GSoC 2026: Improve performance of 2-electron integrals using screening

[PaulWAyers]

Description

Use screening based on the overlap between basis functions to improve performance.

📚 Package Description and Impact

For large molecules, textbook expressions for quantities expanded in Gaussian basis functions (e.g., the electron density) or integrals based on Gaussian basis functions (e.g., the kinetic-energy integral) typically include many negligible terms. By screening out these terms, and only evaluating terms that are nonnegligible, the performance of GBasis can be greatly enhanced.

👷 What will you do?

In GBasis we provide a utility for screening these terms using their overlap, is_overlap_included. A generalization of this approach allows fast evaluation of spatial quantities and 2-electron integrals. Your main goal would be to screen 2-electron integrals.

🏁 Expected Outcomes

1. Implement screening of 2-electron integrals. This will probably involve several tiers, starting with a simple Cauchy-Schwartz screening and extending to Ochsenfeld-style screening.
2. Document and test the new feature.
3. Test the performance of the code.
4. Provide a tutorial notebook demonstration for the new feature.

Required skills	Python, OOP
Preferred skills	Be comfortable with math, physics. Experience with scientific programming, quantum chemistry would be huge plus
Project size	175 hours, Medium
Difficulty	Medium 😉

🙋 Mentors

Marco Martínez-González	mmg870630_at_gmail_dot_com	@marco-2023
Omid Hossainzadeh	ohosseinzade0_at_gmail_dot_com	@ohzde
Esteban Vöhringer-Martinez	estebanvohringer_at_qcmmlab_dot_com	@evohringer
Paul Ayers	ayers_at_mcmaster_dot_ca	@PaulWAyers
Gabriela Sánchez-Díaz	sanchezg_at_mcmaster_dot_ca	@gabrielasd

📝 Notes & References

• Supersedes GSoC 2025: Improve Performance Using Screening #167
• Related to GSoC 2025: Arbitrary-order Overlap Integrals (and evaluations enabled thereby) #183 , [BUG] Big memory usage for density evaluation #121, GSoc 2025: Improve Performance Using Screening #207
• Screening is discussed in Chapter 9 of Molecular Electronic Structure Theory.
• A recent(ish) discussion from the Ochsenfeld group
• Paul's notes on overlap screening
• Paul's notes on screening in general. This includes screening for 2-electron integrals.

🏋️ Stretch Goals

Implement another way to speed up the 2-electron integrals, e.g. density-fitting or Cholesky decomposition.

[lukedev45]

Hello, my name is Luke and this project is extremely interesting to me. I have a BSci in Applied Mathematics and Physics and have a deferred place at Trinity College Dublin for a MSci in Quantum Science and Technology. I think that this project would be extremely helpful for me to learn about the intricacies of quantum chemistry, more particularly screening.

My most proficient programming language is Python and I think my expertise in maths and physics would be beneficial for this project.

Thanks,
Luke

[Ao-chuba]

Hello @PaulWAyers @marco,
This project looks interesting to me , I’ve been analyzing the gbasis codebase/reference notes and Currently, the arrays in ElectronRepulsionIntegral evaluate without any screening before the underlying integral calculations, leading to unnecessary $O(N^4)$ evaluations even when the underlying atomic distributions do not overlap. Based on the project description and the notes provided, my approach to implementing this:

• Tier 1 (Distance-Based Overlap Screening): Leverage the existing is_two_index_overlap_screened logic to immediately screen out negligible $(ab|cd)$ blocks based on distance cutoffs and the tol_screen parameter. We can skip entire evaluations when the product $\phi_a\phi_b$ or $\phi_c\phi_d$ is negligible.
• Tier 2 (Cauchy-Schwarz Screening): Implement the standard CS bound $|(ab|cd)| \le \sqrt{(ab|ab)} \sqrt{(cd|cd)}$. This will involve adding a routine to pre-compute and store the $(ab|ab)$ matrix ($Q$), allowing for $O(1)$ lookup to rigorously bound and screen integrals, reducing the formal scaling to roughly $O(N^2)$.
• Tier 3 (Ochsenfeld Coulomb Screening): Implement the tighter distance dependent bound using the precomputed $Q$ matrix and the Coulomb interactions between atomic centers ($M$). This will properly account for the $1/R$ decay over long distances where CS might still confidently predict falsely significant interactions.

I would love to hear your thoughts for this approach!

[kir943]

Hi @PaulWAyers,

My name is Kiran Jadhav, and I’m currently contributing to gbasis under Winter of Code 5.0 (working on issue #220). While studying the discussion and references for this issue on improving the performance of two-electron integrals using screening, I’ve been trying to understand the intended scope of the optimization.

From what I understand, the primary goal is to introduce screening strategies (such as distance-based screening and Cauchy–Schwarz bounds) to avoid evaluating negligible ERIs. While reading about implementations in other quantum chemistry libraries, I noticed that further improvements are sometimes achieved using approaches like shell-pair screening, precomputed Schwarz matrices, or vectorized evaluation of screening bounds.

Would the intended scope of this issue mainly focus on implementing the screening strategy described here, or would exploring additional optimizations such as shell/block screening or vectorized bound evaluation also be within scope for gbasis?

I’m currently going through the ERI implementation in the repository to better understand where screening could be integrated, so any guidance on the preferred direction would be really helpful.

Thank you!

[Ao-chuba]

@PaulWAyers ,
while planning the Cauchy-Schwarz screening implementation for Tier 2, I discovered an issue in the bounding utilities.
Currently, compute_primitive_upper_bound and compute_contraction_upper_bound in screening.py utilize a 1D normalization factor instead of the required 3D tensor norm, which results in bounds that don't scale correctly with the 3D primitives evaluated in
ElectronRepulsionIntegral . I have opened Issue #246 with the details of the issue. Since mathematical upper bounds are an requirement for Cauchy-Schwarz and Ochsenfeld screening algorithms to function safely (without improperly dropping significant integrals), resolving issue #246 will be a prerequisite to establishing the formal Q matrices necessary for the $O(N^2)$ ERI screening proposed in this project outline.

[San1357]

Hi @PaulWAyers @marco-2023 @ohzde @evohringer @gabrielasd, @msricher

I'm interested in working on this project for GSoC 2026.

I recently completed issue #221 (2-electron integrals using Obara-Saika and Head-Gordon-Pople recursions)
through PR #245 which was merged into master — and as part of that work,
I already implemented Tier 2 (Cauchy-Schwarz screening) in _schwarz_screening.py. So I have a strong foundation to build on for this project.

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What's Already Done

Tier 2 — Cauchy-Schwarz Screening is already implemented and merged (PR #245):

• SchwarzScreener class in _schwarz_screening.py precomputes the Q matrix (sqrt((ab|ab)) bounds for all shell pairs) using |(ab|cd)| ≤ Q_ab * Q_cd
• Integrated into electron_repulsion_integral in electron_repulsion.py with caching — SchwarzScreener is only built once per basis set
• compute_schwarz_bounds runs O(N²) to precompute, then O(1) lookup per quartet — reducing formal scaling from O(N⁴) to O(N²)
• 23 tests pass — verified that screened and unscreened results match, and that extended systems achieve >30% screening

Tier 1 — Distance-Based Screening already exists in screening.py as is_two_index_overlap_screened and is already integrated into 1-electron integrals (overlap.py, kinetic_energy.py, moment.py, momentum.py, angular_momentum.py) — but not yet into the ERI (2-electron) computation loop. This would be a straightforward addition.

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What Remains (Proposed Approach)

1. Tier 1 — Integrate into ERI loop:
Connect is_two_index_overlap_screened as a pre-filter in the ERI shell quartet loop — skip entire shell pairs whose overlap is negligible based on distance cutoff before even computing Schwarz bounds.

2. Tier 3 — Ochsenfeld-style screening:
Based on Paul's notes and the Ochsenfeld paper (J. Chem. Phys. 147, 144101), this requires computing two quantities:

• Q_ab = sqrt((ab|ab)) — already done in SchwarzScreener. This is a shell-pair bound where both electrons are on the same shell pair (ab).
• M_ac = sqrt((aa|cc)) — needs to be added. This is different from Q — here the two electrons are on different shell pairs (a,a) and (c,c), capturing the Coulomb interaction between atomic centers rather than within a shell pair.

Then the full bound is:

|(ab|cd)| ≈ (Q_ab * Q_cd) / sqrt(Q_aa * Q_bb * Q_cc * Q_dd) * max(M_ac * M_bd, M_ad * M_bc)

This tighter bound accounts for the 1/R Coulomb decay over long distances — where Cauchy-Schwarz alone might still pass interactions that are actually negligible. Implementation requires extending SchwarzScreener to precompute the M matrix (an N_atoms × N_atoms array) alongside the existing Q matrix, and updating is_significant to apply the full Ochsenfeld bound instead of just Q_ab * Q_cd.

3. Spatial quantities screening:

• is_two_index_overlap_screened is already used for 1-electron integrals but is not yet generalized for spatial quantities like electron density and density gradient.
• get_points_mask_for_contraction already exists in screening.py and computes a boolean mask for which grid points are within the cutoff radius of a contraction — but this is not yet systematically applied across all spatial evaluation functions.
• The goal is to generalize this — for each contraction shell, skip grid points that are beyond the cutoff radius, and for each grid point, skip contraction shells that are too far away.
• This reduces the cost of spatial evaluations from O(N_shells × N_points) to roughly O(N_points) for sparse systems.

4. Stretch Goal — Density-fitting or Cholesky decomposition:

• If time permits, implement an additional method to speed up 2-electron integrals beyond screening.

• Density-fitting approximates the 4-center ERI as a product of 3-center integrals using an auxiliary basis — reducing scaling from O(N⁴) to O(N³).

• Cholesky decomposition factorizes the ERI tensor directly without an auxiliary basis, achieving similar scaling reduction.

• Both approaches are complementary to screening and would significantly extend GBasis's capabilities for large molecules.

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Looking forward to your feedback before finalizing the proposal.