The density matrix: where the electrons are
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The converged SCF hands you P = 2C_occ C_occᵀ — and for one-electron questions, P is the wavefunction. Any one-electron observable is a trace: the electron count is tr(PS) = N, the dipole is tr(P·d) plus the nuclear part, energies contract P against H and F. The 4-numbers-per-basis-pair summary carries everything a one-electron operator can ask.
The density matrix P = 2C_occ C_occᵀ compresses the wavefunction. What does its trace with the overlap matrix, tr(PS), always equal?
Chemistry's favorite question — whose electrons are they? — has no exact answer, only conventions. Mulliken's: split tr(PS) by basis-function ownership, q_A = Z_A − (PS)_AA. For HeH⁺ this gives q_He = +0.47, q_H = +0.53: helium holds 1.53 of the 2 electrons yet stays positive, because it brought Z = 2 to the table. Read populations and charges, never one without the other.
Mulliken charges are famously basis-set sensitive — add diffuse functions and the numbers can change wildly. What is the structural reason?
Unlike charges, the dipole moment is a measurable observable — for neutral molecules. HeH⁺ carries net charge, so its 0.889 a.u. dipole is defined only relative to a stated origin. Project 11 has you compute both, including the one-line dipole integral the product theorem gives away.
HeH⁺'s dipole moment is 0.889 a.u. with the origin at He. For H₂O the origin never needs stating. What's the difference?