1. The GW Method for Quantum Chemistry applications: Theory, Implementation, and Benchmarks
M.J. van Setten

3. plan
G0W0
As little approximations as possible, i.e., exact frequency treatment
Molecular systems and clusters
Local (Gaussian) basis-set
“general” approach, portable to other codes

4. Theory and implementation

5. KS v.s. Quasi-Particle equation
Electron density parameterized by KS single particle states
Greens function in spectral representation, expressed in terms of quasi-particle states leads to the quasi particle equation:
The quasi-particle energies are the electron removal and addition energies
Vxc:exchange correlation potential
Σ: self-energy
At this point all is exact, now we need a way to calculate Sigma
HEDIN Phys Rev 139, A796, HYBERTSEN and LOUIE PRB 34, 5390

6. G0W0 and linearized QP equation
Calculate the self-energy only once from KS-DFT energies and orbitals
Because of our spectral approach we can now evaluate these last experssions analitically

7. G0W0 and linearized QP equation
Calculate the self-energy only once from KS-DFT energies and orbitals
Spectral representation of the reducible polarization function
from TD-KS response equation
m labels excitations
Because of our spectral approach we can now evaluate these last experssions analitically

8. Reducible response function
transition densities
Generalized Dyson equation
As used in TDHF and TDDFT
RPA or TDDFT version of the orbital rotation Hessians

9. G0W0 and linearized QP equation
Calculate the self-energy only once from KS-DFT energies and orbitals
Linearized Quasi-Particle equation
Because of our spectral approach we can now evaluate these last experssions analitically

10. Matrix elements to be calculated
Implemented in TURBOMOLE (escf)
Very efficient evaluation of electron-electon integrals:
(Contracted) Gaussian basis sets
Resolution of the Identity (RI) method
MVS, F. Weigend, F Evers JCTC in press doi: /ct300648t

11. Test results for molecules

12. Test results and Benchmarking
Test set including:
H2, N2, F2, Li2, Na2, Cs2
NH3, SiH4, SF4
LiH, BF, CO2, H2O
Au2, Au4
Acetone, Acrolein, Ethylene, Isobutane
Methane – Butane, Benzene – Naphthacene
Method comparison
Basis-set convergence
Functional dependence

13. Method comparison ionization potentials

14. Method comparison electron affinities

15. Higher Ionization potentials benzene

16. G0W0 basis-set convergence
Tested for all molecules
In general linear extrapolation possible especially to the DEF2 results
TZVPP in this test set converged within eV
QZVP
TZVP
SVP
TZVPP
Extrapolated results
cc-pV5Z
cc-pVTZ
cc-pVDZ
cc-pVQZ

17. Basis set convergence
Deviation from the extrapolated results

18. G0W0 Functional dependence
Differences between G0W0 results (Δ) are much smaller than deviations between the respective DFT results
Largest differences for LiH, H2O and hydrocarbons with OH groups

19. Approximations within G0W0
Plasmon pole / model response
Analytic continuation / numerical integration
Neglect of semi-core state contributions / pseudo potentials
Linearized quasi particle equation (< 0.1 eV)
RI approximation (< 0.1 eV)
Difference RPA / TDDFT response (~ 0.1 eV)
Off-diagonal elements
> Perturbative correction
Neglect of self-consistency
> Partial self-consistency in the energies

20. Method testing New method, much testing needed
Basis sets
Approximations within GW
Collaboration with FHI Berlin and Berkeley Lab US DoE
F. Caruso and P. Rinke S. Sharifhades and J. Neaton
TURBOMOLE complementary to FHI-aims: Moderate v.s. Massive parallelization Gaussian v.s. numerical basis-sets Analytic v.s. numerical energy integration / analytic continuation
BerkeleyGW: plane waves, supercell, periodic boundary conditions
100 molecule benchmark set: GW100
Open shell systems
Metal cluster benchmark set: alkali, alkaline earth, transition, noble

21. 100 closed shell molecules
GW100
100 closed shell molecules
Simple dimers, noble gas atoms
Oxides, hydrides, fluorides
Small metal clusters
Hydrocarbons, Aromatic molecules,
Comparison between FHI-aims and TURBOMOLE with identical basis sets

22. 100 closed shell molecules
GW100
100 closed shell molecules
Simple dimers, noble gas atoms
Oxides, hydrides, fluorides
Small metal clusters
Hydrocarbons, Aromatic molecules,
Comparison between FHI-aims and TURBOMOLE with identical basis sets

23. Open shell systems Collinear case Matrices are spin-diagonal
Screening is comes from both spin channels

24. Acknowledgements
Prof. F. Evers
Dr. F. Weigend
Center for Functional Nanostructures

25. Summarizing
Formulation of the matrix elements for a G0W0 calculation in a format practical for evaluation in a Gaussian basis set
Effectiveness relies on the efficient evaluation of four- center-integrals
Basis set convergence similar to that of DFT
Already with G0W0 much reduced functional dependence
medium sized molecules are already feasible (without parallelization and symmetry exploitation)

26. Scaling of CPU time GW is part of ESCF
RI reduces effective scaling by more than power of one
Most time is spent at solving the TD-KS response equation
For larger systems not all excitations are needed

27. Further development
Open shell / spin polarized system
Symmetry / optimization / parallelization
Optical excitation spectra via BSE

28. Computational approaches
Density functional theory
Maps the many particle problem into a single particle problem
Scalar effective potential Vxc(r)
Applicable to large systems: > 2000 atoms
Broadly used across many fields of sciences
Description of electronic levels (charge transfer), long range interactions, and band gaps not accurate enough
GW-method
Replace the potential by a self-energy matrix: Vxc(r) → Σ(r,r’,E)
Becoming increasingly available for bulk solids
Significantly improved description of electronic levels (charge transfer), long range interactions, and band gaps
Improved experimental conditions

29. GW approximation in a nutshell
Calculating the self-energy
Perturbative expansion in terms of the screened interaction instead of the bare Coulomb interaction (Hedin equations)
First order
Bare Coulomb interaction
Screened Coulomb interaction

30. Approximations within GW
Plasmon pole / model response
Analytic continuation
Neglect of core state contributions
Linearized quasi particle equation (< 0.1 eV)
RPA response (~ 0.1 eV)
Neglect of off-diagonal elements
Neglect of self-consistency
Vertices

31. Approximations within GW
Plasmon pole / model response
Analytic continuation
Neglect of core state contributions
Linearized quasi particle equation (< 0.1 eV)
RPA response (~ 0.1 eV)
Neglect of off-diagonal elements
Neglect of self-consistency
Vertices

32. Approximations within GW
Plasmon pole / model response
Analytic continuation
Neglect of core state contributions
Linearized quasi particle equation (< 0.1 eV)
RPA response (~ 0.1 eV)
Neglect of off-diagonal elements
Neglect of self-consistency
Vertices
KRESSE PRB 75,
VASP calculation

33. Approximations within GW
Plasmon pole / model response
Analytic continuation
Neglect of core state contributions
Linearized quasi particle equation (< 0.1 eV)
RPA response (~ 0.1 eV)
Neglect of off-diagonal elements
Neglect of self-consistency
Vertices
Full GW self-consistency for N2
Courtesy P. Rinke FHI Berlin
FHI-aims calculation

34. Hedin Equations
Space time notation (the numbers indicate a contracted space, time and spin index)
HEDIN Phys Rev 139, A796

35. Hedin Equations
Space time notation (the numbers indicate a contracted space, time and spin index)
Neglecting the second term in the vertex function leads to the GW approximation for the self-energy Fourier transformed to frequency domain:
HEDIN Phys Rev 139, A796

36. Transport setup (collaboration with A. Bagrets)
A first test of the effect of G0W0 shifts of the QP levels on the transmission-function.
BP86 G0W0
Bare molecule HOMO-LUMO
Molecule in Junction gap (eV)

37. Self-consistence in G
SVP basis

38. Higher ionization energies
H2O (solved qpe)

39. GW for solids: Bandgaps
KRESSE PRB 75,
HEDIN J PHYS, COND MAT. 11 R489 (SHIRLEY)

40. GW + BSE for solids: optical excitations
Macroscopic dielectric functions
Silicon
NaH
ONIDA, Rev Mod Phys 74, 601
VAN SETTEN PRB 83,