{"paper":{"title":"Explicitly Correlated Gaussian Basis Approach to Periodic Systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Closed-form expressions for matrix elements are derived for variational electronic structure calculations of periodic solids using explicitly correlated Gaussian bases.","cross_cats":["physics.chem-ph"],"primary_cat":"quant-ph","authors_text":"Kalman Varga","submitted_at":"2026-05-12T21:50:41Z","abstract_excerpt":"Closed-form expressions for all matrix elements required for variational calculation of the electronic structure of periodic solids have been derived using a basis of explicitly correlated Gaussians (ECGs). Periodic basis functions are constructed by summing shifted correlated Gaussians over all composite lattice translations, where a generalized unfolding theorem reduces the resulting double lattice sum to a single sum through a unified computational framework for overlap, kinetic energy, and Coulomb potential operators. The formalism has been validated through application to an infinite one-"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Closed-form expressions for all matrix elements required for variational calculation of the electronic structure of periodic solids have been derived using a basis of explicitly correlated Gaussians (ECGs).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The generalized unfolding theorem correctly reduces the double lattice sum to a single sum for overlap, kinetic energy, and Coulomb operators in the periodic ECG basis.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives closed-form matrix elements for explicitly correlated Gaussian basis in periodic systems and validates on infinite 1D hydrogen chain.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Closed-form expressions for matrix elements are derived for variational electronic structure calculations of periodic solids using explicitly correlated Gaussian bases.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e604380e4d1596996947794bbb43c36f0d4d3b8d66d934d27b9ec958bda3b537"},"source":{"id":"2605.12781","kind":"arxiv","version":1},"verdict":{"id":"ddb0b68d-5dca-4260-88a1-d3e76c779247","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:43:33.111622Z","strongest_claim":"Closed-form expressions for all matrix elements required for variational calculation of the electronic structure of periodic solids have been derived using a basis of explicitly correlated Gaussians (ECGs).","one_line_summary":"Derives closed-form matrix elements for explicitly correlated Gaussian basis in periodic systems and validates on infinite 1D hydrogen chain.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The generalized unfolding theorem correctly reduces the double lattice sum to a single sum for overlap, kinetic energy, and Coulomb operators in the periodic ECG basis.","pith_extraction_headline":"Closed-form expressions for matrix elements are derived for variational electronic structure calculations of periodic solids using explicitly correlated Gaussian bases."},"references":{"count":161,"sample":[{"doi":"","year":null,"title":"Ewald decomposition (Secs. II H–III). The pe- riodic 1 /r potential is split into a short-range complementary-error-function part evaluated in real space and a long-range smooth part evaluated in reci","work_id":"ea21a56d-d6f6-4cf7-9453-0d2def940bb7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Direct neutral-cell sum (Sec. III E). When the sim- ulation cell is charge-neutral the lattice sum over 1/r converges absolutely when shells are grouped by charge neutrality. Each matrix element re- d","work_id":"1e56dbbe-1d99-4191-afc5-bccb74e04811","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Dirac delta convolution (Sec. III F). The 1 /r ker- nel is the convolution of a Dirac delta density with the Green’s function of the Laplacian. The matrix element of δ(3)(ri − rj) gives the pair-conta","work_id":"c2706d6b-3390-4af3-9048-29bd1525771e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Electron–electron reciprocal-space term The electron–electron reciprocal-space matrix element is (see Appendix D V (ee,G) kl = 4π Ω Skl n−1X i=1 nX j=i+1 X G̸=0 e−G2/4κ2 G2 × X M ωM e iG·PT ij¯rM−σ2 i","work_id":"81544b5a-376b-4d59-98f3-0f877846746e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Electron–electron real-space term For the real-space sum we define the t-augmented non- linear parameter matrix and its determinant ratio, A(t,ij) kl = Akl + t2(ei − ej)(ei − ej)T , (46) det A(t,ij) k","work_id":"988eb193-ae4d-4a7f-8eb6-9ef8b90c808f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":161,"snapshot_sha256":"4d0638ab86fbe4ed8bd746897aa353b0cd8963580818a40010b13ffe09610b9d","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"f2a151f1aa518e91006942366a953e14ee4757d91938216a0eac506879f8bcd4"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}