Localized Resolution of Identity Approach to the Analytical Gradients of Random-Phase Approximation Ground-State Energy: Algorithm and Benchmarks

Machine Learning K-Means Clustering of Interpolative Separable Density Fitting Algorithm for Accurate and Efficient Cubic-Scaling Exact Exchange Plus Random Phase Approximation within Plane Waves

The exact-exchange plus random-phase approximation (EXX+RPA) method has emerged as a crucial tool for precisely characterizing electronic structures in molecular and solid systems. We present an accurate and efficient implementation of EXX+RPA calculations that scale cubically and are conducted within plane waves. Our approach incorporates the interpolative separable density fitting (ISDF) algorithm, effectively mitigating the computational challenges associated with the plane wave basis set. To overcome the constraints of the conventional ISDF algorithm, characterized by the exceptionally high prefactor in QR factorization for interpolation point selection, we introduce an enhanced machine learning K-means method. This method incorporates a novel empirical weight function called "SSM+" for more precise interpolation point selection, capturing physical information more accurately across diverse systems. Our machine learning approach offers a quasiquadratic scaling alternative, effectively replacing the computationally demanding cubic-scaling QRCP algorithm in plane-wave-based EXX+RPA calculations. Furthermore, we enhance the method's capabilities by optimizing GPU acceleration using MATLAB's integrated GPU toolkit. In particular, our approach reduces the computational scaling of χ0 from 3.80 to 2.13 and the overall computational scaling of EXX from 2.74 to 2.10. We achieve a remarkable GPU acceleration speedup of up to 35×. Regarding CPU computation time, the standard quartic-scaling method requires 22 h to compute Si128, while QRCP completes the calculation in only around 1 h, achieving a speedup up to 20×. However, the utilization of the K-means algorithm reduces the time to 800 s, a substantial improvement of 100× compared to the standard algorithm. By employing the K-means algorithm, the computational time for interpolative point calculation using QRCP decreases from 1 h to 1 min, resulting in a 55× speed increase. With this improved algorithm, we successfully computed the dissociation curve of H2 and the equilibrium polyynic geometry of C18 molecules.

Massively parallel implementation of gradients within the random phase approximation: Application to the polymorphs of benzene

The Random-Phase approximation (RPA) provides an appealing framework for semi-local density functional theory. In its Resolution-of-the-Identity (RI) approach, it is a very accurate and more cost-effective method than most other wavefunction-based correlation methods. For widespread applications, efficient implementations of nuclear gradients for structure optimizations and data sampling of machine learning approaches are required. We report a well scaling implementation of RI-RPA nuclear gradients on massively parallel computers. The approach is applied to two polymorphs of the benzene crystal obtaining very good cohesive and relative energies. Different correction and extrapolation schemes are investigated for further improvement of the results and estimations of error bars.

Developing correlation-consistent numeric atom-centered orbital basis sets for krypton: Applications in RPA-based correlated calculations

Localized atomic orbitals are the preferred basis set choice for large-scale explicit correlated calculations, and high-quality hierarchical correlation-consistent basis sets are a prerequisite for correlated methods to deliver numerically reliable results. At present, numeric atom-centered orbital (NAO) basis sets with valence correlation consistency (VCC), designated as NAO-VCC-nZ, are only available for light elements from hydrogen (H) to argon (Ar) [Zhang et al., New J. Phys. 15, 123033 (2013)]. In this work, we extend this series by developing NAO-VCC-nZ basis sets for krypton (Kr), a prototypical element in the fourth row of the periodic table. We demonstrate that NAO-VCC-nZ basis sets facilitate the convergence of electronic total-energy calculations using the Random Phase Approximation (RPA), which can be used together with a two-point extrapolation scheme to approach the complete basis set limit. Notably, the Basis Set Superposition Error (BSSE) associated with the newly generated NAO basis sets is minimal, making them suitable for applications where BSSE correction is either cumbersome or impractical to do. After confirming the reliability of NAO basis sets for Kr, we proceed to calculate the Helmholtz free energy for Kr crystal at the theoretical level of RPA plus renormalized single excitation correction. From this, we derive the pressure-volume (P-V) diagram, which shows excellent agreement with the latest experimental data. Our work demonstrates the capability of correlation-consistent NAO basis sets for heavy elements, paving the way toward numerically reliable correlated calculations for bulk materials.

Analytical Second-Order Properties for the Random Phase Approximation: Nuclear Magnetic Resonance Shieldings

A method for the analytical computation of nuclear magnetic resonance (NMR) shieldings within the direct random phase approximation (RPA) is presented. As a starting point, we use the RPA ground-state energy expression within the resolution-of-the-identity approximation in the atomic-orbital formalism. As has been shown in a recent benchmark study using numerical second derivatives [Glasbrenner, M. J. Chem. Theory Comput. 2022, 18, 192], RPA based on a Hartree-Fock reference shows accuracies comparable to coupled cluster singles and doubles (CCSD) for NMR chemical shieldings. Together with the much lower computational cost of RPA, it has emerged as an accurate method for the computation of NMR shieldings. Therefore, we aim to extend the applicability of RPA NMR to larger systems by introducing analytical second-order derivatives, making it a viable method for the accurate and efficient computation of NMR chemical shieldings.

Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation

A method for the evaluation of analytical frozen-core gradients within the random phase approximation is presented. We outline an efficient way to evaluate the response of the density of active electrons arising only when introducing the frozen-core approximation and constituting the main difficulty, together with the response of the standard Kohn-Sham density. The general framework allows to extend the outlined procedure to related electron correlation methods in the atomic orbital basis that require the evaluation of density responses, such as second-order Møller-Plesset perturbation theory or coupled cluster variants. By using Cholesky decomposed densities─which reintroduce the occupied index in the time-determining steps─we are able to achieve speedups of 20-30% (depending on the size of the basis set) by using the frozen-core approximation, which is of similar magnitude as for molecular orbital formulations. We further show that the errors introduced by the frozen-core approximation are practically insignificant for molecular geometries.