Computational Chemistry Comparison and Benchmark DataBase Release 22 (May 2022) Standard Reference Database 101 National Institute of Standards and Technology
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Tutorial on Electric Quadrupole Moments

What is it?

Formulas

Pictures

Practical considerations

Molecular electric quadrupole moments in general are derived (sometimes it seems in rather involved ways) from some other measured property. They are derived from the measurement of the second virial coefficient of gases (a measure of how non-ideal a gas is) and spectroscopic observations changes in the rotational energy levels when the molecule is in a magnetic or electric field. In both cases the quadrupole moment is a fudge factor that tweaks some other property. There is a physical basis for the expansion used in both cases, but I would not expect the values obtained from two such different measurements, with different random and systematic errors, to agree within the stated uncertainties of each measurement.

Primitive? Traceless?

Relation between cartesian and spherical components

monopole
dipole
quadrupole