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A novel implementation of CCSD analytic gradients using Cholesky decomposition of the two-electron integrals and Abelian point-group symmetry

Luca Melega, Tommaso Nottoli, Jürgen Gauss, Filippo Lipparini

TL;DR

This work tackles the high cost of CCSD analytic gradients for medium-to-large molecules by marrying Cholesky decomposition (CD) of the electron-repulsion integral tensor with explicit Abelian point-group symmetry. The authors develop a symmetry-adapted, two-step CD framework and a Lagrangian-based CCSD gradient formalism (including Lambda and Z-vector equations), enabling on-the-fly contraction of differentiated CD quantities with CC density matrices in the AO basis. Key contributions include a symmetry-aware, RAM-efficient gradient implementation that eliminates the need to store $OV^3$ and $V^4$ intermediates, and a detailed treatment of derivative computation via the RI/CD analogy. Validation on large symmetric systems (e.g., coronene and hexabenzocoronene) demonstrates meaningful speedups and memory savings, with clear paths for extensions to open-shell cases, perturbative triples, and higher derivatives.

Abstract

We present a novel and efficient implementation of coupled-cluster with singles and doubles (CCSD) analytic gradients that combines the Cholesky decomposition (CD) of electron-repulsion integrals with the exploitation of Abelian point-group symmetry. This approach is particularly effective for medium-sized and large symmetric molecular systems. The CD of two-electron integrals is performed using a symmetry-adapted two-step algorithm, while the derivatives of the Cholesky vectors are computed with respect to symmetry-adapted nuclear displacements and contracted on-the-fly with the CCSD density matrices. Geometry optimizations of symmetric systems with several hundreds of basis functions have been carried out to assess the efficiency of our implementation and to quantify the computational gain provided by the exploitation of point-group symmetry.

A novel implementation of CCSD analytic gradients using Cholesky decomposition of the two-electron integrals and Abelian point-group symmetry

TL;DR

This work tackles the high cost of CCSD analytic gradients for medium-to-large molecules by marrying Cholesky decomposition (CD) of the electron-repulsion integral tensor with explicit Abelian point-group symmetry. The authors develop a symmetry-adapted, two-step CD framework and a Lagrangian-based CCSD gradient formalism (including Lambda and Z-vector equations), enabling on-the-fly contraction of differentiated CD quantities with CC density matrices in the AO basis. Key contributions include a symmetry-aware, RAM-efficient gradient implementation that eliminates the need to store and intermediates, and a detailed treatment of derivative computation via the RI/CD analogy. Validation on large symmetric systems (e.g., coronene and hexabenzocoronene) demonstrates meaningful speedups and memory savings, with clear paths for extensions to open-shell cases, perturbative triples, and higher derivatives.

Abstract

We present a novel and efficient implementation of coupled-cluster with singles and doubles (CCSD) analytic gradients that combines the Cholesky decomposition (CD) of electron-repulsion integrals with the exploitation of Abelian point-group symmetry. This approach is particularly effective for medium-sized and large symmetric molecular systems. The CD of two-electron integrals is performed using a symmetry-adapted two-step algorithm, while the derivatives of the Cholesky vectors are computed with respect to symmetry-adapted nuclear displacements and contracted on-the-fly with the CCSD density matrices. Geometry optimizations of symmetric systems with several hundreds of basis functions have been carried out to assess the efficiency of our implementation and to quantify the computational gain provided by the exploitation of point-group symmetry.
Paper Structure (19 sections, 38 equations, 5 figures, 1 algorithm)

This paper contains 19 sections, 38 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theory
  3. Cholesky decomposition of two-electron integrals
  4. CCSD Lagrangian
  5. Stationarity with respect to cluster amplitudes: Lambda equations
  6. CCSD density matrices
  7. Stationarity with respect to orbital rotations: $Z$-vector equations and expressions for the $I_{pq}$ intermediates
  8. CCSD analytic first derivatives
  9. Computational details
  10. Exploitation of Abelian point-group symmetry
  11. Symmetry-adapted two-step CD algorithm
  12. Test calculations and discussion
  13. Analysis of timings
  14. Parallelization speedup analysis
  15. Effect of Abelian point-group symmetry
  16. ...and 4 more sections

Figures (5)

  • Figure 1: Structure of the hexabenzocoronene molecule, as used in its geometry optimization.
  • Figure 2: Gantt chart for the timings (in minutes) for each step necessary for the calculation of CD-CCSD gradients of hexabenzocoronene using the cc-pVDZ basis.
  • Figure 3: Structure of the coronene molecule, as used in its geometry optimization.
  • Figure 4: Gantt chart for the timings (in minutes) for each step necessary for the calculation of CD-CCSD gradients of cc-pVTZ coronene.
  • Figure 5: Speedup graph for CD-CCSD/cc-pVDZ gradient calculations on coronene. The green curve refers to the contraction involving the vvvo block of the two-body density matrix in the evaluation of the molecular gradient, while the purple curve refers to the analogous contraction involving the vvvv block of the two-body density matrix. The ratio between the execution time of the serial code and the execution time of the parallel one is plotted against the number of used OpenMP threads, in the log$_2$ scale.