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

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

Theory and implementation

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

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

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

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

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

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:10.1021/ct300648t

Test results for molecules

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

Method comparison ionization potentials

Method comparison electron affinities

Higher Ionization potentials benzene

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 0.25 eV QZVP TZVP SVP TZVPP Extrapolated results cc-pV5Z cc-pVTZ cc-pVDZ cc-pVQZ

Basis set convergence Deviation from the extrapolated results

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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, 235102 VASP calculation

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

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

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

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 5.27 10.93 HOMO-LUMO Molecule in Junction 3.75 8.72 gap (eV)

Self-consistence in G SVP basis

Higher ionization energies H2O (solved qpe)

GW for solids: Bandgaps KRESSE PRB 75, 235102 HEDIN J PHYS, COND MAT. 11 R489 (SHIRLEY)

GW + BSE for solids: optical excitations Macroscopic dielectric functions Silicon NaH ONIDA, Rev Mod Phys 74, 601 VAN SETTEN PRB 83, 035422