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BSE: update references (#5515)
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Max Graml
GitHub
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3 changed files with 46 additions and 27 deletions
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@ -1,10 +1,11 @@
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# *GW* + Bethe-Salpeter equation
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The Bethe-Salpeter equation (BSE) is a method for computing electronic excitation energies and
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optical absorption spectra. We describe in Sec. [1](#header-theory) the theory and implementation of
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BSE, in Sec. [2](#header-input) the BSE input keywords and in Sec. [3](#header-example) a full CP2K
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input file of a BSE calculation and the corresponding output. For reviews on BSE, see
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\[[](#Blase2018), [](#Blase2020), [](#Bruneval2015), [](#Sander2015)\].
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optical absorption spectra. We repeat the theory and implementation of BSE from \[[](#Graml2026)\]
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in Sec. [1](#header-theory), 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. For
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further references on BSE, see \[[](#Blase2018), [](#Blase2020), [](#Bruneval2015),
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[](#Sander2015)\].
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(header-theory)=
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@ -52,7 +53,7 @@ where $p,q,r,s \in [ 1, N_\mathrm{occ}+N_\mathrm{empty}]$ are KS orbital indices
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the screened Coulomb interaction is always computed from DFT quantitites, i.e. $W_{0}(\omega=0)$
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enters the BSE. $(\mathbf{X}^{(n)},\mathbf{Y}^{(n)})$ with elements $X_{ia}^{(n)}$ and
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$Y_{ia}^{(n)}$ are the eigenvectors of the excitation $n$, which relate to the wave function of the
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electronic excitation \[[](#Blase2020)\],
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electronic excitation,
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$$
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\begin{align}
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@ -237,8 +238,7 @@ where we drop the excitation index $n$ from now on for better readability.
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For each excitation level $n$, we have then several quantities, which allow us to quantify the
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spatial extent of the electron, the hole and their combined two-particle character as combined
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electron-hole pair, i.e. as an exciton. By that, we can often determine the type of the excited
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state, i.e. distinguish between, e.g., valence, Rydberg or charge-transfer states
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\[[](#Mewes2018)\].
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state, i.e. distinguish between, e.g., valence, Rydberg or charge-transfer states.
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First, we define the distance between electron and hole as
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@ -352,7 +352,7 @@ 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 $\Omega^{(n)}$ \[[](#Liu2020)\].
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increasing `ENERGY_CUTOFF_OCC` and observing the effect on $\Omega^{(n)}$ \[[](#Graml2026)\].
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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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@ -394,8 +394,8 @@ optical properties:
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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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\[[](#Knysh2024)\]. Motivated by the discussion in \[[](#Schambeck2024)\], we strongly recommend
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to use BSE@ev*GW*<sub>0</sub>@PBE, i.e. the PBE functional as DFT starting point (see also
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\[[](#Graml2026)\]. Motivated by the discussion in \[[](#Graml2026)\], we strongly recommend to
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use BSE@ev*GW*<sub>0</sub>@PBE, i.e. the PBE functional as DFT starting point (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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$N_\mathrm{empty}$ and thus the size of the matrices $A_{ia,jb}$ and $B_{ia,jb}$. The
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@ -410,11 +410,11 @@ $100 \cdot N_\mathrm{occ}^2 N_\mathrm{empty}^2$ Bytes. You can see $N_\mathrm{oc
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$N_\mathrm{empty}$ and the estimated memory consumption from the BSE output. The BSE implementation
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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 compared its results to
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the BSE implementation in FHI aims \[[](#Liu2020)\]. For our recommended settings, i.e.
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BSE@ev*GW*<sub>0</sub>@PBE with the aug-cc-pVDZ basis set, we have found excellent agreement with
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less than 5 meV mean absolute deviation averaged over the first 10 excitation levels and the 28
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molecules in *Thiel's set* for Singlet excitations.
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We have benchmarked the numerical precision of our BSE implementation in \[[](#Graml2026)\] and
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compared its results to the BSE implementation in FHI aims \[[](#Liu2020)\]. For our recommended
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settings, i.e. BSE@ev*GW*<sub>0</sub>@PBE with the aug-cc-pVDZ basis set, we have found excellent
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agreement with less than 5 meV mean absolute deviation averaged over the first 10 excitation levels
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and the 28 molecules in *Thiel's set* for Singlet excitations.
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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 Γ-only approach and with full *k*-point sampling is work in progress.
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@ -13,6 +13,8 @@
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MODULE bse_print
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USE atomic_kind_types, ONLY: get_atomic_kind
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USE bibliography, ONLY: Graml2026,&
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cite_reference
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USE bse_properties, ONLY: compute_and_print_absorption_spectrum,&
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exciton_descr_type
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USE bse_util, ONLY: filter_eigvec_contrib
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@ -59,6 +61,7 @@ CONTAINS
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INTEGER :: handle
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CALL timeset(routineN, handle)
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CALL cite_reference(Graml2026)
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IF (unit_nr > 0) THEN
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WRITE (unit_nr, *) ' '
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@ -126,8 +129,8 @@ CONTAINS
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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WRITE (unit_nr, '(T2,A4,T7,A41)') 'BSE|', 'sum_jb ( A_ia,jb X_jb^n ) = Ω^n X_ia^n'
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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WRITE (unit_nr, '(T2,A4,T7,A30)') 'BSE|', 'prelim Ref.: Eq. (36) with B=0'
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WRITE (unit_nr, '(T2,A4,T7,A71)') 'BSE|', 'in PRB 92,045209 (2015); http://dx.doi.org/10.1103/PhysRevB.92.045209 .'
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WRITE (unit_nr, '(T2,A4,T7,A14)') 'BSE|', 'Ref.: Eq. (18)'
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WRITE (unit_nr, '(T2,A4,T7,A61)') 'BSE|', 'in PRB 113,205152 (2026); https://doi.org/10.1103/38k2-d55h .'
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ELSE
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WRITE (unit_nr, '(T2,A4,T7,A74)') 'BSE|', '**************************************************************************'
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WRITE (unit_nr, '(T2,A4,T7,A74)') 'BSE|', '* Full ("ABBA") Bethe Salpeter equation (BSE) (i.e. without TDA) *'
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@ -143,6 +146,9 @@ CONTAINS
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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WRITE (unit_nr, '(T2,A4,T7,A62)') 'BSE|', ' sum_jb ( A_ia,jb X_jb^n + B_ia,jb Y_jb^n ) = Ω^n X_ia^n'
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WRITE (unit_nr, '(T2,A4,T7,A62)') 'BSE|', '- sum_jb ( B_ia,jb X_jb^n + A_ia,jb Y_jb^n ) = Ω^n Y_ia^n'
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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WRITE (unit_nr, '(T2,A4,T7,A13)') 'BSE|', 'Ref.: Eq. (8)'
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WRITE (unit_nr, '(T2,A4,T7,A61)') 'BSE|', 'in PRB 113,205152 (2026); https://doi.org/10.1103/38k2-d55h .'
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END IF
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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@ -164,8 +170,8 @@ CONTAINS
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WRITE (unit_nr, '(T2,A4,T7,A)') 'BSE|', 'B_ia,jb = α * v_ia,jb'
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END IF
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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WRITE (unit_nr, '(T2,A4,T7,A35)') 'BSE|', 'prelim Ref.: Eqs. (24-27),(30),(35)'
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WRITE (unit_nr, '(T2,A4,T7,A71)') 'BSE|', 'in PRB 92,045209 (2015); http://dx.doi.org/10.1103/PhysRevB.92.045209 .'
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WRITE (unit_nr, '(T2,A4,T7,A14)') 'BSE|', 'Ref.: Eq. (10)'
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WRITE (unit_nr, '(T2,A4,T7,A61)') 'BSE|', 'in PRB 113,205152 (2026); https://doi.org/10.1103/38k2-d55h .'
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END IF
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IF (.NOT. flag_TDA) THEN
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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@ -382,10 +388,10 @@ CONTAINS
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WRITE (unit_nr, '(T2,A4,T9,A2,T14,A18)') &
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'BSE|', "r:", "position operator."
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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WRITE (unit_nr, '(T2,A4,T7,A28)') &
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'BSE|', "prelim Ref.: Eqs. (23), (24)"
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WRITE (unit_nr, '(T2,A4,T7,A71)') &
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'BSE|', "in J. Chem. Phys. 152, 044105 (2020); https://doi.org/10.1063/1.5123290"
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WRITE (unit_nr, '(T2,A4,T7,A21)') &
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'BSE|', "Ref.: Eqs. (21), (D3)"
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WRITE (unit_nr, '(T2,A4,T7,A69)') &
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'BSE|', "in Phys. Rev. B 113, 205152 (2026); https://doi.org/10.1103/38k2-d55h"
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WRITE (unit_nr, '(T2,A4)') 'BSE|'
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IF (flag_TDA) THEN
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WRITE (unit_nr, '(T2,A4,T7,A55)') 'BSE|', &
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@ -596,10 +602,10 @@ CONTAINS
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prefix_output, "r:", "position operator."
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WRITE (unit_nr, '(T2,A4)') prefix_output
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WRITE (unit_nr, '(T2,A4,T7,A)') prefix_output, &
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'prelim Ref.: Eqs. (15)-(22)'
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'Ref.: Eqs. (28)-(32)'
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WRITE (unit_nr, '(T2,A4,T7,A,A)') prefix_output, &
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'JCTC 2018, 14, 710-725; ', &
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'http://doi.org/10.1021/acs.jctc.7b01145'
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'Phys. Rev. B 113, 205152 (2026); ', &
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'https://doi.org/10.1103/38k2-d55h'
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WRITE (unit_nr, '(T2,A4)') prefix_output
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WRITE (unit_nr, '(T2,A4)') prefix_output
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IF (exc_descr(1)%flag_TDA) THEN
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@ -660,6 +666,12 @@ CONTAINS
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WRITE (unit_nr, '(T2,A4,T15,A)') prefix_output, &
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"R_eh^{μμ'} = COV_eh^{μμ'}/(σ^μ_e σ^μ_h) "
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WRITE (unit_nr, '(T2,A4)') prefix_output
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WRITE (unit_nr, '(T2,A4,T7,A)') prefix_output, &
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'Ref.: Eqs. (33)-(36)'
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WRITE (unit_nr, '(T2,A4,T7,A,A)') prefix_output, &
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'Phys. Rev. B 113, 205152 (2026); ', &
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'https://doi.org/10.1103/38k2-d55h'
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WRITE (unit_nr, '(T2,A4)') prefix_output
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WRITE (unit_nr, '(T2,A4)') prefix_output
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IF (exc_descr(1)%flag_TDA) THEN
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WRITE (unit_nr, '(T2,A4,T7,A,A,A)') prefix_output, &
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@ -93,7 +93,7 @@ MODULE bibliography
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Sander2015, Schreiber2008, vanSetten2015, Setyawan2010, Ahart2024, Knysh2024, &
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Schambeck2024, Mewes2018, Sertcan2024, Drautz2019, Lysogorskiy2021, Bochkarev2024, &
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VazdaCruz2021, Chen2025, Hernandez2025, Marek2025, Hehn2022, Hehn2024, Pasquier2025, &
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Hanasaki2025, Tan2025, Broyden1965, Johnson1988, Kerker1981, &
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Hanasaki2025, Graml2026, Tan2025, Broyden1965, Johnson1988, Kerker1981, &
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FuHo1983, MethfesselPaxton1989, Marzari1999, dosSantos2023, Mermin1965, &
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KuhneHeskeProdan2020, Schreder2021, Schreder2024_1, Schreder2024_2, &
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Shiga2022, Lindh1995, Chai2024a, Rullan2026, Sundararaman2017, Andreussi2019, &
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@ -2093,6 +2093,13 @@ CONTAINS
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source="Phys. Rev. B", volume="112", pages="205130", &
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year=2025, doi="10.1103/v4zv-1pf9")
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CALL add_reference(key=Graml2026, &
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authors=s2a("M. Graml", "J. Wilhelm"), &
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title="Optical excitations in nanographenes from the Bethe-Salpeter equation "// &
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"and time-dependent density functional theory: Absorption spectra and spatial descriptors", &
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source="Phys. Rev. B", volume="113", pages="205152", &
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year=2026, doi="10.1103/38k2-d55h")
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CALL add_reference(key=Hanasaki2025, &
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authors=s2a("K. Hanasaki", "T. Futaii de Jong", "K. Komarov", "R. Kumar", "M. Malis", &
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"J. Mattiat", "L. I. Hernandez-Segura", "L. Schreder", "A. Sinyavskiy", "S. Luber"), &
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