Choice

      Default value:
         RPA

      Options:
         [RPA, BSE, G4W1, G4V1, TDHF]

      Description:
         Sets the expression for the Polarizability used in the GW
         calculation.                      RPA is the Default and
         amounts to a standard GW calculation.
         BSE denotes screening in the Bethe-Salpeter-equation
         formalism.

   "SelfEnergy"
      Type:
         Multiple Choice

      Default value:
         GW

      Options:
         [HF, GW, G3W2, SOSEX, GWGamma, G3W2dynamic, GWGammaInf,
         GWGammaInfDyn]

      Description:
         Controls the form of the self-energy.
         GW is the default and corresponds to the standard GW
         calculation.                      G3W2 is a GW calculation
         plus a perturbative second-order statically screened exchange
         correction (second order expansion in the self-energy). Note,
         that there the self-energy is always static.

   "SelfConsistency"
      Type:
         Multiple Choice

      Default value:
         G0W0

      Options:
         [G0W0, EVGW0, EVGW, QSGW0, QSGW]

      Description:
         Sets the level of self-consistency in a GW calculation.
         G0W0 calculates a one-shot, perturbative correction to the KS
         eigenvalues.                      In evGW and evGW0, the
         quasi-particle energies are updated until self-consistency is
         reached.                      evGW0 requests that the Green's
         function is evaluated self-consistently but not the screened
         interaction.                      In qsGW, the density is
         updated as well, however, the self-energy is mapped to a
         static effective potential and the Dyson equation is solved
         by diagonalization instead of inversion. The results of a
         qsGW are independent of the choice of the underlying
         exchange-correlation functional and are usually the most
         accurate ones.                      The same is done in
         qsGW0, but the screened interaction is not updated.

   "QPHamiltonian"
      Type:
         Multiple Choice

      Default value:
         KSF2

      Options:
         [KSF1, KSF2, SRG, LQSGW]

      Description:
         The quasi-particle Hamiltonian can be constructed in
         different ways.                      KSF1 refers to the
         original construction by Kotani, Van Schilfgaarde anf Faleev
         (KSF) which is also implemented in TURBOMOLE.
         KSF2 refers to an alternative construction by KSF.
         KSF1 is not recommended since it is numerically less stable
         than KSF2 in case analytical continuation is used (the
         default).                      In case the analytical
         integration algorithm is used, only KSF1 is implemented.
         Therefore, make sure that KSF1 is specified. The results are
         typically very similar.                      The QP energies
         at which the matrix elements are evaluated can be tweaked
         further, see the two subsequent keys: However, KSF2 is
         recommended since it typically leads to QP energies with the
         best agreement with experiment.                      Ignored
         when not a quasi-particle self-consistent GW calculation is
         performed

   "AnalyticalIntegration"
      Type:
         Block

      Description:
         Use analytical integration to calculate the self-energy. Very
         slow, unless the system is very small but useful to check the
         accuracy of the frequency integration

      "Enabled"
         Type:
            Bool

         Default value:
            No

         GUI name:
            analytical integration

         Description:
            Enable the calculation of the GW quasi-particle energies
            via analytical integration.

      "SpectralFunctionResolution"
         Type:
            Integer

         Default value:
            800

         Description:
            Number of points at which spectral function is evaluated.

      "TDA"
         Type:
            Bool

         Default value:
            No

         Description:
            Solve the linear response equations in the Tamm-Dancoff
            approximation.

      "eta"
         Type:
            Float

         Default value:
            0.001

         Description:
            Artificial (positive) broadening parameter for evaluation
            of self-energy in analytical integration.
            Ideally should be as small as possible but this might lead
            to convergence issues in partially self-consistent
            approaches.                         In this case, a value
            of up to 0.1 is possible.

   "nIterations"
      Type:
         Integer List

      Default value:
         [10]

      GUI name:
         Number of iterations

      Description:
         The maximum number of iterations within the (partially or
         fully) self-consistent GW calculation has to converge.
         Ignored when Formalism is set to G0W0

   "Converge"
      Type:
         Block

      Description:
         Sets convergence criteria for the GW calculation in self-
         consistent case

      "Density"
         Type:
            Float List

         Default value:
            [1e-08, 1e-05]

         Description:
            First Criterion for self-consistency procedure to
            terminate.                         Criterion is the trace
            of the density matrix. Ignored in non-selfconsistent
            Calculation and in eigenvalue self-consistent GW
            It is possible to run a qsGW calculation with an inner SCF
            loop which updates the static part of the elf-energy only.
            This can be useful to accelerate the convergence in case
            linear mixing is used. It is not recommended to use linear
            mixing, so it is also not recommended to use that inner
            loop as well. The second number in this list specifies the
            convergence criterion for the inner SCF loop.

      "HOMO"
         Type:
            Float

         Default value:
            0.003

         Unit:
            eV

         GUI name:
            HOMO energy convergence

         Description:
            Criterion for self-consistency procedure to terminate.
            The self-consistent GW calculation terminates, when the
            difference between the HOMO QP energies between 2
            consecutive iterations is below this number.
            The LUMO energy converged faster than the HOMO energy so
            when the HOMO energy is converged according to this
            criterion, the LUMO energy will be converged as well.
            In non-selfconsistent Calculation, this criterion is
            ignored.

   "LinearMixing"
      Type:
         Float List

      Description:
         Requests to use linear mixing instead of DIIS and sets the
         mixing parameter for linear mixing of Green's function in
         case of self-consistency.                      It is ignored
         in non-selfconsistent calculation and overwritten by DIIS
         when DIIS is also present.

   "AdaptiveMixing"
      Type:
         Float List

      Description:
         Requests to use adaptive mixing instead of DIIS and sets the
         staring mixing parameter for mixing of Green's function in
         case of self-consistency.                      Adapative
         mixing is recommended in case a qsGW calculation does not
         converge with DIIS.                      It is ignored in
         non-selfconsistent calculation and overwritten by DIIS when
         DIIS is also present.

   "DIIS"
      Type:
         Integer

      Default value:
         10

      Description:
         Requests to use DIIS. This is the Default. Number of
         expansion coefficients can be requested as well. Ignored in
         non-selfconsistent calculation

   "LinearizeQPequations"
      Type:
         Bool

      Default value:
         No

      Description:
         Instead of solving the non-linear QP equations in a G0W0 (or
         evGW calculation) by bisection exacly, linearize them by
         first-order Taylor expansion.                      This is
         not recommended since it does not save computational time
         when used together with analytical continuation (as
         implemented in AMS). It might however be useful for
         benchmarking or for validating results.
         If the results os the linearization differ by a lot (for
         instance, more than 0.1 eV in frontier QP energies) from the
         non-linearized results, this might indicate that the GW
         calculation is not reliable.

   "ScissorShift"
      Type:
         Bool

      Default value:
         No

      Description:
         Only calculate the HOMO and LUMO QP energies and shift the
         remaining QP energies by the same amount.
         This is a rather crude approximation and not recommended.
         It might again be useful for benchmarking purposes.

[1] Arno Förster, Lucas Visscher, *Low-order scaling G0W0 by pair
    atomic density fitting*, Journal of Chemical Theory and
    Computation 16 (12), 7381–7399 (2020)

[2] F. Bruneval, M.A.L., Marques, *Benchmarking the starting points of
    the GW approximation for molecules*, Journal of Chemical Theory
    and Computation 9 (1), 324-329 (2013)

[3] M.J. Van Setten, F. Caruso S. Sharifzadeh X. Ren M. Scheffler F.
    Liu J. Lischner L. Lin J. Deslippe S.G. Louie C. Yang F. Weigend
    J.B. Neaton F. Evers P. Rinke, *GW100: Benchmarking G0W0 for
    Molecular Systems*, Journal of Chemical Theory and Computation 11
    (12), 5665-5687 (2015)

[4] J. Wilhelm, D. Golze, L. Talirz, J. Hutter, C.A. Pignedoli,
    *Toward GW Calculations on Thousands of Atoms*, Journal of
    Physical Chemistry Letters 9 (2), 306-312 (2012)

[5] Arno Förster, Lucas Visscher, *GW100: A Slater-Type Orbital
    Perspective*, Journal of Chemical Theory and Computation 17 (8),
    5080–5097 (2021)

[6] Arno Förster, Lucas Visscher, *Low-Order Scaling Quasiparticle
    Self-Consistent GW for Molecules*, Frontiers in Chemistry 9,
    736591 (2021)

[7] A.L. Kutepov, V.S. Oudovenko, G. Kotliar, *Linearized self-
    consistent quasiparticle GW method: Application to semiconductors
    and simple metals*, Computer Physics Communications 407-414 (2017)

[8] Takao Kotani, Mark Van Schilfgaarde, Sergey V. Faleev,
    *Quasiparticle self-consistent GW method: A basis for the
    independent-particle approximation*, Physical Review B 76 (16)
    1-24 (2007)

[9] A. Förster, L. Visscher, *Exploring the statically screened G3W2
    correction to the GW self-energy : Charged excitations and total
    energies of finite systems*, Physical Review B 105, 125121 (2022)

[10] A. Förster, L. Visscher, *Exploring the statically screened G3W2
     correction to the GW self-energy : Charged excitations and total
     energies of finite systems*, Physical Review B 105, 125121 (2022)

[11] F.Bruneval, A. Förster, *Fully Dynamic G3W2 Self-Energy for
     Finite Systems: Formulas and Benchmark*, Journal of Chemical
     Theory and Computation 20 (8), 3218-3230 (2024)

[12] A. Förster, F.Bruneval,  *Why Does the GW Approximation Give
     Accurate Quasiparticle Energies? The Cancellation of Vertex
     Corrections Quantified*, The Journal of Physical Chemistry
     Letters 15, 12526-12534 (2024)

[13] A. Förster, *Beyond quasi-particle self-consistent GW for
     molecules with vertex corrections*, arXiv:2412.01581 (2024)

[14] A. Förster, E. van Lenthe, E. Spadetto, L. Visscher, *Two-
     Component GW Calculations: Cubic Scaling Implementation and
     Comparison of Vertex-Corrected and Partially Self-Consistent GW
     Variants*, Journal of Chemical Theory and Computation 19, 5958
     (2023)

[15] I.Duchemin, D. Jacquemin, X. Blase, *Combining the GW formalism
     with the polarizable continuum model: A state-specific non-
     equilibrium approach*, The Journal of Chemical Physics 144,
     164106 (2016)

[16] 7. Nottoli, P. Lafiosca, F.E.Q. Rodríguez, F. Egidi, A. Förster,
        C. Cappelli, *GW Approximation Coupled with Classical
        Fluctuating Charges and Dipoles*, Journal of Chemical Theory
        and Computation 22 (2026) 371–384
