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Docs: Some minor touch ups
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Ole Schütt
Ole Schütt
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3 changed files with 46 additions and 38 deletions
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@ -5,8 +5,8 @@ GW for molecules/condensed phase systems with CP2K. In DFT, the energy of a mole
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corresponds to an eigenvalue of the Kohn-Sham matrix. In GW, the procedure for getting the level
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energies is to first perform a DFT calculation (commonly with the PBE or PBE0 functional) to get the
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molecular orbital wavefunctions and then compute a new GW energy for the molecular orbitals of
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interest. For an introduction into the concept of GW, please have a look at [Golze2019](#Golze2019)
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or read Sec. II and the introduction to Sec. III in [Hueser2013](#Hueser2013).
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interest. For an introduction into the concept of GW, please have a look at [](#Golze2019) or read
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Sec. II and the introduction to Sec. III in [](#Hueser2013).
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The GW implementation in CP2K is based on the developments described in [](#Wilhelm2016). The
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computational cost of GW is comparable to RPA and MP2 total energy calculations and therefore high.
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@ -27,14 +27,14 @@ calculation is used for computing the molecular orbitals which can be seen from
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[XC_FUNCTIONAL PBE](#CP2K_INPUT.FORCE_EVAL.DFT.XC.XC_FUNCTIONAL.SECTION_PARAMETERS). The input
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parameters for G0W0 are commented below. While the calculation is running, you can look up the
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G0W0@PBE value for the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular
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orbital (LUMO) in [vanSetten2015](#vanSetten2015) \[Table 2 and 3 (column AIMS-P16/TM-no RI,
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molecule index 76) which should be -11.97 eV and 2.37 eV for HOMO and LUMO, respectively\]. CP2K
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should be able to exactly reproduce these values. In the output of CP2K, the G0W0@PBE results are
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listed after the SCF after the headline *GW quasiparticle energies*.
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orbital (LUMO) in [](#vanSetten2015) \[Table 2 and 3 (column AIMS-P16/TM-no RI, molecule index 76)
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which should be -11.97 eV and 2.37 eV for HOMO and LUMO, respectively\]. CP2K should be able to
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exactly reproduce these values. In the output of CP2K, the G0W0@PBE results are listed after the SCF
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after the headline *GW quasiparticle energies*.
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For checking the basis set convergence, we refer to a detailed analysis in [](#Wilhelm2016) in Fig.
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2 for benzene. An extensive table of basis set extrapolated GW levels can be found in
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[vanSetten2015](#vanSetten2015) (column EXTRA).
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[](#vanSetten2015) (column EXTRA).
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```none
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&FORCE_EVAL
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@ -169,19 +169,19 @@ with 1/N_basis -11.97 +/- 0.02 1.71 +/- 0.29
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GW100 -12.05 2.01
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```
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For the extrapolation, two schemes have been used as described in [vanSetten2015](#vanSetten2015)
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and its supporting information. The first scheme employs a linear fit on the HOMO or LUMO values
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when they are plotted against the inverse cardinal number $N_\text{card}$ of the basis set while the
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second scheme extrapolates versus the inverse number of basis functions $N_\text{basis}$ which can
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be computed as sum of the number of occupied orbitals and the number of virtual orbitals as printed
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in RI_INFO in the output. You can check the extrapolation from the table above with your tool of
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For the extrapolation, two schemes have been used as described in [](#vanSetten2015) and its
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supporting information. The first scheme employs a linear fit on the HOMO or LUMO values when they
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are plotted against the inverse cardinal number $N_\text{card}$ of the basis set while the second
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scheme extrapolates versus the inverse number of basis functions $N_\text{basis}$ which can be
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computed as sum of the number of occupied orbitals and the number of virtual orbitals as printed in
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RI_INFO in the output. You can check the extrapolation from the table above with your tool of
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choice.
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The basis set extrapolated values from the table above deviate from the values reported in
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[vanSetten2015](#vanSetten2015), probably because only two basis sets (def2-TZVP, def2-QZVP) have
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been used in [vanSetten2015](#vanSetten2015) for the extrapolation. The extrapolation for the LUMO
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is not working well because one would need much more diffuse functions to represent unbound
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electronic levels (with positive energy).
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been used in [](#vanSetten2015) for the extrapolation. The extrapolation for the LUMO is not working
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well because one would need much more diffuse functions to represent unbound electronic levels (with
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positive energy).
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Often, the HOMO-LUMO gap is of interest. In this case, augmented basis sets (e.g. from the EMSL
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database) can offer an alternative for very fast basis set convergence, see also Fig. 2b in
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@ -204,9 +204,17 @@ The G0W0@PBE HOMO value of the H2O molecule (~ -12.0 eV) is not in good agreemen
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experimental ionization potential (12.62 eV). Benchmarks on molecules and solids indicate that
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self-consistency of eigenvalues in the Green's function G improves the agreement between the GW
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calculation and experiment. This scheme is called GW0@PBE and is our favorite GW flavor so far
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(experience based on nano-sized aromatic molecules). You can also check
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[this video](https://www.youtube.com/watch?v=1vUuethWhbs&t=5563s) or [Golze2019](#Golze2019) on
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which DFT starting functional and which self-consistency scheme could be good for your system.
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(experience based on nano-sized aromatic molecules). You can also check the following video or
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[](#Golze2019) on which DFT starting functional and which self-consistency scheme could be good for
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your system.
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```{youtube} 1vUuethWhbs
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---
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url_parameters: ?start=5563
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align: center
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privacy_mode:
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---
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```
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You can run GW0 calculations in CP2K by putting
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@ -2,19 +2,18 @@
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In this section, we discuss the basics for computing optical properties of molecules using the
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Bethe-Salpeter equation (BSE) in CP2K. The BSE enables the computation of electronic excitation
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energies and optical absorption spectra, for a review, see \[[Blase2018](#Blase2018),
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[Blase2020](#Blase2020), [Bruneval2015](#Bruneval2015), [Sander2015](#Sander2015)\]. In this howto,
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we describe in Sec. [1](#header-theory) the theory and implementation of BSE, in Sec.
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[2](#header-input) the BSE input keywords and in Sec. [3](#header-example) a full CP2K input file of
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a BSE calculation and the corresponding output.
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energies and optical absorption spectra, for a review, see \[[](#Blase2018), [](#Blase2020),
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[](#Bruneval2015), [](#Sander2015)\]. In this howto, we describe in Sec. [1](#header-theory) the
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theory and implementation of BSE, in Sec. [2](#header-input) the BSE input keywords and in Sec.
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[3](#header-example) a full CP2K input file of a BSE calculation and the corresponding output.
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(header-theory)=
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## 1. Theory and implementation of BSE
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A central goal of a BSE calculation is to compute electronic excitation energies
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$\Omega^{(n)}, n=1,2,\ldots$ (cf. Refs. \[[Blase2018](#Blase2018), [Blase2020](#Blase2020)\] for
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more usecases and details).
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$\Omega^{(n)}, n=1,2,\ldots$ (cf. Refs. \[[](#Blase2018), [](#Blase2020)\] for more usecases and
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details).
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The following ingredients are necessary for computing $\Omega^{(n)}$:
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@ -24,7 +23,7 @@ The following ingredients are necessary for computing $\Omega^{(n)}$:
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- $GW$ eigenvalues $\varepsilon_i^{GW}$ and $\varepsilon_a^{GW}$ of corresponding KS orbitals.
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In CP2K, it is possible to use $G_0W_0$, ev$GW_0$ or ev$GW$ eigenvalues, see details in [GW] and in
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Ref. \[[Golze2019](#Golze2019)\], i.e. we perform BSE@$G_0W_0$/ev$GW_0$/ev$GW$@DFT. Thus, also input
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Ref. \[[](#Golze2019)\], i.e. we perform BSE@$G_0W_0$/ev$GW_0$/ev$GW$@DFT. Thus, also input
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parameters for a DFT and $GW$ calculation can be given (see full input in Sec.
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[3.1](#header-input-file)). We obtain optical properties from BSE solving the following generalized
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eigenvalue problem that involves the block matrix $ABBA$:
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@ -33,7 +32,7 @@ $$\left( \begin{array}{cc}A & B\\B & A\end{array} \right)\left( \begin{array}{
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We abbreviate $A$ and $B$ as matrices with index $A_{ia,jb}$, i.e. they have
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$N_\mathrm{occ}N_\mathrm{empty}$ rows and $N_\mathrm{occ}N_\mathrm{empty}$ columns. The entries of
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$A$ and $B$ are given by \[[Blase2018](#Blase2018)\]
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$A$ and $B$ are given by \[[](#Blase2018)\]
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$$ \begin{align}
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A_{ia,jb} &= (\varepsilon_a^{GW}-\varepsilon_i^{GW})\delta_{ij}\delta_{ab} + \alpha^\mathrm{S/T}
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@ -47,7 +46,7 @@ interaction and $W_{pq,rs}(\omega=0)$ the statically ($\omega=0$) screened Coulo
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where $p,q,r,s \in [ 1, N_\mathrm{occ}+N_\mathrm{empty}]$ are KS orbital indices.
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$(\mathbf{X}^{(n)},\mathbf{Y}^{(n)})$ with elements $X_{ia}^{(n)}$ and $Y_{ia}^{(n)}$ are the
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eigenvectors of the excitation $n$ which relate to the wavefunction of the electronic excitation
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\[[Blase2020](#Blase2020)\],
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\[[](#Blase2020)\],
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$$ \begin{align}
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\Psi_\text{excitation}^{(n)}(\mathbf{r}_e,\mathbf{r}_h) = \sum_{ia} X_{ia}^{(n)} \varphi_i(\mathbf{r}_h) \varphi_a(\mathbf{r}_e) + Y_{ia}^{(n)} \varphi_i(\mathbf{r}_e) \varphi_a(\mathbf{r}_h) \quad ,
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@ -114,7 +113,7 @@ In the upper GW/BSE section, the following keywords have been used:
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Setting a small `ENERGY_CUTOFF_OCC` drastically reduces the computation time and the memory
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consumption, but also might affect the computed excitation energies $\Omega^{(n)}$. Recommended to
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use for large systems with more than 30 atoms, but we recommend a careful convergence test by
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increasing `ENERGY_CUTOFF_OCC` and observing the effect on $\Omega^{(n)}$ \[[Liu2020](#Liu2020)\].
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increasing `ENERGY_CUTOFF_OCC` and observing the effect on $\Omega^{(n)}$ \[[](#Liu2020)\].
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- [ENERGY_CUTOFF_EMPTY](#CP2K_INPUT.FORCE_EVAL.DFT.XC.WF_CORRELATION.RI_RPA.GW.BSE.ENERGY_CUTOFF_EMPTY)
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$E_\text{cut}^\text{empty}$: Analogous to `ENERGY_CUTOFF_OCC`, but for the empty states, i.e. only
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@ -126,9 +125,9 @@ excitation energies:
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- [XC_FUNCTIONAL](#CP2K_INPUT.FORCE_EVAL.DFT.XC.XC_FUNCTIONAL): Choose between one of the available
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xc-functionals. The starting point can have a profound influence on the excitation energies
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\[[Bruneval2015](#Bruneval2015), [Gui2018](#Gui2018), [Jacquemin2017](#Jacquemin2017)\]. We either
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recommend the PBE functional as DFT starting point when using BSE@ev$GW_0$@PBE or the PBE0
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functional, when using BSE@$G_0W_0$@PBE0 (see also
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\[[Bruneval2015](#Bruneval2015), [](#Gui2018), [](#Jacquemin2017)\]. We either recommend the PBE
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functional as DFT starting point when using BSE@ev$GW_0$@PBE or the PBE0 functional, when using
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BSE@$G_0W_0$@PBE0 (see also
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[SELF_CONSISTENCY](#CP2K_INPUT.FORCE_EVAL.DFT.XC.WF_CORRELATION.RI_RPA.GW.SELF_CONSISTENCY)).
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- [BASIS_SET](#CP2K_INPUT.FORCE_EVAL.SUBSYS.KIND.BASIS_SET): Specify the basis set, which affects
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@ -144,7 +143,7 @@ $N_\mathrm{empty}$ and the estimated memory consumption from the BSE output. The
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is well parallelized, i.e. you can use several nodes that can provide the memory.
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We have benchmarked the numerical precision of our BSE implementation and we found excellent
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agreement within only 10 meV compared to the BSE implementation in FHI aims \[[Liu2020](#Liu2020)\].
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agreement within only 10 meV compared to the BSE implementation in FHI aims \[[](#Liu2020)\].
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The current BSE implementation in CP2K works for molecules. The inclusion of periodic boundary
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conditions in a $\Gamma$-only approach and with full $k$-point sampling is work in progress.
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@ -243,7 +242,7 @@ The basis sets `aug-cc-pVDZ` and `aug-cc-pVDZ-RIFIT` in `BASIS-aug` can be obtai
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Set Exchange Library:
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<a href="https://www.basissetexchange.org/basis/aug-cc-pvdz/format/cp2k/?version=1&elements=1" target="_blank">aug-cc-pVDZ</a>,
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<a href="https://www.basissetexchange.org/basis/aug-cc-pvdz-rifit/format/cp2k/?version=1&elements=1" target="_blank">aug-cc-pVDZ-RIFIT</a>.
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The geometry for $\mathrm{H}_2$ was taken from [vanSetten2015](#vanSetten2015).
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The geometry for $\mathrm{H}_2$ was taken from [](#vanSetten2015).
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### 3.2 Output
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@ -516,8 +516,9 @@ excited state for the Oxygen 1s is degenerate: there are two available excited s
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the $xy$-plane. That is why a third projection is requested which uses the other excited state
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proposed by the XAD_TDP module. Therefore, the excited state should be understood as the sum over
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the two equivalent excited states:
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\$$
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$$
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n_{Exc}(t) = \rho_{\omega}(t) + \rho_{\omega'}(t)
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$$
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where $\\omega'$ stands for the other equivalent excited state, with $\\omega=\\omega'=529\$
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eV.
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where $\omega'$ stands for the other equivalent excited state, with $\omega=\omega'=529$ eV.
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