DipolmomentOperator

The dipolmomentOperator, commonly referred to as the dipole moment operator, is a quantum-mechanical vector operator that represents the distribution of electrical charges in a system. It is defined as μ̂ = ∑_i q_i r_i, where the sum runs over all charged particles, q_i is the charge of particle i, and r_i is its position operator relative to a chosen origin.

In a molecular system with electrons and nuclei, this becomes μ̂ = -e ∑_{electrons} r_i + ∑_{nuclei A} Z_A e R_A, where e is the elementary charge, r_i are electronic coordinates, and R_A are nuclear coordinates with Z_A the nuclear charge. For a neutral molecule, the total charge is zero, and the permanent dipole moment is origin independent; for charged systems the result depends on the chosen origin.

In the Born–Oppenheimer approximation, nuclei are treated as fixed while electrons move in their field. The electronic dipole moment function μ̂_e(R) is evaluated at fixed nuclear geometry R, and the total dipole moment at that geometry is μ_tot(R) = μ̂_e(R) + ∑_A Z_A e R_A. The expectation value ⟨Ψ| μ̂ |Ψ⟩ gives the permanent dipole moment of the molecular state.

The dipole moment operator also mediates transitions between states through matrix elements ⟨Ψ_f| μ̂ |Ψ_i⟩, which determine the intensity of infrared and microwave transitions. As a vector operator, μ̂ transforms as a first-rank tensor under rotations, with components obeying standard angular-momentum commutation relations.

Applications include predicting and interpreting molecular polarity, spectroscopy selection rules, and transition strengths. In practice, both electronic and nuclear contributions are included, and dipole moments are often reported in Debye units.