10.2.3 Multipole Moments

This section discusses how to compute arbitrary electrostatic multipole moments for an entire molecule, including both ground- and excited-state electron densities. Occasionally, however, it is useful to decompose the electronic part of the multipole moments into contributions from individual MOs. This decomposition is especially useful for systems containing unpaired electrons, ¹³⁶⁸ Williams C. F., Herbert J. M.
J. Phys. Chem. A
(2008), 112, pp. 6171. Link where the first-order moments $⟨x⟩$, $⟨y⟩$, and $⟨z⟩$ characterize the centroid (mean position) of the half-filled MO, and the second-order moments determine its radius of gyration, $R_g$, which characterizes the size of the MO. Upon setting PRINT_RADII_GYRE = TRUE, Q-Chem will print out centroids and radii of gyration for each occupied MO and for the overall electron density of the Hartree-Fock or Kohn-Sham reference determinant. If CIS or TDDFT excited states are requested, then this keyword will also print out the centroids and radii of gyration for each excited-state electron density.

Note:  These keywords only apply to SCF, DFT, CIS, and TDDFT calculations. To compute these quantities for correlated wavefunctions, use keywords that specify properties calculations for the corresponding method, as described in Chapters 6 and 7. For example, to compute CCSD or EOM-CCSD multipole moments and other properties, use CC_REF_PROP = TRUE and CC_EOM_PROP = TRUE.

Q-Chem can compute Cartesian multipole moments of the charge density to arbitrary order, both for the ground state and for excited states calculated using the CIS or TDDFT methods.