Low-Scaling Algorithm for the Random Phase Approximation Using Tensor Hypercontraction with k-point Sampling.

Area of Science:

• Computational Chemistry
• Quantum Many-Body Physics
• Electronic Structure Theory

Background:

• Accurate calculation of electron repulsion integrals (ERIs) is crucial for many-body perturbation theories.
• Traditional methods face significant computational scaling challenges with increasing system size and k-point sampling.
• Random Phase Approximation (RPA) is a powerful tool for electronic structure but is computationally demanding.

Purpose of the Study:

• To develop a low-scaling algorithm for RPA calculations with k-point sampling.
• To implement this algorithm within the tensor hypercontraction (THC) framework for electron repulsion integrals (ERIs).
• To enable accurate and efficient electronic structure calculations for large-scale systems.

Main Methods:

• Utilized a revised interpolative separable density fitting (ISDF) procedure for THC factorization.
• Employed a momentum-dependent auxiliary basis for single-particle Bloch orbitals.
• Developed a formulation that systematically controls accuracy via the number of interpolating points, avoiding preoptimization.

Main Results:

• Achieved a linear scaling with the number of k-points and cubic scaling with system size for the RPA algorithm.
• Demonstrated rapid convergence of ERI and RPA energy errors with the size of the THC auxiliary basis.
• The algorithm operates without assumptions on orbital sparsity or locality.

Conclusions:

• The presented low-scaling RPA algorithm using THC is a robust and promising approach for large-scale electronic structure calculations.
• This method paves the way for efficient algorithms in higher-order many-body perturbation theories.
• The systematic control over accuracy and efficient scaling make it suitable for complex systems.