{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"bb664b54a33a2bab396ae91b4b9858583f66f616c2389a3814c5a826b75177bc","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"29b47955df16877356bb7d1a7fe358023aae59fa3b06344e05cfc389ab55a5c2","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.ionization.molecular-excess","kind":"well-posed","literature_status":"open","n":"1","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":null,"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":"139d0cc04cd3b3dbd331e014add418ba05f5b21aa4fd7ade07403261b548b505"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"5f7d53e831f96b4653ba8a968eaca31502bd6c28757c0a0669c7eeb80d2d13e2","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"000fa9249444507428c339b7efb5242afed0a74353444a47d9a62b89cc97636f","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"B-Z9qSu8azMd8amO-rMXrxvgzw-nhECJjh5HMtvjae2ss6YtFa9jztCFr2ocYTnBuSkWyYyQut2gRY4Fv5JKAQ"},"schema":"pubphys.envelope/1"},"record_hash":"5f7d53e831f96b4653ba8a968eaca31502bd6c28757c0a0669c7eeb80d2d13e2","leaf_index":1626}