# 11.2.2 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,987 where the first-order moments $\langle x\rangle$, $\langle y\rangle$, and $\langle z\rangle$ 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/EOM-CCSD multipole moments and other properties, use CC_REF_PROP=TRUE and CC_EOM_PROP=TRUE.

Controls printing of MO centroids and radii of gyration.
TYPE:
LOGICAL/INTEGER
DEFAULT:
FALSE
OPTIONS:
TRUE (or 1) Print the centroid and radius of gyration for each occupied MO and each density. 2 Print centroids and radii of gyration for the virtual MOs as well. FALSE (or 0) Do not calculate these quantities.
RECOMMENDATION:
None

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.

MULTIPOLE_ORDER
Determines highest order of multipole moments to print if wave function analysis requested.
TYPE:
INTEGER
DEFAULT:
4
OPTIONS:
$n$ Calculate moments to $n$th order.
RECOMMENDATION:
Use the default unless higher multipoles are required.

CIS_MOMENTS
Controls calculation of excited-state (CIS or TDDFT) multipole moments
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not calculate excited-state moments. TRUE Calculate moments for each excited state.
RECOMMENDATION:
Set to TRUE if excited-state moments are desired. (This is a trivial additional calculation.) The MULTIPOLE_ORDER controls how many multipole moments are printed.