Session B44: Real-Space Methods for the Electronic Structure Problem II

Real space and real time electron dynamics simulations for attosecond physics in solids

Real-space and real-time electron dynamics simulation based on the time-dependent density functional theory is a powerful tool to analyze complex and highly-nonlinear interactions of light with solids. To investigate laser-induced ultrafast electron dynamics in solids, we developed a numerical technique to simulate pump-probe experiments [1]. Recently, we applied the numerical pump-probe simulations for the attosecond transient absorption spectroscopy and studied the light-induced ultrafast electron dynamics in solids [2,3]: First, we investigated laser-induced electron dynamics in GaAs with the first-principles simulations. As a result, we found an important role of the light-induced intraband transition in transient optical properties of optically-driven semiconductors [2]. Then, we investigated ultrafast electron dynamics in Titanium, and the first-principles simulations provided microscopic insight into laser-induced electron-localization dynamics in transition metals [3].

[1] S. A. Sato, K. Yabana, Y. Shinohara, T. Otobe, G. F. Bertsch, Phys. Rev. B 89, 064304 (2014).
[2] F. Schlaepfer, M. Lucchini, S. A. Sato, M. Volkov, L. Kasmi, N. Hartmann, A. Rubio, L. Gallmann, U. Keller, Nature Physics 14, 560 (2018).
[3] M. Volkov, S. A. Sato, F. Schlaepfer, L. Kasmi, N. Hartmann, M. Lucchini, L. Gallmann, A. Rubio, U. Keller, Nature Physics (2019): https://doi.org/10.1038/s41567-019-0602-9   

Speeding up real-space based first-principles methods for excited-states properties using interpolative separable density fitting

Density fitting methods are a class of algorithms that provide low-rank approximations for products of orbital pairs. However, most of its applications in first-principles codes are based on localized basis sets. A recently proposed interpolative separable density fitting (ISDF) method does not rely on predefined auxiliary basis and is formulated in real space representation. We employ the ISDF method to significantly reduce the cost of linear response time-dependent density functional theory (LR-TDDFT) and GW calculations. In our implementation, we exploit the symmetry property of a system to effectively reduce the number of auxiliary basis and thus the computational cost. Our benchmarks show the cost for constructing auxiliary basis and interpolation coefficients are negligible compared to the total computational cost. Compared to a conventional “brutal-force” approach, the cost for evaluating all kernel matrix elements in LR-TDDFT and GW calculations is reduced by up to three orders of magnitude. The accuracy of our implementation is benchmarked with the GW100 set.
  

Ab initio stochastic time-domain formulation of the Bethe-Salpeter equation

We present a reduced scaling formulation of the Bethe-Salpeter equation (BSE) through a combination of a time-dependent approach and stochastic representations, which allows efficient ab initio calculations of optical absorption spectra for semiconductor nanoparticles. The linear response of the system to external fields is simulated by propagating quasiparticle orbital dynamics in real time, followed by Fourier transforming the dipole-dipole correlation function to obtain the absorption spectrum. The spatially dependent screening of the system is described within the random phase approximation (RPA) and combines several efficient stochastic techniques, including factorization with stochastic representations, and time-dependent Hartree propagation of stochastic occupied orbitals. The computational cost of the new BSE formulation is quadratic scaling with respect to system size, which is a significant improvement compared with the conventional symplectic eigenvalue representation of the BSE. We discuss preliminary results from the applications of the method to silicon and CdSe nanocrystals.   

Fast real-time time-dependent density functional theory calculations using higher-order finite element methods

We present a computationally efficient approach to solve the time-dependent Kohn-Sham equations in real time using higher order finite-element spatial discretization, applicable to both pseudopotential and all-electron calculations. To this end, we develop an apriori mesh adaption technique, based on the semidiscrete (discrete in space but continuous in time) error estimate on the time-dependent Kohn-Sham orbitals, to construct an efficient finite-element discretization. Subsequently, we obtain the full-discrete error estimate to guide our choice of the time step. Importantly, for the all-electron case, we present an efficient mixed basis, termed as enriched finite element basis, that combines the efficiency of atomic-orbitals-type basis to capture the sharp variations of the electronic fields closer to the atoms along with the completeness and spatial-adaptivity of the finite element basis. We demonstrate significant savings afforded by our approach over the finite-difference and Gaus-
sian basis based approaches, for pseudopotential and all-electron calculations, respectively. Additionally, we discuss schemes to accelerate the time-evolution by constructing dynamic subspace based approaches, informed by time-dependent perturbation theory.   

Recent Advances in the Development of the Octopus Code for GPUs

Since the end of Moore's law, using state-of-the-art supercomputers has become more and more difficult because simulation codes need to expose more and more parallelism to benefit from larger machines. Moreover, supercomputers have become heterogeneous as the increase in CPU performance has slowed down: all of the first three exascale machines to be built in the US will heavily rely on GPUs to achieve their compute performance. Thus, simulation codes in all areas, also real-space DFT codes, will need to utilize GPUs efficiently in order to use these next-generation machines. In this contribution, we report on recent advances in the development of the real-space DFT code Octopus for GPUs. Those developments include porting more algorithms to GPUs, both for ground-state and time-dependent calculations, improving the data handling when copying data to and from the GPU and improving the overlap of communication with computation. We will show how these advances improve the speed of the code and its scalability and apply the code to some larger problem sizes on GPUs.   

Excitonic effects from real-time parameter-free hybrid functions

Hybrid functionals are know to provide excellent band-gap of semiconductors compared to local and semi-local functional. Because hybrid functionals can be constructed to include the long-range part of the screened Coulomb interaction, they contains the basic ingredients that can produce the bound excitons in optical spectra of semiconductors and insulators. So far, the construction of such parameter-free functional requires complex procedures such that optimally-tune parameters [], or fitting of the band-gap on GW calculations []. This hampers the predictive power of such a functional and ancors it to the realm of optical properties of materials at equilibrium.
Because real-time time-dependent density functional theory (TDDFT) does not rely on perturbation theory, it has the capability to study materials out of their equilibrium or perturbed by strong fields. It is therefore of tremendous importance to find hybrid functionals that do not requires auxiliaryt calculations to adjust parameters, in order to bring their predictive power to nonlinear and out-of-equilibrium optical properties.

[1] Refaely-Abramson et al., Phys. Rev. B 92, 081204(R) (2015)
[2] Wing et al., Phys. Rev. Materials 3, 064603 (2019)   

Ab-initio photo-ionization dynamics without continuum states

Photo-ionization underpins a range of spectroscopies central to the study of structural and dynamical properties of matter in the gas and solid phases. In solid state physics angular resolved photoelectron spectroscopy (ARPES) and time-resolved (tr) ARPES are the most prominent techniques. Leveraging the flexibility offered by real-space methods we developed a technique, based on the real-time formulation of time-dependent density functional theory (TDDFT), to simulate ARPES and tr-ARPES ab-initio without explicit reference to continuum states [1]. I will present the theory, the algorithm involved in the implementation and some of the most representative applications and predictions.

[1] U. De Giovannini, H Hübener, A. Rubio, JCTC. 13, 265 (2017).   

Efficient electromagnetic potential calculations within ground state and time dependent density functional theory.

Electromagnetic potentials play a fundamental role in ground and excited state calculations of Density Functional Theory (DFT). One method for the calculation of such potentials is to directly solve their corresponding differential equations. In this work, the equivalent integral expressions of those potentials are evaluated within their spectral representation. Such integral expressions play a critical role in handling problems ranging from hybrid functionals to the calculation of retarded potentials within the time-dependent Kohn-Sham equation. The simplest of these are FFT procedures used to evaluate the static Poisson integral within ground-state calculations. We extend the use of these integral expressions to time-dependent retarded electromagnetic potentials in both the Lorenz and Coulomb gauges and demonstrate the efficiency of the approach. We use this method for the calculation of several electronic structures using a real-time real-space TDDFT approach. Finally, the various gauge-fixing conditions are compared to assure alignment in the limit of small magnetic fields and the advantages of the Lorenz gauge are outlined.   

Real-space approach to the calculation for ionization potentials, exciton, and biexciton binding energies in quantum dots using explictly-correlated electron-hole interaction kernel method

Inclusion of unoccupied states is the leading computational bottleneck for calculation of excited states for large chemical systems. In this work, we present the geminal-screened electron-hole interaction kernel (GSIK) method to address this problem. The GSIK is a real-space r12 method that avoids unoccupied orbitals for constructing the electron-hole interaction kernel by performing an infinite-order diagrammatic summation of particle-hole excitations and deriving a renormalized real-space electron-hole correlator operator. The GSIK method also bypasses the computational expensive AO-to-MO integral transformation by computing all integrals directly in the real-space numerically using permutation sampling Monte Carlo method. These two features allow GSIK method to be used for chemical systems where inclusion of a large number of unoccupied orbitals will be computationally prohibitive. In this work, the GISK method was applied to investigate exciton binding energies, biexciton binding energies, and ionization potentials for large semiconductor (Pb₁₄₀S₁₄₀, Pb₁₄₀Se₁₄₀, Cd₁₄₄Se₁₄₄) nanoparticles. The results from these calculations demonstrate the efficacy of the GSIK method for capturing electron-hole correlation in large clusters and nanoparticles.   

Recent developments in the Octopus code for strong light-matter coupling

The Octopus code [1,2] is a finite-differences real-space code designed to fully take advantage of the flexibility and versatility of real-space grids and provide developers with a framework to easily implement and test new ideas and methods in the field of electronic excited states properties and dynamics, while ensuring optimal execution performance and parallelization.

In this talk I will give an overview of the recent advances in the Octopus code regarding several novel approaches in the field of strong light-matter interactions [3]. Such new methods are essential to correctly describe the coupling of light to chemical systems, quantum materials, or nanoplasmonic systems, among others, when the electron-photon interaction has to be considered explicitly.

[1] A. Castro et al, "octopus: a tool for the application of time-dependent density functional theory", Phys. Stat. Sol. B 243 2465-2488 (2006)
[2] X. Andrade et al, "Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems", Phys. Chem. Chem. Phys. 17 31371-31396 (2015)
[3] N. Tancogne-Dejean et al, "Octopus, a framework for exploring the ultrafast quantum electron-ion dynamics in extended and finite systems", to be submitted.