{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"34f9cc3ff889cab6245338e60e006a08291f3b913c0093bd0f5a14f32dad6fe1","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["5f7d53e831f96b4653ba8a968eaca31502bd6c28757c0a0669c7eeb80d2d13e2","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"1a52bb439384462dbbace04e791661e6a3307353067d604d35d8e106af076f0d","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.ionization.molecular-excess","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"A molecule with several nuclei can also hold extra electrons. Show that the number of extra electrons is at most a fixed constant times the number of nuclei, whatever the nuclear charges and positions.","posed_since":"1984","precise":"For $K$ static nuclei with charges $Z_1, ..., Z_K$ at positions $R_1, ..., R_K$, $Z_{\\mathrm{tot}} = \\sum_k Z_k$, and $N$ electrons with Coulomb interactions, let $N_c$ be the largest N for which the electronic ground state is bound. Prove $N_c \\le Z_{\\mathrm{tot}} + C K$ with C independent of the $Z_{k}$, $R_{k}$ and K, or give a counterexample family.","problem_ref":"5f7d53e831f96b4653ba8a968eaca31502bd6c28757c0a0669c7eeb80d2d13e2","references":"","settled_by":"A proof of the bound uniform in nuclear charges and positions, or an explicit configuration violating it.","status_note":"Lieb's 1984 bound $N_c < 2 Z_{\\mathrm{tot}} + K$ holds for all nuclear positions.","title":"Excess charge of molecules bounded by a constant per nucleus","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"eb6d7e8809990bba533e92f30d795428c1e458fc05e51630693a59108468ce9c","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"99b709aaeca835cdf1751c72d47d1bbdc675927e1984015ef5a4093bab722df0","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"fbHZ6cIKcoJ0BDvTU-dy_CqoeOU5NSkxlkKDBnRN4h1xU9FEY-Y89lHytyidLaaIznSRr2-4oCu-PCx048uPAg"},"schema":"pubphys.envelope/1"},"record_hash":"eb6d7e8809990bba533e92f30d795428c1e458fc05e51630693a59108468ce9c","leaf_index":3424}