Coupling the Qbox and WEST codes H. Ma1,2, N. L. Nguyen1, G. Galli1,2,3, M. Govoni1,3, F. Gygi4 1Institute for Molecular Engineering, University of Chicago, Chicago, IL 60637 2Department of Chemistry, University of Chicago, Chicago, IL 60637 3Materials Science Division, Argonne National Laboratory, Lemont, IL 60439 4Department of Computer Science, University of California Davis, Davis, CA 95616.

GW calculations beyond the random IntroductionIntroduction Bethe-Salpeter Equation phase approximation

Most GW calculations are performed within the random phase approximation (RPA) where the exchange- Solution of the Bethe-Salpeter equation (BSE) provides valuable insight into the optical spectra of FPMD simulations of all correlation kernel 푓 is neglected in the density response function 휒. Based on finite-field calculations and solids. We developed an efficient approach to solve the BSE, based on the Liouville-Lanczos inorganic nanoparticles (InAs xc performed by coupling Qbox and WEST, 휒 can be evaluated beyond the RPA in a straightforward manner. in SnS matrix). Scalise et al.. algorithm and finite-field calculations. This approach avoids the explicit summation over empty electronic states and altogether the calculation of the dielectric matrix. The screened Coulomb interaction W is Qbox is a massively parallel implementation of first-principles Nature Nano. 2018. Eigenvalues of 휒 for Ar, SiH and CO molecules computed within and beyond the RPA: 4 2 obtained from finite-field calculations, using the coupling between the Qbox and WEST codes. (FPMD) based on the plane-wave formalism. FPMD simulations of We verified our BSE approach by computing the Code features: vibrational spectra and ionic singlet excitation energies (Δ) of the 28 molecules of conductivity of water in • Efficient hybrid DFT calculations using the recursive subspace extreme conditions. Rozsa et the Thiel’s set: bisection (RSB) technique al.. PNAS. 2018. • Simulation in arbitrary electric field • Calculation of vibrational spectra (IR, Raman) and ionic FPMD simulations of water and salt solutions with hybrid conductivity (eV) functionals. Gaiduk et al. • Calculation of thermal conductivity Δ JPCL. 2017; JPCL. 2018.

The kernel 푓xc can be computed by inverting the Dyson equation connecting FPMD+GW calculations of 휒0 and 휒. Semi-local and hybrid functionals are treated on an equal footing: electron affinity of water. −1 −1 Gaiduk et al. Nature Comm. 푓xc = 휒0 − 휒 − 푣c 2018. WEST is a massively parallel code for many body perturbation The 푓 matrices for Ar, SiH4 and CO molecules in the space of 휒 theory calculations. xc 2 0 Δ (quantum chemistry) (eV) FPMD+GW simulations of eigenpotentials: photoanodes (WO ) for water By utilizing compact representations of Kohn-Sham wavefunctions (bisection orbitals) as implemented in Code features: 3 Comparison between finite-field splitting. Gerosa et al. Nature the Qbox code, the computational cost of solving the BSE may be significantly reduced by neglecting pairs • Large-scale GW calculation without explicit calculation of and analytical evaluation of 푓 at Mat. 2018 xc of orbitals with small overlap. empty electronic states the LDA level. Small mean relative • Low-rank decomposition of dielectric matrix Optical absorption spectra of the C60 : difference (훥푓푥푐 ) indicates a good • Scalar-relativistic and full-relativistic calculations GW calculations of defects in semiconductors for qubit accuracy of the finite-field • GW starting from semi-local and hybrid DFT design. Seo et al. Sci. Rep. approach adopted to compute 푓xc. • GW with full frequency integration 2016; Nature Comm. 2016; Orbital overlap: PRM. 2017. 2 2 ׬ |휑푖| |휑푗| 푑푟 GW calculations beyond the RPA may be performed using 푂푖푗 = 푓 computed from coupling the Qbox and WEST codes: ′ 푑휔 ′ ′ ′ ′ 4 4 xc 훴 푟, 푟 ; 휔 = න 퐺 푟, 푟 ; 휔 + 휔 푊(푟, 푟 ; 휔 ) ׬ |휑푖| 푑푟 ׬ |휑푗| 푑푟 푥푐 2휋 퐺 푊RPA 휒 = 휒 + 휒 푣 휒 0 0 RPA 0 0 c 1 Qbox-WEST coupling 퐺 푟, 푟′; 휔 = 푟 푟′ fxc 휔 − 퐻 퐺0푊0 휒 = 휒0 + 휒0(푣c + 푓xc)휒 ′ ′′ ′′ ′′ ′ −1 −1 푊 푟, 푟 ; 휔 = න 푑푟 [훿 푟, 푟 + 휒 푟, 푟 ; 휔 ] 푣푐(푟′′, 푟 ) Coupling the Qbox and WEST codes: 퐺0푊0Γ0 휒Γ = [푣c − 푣c휒0(푣c + 푓xc)] −푣c • WEST performs iterative diagonalization of response functions within or beyond the random phase approximation. By combining FPMD simulations with BSE and GW calculations, we computed the absorption spectra of ice • Qbox in client server mode serves as an efficient DFT engine Vertical ionization potential (VIP) and vertical electron affinity (VEA) for corrections for and water close to ambient conditions. In particular, we computed the imaginary part of the macroscopic performing calculations in finite electric fields. molecules: GW100 set the VBM and CBM of solids dielectric constant (퐈퐦[휺푴]) for water and ice by averaging over multiple snapshots of FPMD trajectories. • The scheme is readily applicable to calculations. • Orbital localization techniques implemented in Qbox reduce the Good agreement is found between computational cost. GW-BSE and experiment both for • The coupling can be performed in parallel, with WEST coupled the relative energy positions and simultaneously with multiple copies of Qbox. intensities of the peaks over a wide range of energy. The coupling between Qbox and WEST relies on a finite-field approach to evaluate density response functions (휒0, 휒RPA and 휒).

휒 = 휒0 + 휒0(푣c + 푓xc)휒 휒RPA = 휒0 + 휒0푣c휒

PBE400: www.quantum- simulation.org/reference/h2o/pbe400/s3 2/index.htm PIMD: path integral MD. A. Gaiduk, T. A. Pham, M. Govoni, F. Paesani, G. Galli. Nature Comm. 2018 sc-hybrids: J. Skone, M. Govoni, G. Galli. PRB. 2014 Bisection algorithm: F. Gygi. PRL. 2009

We developed a scheme to renormalize 푓xc to accelerate the convergence of 퐺0푊0Γ0 calculations: References and Acknowledgements

Code coupling verification: M. Govoni, G. Galli. JCTC. 2015. 11(6), 2680-2696. P. Scherpelz, I. Hamada, M. Govoni, G. Galli. JCTC. 2017. 12(8), 3523-3544. M. Govoni, G. Galli. JCTC. 2018. 14(4), 1895-1909. WEST was coupled to The eigenvalues of the renormalized H. Ma, M. Govoni, F.Gygi, G.Galli. 2018 (submitted). 240 Qbox instances to 푓xc are larger than or equal to the L. N. Nguyen, H. Ma, M. Govoni, F.Gygi, G.Galli. 2018 (submitted). perform the calculation. negative of the bare Coulomb interaction, thus guaranteeing the Work supported by MICCoM, as part of the Computational Materials Sciences Program funded by the U.S. rapid convergence of 퐺0푊0Γ0 Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division quasiparticle energies. through Argonne National Laboratory, under contract number DE-AC02-06CH11357. We used computational resources from the National Energy Research Scientific Computing Center (NERSC), Argonne Leadership 푓xc, 휒Γ and quasiparticle energies for the SiH4 molecule and bulk Si. Computing Facility and University of Chicago Research Computing Center. First 2048 eigenvalues of the density response function for a Si147H100 nanoparticle.