Quantum-mechanical sum rules and gauge invariance: A study of the HF molecule

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Abstract
The perturbed Hamiltonian for magnetic dipole transitions is rewritten in terms of the torque operator, instead of the angular momentum operator, which, owing to its nondifferential form, permits tactical advantages in actual calculations of magnetic susceptibility. The translational gauge invariance of the magnetic properties is used to obtain a large series of sum rules involving linear and angular momenta and torque, force, and position operators. These are found to be very general quantum-mechanical relations, restating in a synthetic and unitary form the Thomas-Reiche-Kuhn sum rule, the basic operator commutation properties, the hypervirial theorem, and the conservation of the current-density vector, which are reduced to the same theoretical framework. Accurate calculations of the magnetic properties of the HF molecule, based on the equation-of-motion approach, reveal that the gauge-invariant sum rules can be used for rigorous tests of the quality of approximate molecular wave functions.