Interpolative Separable Density Fitting Research Articles

Use of Multigrids to Reduce the Cost of Performing Interpolative Separable Density Fitting.

In this article, we present an interpolative separable density fitting (ISDF)-based algorithm to calculate the exact exchange in periodic mean field calculations. In the past, decomposing the two-electron integrals into the tensor hypercontraction (THC) form using ISDF was the most expensive step of the entire mean field calculation. Here, we show that by using a multigrid-ISDF algorithm, both the memory and the CPU cost of this step can be reduced. The CPU cost is brought down from cubic scaling to quadratic scaling with a low computational prefactor which reduces the cost by almost 2 orders of magnitude. Thus, in the new algorithm, the cost of performing ISDF is largely negligible compared to other steps. Along with the CPU cost, the memory cost of storing the factorized two-electron integrals is also reduced by a factor of up to 35. With the current algorithm, we can perform Hartree-Fock calculations on a diamond supercell containing more than 17,000 basis functions and more than 1500 electrons on a single node with no disk usage. For this calculation, the cost of constructing the exchange matrix is only a factor of 4 slower than the cost of diagonalizing the Fock matrix. Augmenting our approach with linear scaling algorithms can further speed up the calculations.

Low-Scaling Algorithms for GW and Constrained Random Phase Approximation Using Symmetry-Adapted Interpolative Separable Density Fitting.

We present low-scaling algorithms for GW and constrained random phase approximation based on a symmetry-adapted interpolative separable density fitting (ISDF) procedure that incorporates the space-group symmetries of crystalline systems. The resulting formulations scale cubically, with respect to system size, and linearly with the number of k-points, regardless of the choice of single-particle basis and whether a quasiparticle approximation is employed. We validate these methods through comparisons with published literature and demonstrate their efficiency in treating large-scale systems through the construction of downfolded many-body Hamiltonians for carbon dimer defects embedded in hexagonal boron nitride supercells. Our work highlights the efficiency and general applicability of ISDF in the context of large-scale many-body calculations with k-point sampling beyond density functional theory.

Machine Learning K-Means Clustering in Interpolative Separable Density Fitting Algorithm: Advancing Accurate and Efficient Cubic-Scaling Density Functional Perturbation Theory Calculations within Plane Waves.

Density functional perturbation theory (DFPT) is a crucial tool for accurately describing lattice dynamics. The adaptively compressed polarizability (ACP) method reduces the computational complexity of DFPT calculations from O(N4) to O(N3) by combining the interpolative separable density fitting (ISDF) algorithm. However, the conventional QR factorization with column pivoting (QRCP) algorithm, used for selecting the interpolation points in ISDF, not only incurs a high cubic-scaling computational cost, O(N3), but also leads to suboptimal convergence. This convergence issue is particularly pronounced when considering the complex interplay between the external potential and atomic displacement in ACP-based DFPT calculations. Here, we present a machine learning K-means clustering algorithm to select the interpolation points in ISDF, which offers a more efficient quadratic-scaling O(N2) alternative to the computationally intensive cubic-scaling O(N3) QRCP algorithm. We implement this efficient K-means-based ISDF algorithm to accelerate plane-wave DFPT calculations in KSSOLV, which is a MATLAB toolbox for performing Kohn-Sham density functional theory calculations within plane waves. We demonstrate that this K-means algorithm not only offers comparable accuracy to QRCP in ISDF but also achieves better convergence for ACP-based DFPT calculations. In particular, K-means can remarkably reduce the computational cost of selecting the interpolation points by nearly 2 orders of magnitude compared to QRCP in ISDF.

Complex-Valued K-Means Clustering of Interpolative Separable Density Fitting Algorithm for Large-Scale Hybrid Functional Enabled Ab Initio Molecular Dynamics Simulations within Plane Waves.

K-means clustering, as a classic unsupervised machine learning algorithm, is the key step to select the interpolation sampling points in interpolative separable density fitting (ISDF) decomposition for hybrid functional electronic structure calculations. Real-valued K-means clustering for accelerating the ISDF decomposition has been demonstrated for large-scale hybrid functional enabled ab initio molecular dynamics (hybrid AIMD) simulations within plane-wave basis sets where the Kohn-Sham orbitals are real-valued. However, it is unclear whether such K-means clustering works for complex-valued Kohn-Sham orbitals. Here, we propose an improved weight function defined as the sum of the square modulus of complex-valued Kohn-Sham orbitals in K-means clustering for hybrid AIMD simulations. Numerical results demonstrate that the K-means algorithm with a new weight function yields smoother and more delocalized interpolation sampling points, resulting in smoother energy potential, smaller energy drift, and longer time steps for hybrid AIMD simulations compared to the previous weight function used in the real-valued K-means algorithm. In particular, we find that this improved algorithm can obtain more accurate oxygen-oxygen radial distribution functions in liquid water molecules and a more accurate power spectrum in crystal silicon dioxide compared to the previous K-means algorithm. Finally, we describe a massively parallel implementation of this ISDF decomposition to accelerate large-scale complex-valued hybrid AIMD simulations containing thousands of atoms (2,744 atoms), which can scale up to 5,504 CPU cores on modern supercomputers.

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.

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

We present a low-scaling algorithm for the random phase approximation (RPA) with k-point sampling in the framework of tensor hypercontraction (THC) for electron repulsion integrals (ERIs). The THC factorization is obtained via a revised interpolative separable density fitting (ISDF) procedure with a momentum-dependent auxiliary basis for generic single-particle Bloch orbitals. Our formulation does not require preoptimized interpolating points or auxiliary bases, and the accuracy is systematically controlled by the number of interpolating points. The resulting RPA algorithm scales linearly with the number of k-points and cubically with the system size without any assumption on sparsity or locality of orbitals. The errors of ERIs and RPA energy show rapid convergence with respect to the size of the THC auxiliary basis, suggesting a promising and robust direction to construct efficient algorithms of higher order many-body perturbation theories for large-scale systems.

Even Faster Exact Exchange for Solids via Tensor Hypercontraction.

Hybrid density functional theory (DFT) remains intractable for large periodic systems due to the demanding computational cost of exact exchange. We apply the tensor hypercontraction (THC) (or interpolative separable density fitting) approximation to periodic hybrid DFT calculations with Gaussian-type orbitals using the Gaussian plane wave approach. This is done to lower the computational scaling with respect to the number of basis functions (N) and k-points (Nk) at a fixed system size. Additionally, we propose an algorithm to fit only occupied orbital products via THC (i.e., a set of points, NISDF) to further reduce computation time and memory usage. This algorithm has linear scaling cost with k-points, no explicit dependence of NISDF on basis set size, and overall cubic scaling with unit cell size. Significant speedups and reduced memory usage may be obtained for moderately sized k-point meshes, with additional gains for large k-point meshes. Adequate accuracy can be obtained using THC-oo-K for self-consistent calculations. We perform illustrative hybrid density function theory calculations on the benzene crystal in the basis set and thermodynamic limits to highlight the utility of this algorithm.

Low-rank approximations for accelerating plane-wave hybrid functional calculations in unrestricted and noncollinear spin density functional theory.

The adaptively compressed exchange (ACE) operator combined with interpolative separable density fitting (ISDF) decomposition has been utilized to accelerate plane-wave hybrid functional calculations for restricted Kohn-Sham density functional theory (DFT), but the neglect of spin degree of freedom has limited its application in the exploration of systems where the spin property of the electron is critical. Herein, we derive the ACE-ISDF formulation for hybrid functional calculations in both unrestricted and noncollinear spin DFT with plane waves and periodic boundary conditions. We proposed an improved ISDF algorithm for the sum of Kohn-Sham orbital pairs to further reduce the computational cost for the spin-noncollinear case. Numerical results demonstrate that these improved ACE-ISDF low-rank approximations can not only significantly reduce the computational time by two orders of magnitude compared with conventional plane-wave hybrid functional calculations but also lead to a good convergence behavior when a moderate rank parameter is set, even for complex periodic magnetic systems. By using these ACE-ISDF approximations, we investigate the electronic and magnetic properties of two-dimensional periodic ferromagnetic semiconductors consisting of triangular zigzag graphene quantum dots and transition metal atoms. Our computational results showcase that hybrid functional calculations in spin DFT can provide not only accurate electronic structures but also accurate magnetic order temperature of ferromagnetic semiconductors compared to local or semilocal functional calculations.

Low-rank approximations to accelerate hybrid functional enabled real-time time-dependent density functional theory within plane waves

Real-time time-dependent density functional theory (RT-TDDFT) is a powerful tool for predicting excited-state dynamics. Herein, we combine the adaptively compressed exchange (ACE) operator with interpolative separable density fitting (ISDF) algorithm to accelerate the hybrid functional calculations in RT-TDDFT (hybrid RT-TDDFT) dynamics simulations for molecular and periodic systems within plane waves. Under this low-rank representation, we demonstrate that the ACE-ISDF enabled hybrid RT-TDDFT can yield accurate excited-state dynamics, but much faster than conventional calculations. Furthermore, we describe a massively parallel implementation of ACE-ISDF enabled hybrid RT-TDDFT dynamics simulations containing thousands of atoms (1728 atoms), which can scale up to 3456 central processing unit cores on modern supercomputers.

Interpolative Separable Density Fitting for Accelerating Two-Electron Integrals: A Theoretical Perspective.

Low-rank approximations have long been considered an efficient way to accelerate electronic structure calculations associated with the evaluation of electron repulsion integrals (ERIs). As an accurate and efficient algorithm for compressing the ERI tensor, the interpolative separable density fitting (ISDF) decomposition has recently attracted great attention in this context. In this perspective, we introduce the ISDF decomposition from the theoretical aspects and technique details. The ISDF decomposition can construct a fully separable low-rank approximation (tensor hypercontraction factorization) of ERIs in real space with a cubic cost, offering great flexibility for accelerating high-scaling electronic structure calculations. We review the typical applications of ISDF in hybrid functionals, time-dependent density functional theory, and GW approximation. Finally, we discuss the promising directions for future development of ISDF.

Fast Exchange with Gaussian Basis Set Using Robust Pseudospectral Method.

In this article, we present an algorithm to efficiently evaluate the exchange matrix in periodic systems when a Gaussian basis set with pseudopotentials is used. The usual algorithm for evaluating exchange matrix scales cubically with the system size because one has to perform O(N2) fast Fourier transform (FFT). Here, we introduce an algorithm that retains the cubic scaling but reduces the prefactor significantly by eliminating the need to do FFTs during each exchange build. This is accomplished by representing the products of Gaussian basis function using a linear combination of an auxiliary basis the number of which scales linearly with the size of the system. We store the potential due to these auxiliary functions in memory, which allows us to obtain the exchange matrix without the need to do FFT, albeit at the cost of additional memory requirement. Although the basic idea of using auxiliary functions is not new, our algorithm is cheaper due to a combination of three ingredients: (a) we use a robust pseudospectral method that allows us to use a relatively small number of auxiliary basis to obtain high accuracy; (b) we use occ-RI exchange, which eliminates the need to construct the full exchange matrix; and (c) we use the (interpolative separable density fitting) ISDF algorithm to construct these auxiliary basis sets that are used in the robust pseudospectral method. The resulting algorithm is accurate, and we note that the error in the final energy decreases exponentially rapidly with the number of auxiliary functions.

Low-Rank Approximations Accelerated Plane-Wave Hybrid Functional Calculations with k-Point Sampling.

The low-rank approximations of the adaptively compressed exchange (ACE) operator and interpolative separable density fitting (ISDF) algorithms significantly reduce the computational cost and memory usage of hybrid functional calculations in real space, but the lack of k-point sampling hinders their implementation in reciprocal space for periodic systems with the plane-wave basis set. Here, we combine the ACE operator and ISDF decomposition into a new ACE-ISDF algorithm for periodic systems in reciprocal space with k-point sampling. On the basis of the ACE-ISDF algorithm with the improved reciprocal space ACE operator and k-point Fourier convolution, the time complexity of the hybrid functional calculation is reduced from to (Ne and Nk are the number of electrons and k-points, respectively) with a much smaller prefactor and much lower memory consumption compared to the standard method for periodic systems with a plane-wave basis set.

Realizing Effective Cubic-Scaling Coulomb Hole Plus Screened Exchange Approximation in Periodic Systems via Interpolative Separable Density Fitting with a Plane-Wave Basis Set.

The GW approximation is an effective way to accurately describe the single-electron excitations of molecules and the quasiparticle energies of solids. However, a perceived drawback of the GW calculations is their high computational cost and large memory usage, which limit their applications to large systems. Herein, we demonstrate an accurate and effective low-rank approximation to accelerate non-self-consistent GW (G0W0) calculations under the static Coulomb hole plus screened exchange (COHSEX) approximation for periodic systems. Our approach is to adopt the interpolative separable density fitting (ISDF) decomposition and Cauchy's integral to construct low-rank representations of the dielectric matrix ϵ and self-energy matrix Σ. This approach reduces the number of floating point operations from to and requires a much smaller memory footprint. Two methods are used to select the interpolation points in ISDF, including the standard QR factorization with column pivoting (QRCP) procedure and the machine learning K-means clustering (K-means) algorithm. We demonstrate that these two methods can yield similar accuracy for both molecules and solids at much lower computational cost. In particular, K-means clustering can significantly reduce the computational cost of selecting the interpolation points by an order of magnitude compared to QRCP, resulting in an overall speedup factor of about ten times ISDF accelerated the static COHSEX calculations compared with conventional COHSEX approximation.

Machine Learning K-Means Clustering Algorithm for Interpolative Separable Density Fitting to Accelerate Hybrid Functional Calculations with Numerical Atomic Orbitals.

The interpolative separable density fitting (ISDF) is an efficient and accurate low-rank decomposition method to reduce the high computational cost and memory usage of the Hartree-Fock exchange (HFX) calculations with numerical atomic orbitals (NAOs). In this work, we present a machine learning K-means clustering algorithm to select the interpolation points in ISDF, which offers a much cheaper alternative to the expensive QR factorization with column pivoting (QRCP) procedure. We implement this K-means-based ISDF decomposition to accelerate hybrid functional calculations with NAOs in the HONPAS package. We demonstrate that this method can yield a similar accuracy for both molecules and solids at a much lower computational cost. In particular, K-means can remarkably reduce the computational cost of selecting the interpolation points by nearly two orders of magnitude compared to QRCP, resulting in a speedup of ∼10 times for ISDF-based HFX calculations.

Interpolative Separable Density Fitting Decomposition for Accelerating Hartree-Fock Exchange Calculations within Numerical Atomic Orbitals.

The high cost associated with the evaluation of Hartree-Fock exchange (HFX) makes hybrid functionals computationally challenging for large systems. In this work, we present an efficient way to accelerate HFX calculations with numerical atomic basis sets. Our approach is based on the recently proposed interpolative separable density fitting (ISDF) decomposition to construct a low-rank approximation of the HFX matrix, which avoids explicit calculations of the electron repulsion integrals (ERIs) and significantly reduces the computational cost. We implement the ISDF method for hybrid functional (PBE0) calculations in the HONPAS package. We take benzene and polycyclic aromatic hydrocarbon molecules as examples and demonstrate that hybrid functionals with ISDF yield quite promising results at a significantly reduced computational cost. Especially, the ISDF approach reduces the total cost of the evaluating HFX matrix by nearly 2 orders of magnitude compared to conventional approaches of direct evaluation of ERIs.

Fast optical absorption spectra calculations for periodic solid state systems

We present a method to construct an efficient approximation to the bare exchange and screened direct interaction kernels of the Bethe-Salpeter Hamiltonian for periodic solid state systems via the interpolative separable density fitting technique. We show that the cost of constructing the approximate Bethe-Salpeter Hamiltonian scales nearly optimally as $\mathcal{O}(N_k)$ with respect to the number of samples in the Brillouin zone $N_k$. In addition, we show that the cost for applying the Bethe-Salpeter Hamiltonian to a vector scales as $\mathcal{O}(N_k \log N_k)$. Therefore the optical absorption spectrum, as well as selected excitation energies can be efficiently computed via iterative methods such as the Lanczos method. This is a significant reduction from the $\mathcal{O}(N_k^2)$ and $\mathcal{O}(N_k^3)$ scaling associated with a brute force approach for constructing the Hamiltonian and diagonalizing the Hamiltonian respectively. We demonstrate the efficiency and accuracy of this approach with both one-dimensional model problems and three-dimensional real materials (graphene and diamond). For the diamond system with $N_k=2197$, it takes $6$ hours to assemble the Bethe-Salpeter Hamiltonian and $4$ hours to fully diagonalize the Hamiltonian using $169$ cores when the brute force approach is used. The new method takes less than $3$ minutes to set up the Hamiltonian and $24$ minutes to compute the absorption spectrum on a single core.

Accelerating Time-Dependent Density Functional Theory and GW Calculations for Molecules and Nanoclusters with Symmetry Adapted Interpolative Separable Density Fitting

Computing integrals over orbital pairs is one of the most costly steps in many popular first-principles methods used by the quantum chemistry and condensed matter physics community. Here, we employ a recently proposed interpolative separable density fitting method (ISDF) to significantly reduce the cost of steps involving orbital pairs in linear response time-dependent density functional theory and GW calculations. In our implementation, we exploit the symmetry property of a system to effectively reduce the number of interpolation points and thus the computational cost. The performance of ISDF is illustrated with calculations on the GW100 set and silicon nanoclusters. We demonstrate the cost for constructing auxiliary basis and interpolation coefficients are negligible compared to the total cost. Compared to the conventional brute-force approach, the computation cost for evaluating all kernel matrix elements is reduced by nearly 3 orders of magnitude. The total cost for GW calculations can be reduced by four to eight times, depending on the system size.

Improved Grid Optimization and Fitting in Least Squares Tensor Hypercontraction

A new method for generating fitting grids for least-squares tensor hypercontraction (LS-THC) is presented. This method draws inspiration from the related interpolative separable density fitting (ISDF) technique, but uses only a pivoted Cholesky decomposition of the metric matrix, S, already computed as a matter of course in LS-THC. The size and quality of the resulting grid is controlled by a user-defined cutoff parameter and the size of the starting grid. Additionally, the Cholesky-based method provides an alternative and possible more numerically stable method for performing the least-squares fit. The quality of the grids produced is evaluated for LS-DF-THC-MP2 calculations on retinal and benzene, the former with a large starting grid and small cc-pVDZ basis set, and the latter with a wide range of grids and basis sets. The error and grid size is found to be well-controlled by either the cutoff parameter (with a large starting grid) or the starting grid size (with a tight cutoff) and highly predictable. The Cholesky-based method is also able to generate unique grids tailored to different charge distributions, for example the (ab|, (ai|, and (ij| distributions that arise in the molecular orbital integrals. While only the (ai| grid directly affects the MP2 energy, the relative sizes of the other grids are examined.

Accelerating Excitation Energy Computation in Molecules and Solids within Linear-Response Time-Dependent Density Functional Theory via Interpolative Separable Density Fitting Decomposition.

We present an efficient way to compute the excitation energies in molecules and solids within linear-response time-dependent density functional theory (LR-TDDFT). Conventional methods to construct and diagonalize the LR-TDDFT Hamiltonian require ultrahigh computational cost, limiting its optoelectronic applications to small systems. Our new method is based on the interpolative separable density fitting (ISDF) decomposition combined with implicitly constructing and iteratively diagonalizing the LR-TDDFT Hamiltonian and only requires low computational cost to accelerate the LR-TDDFT calculations in the plane-wave basis sets under the periodic boundary condition. We show that this method accurately reproduces excitation energies in a fullerene (C60) molecule and bulk silicon Si64 system with significantly reduced computational cost compared to conventional direct and iterative calculations. The efficiency of this ISDF method enables us to investigate the excited-state properties of liquid water absorption on MoS2 and phosphorene by using the LR-TDDFT calculations. Our computational results show that an aqueous environment has a weak effect on low excitation energies but a strong effect on high excitation energies of 2D semiconductors for photocatalytic water splitting.

Systematically Improvable Tensor Hypercontraction: Interpolative Separable Density-Fitting for Molecules Applied to Exact Exchange, Second- and Third-Order Møller-Plesset Perturbation Theory.

We present a systematically improvable tensor hypercontraction (THC) factorization based on interpolative separable density fitting (ISDF). We illustrate algorithmic details to achieve this within the framework of Becke's atom-centered quadrature grid. A single ISDF parameter cISDF controls the trade-off between accuracy and cost. In particular, cISDF sets the number of interpolation points used in THC, NIP = cISDF × NX with NX being the number of auxiliary basis functions. In conjunction with the resolution-of-the-identity (RI) technique, we develop and investigate the THC-RI algorithms for cubic-scaling exact exchange for Hartree-Fock and range-separated hybrids (e.g., ωB97X-V) and quartic-scaling second- and third-order Møller-Plesset theory (MP2 and MP3). These algorithms were evaluated over the W4-11 thermochemistry (atomization energy) set and A24 noncovalent interaction benchmark set with standard Dunning basis sets (cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ). We demonstrate the convergence of THC-RI algorithms to numerically exact RI results using ISDF points. Based on these, we make recommendations on cISDF for each basis set and method. We also demonstrate the utility of THC-RI exact exchange and MP2 for larger systems such as water clusters and C20. We stress that more challenges await in obtaining accurate and numerically stable THC factorization for wave function amplitudes as well as for the space spanned by virtual orbitals in large basis sets and implementing sparsity-aware THC-RI algorithms.