The Conundrum of Diffuse Basis Sets: A Blessing for Accuracy yet a Curse for Sparsity

Henryk Laqua; Linus Bjarne Dittmer; Martin Head-Gordon

REVIEW 3 major objections 5 minor 79 references

The Conundrum of Diffuse Basis Sets: A Blessing for Accuracy yet a Curse for Sparsity

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper claims that the sparsity loss caused by diffuse basis sets in quantum chemistry is a basis-set artifact: the inverse overlap matrix $S^{-1}$ delocalizes the one-particle density matrix, and a perturbative CABS-singles correction…

desk verdict Useful new explanation for the diffuse-basis sparsity curse via S^-1; the causal link is a hypothesis, not a proof, but the paper is worth engaging and the remedy is practical. read the letter →

arxiv 2502.04631 v1 pith:3PZDAQ5X submitted 2025-02-07 physics.chem-ph

Henryk Laqua , Linus Bjarne Dittmer , Martin Head-Gordon This is my paper · ORCID

classification physics.chem-ph

keywords diffusebasissetsone-particledensitymatrixsparsityinverseoverlapcontra-variantfunctionslinear-scalingelectronicstructurenon-covalentinteractionsCABSsinglescorrection

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quantum-chemistry calculations need diffuse basis functions to describe non-covalent interactions accurately, but these same functions destroy the sparsity of the one-particle density matrix (1-PDM), the property that makes linear-scaling methods fast. The paper argues that this 'curse of sparsity' is purely a basis-set artifact, not a property of the electrons: even the real-space 1-PDM shows artificial slow-decaying tails when diffuse functions are present. The cause is traced to the inverse overlap matrix $S^{-1}$, the object that defines the contra-variant dual basis, which is much less local than the overlap matrix itself. A model of a non-interacting helium chain shows the decay rate of $S^{-1}$ is set by the diffuseness and local incompleteness of the basis, so small diffuse sets suffer most. The paper then shows that keeping diffuse functions out of the self-consistent field (SCF) step and adding them only through a complementary auxiliary basis set (CABS) singles correction, together with compact pruned basis sets, recovers non-covalent accuracy at lower cost.

What carries the argument

The load-bearing objects are the contra-variant basis functions $\tilde{\chi} = S^{-1}\chi$, the inverse overlap matrix $S^{-1}$, and the idempotency relation $PSP=P$, which in a minimal basis gives $P=S^{-1}$. The model system carries the quantitative argument: an infinite chain of non-interacting helium atoms, each with a compact function $\chi_A$ and a diffuse augmenting function $\Delta_A$ that overlaps only nearest neighbors, where the SCF-optimized mixing coefficient $\omega$ is approximated by its first-order energy gradient. This yields a tridiagonal Toeplitz overlap matrix $S = N^2\operatorname{trid}(s, 1+s^2, s)$, whose exact inverse $(S^{-1})_{\mu\nu}=(-s)^{|\mu-\nu|}$ supplies the exponential decay law. The parameter $s$ is proportional to the product of the nearest-neighbor overlap $S^{AB}_{\Delta\chi}$ and the intra-atomic Fock matrix element $F^{AA}_{\Delta\chi}$, which measures how much the diffuse function cures local incompleteness; this product quantifies how the curse scales with diffuseness and local incompleteness.

What would settle it

For a fixed insulating chain, compute the real-space 1-PDM tail in two diffuse bases with identical physical system: one standard augmented basis and one where the diffuse functions have been Gram-Schmidt orthogonalized against all neighboring compact functions so that $S^{-1}$ remains banded. If the long-range oscillations persist in the orthogonalized basis, then $S^{-1}$ non-locality is not the cause; if they vanish, the paper's causal claim is confirmed.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the loss of 1-PDM sparsity seen with diffuse basis sets is a basis-set artifact. Because the converged density matrix is represented in the contra-variant dual basis, $P_{\mu\nu} = \int\int \tilde{\chi}_\mu(r)\rho(r,r')\tilde{\chi}_\nu(r')$, its locality is governed by $S^{-1}$, not $S$. In a minimal basis with no extra unoccupied orbitals, idempotency forces $P = S^{-1}$; the authors argue that $S^{-1}$ remains the diagnostic for the optimization manifold more generally, and its sparsity pattern matches that of $P$. For the model system of an infinite non-interacting chain of helium atoms, a minimal local basis augmented by diffuse functions with only nearest-neighbor overlap produces a tridiagonal Toeplitz overlap whose exact inverse is $(S^{-1})_{\mu\nu}=(-s)^{|\mu-\nu|}$, giving exponential but oscillatory delocalization with rate set by $s \propto F^{AA}_{\Delta\chi}S^{AB}_{\Delta\chi}$. Thus adding diffuse functions without improving local completeness makes the tail decay arbitrarily slow, explaining why aug-cc-pVDZ oscillates most strongly and why no complete-basis limit is approached in the tail. The paper's constructive claim is that compact, high-angular-momentum-pruned bases combined with a CABS singles correction recover nearly basis-set-limit non-covalent interaction energies while keeping the 1-PDM sparse.

Load-bearing premise

The load-bearing premise is that the delocalization of the inverse overlap matrix, not some other mechanism, is what makes the converged density matrix non-local in realistic calculations; the exact equality $P = S^{-1}$ is proven only for minimal basis sets with no extra unoccupied orbitals.

Editorial extensions

If this is right

Real-space 1-PDM tails from diffuse bases oscillate and decay far more slowly than the physical exponential decay expected for insulators; the asymptotic decay rate has no well-defined basis-set limit when diffuse functions are included.
Small, diffuse basis sets such as aug-cc-pVDZ suffer most, because the diffuse functions add non-local degrees of freedom without fixing the local incompleteness that the SCF tries to compensate.
Adding very diffuse functions without improving local completeness can make the artificial decay rate arbitrarily small, so larger and more diffuse basis sets can be worse than smaller ones for sparsity.
The CABS singles correction with compact pruned basis sets restores near-basis-set-limit non-covalent accuracy: prune-cc-pV5Z → aug-cc-pVTZ reaches 0.036 kcal/mol RMSD on S22, and prune-cc-pV6Z matches aug-cc-pV6Z on the ASCDB NCI subset at substantially lower cost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: linear-scaling methods should screen the sparsity of $S^{-1}$, not just $S$, before trusting a sparse density matrix, since a non-local contra-variant basis can spoil the 1-PDM even when the overlap matrix looks local.
Extension: a testable design principle is to block-orthogonalize diffuse functions against the compact functions of neighboring atoms so that $S^{AB}_{\Delta\chi}\to 0$; the model predicts this removes the artificial delocalization while retaining radial accuracy.
Extension: quantities built from the density-matrix tail, such as localized-orbital tails or fragment-embedding densities, may inherit the same artifact, so compact non-augmented references could be more reliable for tail properties than augmented supersets.
Extension: the mechanism suggests that the conventional wisdom that larger basis sets are always better fails specifically for the far off-diagonal of the density matrix, and benchmarks of linear-scaling methods should report the sparsity of $S^{-1}$ alongside timing.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, what's actually new: the paper identifies a real puzzle — diffuse basis sets destroy 1-PDM sparsity out of proportion to their spatial extent — and proposes that the culprit is the non-locality of the inverse overlap matrix, i.e., of the contra-variant basis, not the basis functions themselves. To my knowledge that specific mechanism has not been offered before. The toy model of a helium chain with nearest-neighbor overlap cleanly shows how a local S can have an S^{-1} that decays exponentially with a rate controlled by diffuseness and local incompleteness. The pruned-basis/CABS remedy is practical and benchmarked on S22 and ASCDB, with timings that make the case concrete. The paper does real work: no fitted parameters, data and basis set files in the supplementary material, and the computational details are transparent.

The soft spots are real but contained. Eq. (19) is wrong as printed: the inverse of trid(s,1+s^2,s) carries a prefactor 1/(1-s^2). The decay rate is unaffected, so the model's conclusion survives, but the formula needs fixing. More substantively, the causal link between S^{-1} and P is asserted, not derived, for realistic non-minimal basis sets. Section IV explicitly says 'arguably not just coincidental, but causal,' and the toy model only applies where P=S^{-1} (minimal basis, no virtual space). The sparsity-pattern similarity in Fig. 5 vs Fig. 1 is suggestive, but it's not a proof. I'd like the paper to present this as a well-motivated hypothesis — which is what it is — rather than an established mechanism. That doesn't sink the paper, because the practical remedy is validated empirically and the diagnostic value of S^{-1} sparsity stands regardless.

One wording nit: the abstract says the curse 'becomes worse for larger basis sets.' In fact, the worst case is small augmented basis sets like aug-cc-pVDZ (Fig. 4), so 'basis sets with diffuse functions' or 'more diffuse basis sets' would be more accurate.

Overall: this is a solid, useful paper. The central claim holds up as a hypothesis, and the paper is honest about the limits of the model. The benchmarks are carefully done, and the CABS/pruned-basis results give practitioners something actionable. It deserves serious peer review, with requests for the prefactor fix, a softer causal claim, and a tightened abstract. I would bring it to a reading group if you're interested in linear-scaling methods or basis set design.

Referee Report

3 major / 5 minor

Summary. The paper investigates why diffuse atomic-orbital basis sets destroy the sparsity of the one-particle density matrix (1-PDM) in SCF calculations, despite the expected exponential decay of the exact 1-PDM for insulators. The authors show that the effect persists in a real-space representation, and argue that it is caused by the non-locality of the contra-variant basis functions, quantified by the inverse overlap matrix S^{-1}. They introduce a model of an infinite non-interacting helium chain with a minimal augmented basis, where P = S^{-1} exactly, and derive an exponential decay rate for S^{-1} proportional to the diffuseness and local incompleteness of the basis. They also propose a remedy: compact pruned basis sets combined with a CABS singles correction, and report benchmark results (ASCDB, S22) indicating that this restores non-covalent accuracy at lower cost. The central claim is that the sparsity curse is 'solely a basis set artifact.'

Significance. If the central claim were fully established, the paper would provide a useful mechanistic explanation for a well-known practical problem in linear-scaling electronic structure theory, and the proposed pruned-basis/CABS strategy could be a practical remedy. The benchmarks are carefully performed and the qualitative picture is consistent across figures and tables. The toy model is valuable as an analytically solvable illustration of how a local basis can nonetheless produce a non-local S^{-1}. The paper ships reproducible data (supplementary material with data points and basis set files) and does not fit parameters to the benchmark data. However, the causal link between S^{-1} and P in realistic non-minimal basis sets is asserted rather than derived, and the quantitative inverse formula in Eq. (19) is incorrect as printed; these issues currently weaken the central claim.

major comments (3)

[Section V, Eq. (19)] The stated inverse of S = trid(s, 1+s^2, s) as (S^{-1})_{mu nu} = (-s)^{|mu-nu|} omits the prefactor 1/(1-s^2). For example, a 2x2 block gives the off-diagonal element -s/(1+s^2+s^4). The correct infinite-limit formula is (S^{-1})_{mu nu} = (-s)^{|mu-nu|}/(1-s^2). Since this equation is the quantitative basis for the claimed exponential decay rate proportional to s, the printed formula should be corrected and the proportionality statements in Eqs. (18) and Section VI re-examined.
[Section IV and Section VI] The paper's central explanation—that non-locality of S^{-1} causes non-locality of the converged P—is only proven in the minimal-basis case where P = S^{-1} (Eq. 4). In the realistic non-minimal basis sets used in Figs. 1 and 5, P is not equal to S^{-1}; it is a nonlinear functional of the Fock matrix with a virtual space. The sparsity-pattern similarity is suggestive but not probative. Moreover, the model in Section V explicitly retains the minimal-basis condition (stated in Section VI: 'the augmented basis is still minimal and therefore P = S^{-1} still holds'), so it cannot resolve whether the non-locality of P is caused by S^{-1} or merely coincides with it when no virtual space exists. Please either provide a derivation or a numerical causality test for non-minimal basis sets, or soften the abstract's 'solely a basis set artifact' claim to a diagnostic correlation.
[Section VII, Tables II and III] The proposed remedy (pruned compact basis sets with CABS singles) is evaluated only in terms of energetic accuracy and timings, not in terms of the sparsity of the resulting 1-PDM. Since the paper's central concern is sparsity, the reader cannot assess whether the remedy actually lifts the curse; the timing improvements could partly reflect the smaller basis size rather than improved sparsity. Please report sparsity metrics (e.g., number of significant P or rho(r,r') elements above a threshold) for the pruned basis sets with and without CABS.

minor comments (5)

[Section VII] The word 'diffuese' should be 'diffuse' in the paragraph following Table III.
[Section V, Eqs. (16) and (17)] The relationship between t and s is not stated; since S = N^2 trid(s, 1+s^2, s), it would be helpful to give s = t/N^2 explicitly.
[Section III, Figs. 2 and 3] The real-space 1-PDM is sampled on a sparse grid (one point per atom). Please state the grid density and discuss its adequacy for sparsity counts, since the off-diagonal tail may be sensitive to the sampling.
[Section IV, Eq. (3)] In the definition of the contra-variant dual, the summation index should be nu (or another dummy) rather than mu, which conflicts with the free index mu as written.
[Section V, Eqs. (8)-(10)] The assumptions that the augmentation function overlaps only nearest neighbors and is block-orthogonal to the minimal basis are stated without discussion of their relation to actual Gaussian basis sets; a sentence connecting these assumptions to the numerical observations would improve the model's credibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the model derivation is self-contained and the P=S^{-1} step is an explicit minimal-basis theorem, not a fitted or imported result.

full rationale

The paper's central derivation chain is not circular. The real-space 1-PDM sparsity loss is an observed numerical fact (Figs. 2-4); the authors then hypothesize that the non-locality of S^{-1} provides the parameter space for the optimization (Section IV, Eqs. (2)-(3)). This hypothesis is explicitly flagged as an argument ('arguably not just coincidental, but causal'), not derived from the benchmark data, so there is no fitted-parameter-as-prediction reduction. The mathematical model in Sections V-VI is self-contained: the tridiagonal overlap matrix is constructed from explicitly stated overlap assumptions (Eqs. (7)-(11)), omega is set by the first-order gradient condition (Eq. (13)), and the exponential decay of S^{-1} follows from the known inverse of a tridiagonal Toeplitz matrix (Eq. (19)). The use of P=S^{-1} is restricted to the model where the basis remains minimal, which is explicitly stated ('because the augmented basis is still minimal and therefore P=S^{-1} (eq. (4)) still holds'); this is a theorem, not an input fitted to the target phenomenon. No parameters are fitted to the benchmark sparsity data. Self-citations (e.g., ref. 56 for the standard contravariant-dual definition, ref. 40 for omegaB97X-V) are routine and non-load-bearing. A correctable typo in Eq. (19) (omitted prefactor 1/(1-s^2)) affects precision, not the exponential-decay conclusion, and is not a circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters are used in the central argument: omega is derived from the Fock gradient, and the decay rate s is a function of overlap and Fock matrix elements. The main assumptions are the idealizations of the helium-chain model and the hypothesis that S^{-1} non-locality causes P non-locality. No new physical entities are introduced; the prune-cc-pVXZ basis sets are computational constructs provided as supplemental files.

assumptions (6)

domain assumption The exact one-particle density matrix of the infinite non-interacting helium chain is atom-block diagonal.
Section V: the exact real-space PDM is atom-block diagonal, serving as the reference against which the basis-set artifact is measured.
ad hoc to paper The diffuse augmentation function Delta_A overlaps only nearest-neighbor basis functions and is on-site block-orthogonal to the minimal basis.
Section V, eqs. (8)-(10): these conditions define the toy model; the authors note real Gaussians overlap farther, which would make the artifact stronger, so the assumption is conservative.
domain assumption The mixing coefficient omega is small and given by the negative first-order SCF gradient, omega = -2 F^AA_DeltaChi.
Appendix B derives this via Pulay's gradient formula assuming translational symmetry and a minimal basis; it ignores the Hessian and higher-order terms.
standard math In a minimal basis with no virtual space, the density matrix equals the inverse overlap, P = S^{-1}.
Section IV, from idempotency PSP=P; exact for a minimal basis.
ad hoc to paper Non-locality of S^{-1} is causal for the non-locality of the converged density matrix in non-minimal basis sets.
Section IV: 'arguably not just coincidental, but causal'; this is the central hypothesis, supported by sparsity-pattern similarity and the toy model, not by a general derivation.
standard math The tridiagonal Toeplitz inverse decays as (-s)^{|mu-nu|}.
Section V cites Demko et al.; the exact inverse carries a 1/(1-s^2) prefactor, but the exponential rate is s.

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Pith. "Pith review of The Conundrum of Diffuse Basis Sets: A Blessing for Accuracy yet a Curse for Sparsity." pith.science (2026). https://pith.science/paper/3PZDAQ5X

@misc{pith2026250204631,
  author       = {Pith},
  title        = {Pith review of: The Conundrum of Diffuse Basis Sets: A Blessing for Accuracy yet a Curse for Sparsity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3PZDAQ5X}},
  note         = {Machine review of arXiv:2502.04631}
}

abstract

Diffuse atomic orbital basis sets have proven to be essential to obtain accurate interaction energies, especially in regard to non-covalent interactions. However, they also have a detrimental impact on the sparsity of the one-particle density matrix (1-PDM), to a degree stronger than the spatial extent of the basis functions alone could explain. This is despite the fact that the matrix elements of the 1-PDM of insulators (systems with significant HOMO-LUMO gaps) are expected to decay exponentially with increasing real-space distance from the diagonal and the asymptotic decay rate is expected to have a well-defined basis set limit. The observed low sparsity of the 1-PDM appears to be independent of representation and even persists after projecting the 1-PDM onto a real-space grid, leading to the conclusion that this "curse of sparsity" is solely a basis set artifact, which, counterintuitively, becomes worse for larger basis sets, seemingly contradicting the notion of a well-defined basis set limit. We show that this is a consequence of the low locality of the contra-variant basis functions as quantified by the inverse overlap matrix $\mathbf{S}^{-1}$ being significantly less sparse than its covariant dual. Introducing the model system of an infinite non-interacting chain of helium atoms, we are able to quantify the exponential decay rate to be proportional to the diffuseness as well as local incompleteness of the basis set, meaning small and diffuse basis sets are affected the most. Finally, we propose one solution to the conundrum in the form of the complementary auxiliary basis set (CABS) singles correction in combination with compact, low l-quantum-number basis sets, showing promising results for non-covalent interactions.

Figures

Figures reproduced from arXiv: 2502.04631 by the authors.

**Figure 2.** FIG. 2: Element-wise sparsity-pattern (each pixel corresponds to one matrix element; [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3: Element-wise sparsity counts of real-space density matrix [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: One row ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5: Block-sparsity-pattern (32 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

79 extracted references · 44 canonical work pages

[1]

Kohn ,\ title title Density Functional and Density Matrix Method Scaling Linearly with the Number of Atoms , \ https://doi.org/10.1103/PhysRevLett.76.3168 journal journal Phys

author author W. Kohn ,\ title title Density Functional and Density Matrix Method Scaling Linearly with the Number of Atoms , \ https://doi.org/10.1103/PhysRevLett.76.3168 journal journal Phys. Rev. Lett. \ volume 76 ,\ pages 3168--3171 ( year 1996 ) NoStop

doi:10.1103/physrevlett.76.3168
[2]

author author J. C. \ Burant , author G. E. \ Scuseria ,\ and\ author M. J. \ Frisch ,\ title title A linear scaling method for Hartree - Fock exchange calculations of large molecules. \ https://doi.org/10.1063/1.472627 journal journal J. Chem. Phys. \ volume 105 ,\ pages 8969--8972 ( year 1996 ) NoStop

work page doi:10.1063/1.472627
[3]

Schwegler , author M

author author E. Schwegler , author M. Challacombe ,\ and\ author M. Head-Gordon ,\ title title Linear scaling computation of the Fock matrix. II . Rigorous bounds on exchange integrals and incremental Fock build. \ https://doi.org/10.1063/1.473833 journal journal J. Chem. Phys. \ volume 106 ,\ pages 9708--9717 ( year 1997 ) NoStop

doi:10.1063/1.473833
[4]

Ochsenfeld , author C

author author C. Ochsenfeld , author C. A. \ White ,\ and\ author M. Head-Gordon ,\ title title Linear and sublinear scaling formation of Hartree - Fock -type exchange matrixes. \ https://doi.org/10.1063/1.476741 journal journal J. Chem. Phys. \ volume 109 ,\ pages 1663--1669 ( year 1998 ) NoStop

doi:10.1063/1.476741
[5]

Neese , author F

author author F. Neese , author F. Wennmohs , author A. Hansen ,\ and\ author U. Becker ,\ title title Efficient, approximate and parallel Hartree - Fock and hybrid DFT calculations. A 'chain-of-spheres' algorithm for the Hartree - Fock exchange. \ https://doi.org/10.1016/j.chemphys.2008.10.036 journal journal Chem. Phys. \ volume 356 ,\ pages 98--109 ( y...

doi:10.1016/j.chemphys.2008.10.036
[6]

Laqua , author J

author author H. Laqua , author J. Kussmann ,\ and\ author C. Ochsenfeld ,\ title title Efficient and Linear - Scaling Seminumerical Method for Local Hybrid Density Functionals . \ https://doi.org/10.1021/acs.jctc.8b00062 journal journal J. Chem. Theory Comput. \ volume 14 ,\ pages 3451--3458 ( year 2018 a ) NoStop

doi:10.1021/acs.jctc.8b00062
[7]

Laqua , author T

author author H. Laqua , author T. H. \ Thompson , author J. Kussmann ,\ and\ author C. Ochsenfeld ,\ title title Highly Efficient , Linear - Scaling Seminumerical Exact - Exchange Method for Graphic Processing Units . \ https://doi.org/10.1021/acs.jctc.9b00860 journal journal J. Chem. Theory Comput. \ volume 16 ,\ pages 1456--1468 ( year 2020 ) NoStop

doi:10.1021/acs.jctc.9b00860
[8]

Helmich-Paris , author B

author author B. Helmich-Paris , author B. de Souza , author F. Neese ,\ and\ author R. Izsak ,\ title title An improved chain of spheres for exchange algorithm , \ https://doi.org/10.1063/5.0058766 journal journal J. Chem. Phys. \ volume 155 ,\ pages 104109 ( year 2021 ) NoStop

doi:10.1063/5.0058766

Show all 79 references

[9]

Challacombe ,\ title title A simplified density matrix minimization for linear scaling self-consistent field theory , \ https://doi.org/10.1063/1.477969 journal journal J

author author M. Challacombe ,\ title title A simplified density matrix minimization for linear scaling self-consistent field theory , \ https://doi.org/10.1063/1.477969 journal journal J. Chem. Phys. \ volume 110 ,\ pages 2332--2342 ( year 1999 ) NoStop

doi
[10]

author author A. M. N. \ Niklasson , author V. Weber ,\ and\ author M. Challacombe ,\ title title Nonorthogonal density-matrix perturbation theory. \ https://doi.org/10.1063/1.1944725 journal journal J. Chem. Phys. \ volume 123 ,\ pages 044107 ( year 2005 ) NoStop

doi
[11]

author author D. K. \ Jordan \ and\ author D. A. \ Mazziotti ,\ title title Comparison of two genres for linear scaling in density functional theory: Purification and density matrix minimization methods. \ https://doi.org/10.1063/1.1853378 journal journal J. Chem. Phys. \ volu...

doi
[12]

author author E. H. \ Rubensson , author E. Rudberg ,\ and\ author P. Salek ,\ title title Density matrix purification with rigorous error control. \ https://doi.org/10.1063/1.2826343 journal journal J. Chem. Phys. \ volume 128 ,\ pages 074106 ( year 2008 ) NoStop

doi
[13]

Suryanarayana ,\ title title Optimized purification for density matrix calculation

author author P. Suryanarayana ,\ title title Optimized purification for density matrix calculation. \ https://doi.org/10.1016/j.cplett.2012.10.090 journal journal Chem. Phys. Lett. \ volume 555 ,\ pages 291--295 ( year 2013 ) NoStop

2012 doi
[14]

author author D. S. \ Lambrecht , author B. Doser ,\ and\ author C. Ochsenfeld ,\ title title Rigorous integral screening for electron correlation methods , \ https://doi.org/10.1063/1.2079987 journal journal J. Chem. Phys. \ volume 123 ,\ pages 184102 ( year 2005 ) NoStop

doi
[15]

author author S. A. \ Maurer , author D. S. \ Lambrecht , author J. Kussmann ,\ and\ author C. Ochsenfeld ,\ title title Efficient distance-including integral screening in linear-scaling Moller - Plesset perturbation theory , \ https://doi.org/10.1063/1.4770502 journal journal...

doi
[16]

Glasbrenner , author D

author author M. Glasbrenner , author D. Graf ,\ and\ author C. Ochsenfeld ,\ title title Efficient Reduced - Scaling Second - Order Moller - Plesset Perturbation Theory with Cholesky - Decomposed Densities and an Attenuated Coulomb Metric , \ https://doi.org/10.1021/acs.jctc....

doi
[17]

Wang , author A

author author Z. Wang , author A. Aldossary ,\ and\ author M. Head-Gordon ,\ title title Sparsity of the electron repulsion integral tensor using different localized virtual orbital representations in local second-order Moller - Plesset theory , \ https://doi.org/10.1063/5.013...

doi
[18]

Shi , author Z

author author T. Shi , author Z. Wang , author A. Aldossary , author Y. Liu , author X. S. \ Li ,\ and\ author M. Head-Gordon ,\ title title Local Second Order Moeller - Plesset Theory with a Single Threshold Using Orthogonal Virtual Orbitals : A Distributed Memory Implementat...

doi
[19]

author author H. F. \ Schurkus \ and\ author C. Ochsenfeld ,\ title title Communication: An effective linear-scaling atomic-orbital reformulation of the random-phase approximation using a contracted double- Laplace transformation , \ https://doi.org/10.1063/1.4939841 journal j...

doi
[20]

Luenser , author H

author author A. Luenser , author H. F. \ Schurkus ,\ and\ author C. Ochsenfeld ,\ title title Vanishing- Overhead Linear - Scaling Random Phase Approximation by Cholesky Decomposition and an Attenuated Coulomb - Metric , \ https://doi.org/10.1021/acs.jctc.6b01235 journal jour...

doi
[21]

Graf , author M

author author D. Graf , author M. Beuerle , author H. F. \ Schurkus , author A. Luenser , author G. Savasci ,\ and\ author C. Ochsenfeld ,\ title title Accurate and Efficient Parallel Implementation of an Effective Linear - Scaling Direct Random Phase Approximation Method , \ ...

doi
[22]

Drontschenko , author D

author author V. Drontschenko , author D. Graf , author H. Laqua ,\ and\ author C. Ochsenfeld ,\ title title Lagrangian- Based Minimal - Overhead Batching Scheme for the Efficient Integral - Direct Evaluation of the RPA Correlation Energy , \ https://doi.org/10.1021/acs.jctc.1...

doi
[23]

Meyer ,\ title title PNO - CI [pseudonatural orbitals-configuration interaction] studies of electron correlation effects

author author W. Meyer ,\ title title PNO - CI [pseudonatural orbitals-configuration interaction] studies of electron correlation effects. I . Configuration expansion by means of nonorthogonal orbitals, and application to the ground state and ionized states of methane , \ http...

doi
[24]

Ahlrichs , author F

author author R. Ahlrichs , author F. Driessler , author H. Lischka , author V. Staemmler ,\ and\ author W. Kutzelnigg ,\ title title PNO [pair natural orbital]- CL and CEPA [coupled electron pair approximation]- PNO calculations of molecular systems. II . Molecules berryllium...

doi
[25]

Pulay ,\ title title Localizability of dynamic electron correlation , \ https://doi.org/10.1016/0009-2614(83)80703-9 journal journal Chem

author author P. Pulay ,\ title title Localizability of dynamic electron correlation , \ https://doi.org/10.1016/0009-2614(83)80703-9 journal journal Chem. Phys. Lett. \ volume 100 ,\ pages 151 ( year 1983 ) NoStop

1983 doi
[26]

Schutz , author G

author author M. Schutz , author G. Hetzer ,\ and\ author H.-J. \ Werner ,\ title title Low-order scaling local electron correlation methods. I . Linear scaling local MP2 , \ https://doi.org/10.1063/1.479957 journal journal J. Chem. Phys. \ volume 111 ,\ pages 5691--5705 ( yea...

doi
[27]

Schuetz ,\ title title A new, fast, semi-direct implementation of linear scaling local coupled cluster theory , \ https://doi.org/10.1039/b203994j journal journal Phys

author author M. Schuetz ,\ title title A new, fast, semi-direct implementation of linear scaling local coupled cluster theory , \ https://doi.org/10.1039/b203994j journal journal Phys. Chem. Chem. Phys. \ volume 4 ,\ pages 3941--3947 ( year 2002 ) NoStop

doi
[28]

Neese , author A

author author F. Neese , author A. Hansen ,\ and\ author D. G. \ Liakos ,\ title title Efficient and accurate approximations to the local coupled cluster singles doubles method using a truncated pair natural orbital basis , \ https://doi.org/10.1063/1.3173827 journal journal J...

doi
[29]

Riplinger \ and\ author F

author author C. Riplinger \ and\ author F. Neese ,\ title title An efficient and near linear scaling pair natural orbital based local coupled cluster method , \ https://doi.org/10.1063/1.4773581 journal journal J. Chem. Phys. \ volume 138 ,\ pages 034106 ( year 2013 ) NoStop

doi
[30]

Schwilk , author Q

author author M. Schwilk , author Q. Ma , author C. Koeppl ,\ and\ author H.-J. \ Werner ,\ title title Scalable Electron Correlation Methods . 3. Efficient and Accurate Parallel Local Coupled Cluster with Pair Natural Orbitals ( PNO - LCCSD ) , \ https://doi.org/10.1021/acs.j...

doi
[31]

Sacchetta , author D

author author F. Sacchetta , author D. Graf , author H. Laqua , author M. A. \ Ambroise , author J. Kussmann , author A. Dreuw ,\ and\ author C. Ochsenfeld ,\ title title Effective sub-quadratic scaling atomic-orbital reformulation of scaled opposite-spin RI - CC2 ground-state...

doi
[32]

Sandler , author J

author author I. Sandler , author J. Chen , author M. Taylor , author S. Sharma ,\ and\ author J. Ho ,\ title title Accuracy of DLPNO - CCSD ( T ): Effect of Basis Set and System Size , \ https://doi.org/10.1021/acs.jpca.0c11270 journal journal J. Phys. Chem. A \ volume 125 ,\...

doi
[33]

Gray \ and\ author J

author author M. Gray \ and\ author J. M. \ Herbert ,\ title title Assessing the domain-based local pair natural orbital ( DLPNO ) approximation for non-covalent interactions in sizable supramolecular complexes , \ https://doi.org/10.1063/5.0206533 journal journal J. Chem. Phy...

doi
[34]

author author M. S. \ Marshall , author L. A. \ Burns ,\ and\ author C. D. \ Sherrill ,\ title title Basis set convergence of the coupled-cluster correction, CCSD ( T ) MP2 Best practices for benchmarking non-covalent interactions and the attendant revision of the S22 , NBC10 ...

doi
[35]

Papajak , author J

author author E. Papajak , author J. Zheng , author X. Xu , author H. R. \ Leverentz ,\ and\ author D. G. \ Truhlar ,\ title title Perspectives on Basis Sets Beautiful : Seasonal Plantings of Diffuse Basis Functions , \ https://doi.org/10.1021/ct200106a journal journal J. Chem...

doi
[36]

Witte , author J

author author J. Witte , author J. B. \ Neaton ,\ and\ author M. Head-Gordon ,\ title title Push it to the limit: Characterizing the convergence of common sequences of basis sets for intermolecular interactions as described by density functional theory , \ https://doi.org/10.1...

doi
[37]

author author P. R. \ Nagy \ and\ author M. Kallay ,\ title title Approaching the Basis Set Limit of CCSD ( T ) Energies for Large Molecules with Local Natural Orbital Coupled - Cluster Methods , \ https://doi.org/10.1021/acs.jctc.9b00511 journal journal J. Chem. Theory Comput...

doi
[38]

Gray \ and\ author J

author author M. Gray \ and\ author J. M. \ Herbert ,\ title title Comprehensive Basis - Set Testing of Extended Symmetry - Adapted Perturbation Theory and Assessment of Mixed - Basis Combinations to Reduce Cost , \ https://doi.org/10.1021/acs.jctc.1c01302 journal journal J. C...

doi
[39]

Morgante \ and\ author R

author author P. Morgante \ and\ author R. Peverati ,\ title title Statistically representative databases for density functional theory via data science. \ https://doi.org/10.1039/c9cp03211h journal journal Phys. Chem. Chem. Phys. \ volume 21 ,\ pages 19092--19103 ( year 2019 ) NoStop

doi
[40]

Mardirossian \ and\ author M

author author N. Mardirossian \ and\ author M. Head-Gordon ,\ title title B97X - V : A 10-parameter, range-separated hybrid, generalized gradient approximation density functional with nonlocal correlation, designed by a survival-of-the-fittest strategy. \ https://doi.org/10.10...

doi
[41]

Weigend \ and\ author R

author author F. Weigend \ and\ author R. Ahlrichs ,\ title title Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn : Design and assessment of accuracy , \ https://doi.org/10.1039/B508541A journal journal Phys. Chem. Chem....

doi
[42]

author author T. H. J. \ Dunning ,\ title title Gaussian basis sets for use in correlated molecular calculations. I . The atoms boron through neon and hydrogen , \ https://doi.org/10.1063/1.456153 journal journal J. Chem. Phys. \ volume 90 ,\ pages 1007 ( year 1989 ) NoStop

doi
[43]

author author D. E. \ Woon \ and\ author T. H. J. \ Dunning ,\ title title Gaussian basis sets for use in correlated molecular calculations. III . The atoms aluminum through argon , \ https://doi.org/10.1063/1.464303 journal journal J. Chem. Phys. \ volume 98 ,\ pages 1358 ( y...

doi
[44]

author author D. E. \ Woon \ and\ author T. H. J. \ Dunning ,\ title title Gaussian basis sets for use in correlated molecular calculations. IV . Calculation of static electrical response properties , \ https://doi.org/10.1063/1.466439 journal journal J. Chem. Phys. \ volume 1...

doi
[45]

author author N. B. \ Balabanov \ and\ author K. A. \ Peterson ,\ title title Systematically convergent basis sets for transition metals. I . All -electron correlation consistent basis sets for the 3d elements Sc - Zn , \ https://doi.org/10.1063/1.1998907 journal journal J. Ch...

doi
[46]

author author B. P. \ Pritchard , author D. Altarawy , author B. Didier , author T. D. \ Gibson ,\ and\ author T. L. \ Windus ,\ title title New Basis Set Exchange : An Open , Up -to- Date Resource for the Molecular Sciences Community , \ https://doi.org/10.1021/acs.jcim.9b007...

doi
[47]

author author S. F. \ Boys \ and\ author F. Bernardi ,\ title title The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors , \ https://doi.org/10.1080/00268977000101561 journal journal Mol. Phys. \ vol...

doi
[48]

author author F. B. \ van Duijneveldt , author J. G. C. M. \ van Duijneveldt-van de Rijdt ,\ and\ author J. H. \ van Lenthe ,\ title title State of the Art in Counterpoise Theory , \ https://doi.org/10.1021/cr00031a007 journal journal Chem. Rev. (Washington, D. C.) \ volume 94...

doi
[49]

author author M. K. \ Kesharwani , author A. Karton , author N. Sylvetsky ,\ and\ author J. M. L. \ Martin ,\ title title The S66 Non - Covalent Interactions Benchmark Reconsidered Using Explicitly Correlated Methods Near the Basis Set Limit , \ https://doi.org/10.1071/ch17588...

doi
[50]

Gray , author P

author author M. Gray , author P. E. \ Bowling ,\ and\ author J. M. \ Herbert ,\ title title Systematic Evaluation of Counterpoise Correction in Density Functional Theory , \ https://doi.org/10.1021/acs.jctc.2c00883 journal journal J. Chem. Theory Comput. \ volume 18 ,\ pages ...

doi
[51]

Baer \ and\ author M

author author R. Baer \ and\ author M. Head-Gordon ,\ title title On the sparsity of the density matrix in Kohn Sham Density Functional Theory and an assessment of linear system-size scaling methods , \ @noop journal journal PHYSICAL REVIEW LETTERS \ ( year 1997 ) NoStop

1997
[52]

author author P. E. \ Maslen , author C. Ochsenfeld , author C. A. \ White , author M. S. \ Lee ,\ and\ author M. Head-Gordon ,\ title title Locality and Sparsity of Ab Initio One - Particle Density Matrixes and Localized Orbitals , \ https://doi.org/10.1021/jp972919j journal ...

doi
[53]

Ismail-Beigi \ and\ author T

author author S. Ismail-Beigi \ and\ author T. A. \ Arias ,\ title title Locality of the Density Matrix in Metals , Semiconductors , and Insulators , \ https://doi.org/10.1103/PhysRevLett.82.2127 journal journal Phys. Rev. Lett. \ volume 82 ,\ pages 2127--2130 ( year 1999 ) NoStop

doi
[54]

Jedrzejewski \ and\ author T

author author J. Jedrzejewski \ and\ author T. Krokhmalskii ,\ title title Exact results for spatial decay of the one-body density matrix in low-dimensional insulators , \ https://doi.org/10.1103/physrevb.70.153102 journal journal Phys. Rev. B: Condens. Matter Mater. Phys. \ v...

doi
[55]

author author E. H. \ Rubensson \ and\ author E. Rudberg ,\ title title Bringing about matrix sparsity in linear‐scaling electronic structure calculations , \ https://doi.org/10.1002/jcc.21723 journal journal J Comput Chem \ volume 32 ,\ pages 1411--1423 ( year 2011 ) NoStop

doi
[56]

Head-Gordon , author M

author author M. Head-Gordon , author M. Lee , author P. Maslen , author T. Van Voorhis ,\ and\ author S. Gwaltney ,\ title Modern methods and algorithms of quantum chemistry , \ \ ( publisher John von Neumann Institute for Computing ,\ address Jülich ,\ year 2000 )\ Chap.\ ch...

2000
[57]

McWeeny ,\ title title Hartree- Fock theory with nonorthogonal basis functions , \ https://doi.org/10.1103/physrev.114.1528 journal journal Phys

author author R. McWeeny ,\ title title Hartree- Fock theory with nonorthogonal basis functions , \ https://doi.org/10.1103/physrev.114.1528 journal journal Phys. Rev. \ volume 114 ,\ pages 1528 ( year 1959 ) NoStop

1959 doi
[58]

Pulay ,\ title title Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules

author author P. Pulay ,\ title title Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules. I . Theory . \ https://doi.org/10.1080/00268976900100941 journal journal Mol. Phys. \ volume 17 ,\ pages 197--204 ( year 1969 ) NoStop

doi
[59]

Demko , author W

author author S. Demko , author W. F. \ Moss ,\ and\ author P. W. \ Smith ,\ title title Decay Rates for Inverses of Band Matrices , \ https://doi.org/10.2307/2008290 journal journal Mathematics of Computation \ volume 43 ,\ pages 491--499 ( year 1984 ) NoStop

doi
[60]

Liang \ and\ author M

author author W. Liang \ and\ author M. Head-Gordon ,\ title title Approaching the Basis Set Limit in Density Functional Theory Calculations Using Dual Basis Sets without Diagonalization , \ https://doi.org/10.1021/jp0374713 journal journal J. Phys. Chem. A \ volume 108 ,\ pag...

doi
[61]

author author R. P. \ Steele , author Y. Shao , author R. A. J. \ DiStasio ,\ and\ author M. Head-Gordon ,\ title title Dual- Basis Analytic Gradients . 1. Self - Consistent Field Theory , \ https://doi.org/10.1021/jp065444h journal journal J. Phys. Chem. A \ volume 110 ,\ pag...

doi
[62]

author author R. P. \ Steele , author R. A. J. \ Di Stasio ,\ and\ author M. Head-Gordon ,\ title title Non- Covalent Interactions with Dual - Basis Methods : Pairings for Augmented Basis Sets , \ https://doi.org/10.1021/ct900058p journal journal J. Chem. Theory Comput. \ volu...

doi
[63]

author author R. P. \ Steele , author M. Head-Gordon ,\ and\ author J. C. \ Tully ,\ title title Ab Initio Molecular Dynamics with Dual Basis Set Methods , \ https://doi.org/10.1021/jp107342g journal journal J. Phys. Chem. A \ volume 114 ,\ pages 11853--11860 ( year 2010 ) NoStop

doi
[64]

Hegely , author P

author author B. Hegely , author P. R. \ Nagy ,\ and\ author M. Kallay ,\ title title Dual Basis Set Approach for Density Functional and Wave Function Embedding Schemes , \ https://doi.org/10.1021/acs.jctc.8b00350 journal journal J. Chem. Theory Comput. \ volume 14 ,\ pages 46...

doi
[65]

Csoka \ and\ author M

author author J. Csoka \ and\ author M. Kallay ,\ title title Speeding up Hartree - Fock and Kohn - Sham calculations with first-order corrections , \ https://doi.org/10.1063/5.0041276 journal journal J. Chem. Phys. \ volume 154 ,\ pages 164114 ( year 2021 ) NoStop

doi
[66]

author author T. B. \ Adler , author G. Knizia ,\ and\ author H.-J. \ Werner ,\ title title A simple and efficient CCSD ( T )- F12 approximation , \ https://doi.org/10.1063/1.2817618 journal journal J. Chem. Phys. \ volume 127 ,\ pages 221106 ( year 2007 ) NoStop

doi
[67]

author author R. A. \ Shaw \ and\ author J. G. \ Hill ,\ title title Approaching the Hartree - Fock Limit through the Complementary Auxiliary Basis Set Singles Correction and Auxiliary Basis Sets , \ https://doi.org/10.1021/acs.jctc.7b00140 journal journal J. Chem. Theory Comp...

doi
[68]

Urban , author H

author author L. Urban , author H. Laqua ,\ and\ author C. Ochsenfeld ,\ title title Highly Efficient and Accurate Computation of Multiple Orbital Spaces Spanning Fock Matrix Elements on Central and Graphics Processing Units for Application in F12 Theory , \ https://doi.org/10...

doi
[69]

Jurecka , author J

author author P. Jurecka , author J. Sponer , author J. Cerny ,\ and\ author P. Hobza ,\ title title Benchmark database of accurate ( MP2 and CCSD ( T ) complete basis set limit) interaction energies of small model complexes, DNA base pairs, and amino acid pairs. \ https://doi...

doi
[70]

Grafova , author M

author author L. Grafova , author M. Pitonak , author J. Rezac ,\ and\ author P. Hobza ,\ title title Comparative Study of Selected Wave Function and Density Functional Methods for Noncovalent Interaction Energy Calculations Using the Extended S22 Data Set . \ https://doi.org/...

doi
[71]

Kussmann \ and\ author C

author author J. Kussmann \ and\ author C. Ochsenfeld ,\ title title Pre-selective screening for matrix elements in linear-scaling exact exchange calculations. \ https://doi.org/10.1063/1.4796441 journal journal J. Chem. Phys. \ volume 138 ,\ pages 134114 ( year 2013 ) NoStop

doi
[72]

Kussmann \ and\ author C

author author J. Kussmann \ and\ author C. Ochsenfeld ,\ title title Preselective Screening for Linear - Scaling Exact Exchange - Gradient Calculations for Graphics Processing Units and General Strong - Scaling Massively Parallel Calculations , \ @noop journal journal J Chem T...

2015
[73]

Weigend ,\ title title A fully direct RI - HF algorithm: Implementation , optimised auxiliary basis sets, demonstration of accuracy and efficiency

author author F. Weigend ,\ title title A fully direct RI - HF algorithm: Implementation , optimised auxiliary basis sets, demonstration of accuracy and efficiency. \ https://doi.org/10.1039/b204199p journal journal Phys. Chem. Chem. Phys. \ volume 4 ,\ pages 4285--4291 ( year...

doi
[74]

Neese ,\ title title An improvement of the resolution of the identity approximation for the formation of the Coulomb matrix , \ https://doi.org/10.1002/jcc.10318 journal journal J

author author F. Neese ,\ title title An improvement of the resolution of the identity approximation for the formation of the Coulomb matrix , \ https://doi.org/10.1002/jcc.10318 journal journal J. Comput. Chem. \ volume 24 ,\ pages 1740--1747 ( year 2003 ) NoStop

doi
[75]

Laqua , author J

author author H. Laqua , author J. Kussmann ,\ and\ author C. Ochsenfeld ,\ title title Accelerating seminumerical Fock -exchange calculations using mixed single- and double-precision arithmethic. \ https://doi.org/10.1063/5.0045084 journal journal J. Chem. Phys. \ volume 154 ...

doi
[76]

Kussmann , author H

author author J. Kussmann , author H. Laqua ,\ and\ author C. Ochsenfeld ,\ title title Highly Efficient Resolution -of- Identity Density Functional Theory Calculations on Central and Graphics Processing Units . \ https://doi.org/10.1021/acs.jctc.0c01252 journal journal J. Che...

doi
[77]

Weigend ,\ title title Hartree-fock exchange fitting basis sets for H to Rn , \ https://doi.org/10.1002/jcc.20702 journal journal J

author author F. Weigend ,\ title title Hartree-fock exchange fitting basis sets for H to Rn , \ https://doi.org/10.1002/jcc.20702 journal journal J. Comput. Chem. \ volume 29 ,\ pages 167--175 ( year 2007 ) NoStop

2007 doi
[78]

Gulde , author P

author author R. Gulde , author P. Pollak ,\ and\ author F. Weigend ,\ title title Error- Balanced Segmented Contracted Basis Sets of Double - to Quadruple - Valence Quality for the Lanthanides , \ https://doi.org/10.1021/ct300302u journal journal J. Chem. Theory Comput. \ vol...

doi
[79]

Laqua , author J

author author H. Laqua , author J. Kussmann ,\ and\ author C. Ochsenfeld ,\ title title An improved molecular partitioning scheme for numerical quadratures in density functional theory. \ https://doi.org/10.1063/1.5049435 journal journal J. Chem. Phys. \ volume 149 ,\ pages 20...

doi

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Kohn ,\ title title Density Functional and Density Matrix Method Scaling Linearly with the Number of Atoms , \ https://doi.org/10.1103/PhysRevLett.76.3168 journal journal Phys

author author W. Kohn ,\ title title Density Functional and Density Matrix Method Scaling Linearly with the Number of Atoms , \ https://doi.org/10.1103/PhysRevLett.76.3168 journal journal Phys. Rev. Lett. \ volume 76 ,\ pages 3168--3171 ( year 1996 ) NoStop

doi:10.1103/physrevlett.76.3168

[2] [2]

author author J. C. \ Burant , author G. E. \ Scuseria ,\ and\ author M. J. \ Frisch ,\ title title A linear scaling method for Hartree - Fock exchange calculations of large molecules. \ https://doi.org/10.1063/1.472627 journal journal J. Chem. Phys. \ volume 105 ,\ pages 8969--8972 ( year 1996 ) NoStop

work page doi:10.1063/1.472627

[3] [3]

Schwegler , author M

author author E. Schwegler , author M. Challacombe ,\ and\ author M. Head-Gordon ,\ title title Linear scaling computation of the Fock matrix. II . Rigorous bounds on exchange integrals and incremental Fock build. \ https://doi.org/10.1063/1.473833 journal journal J. Chem. Phys. \ volume 106 ,\ pages 9708--9717 ( year 1997 ) NoStop

doi:10.1063/1.473833

[4] [4]

Ochsenfeld , author C

author author C. Ochsenfeld , author C. A. \ White ,\ and\ author M. Head-Gordon ,\ title title Linear and sublinear scaling formation of Hartree - Fock -type exchange matrixes. \ https://doi.org/10.1063/1.476741 journal journal J. Chem. Phys. \ volume 109 ,\ pages 1663--1669 ( year 1998 ) NoStop

doi:10.1063/1.476741

[5] [5]

Neese , author F

author author F. Neese , author F. Wennmohs , author A. Hansen ,\ and\ author U. Becker ,\ title title Efficient, approximate and parallel Hartree - Fock and hybrid DFT calculations. A 'chain-of-spheres' algorithm for the Hartree - Fock exchange. \ https://doi.org/10.1016/j.chemphys.2008.10.036 journal journal Chem. Phys. \ volume 356 ,\ pages 98--109 ( y...

doi:10.1016/j.chemphys.2008.10.036

[6] [6]

Laqua , author J

author author H. Laqua , author J. Kussmann ,\ and\ author C. Ochsenfeld ,\ title title Efficient and Linear - Scaling Seminumerical Method for Local Hybrid Density Functionals . \ https://doi.org/10.1021/acs.jctc.8b00062 journal journal J. Chem. Theory Comput. \ volume 14 ,\ pages 3451--3458 ( year 2018 a ) NoStop

doi:10.1021/acs.jctc.8b00062

[7] [7]

Laqua , author T

author author H. Laqua , author T. H. \ Thompson , author J. Kussmann ,\ and\ author C. Ochsenfeld ,\ title title Highly Efficient , Linear - Scaling Seminumerical Exact - Exchange Method for Graphic Processing Units . \ https://doi.org/10.1021/acs.jctc.9b00860 journal journal J. Chem. Theory Comput. \ volume 16 ,\ pages 1456--1468 ( year 2020 ) NoStop

doi:10.1021/acs.jctc.9b00860

[8] [8]

Helmich-Paris , author B

author author B. Helmich-Paris , author B. de Souza , author F. Neese ,\ and\ author R. Izsak ,\ title title An improved chain of spheres for exchange algorithm , \ https://doi.org/10.1063/5.0058766 journal journal J. Chem. Phys. \ volume 155 ,\ pages 104109 ( year 2021 ) NoStop

doi:10.1063/5.0058766