Within the framework of many-body perturbation theory based on Green's functions, the GW approximation has emerged as a pivotal method for computing quasiparticle energies and excitation spectra. However, its high computational cost and steep scaling present significant challenges for applications to large molecular systems. In this work, we extend the block tensor decomposition (BTD) algorithm, recently developed in our previous work [Zhang et al., J. Chem. Phys. 163, 174109 (2025)] for low-rank tensor compression, to enable a formally O(N3)-scaling GW algorithm. By integrating BTD with an imaginary-time GW formalism and introducing a real space screening strategy for the polarizability, we achieve an observed scaling of approximately O(N2) in test systems. Key parameters of the algorithm are optimized on the S66 dataset using the JADE algorithm, ensuring a balanced compromise between accuracy and efficiency. Our BTD-based random phase approximation also exhibits O(N2) scaling, and eigenvalue-self-consistent GW calculations become feasible for systems with over 3000 basis functions. This work establishes BTD as an efficient and scalable approach for large-scale GW calculations in molecular systems.
Within the framework of many-body perturbation theory based on Green's functions, the GW approximation has emerged as a pivotal method for computing quasiparticle energies and excitation spectra. However, its high computational cost and steep scaling present significant challenges for applications to large molecular systems. In this work, we extend the block tensor decomposition (BTD) algorithm, recently developed in our previous work [Zhang et al., J. Chem. Phys. 163, 174109 (2025)] for low-rank tensor compression, to enable a formally O(N3)-scaling GW algorithm. By integrating BTD with an imaginary-time GW formalism and introducing a real space screening strategy for the polarizability, we achieve an observed scaling of approximately O(N2) in test systems. Key parameters of the algorithm are optimized on the S66 dataset using the JADE algorithm, ensuring a balanced compromise between accuracy and efficiency. Our BTD-based random phase approximation also exhibits O(N2) scaling, and eigenvalue-self-consistent GW calculations become feasible for systems with over 3000 basis functions. This work establishes BTD as an efficient and scalable approach for large-scale GW calculations in molecular systems.
Within the framework of many-body perturbation theory based on Green's functions, the GW approximation has emerged as a pivotal method for computing quasiparticle energies and excitation spectra. However, its high computational cost and steep scaling present significant challenges for applications to large molecular systems. In this work, we extend the block tensor decomposition (BTD) algorithm, recently developed in our previous work [J. Chem. Phys. 163, 174109 (2025)] for low-rank tensor compression, to enable a formally O(N3)-scaling GW algorithm. By integrating BTD with an imaginary-time GW formalism and introducing a real space screening strategy for the polarizability, we achieve an observed scaling of approximately O(N2) in test systems. Key parameters of the algorithm are optimized on the S66 dataset using the JADE algorithm, ensuring a balanced compromise between accuracy and efficiency. Our BTD-based random phase approximation also exhibits O(N2) scaling, and eigenvalue-self-consistent GW calculations become feasible for systems with over 3000 basis functions. This work establishes BTD as an efficient and scalable approach for large-scale GW calculations in molecular systems.
Block tensor decomposition (BTD) and canonical polyadic decomposition (CPD) are combined into a unified O(N3)-scaling framework for second-order perturbation theory (PT2), demonstrated on MP2 and renormalized PT2 (rPT2). BTD constructs the tensor hyper-contraction kernel at O(N3) via a dual-grid scheme; CPD factorizes the exchange channel through a block-based two-stage ALS. An asymmetric half-kernel design applies bare Coulomb to one vertex and coupling-constant-averaged screening to the other, capturing the SOSEX component of rPT2 without a frequency-dependent CPD. For MP2, BTD-CPD reproduces canonical RI-MP2 to 0.058~kcal/mol per heavy atom. For rPT2@PBE0 on the S66x8 benchmark, the mean absolute error is 0.36~kcal/mol (ME $-$0.19, RMSE 0.46) over 528 data points. The CPD-compressed intermediates yield O(N2) storage alongside O(N3) scaling.
The GW formalism, a Green’s function many-body perturbation theory, is growing in popularity for the description of the electronic properties of condensed matter systems in solid-state physics, and more recently chemistry. Unfortunately, its application to complex systems of interest in nanosciences, chemistry, or even biology, is hampered by the large associated computing cost, in particular in the case of disordered systems, or systems immersed in an opened environment (a solvent, a molecular medium, an electrode, etc.) The goal of the present PhD thesis is to focus on the development of multiscale techniques, merging high-level many-body treatments of the subsystem of interest, with a simplified but fully ab initio description of the electrostatic and dielectric environment. Such approaches aim to go beyond classical parametrized models, mainly developed in the quantum chemistry community, which are based on a continuum (“polarizable continuum model”) or discrete (QM/MM) description of the environment.To reach such a goal, we adopt a divide-and-conquer fragmentation scheme for the environment, particularly suited to molecular systems. This leads to a block-diagonal non- interacting electron susceptibility, decreasing the algorithmic complexity from quartic to cubic. To reduce the prefactor associated with the inversion of the Dyson equation for the screened Coulomb potential W, we have further developed a compression algorithm for the susceptibility operator. The automatic computation of an extremely compact polarization basis set allows a large reduction of the size of the susceptibility blocks, associated to the fragments in the environment. Such a method enables us to compute the dielectric response of systems made of several hundred thousand atoms, with an excellent accuracy when it comes to reproduce the effect of the environment as a response to an excitation in the immersed subsystem. This approach is presented through the study of fullerene bulk, surface and subsurface crystals.While the GW formalism is dynamical, with a frequency-dependent screened Coulomb potential W, a first study is done adopting a static approximation (low-frequency limit) for the screening properties of the environment. Such an approach follows the traditional semi-empirical models of a polarizable environment. This PhD thesis assesses the validity of such an approximation, which assumes an instantaneous response (adiabatic limit) of the environment to an electronic excitation, thanks to an explicit comparison with a fully dynamical dielectric response of the environment. The study of a surface of fullerenes, as well as a water molecule inside a metallic carbon nanotube, show that a static description of the environment leads to errors on the polarization energy below 10%, provided that the “folding” of the environment is treated in a proper way.The fragment approach is also applied to covalent insulator crystals, and more particularly to hexagonal boron nitride (h-BN). We explain how to compute the energy levels of point defects in h-BN, in the true dilute limit, and we give the asymptotic scaling laws for the renormalization of these energy levels, from the monolayer to a (n)-layer system. This study highlights thus the possibility to apply the fragment approach to covalent insulator systems, a possibility hinging probably on the short range behavior of the susceptibility in these systems.All of these developments, extending ab initio many-body methods to increasingly complex systems, have been implemented in the massively parallel code beDeft, dedicated to the study of the electronic properties of large scale systems.
We present a many-body GW formalism for quantum subsystems embedded in discrete polarizable environments containing up to several hundred thousand atoms described at a fully ab initio random phase approximation level. Our approach is based on a fragment approximation in the construction of the Green's function and independent-electron susceptibilities. Further, the environing fragments susceptibility matrices are reduced to a minimal but accurate representation preserving low order polarizability tensors through a constrained minimization scheme. This approach dramatically reduces the cost associated with inverting the Dyson equation for the screened Coulomb potential W, while preserving the description of short to long-range screening effects. The efficiency and accuracy of the present scheme is exemplified in the paradigmatic cases of fullerene bulk, surface, subsurface, and slabs with varying number of layers.
Electronic structure calculations based on many-body perturbation theory (e.g. GW or the random-phase approximation (RPA)) require function evaluations in the complex time and frequency domain, for example inhomogeneous Fourier transforms or analytic continuation from the imaginary axis to the real axis. For inhomogeneous Fourier transforms, the time-frequency component of the GreenX library provides time-frequency grids that can be utilized in low-scaling RPA and GW implementations. In addition, the adoption of the compact frequency grids provided by our library also reduces the computational overhead in RPA implementations with conventional scaling. In this work, we present low-scaling GW and conventional RPA benchmark calculations using the GreenX grids with different codes (FHI-aims, CP2K and ABINIT) for molecules, two-dimensional materials and solids. Very small integration errors are observed when using 30 time-frequency points for our test cases, namely <10−8 eV/electron for the RPA correlation energies, and 10 meV for the GW quasiparticle energies.
Electronic structure calculations based on many-body perturbation theory [e.g., GW or the random-phase approximation (RPA)] require function evaluations in the complex time and frequency domain, for example, inhomogeneous Fourier transforms or analytic continuation from the imaginary axis to the real axis. For inhomogeneous Fourier transforms, the time-frequency component of the GreenX library provides time-frequency grids that can be utilized in low-scaling RPA and GW implementations. In addition, the adoption of the compact frequency grids provided by our library also reduces the computational overhead in RPA implementations with conventional scaling. In this paper, we present low-scaling GW and conventional RPA benchmark calculations using the GreenX grids with different codes (FHI-aims, CP2K, and ABINIT) for molecules, two-dimensional materials and solids. Very small integration errors are observed when using 30 time-frequency points for our test cases, namely $<{10}^{\ensuremath{-}8}$ eV/electron for the RPA correlation energies, and $\ensuremath{\le}10$ meV for the GW quasiparticle energies.