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eT 1.0: an open source electronic structure program with emphasis on coupled cluster and multilevel methods

Sarai D. Folkestad, Eirik F. Kjønstad, Rolf H. Myhre, Josefine H. Andersen, Alice Balbi, Sonia Coriani, Tommaso Giovannini, Linda Goletto, Tor S. Haugland, Anders Hutcheson, Ida-Marie Høyvik, Torsha Moitra, Alexander C. Paul, Marco Scavino, Andreas S. Skeidsvoll, Åsmund H. Tveten, Henrik Koch

arXiv:2002.05631v1physics.chem-ph

TL;DR

Established electronic-structure programs can be difficult to modify or extend with new methods. eT combines coupled-cluster and multilevel methods in an electronic-structure program; an example reports experimental solvatochromism while other approaches give errors of 40–50%.

  • Problem

    Established programs can be challenging to modify and optimize, or to integrate new methods and algorithms.

    Source: §I · p. 2

  • Method

    eT combines coupled-cluster methods with Cholesky-based integrals and multilevel HF and CC2 capabilities.

    Source: §I · p. 2

  • Results

    In the solvatochromism example, one approach gives experimental solvatochromism, while other approaches give errors of 40–50%.

    Source: D. Multilevel and multiscale methods · p. 19

  • Takeaways & Limitations

    The program provides an electronic-structure package with coupled-cluster and multilevel capabilities.

    Source: §I · p. 2

Abstract

from arXiv · show

The eT program is an open source electronic structure package with emphasis on coupled cluster and multilevel methods. It includes efficient spin adapted implementations of ground and excited singlet states, as well as equation of motion oscillator strengths, for CCS, CC2, CCSD, and CC3. Furthermore, eT provides unique capabilities such as multilevel Hartree-Fock and multilevel CC2, real-time propagation for CCS and CCSD, and efficient CC3 oscillator strengths. With a coupled cluster code based on an efficient Cholesky decomposition algorithm for the electronic repulsion integrals, eT has similar advantages as codes using density fitting, but with strict error control. Here we present the main features of the program and demonstrate its performance through example calculations. Because of its availability, performance, and unique capabilities, we expect eT to become a valuable resource to the electronic structure community.

I. INTRODUCTION

Established electronic-structure programs can be challenging to modify or extend with new methods. eT is an open-source program that grew from a Cholesky-based coupled-cluster code into a broader package with standard and multilevel methods.

  • Existing programs can be challenging to modify and optimize, or to integrate new methods and algorithms.Source: §I · p. 2
  • The coupled-cluster code begun in 2016 evolved into the eT electronic-structure program.Source: §I · p. 2
  • eT is released as open source under the GNU General Public License 3.Source: §I · p. 2
  • The first version combines optimized Hartree-Fock with standard coupled-cluster methods and efficient implementations based on Cholesky-decomposed integrals.Source: §I · p. 2
  • eT includes multilevel HF [23] and CC2 [24,25], as well as explicitly time-dependent CCS and CCSD methods.Source: §I · p. 2
  • Its coupled-cluster models can be used in QM/MM calculations or combined with the polarizable continuum model.Source: §I · p. 2

II. PROGRAM FEATURES

eT implements coupled-cluster models and associated properties, with time-dependent propagation for CCS and CCSD. Its post-Hartree–Fock methods use Cholesky vectors to construct electron-repulsion integrals, with a two-step decomposition designed to reduce storage and computational cost.

A. Coupled cluster methods

  • eT provides CCS, CC2, CCSD, CCSD(T), and CC3, with CCSD(T) limited to ground-state energies.Source: A. Coupled cluster methods · p. 3
  • Except for CCSD(T), the methods have spin-adapted ground- and excited-singlet implementations, and support moments, EOM oscillator strengths, and selected polarizabilities.Equation-of-motion polarizabilities are available at CCS, CC2, and CCSD levels.Source: A. Coupled cluster methods · p. 3
  • CCS and CCSD time-dependent equations can be propagated with Euler, RK4, or Gauss-Legendre integrators.Source: A. Coupled cluster methods · pp. 3–4

B. Cholesky decomposition for the electronic repulsion integrals

C. Hartree-Fock

eT implements screened Hartree–Fock methods and multilevel approaches that combine different treatments across molecular regions. The multilevel methods include active-space optimization, reduced-space coupled cluster, and environmental embedding options.

C. Hartree-Fock

  • RHF and UHF use integral-direct implementations with Coulomb/exchange screening and permutation symmetry, SAD guesses, and Roothaan–Hall SCF accelerated by DIIS or CROP.Density differences are used to construct the Fock matrix.Source: C. Hartree-Fock · p. 4

D. Multilevel and multiscale methods

  • MLHF optimizes only active molecular orbitals, with active virtuals built from projected atomic orbitals centered on active atoms.The PAOs are orthonormalized before use.Source: D. Multilevel and multiscale methods · pp. 4–5
  • The most expensive MLHF step, constructing the inactive two-electron Fock contribution, is needed only once because the inactive orbitals are frozen.The iterative cost is dominated by constructing the active two-electron contribution.Source: D. Multilevel and multiscale methods · p. 5
  • MLCC2 treats active orbitals at the CC2 level and inactive orbitals at CCS, with active spaces built from correlated natural transition orbitals or Cholesky-based choices.The listed alternatives include Cholesky orbitals and Cholesky occupied orbitals with PAOs spanning the virtual space.Source: D. Multilevel and multiscale methods · p. 5
  • Reduced-space coupled-cluster calculations use Cholesky orbitals for the occupied space and PAOs for the virtual space, supporting descriptions of localized properties.The program also implements frozen orbitals for all coupled-cluster methods.Source: D. Multilevel and multiscale methods · pp. 4–5
  • eT offers fixed-charge electrostatic QM/MM embedding and QM/FQ, where MM charges respond to electronegativities and the QM potential and mutually polarize with the QM part.In QM/FQ, the charges enter the QM Hamiltonian through a term nonlinear in the QM density.Source: D. Multilevel and multiscale methods · p. 5
  • PCM embedding describes the environment as a homogeneous continuum dielectric in a molecule-shaped cavity that mutually polarizes with the QM solute.The QM/PCM and QM/FQ implementations add terms to the Fock matrix; coupled-cluster-level terms can also be considered.Source: D. Multilevel and multiscale methods · pp. 5–6

E. Spectroscopic properties and response methods

eT provides response and spectroscopic methods across several coupled-cluster levels, including core-excitation, oscillator-strength, and real-time approaches. It also implements valence ionization potentials for selected methods.

E. Spectroscopic properties and response methods

  • Core-valence separation (CVS) is implemented as a projection for CCS, CC2, MLCC2, and CCSD, while CC3 omits noncontributing amplitudes and excitation-vector elements.The program supports UV/Vis and X-ray absorption spectra; the CC3 restriction reduces iterative computational scaling.Source: E. Spectroscopic properties and… · p. 6
  • EOM oscillator strengths are available for CCS, CC2, CCSD, and CC3, and linear-response oscillator strengths are available at CCS.An asymmetric Lanczos algorithm can directly obtain energies and EOM oscillator strengths for CCS, CC2, and CCSD.Source: E. Spectroscopic properties and… · p. 6
  • Real-time propagation provides a nonperturbative route to absorption spectra by Fourier transforming the dipole moment after an initial excitation pulse.The passage describes real-time propagation for CCS and CCSD.Source: E. Spectroscopic properties and… · p. 6
  • Valence ionization potentials are implemented for CCS, CC2, and CCSD using a noninteracting bath orbital and projection of excitation-vector components that do not involve it.The projection is described as similar to the CVS projection.Source: E. Spectroscopic properties and… · p. 6

III. ILLUSTRATIVE APPLICATIONS AND PERFORMANCE TESTS

The illustrative applications section specifies convergence and integral-accuracy thresholds and notes the calculation setup used for its examples. The supplied passages also identify a rifampicin excitation-energy table but do not provide its numerical entries.

III. ILLUSTRATIVE APPLICATIONS AND PERFORMANCE TESTS

  • Hartree–Fock convergence uses the maximum gradient norm, while coupled-cluster residual thresholds use the residual vectors’ L2 norm.Energy thresholds refer to the energy change from the previous iteration.Source: §III · p. 6
  • The Cholesky threshold is the largest absolute error on the electron-repulsion-integral matrix diagonal and bounds the error of all matrix elements.Source: §III · p. 6
  • Coupled-cluster calculations used either Cholesky vectors or electron-repulsion integrals in memory, and the geometries are available from Ref. 72.Source: §III · pp. 6–7
  • Table I reports the lowest FC-CC2/aug-cc-pVDZ excitation energy of rifampicin and identifies the number of frozen orbitals.Source: Table I · p. 7

A. Coupled cluster methods

eT provides coupled-cluster calculations across CCS, CC2, CCSD, and CC3, including excitation energies and oscillator strengths. Examples compare its accuracy and timings with other programs and show that Cholesky approximation errors can remain within the expected FC-CCSD accuracy.

A. Coupled cluster methods

B. Cholesky decomposition

For gramicidin A, eT’s Cholesky decomposition is generally small or negligible in cost relative to constructing a Fock matrix. The section compares decomposition times across basis sets and thresholds with the time for a Hartree–Fock iteration.

B. Cholesky decomposition

C. Hartree-Fock

eT models systems with several hundred atoms using Hartree–Fock. In the reported amylose and gramicidin comparisons, QChem is about twice as fast, while tight-threshold eT energies can be reproduced in LSDalton.

C. Hartree-Fock

  • eT’s Hartree–Fock calculations can model systems with several hundred atoms, including amylose and gramicidin.Source: C. Hartree-Fock · p. 13
  • The eT–QChem comparison is complicated by accuracy thresholds beyond the gradient and energy thresholds, including screening and integral accuracy.Source: C. Hartree-Fock · p. 13
  • QChem outperforms eT by about a factor of two, although the programs’ energies converge to slightly different results.Source: C. Hartree-Fock · p. 14, Table X · p. 14
  • For amylose, the tightest thresholds give an energy difference of 2 · 10^-7 Eh, and tight-threshold eT results can be reproduced in LSDalton 2018.3.Source: C. Hartree-Fock · p. 14

D. Multilevel and multiscale methods

The section demonstrates multilevel methods for excitation energies and Hartree–Fock calculations, using reduced active regions to lower computational cost. Across the examples, accuracy and savings depend on the method, active-region size, and system.

SO2–water excitation energies

  • The SO2–water calculations define three nested active regions: SO2 plus 21 waters for MLHF, five waters for reduced-space coupled cluster, and SO2 alone for MLCC2.The reduced-space and MLCC2 calculations use the corresponding active regions to define their orbital spaces.Source: D. Multilevel and multiscale methods · p. 15

Multilevel CC2

  • Within the expected CC2 error of about 0.1 eV, reduced-space and full-space FC-CC2 excitation energies agree.The comparison includes full and reduced-space FC-CC2, using HF and MLHF reference wave functions.Source: D. Multilevel and multiscale methods · p. 15

MLHF performance

  • Eightfold iteration-time reductions for amylose occur with cc-pVDZ and with aug-cc-pVDZ on active atoms; the latter timing comparison is reported against HF.For gramicidin, MLHF takes one third of the HF iteration time with cc-pVDZ and about two thirds with aug-cc-pVDZ on active atoms.Source: Table XII · p. 16, D. Multilevel and multiscale methods · pp. 16–17
  • For amylose, the smaller active region and linear chain structure are identified as reasons for the larger MLHF timing savings.For gramicidin, the comparatively large active region and compact shape are identified as reasons for smaller savings.Source: D. Multilevel and multiscale methods · p. 17

PNA in aqueous solution

E. Modelling spectroscopies

The section compares Davidson, Lanczos, and real-time approaches for computing absorption spectra. Lanczos reaches convergence in the low-energy spectrum with a smaller reduced space, while the reported peak positions match Davidson’s in the illustrated spectra.

Davidson and Lanczos

  • The low-energy UV/Vis spectrum converges with a smaller reduced space using Lanczos than with Davidson.The illustrated water spectrum is CCSD/aug-cc-pCVDZ; it is artificially extended beyond the 12.3 eV ionization potential to show convergence behavior.Source: E. Modelling spectroscopies · p. 19
  • The oxygen-edge X-ray absorption spectra show the same overall Davidson–Lanczos behavior as the UV/Vis spectra.The X-ray calculations use a CVS projection.Source: E. Modelling spectroscopies · p. 19

Real-time propagation

  • Peak positions match between Davidson and real-time propagation, but intensities differ with pulses centered on the first excitation energy.The first peak is scaled to match Davidson intensity; the comparison uses specified pulse frequency distributions.Source: E. Modelling spectroscopies · p. 19

IV. CONCLUDING REMARKS

eT 1.0 is presented as an optimized open-source electronic-structure program with coupled-cluster capabilities and new time-dependent and multilevel features. The authors highlight its Cholesky-based implementation and its potential use for development and production calculations.

Program capabilities

  • The authors describe eT as an optimized open-source electronic-structure program with Cholesky decomposition at its core.They state that their implementation is faster and less storage-intensive than that of any other program.Source: §IV · pp. 19–20
  • eT features the only spin-adapted closed-shell implementation of time-dependent coupled-cluster theory, according to the authors.The conclusion also calls its CC3 implementation the fastest for calculating ground- and excited-state energies and EOM oscillator strengths, to the authors’ knowledge.Source: §IV · pp. 19–20

Multilevel methods

  • MLHF and MLCC2 extend treatable system size without sacrificing accuracy for intensive properties such as excitation energies.The paper identifies these multilevel methods as part of eT’s capabilities.Source: §IV · p. 20

Outlook

  • The authors expect eT to serve both as a development platform and for production calculations in the quantum-chemistry community.They state that the program will continue to expand, focusing on molecular properties and multilevel methods.Source: §IV · p. 21
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