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Interpolative separable density fitting on adaptive real space grids

Hai Zhu, Chia-Nan Yeh, Miguel A. Morales, Leslie Greengard, Shidong Jiang, Jason Kaye

TL;DR

This work develops a cubic-scaling framework to compress the four-index electron repulsion integral tensor for arbitrary smooth single-particle bases by marrying interpolative separable density fitting (ISDF) with adaptive real-space grids solved by the DMK Poisson solver. The key theoretical advance is proving that an adaptive grid resolving pair densities can be constructed from a grid that resolves the single-particle basis with only a constant-factor overhead, enabling $R = \mathcal{O}(N)$ auxiliary functions and $M = \mathcal{O}(N)$ grid points. Numerically, the method achieves chemical accuracy across all-electron basis sets, remains competitive for highly localized orbitals, and scales cubically with system size, making core-level excitations and GW-type calculations tractable on larger systems. The framework is adaptable to periodic systems and non-Gaussian basis sets, providing a practical route to scalable many-body electronic structure simulations with general basis functions.

Abstract

We generalize the interpolative separable density fitting (ISDF) method, used for compressing the four-index electron repulsion integral (ERI) tensor, to incorporate adaptive real space grids for potentially highly localized single-particle basis functions. To do so, we employ a fast adaptive algorithm, the recently-introduced dual-space multilevel kernel-splitting method, to solve the Poisson equation for the ISDF auxiliary basis functions. The adaptive grids are generated using a high-order accurate, black-box procedure that satisfies a user-specified error tolerance. Our algorithm relies on the observation, which we prove, that an adaptive grid resolving the pair densities appearing in the ERI tensor can be straightforwardly constructed from one that resolves the single-particle basis functions, with the number of required grid points differing only by a constant factor. We find that the ISDF compression efficiency for the ERI tensor with highly localized basis sets is comparable to that for smoother basis sets compatible with uniform grids. To demonstrate the performance of our procedure, we consider several molecular systems with all-electron basis sets which are intractable using uniform grid-based methods. Our work establishes a pathway for scalable many-body electronic structure simulations with arbitrary smooth basis functions, making simulations of phenomena like core-level excitations feasible on a large scale.

Interpolative separable density fitting on adaptive real space grids

TL;DR

This work develops a cubic-scaling framework to compress the four-index electron repulsion integral tensor for arbitrary smooth single-particle bases by marrying interpolative separable density fitting (ISDF) with adaptive real-space grids solved by the DMK Poisson solver. The key theoretical advance is proving that an adaptive grid resolving pair densities can be constructed from a grid that resolves the single-particle basis with only a constant-factor overhead, enabling auxiliary functions and grid points. Numerically, the method achieves chemical accuracy across all-electron basis sets, remains competitive for highly localized orbitals, and scales cubically with system size, making core-level excitations and GW-type calculations tractable on larger systems. The framework is adaptable to periodic systems and non-Gaussian basis sets, providing a practical route to scalable many-body electronic structure simulations with general basis functions.

Abstract

We generalize the interpolative separable density fitting (ISDF) method, used for compressing the four-index electron repulsion integral (ERI) tensor, to incorporate adaptive real space grids for potentially highly localized single-particle basis functions. To do so, we employ a fast adaptive algorithm, the recently-introduced dual-space multilevel kernel-splitting method, to solve the Poisson equation for the ISDF auxiliary basis functions. The adaptive grids are generated using a high-order accurate, black-box procedure that satisfies a user-specified error tolerance. Our algorithm relies on the observation, which we prove, that an adaptive grid resolving the pair densities appearing in the ERI tensor can be straightforwardly constructed from one that resolves the single-particle basis functions, with the number of required grid points differing only by a constant factor. We find that the ISDF compression efficiency for the ERI tensor with highly localized basis sets is comparable to that for smoother basis sets compatible with uniform grids. To demonstrate the performance of our procedure, we consider several molecular systems with all-electron basis sets which are intractable using uniform grid-based methods. Our work establishes a pathway for scalable many-body electronic structure simulations with arbitrary smooth basis functions, making simulations of phenomena like core-level excitations feasible on a large scale.
Paper Structure (14 sections, 2 theorems, 30 equations, 7 figures, 2 tables)

This paper contains 14 sections, 2 theorems, 30 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. THC decomposition via ISDF using adaptive grids
  3. Interpolative separable density fitting
  4. Adaptive real space grids and the solution of the Poisson equation
  5. Summary of the algorithm and computational complexity
  6. Numerical results
  7. Accuracy of adaptive grid representation
  8. Accuracy of ISDF truncation
  9. Increasing basis set size and locality
  10. Comparison with uniform grid approach, and timings
  11. Correlated electron structure using GW
  12. Conclusion
  13. Interpolative decomposition of pair densities
  14. Approximation error of pair densities

Key Result

Lemma 1

Let $l_i(x)$, $i=1,\ldots,n$, be the Lagrange polynomials for the interpolation nodes $x_i=\cos((i-0.5)\pi/n)$, the Chebyshev nodes of the first kind on $[-1,1]$. Let Then Furthermore, let $f\in C^n[-1,1]$ and $p^C$ be its Chebyshev interpolating polynomial of degree less than $n$. Let $p^*$ be a best polynomial approximation of $f$ in $P$, the space of polynomials of degree less than $n$, i.e.,

Figures (7)

  • Figure 1: Adaptive octree resolving single-particle orbitals of $(\mathrm{N}\mathrm{H}_{3})_{2}$ using the aug-cc-pVTZ basis set. Top: three-dimensional view of the octree showing the adaptively refined boxes near nuclei. Bottom: slice view of the same octree. Colors indicate different refinement levels.
  • Figure 2: Maximum $L^2$ error of adaptive octree grid representation of all pair densities $\rho_{ij}$, versus adaptive grid tolerance, for the $(\mathrm{N}\mathrm{H}_{3})_{2}$ example. We show the error of the interpolant on the original adaptive grid used to resolve the single-particle orbitals (blue, no upsampling), as well as for a grid upsampled by a factor of $1.5$ in each dimension (orange, with upsampling). We use the polynomial order $n = \log_{10}(\varepsilon^{-1})+1$ in the octree grid construction.
  • Figure 3: Maximum error of ERI tensor $V_{ijkl}$ for $(\mathrm{NH}_{3})_{2}$, varying the ISDF truncation rank $R$ of the pair densities and the adaptive grid tolerance $\varepsilon$.
  • Figure 4: Maximum error of the ERI tensor $V_{ijkl}$ for $(\mathrm{NH}_{3})_{2}$, varying the ISDF truncation rank of the pair densities, using all-electron cc-PVXZ basis sets with X = D, T, and Q.
  • Figure 5: Maximum error of the ERI tensor $V_{ijkl}$ for chalcogen hydrides with increasing atomic numbers, varying the ISDF truncation rank of the pair densities. The largest GTO exponents $\sigma_{\mathrm{max}}$ for each system are shown in parentheses.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Remark 1
  • Lemma 1
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3