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. Especially important fix 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 G₀W₀ is implemented. In AMS2021 evGW and evGW₀ are implemented. In AMS2022 qsGW, qsGW₀, and G3W2 are implemented. In AMS2023 GW (G₀W₀, evGW, evGW₀, qsGW, qsGW₀, and G3W2) with spin-orbit coupling is implemented.

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 e.g. very accurate ionization potentials and electron affinities which gives 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. GW should not be used in combination with solvent models, like COSMO, or other environments.

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 G₀W₀ 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 make these approaches a factor of 6 to 12 more expensive than G₀W₀.

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 as 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. 11.

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 G₀W₀ 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 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 (evGW₀), 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 (qsGW₀), 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 used 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 G₀W₀ 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 in 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

is the default and calculates 5 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 keep 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 G₀W₀ method remains in the evGW₀ method and it is generally not recommended.

quasiparticle self-consistent GW¶

An qsGW calculation is requested by

GW
  SelfConsistency qsGW
End

One can also only iterate G by keep 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 G₀W₀ method remains in the qsGW₀ 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 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, three variants can be used. KSF1 and KSF2 are 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. The variant denoted as Kutepov has been suggested bu Kutepov et al. 8 and uses a first-order expansion around the chemical potential.
FixedGrids: Per 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 dependent. 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 to adjust this parameter according to your requirements, for example the experimental resolution you would like to match.

For qsGW (qsGW₀), the change in the norm of the density matrix is used as an additional criterion to control convergence. In evGW (evGW₀) 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 of DIIS and request 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 are 10 expansion vectors. In case of difficulties converging

Second-order self-energy¶

Starting from ADF2022, it is possible to go beyond the GW approximation for the self-energy (expansion in screened interaction to first order) and also take into account the next term (expansion in the screened interaction to second order). The algorithm has been described in this paper. .. 10

This can be used with all variants of self-consistency.
The second-order self-energy is always applied as a perturbative correction of the GW quasiparticle energies, using a statically screened interaction W. More precisely,

\[\Sigma^{GW} = G(\omega) * W (\omega)\]

\[\Sigma^{GW+G3W2} = G(\omega) * W(\omega) + G(\omega) * W (\omega = 0) * G(\omega) * G(\omega) * W (\omega = 0)\]

The second-order self-energy is activated by typing

GW
  selfenergy G3W2
End

It has been shown that the second-order correction is especially accurate for electron affinities and HOMO-LUMO gaps. 10 Since it effectively scales as the fourth power of the system size, it should not be used for systems much larger than 50 atoms.

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 of for the property of interest. Details about basis set convergence can be found in this paper 10

Basis sets

All-electron:  All-electron calculations are always recommended.
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. They are however often necessary for an accurate description of the electron affinity for molecules with LUMO above the vacuum level.

Ionization Potentials: TZ2P 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 do 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 to perform an extrapolation to the complete basis set limit: For this, perform 2 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 10

Numerical aspects¶

According to the choice of basis set, the recommended numerical settings can differ. For a discussion, see the MBPT page.
qsGW has the highest requirements on the numerical parameters than evGW and G₀W₀. 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 rather inaccurate.
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.
For optimal accuracy, all electron basis sets hould be used. Note, that frozen cores do not lead to large computational savings in case of GW calculations.

Choosing the KS reference¶

As for ground-state properties, it is far from trivial to recommend a universal functional. However, for G₀W₀ calculations, a few general guidelines can be offered. A good discussion is found in 2. It should be noted, that GGA functional should not be used, even though a PBE starting point is a popular choice. We rather recommend to use 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.

GW key¶

GW
   nStates integer
   SelfEnergy [HF | GW | G3W2]
   SelfConsistency [G0W0 | EVGW0 | EVGW | QSGW0 | QSGW]
   QPHamiltonian [KSF1 | KSF2]
   nIterations integer_list
   Converge
      Density float_list
      HOMO float
   End
   FixedGrids Yes/No
   preconditionQSGW Yes/No
   OffDiagonalEFermi Yes/No
   LinearMixing float_list
   AdaptiveMixing float_list
   DIIS integer
   LinearizeQPequations Yes/No
   ScissorShift Yes/No
End

GW

Type: Block
Description: Instruct ADF to perform a G0W0 calculation.

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’

SelfEnergy

Type: Multiple Choice
Default value: GW
Options: [HF, GW, G3W2]
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 a on-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]
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. 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 recommened 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

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 recommened 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.

FixedGrids

Type: Bool
Default value: No
Description: In a self-consistent GW calculation, do not recalculate Grids. Can be useful in case of convergence problems. Only relevant for qsGW and qsGW0. In case of evGW and evGW0, the grids are always kept fixed.

preconditionQSGW

Type: Bool
Default value: No
Description: If true, the QSGW equations are solved but prior to each diagonalization, i.e. a G0W0 calculation is performed to find the optimal QP energies at which to analytically continue the self-energy. This is in principle a more consistent construction than KSF1 or KSF2 since the diagonal elements are consistent with G0W0. In KSF1 and KSF2, the diagonal elements are evaluated at the QP energies from the previous iteration which is equivalent to a zeroth-order Taylor expansion of the diaognal elements around the previous QP energies.Enabling this option typically leads to slightly lower QP energies.

OffDiagonalEFermi

Type: Bool
Default value: No
Description: Analytically continue the off-diagonal elements of the KSF2 qsGW Hamiltonian at the Fermi-energy instead of omega=0. Typically leads to slightly lower QP energies, i.e. higher ionization potentials. The HOMO-LUMO gaps are typically not affected.

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 recommenened 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 approximaiton and not recommended. It might again be useful for benchmarking purposes.

References

1(1,2): 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(1,2): 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(1,2,3,4): A. Förster, L. Visscher, EGW100: A Slater-Type Orbital Perspective, Journal of Chemical Theory and Computation 17(8), 5080-5097 (2021)
11: 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, 1958 (2023)

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GW¶