10.2 Wave Function Analysis 10.2.8 Intrinsic Atomic Orbitals 10.2.10 Excited-State Analysis for CIS and TDDFT

10.2.9 Atomic dipoles and quadrupoles

(September 1, 2024)

It can sometimes be desirable to go beyond atomic partial charges to calculate atomic dipoles and quadrupoles. This is useful in particular for certain types of force fields, which use these terms to more accurately model charge anisotropy. QChem provides the ability to calculate atomic dipoles and quadrupoles, using a method inspired by the formulations of Ref. ¹¹⁹⁴ Sokalski W. A., Poirier R. A.
Chem. Phys. Lett.
(1983), 98, pp. 86. Link . We simply evaluate atomic contributions to the multipole moment by assigning contributions from each atom’s basis functions. However, we avert any basis set dependence and delocalization errors by using intrinsic atomic orbitals, ⁶⁶⁰ Knizia G.
J. Chem. Theory Comput.
(2013), 9, pp. 4834. Link which have been shown to have excellent convergence with basis set. ³⁷ Aldossary A., Head-Gordon M.
J. Chem. Phys.
(2022), 157, pp. 094102. Link The atomic quadrupoles available from this method are the traceless quadrupoles, which are the more commonly used type in force field contexts.

We calculate atomic dipoles and (traceless) quadrupoles using the following equations:

\mu_{A}^{u}=\sum_{\alpha\ \mathrm{on}\ A}\sum_{\beta}2P_{\alpha\beta}^{IAO}% \langle\chi_{\alpha}|u_{A}-\hat{u}|\chi_{\beta}\rangle

(10.25)

Q_{A}^{uv}=\sum_{\alpha\ \mathrm{on}\ A}\sum_{\beta}2P_{\alpha\beta}^{IAO}% \langle\chi_{\alpha}|\frac{3}{2}(u_{A}-\hat{u})(v_{A}-\hat{v})-\frac{1}{2}||r_% {A}-\hat{r}||^{2}\delta_{uv}|\chi_{\beta}\rangle

(10.26)

Here A is the atom index, $r$ is the vector $(x,y,z)$ , and $u=x,y,$ or $z$ . $\delta$ is the Kronecker delta function, $\chi$ represents an intrinsic atomic orbital, $\alpha$ and $\beta$ are the IAO indices, $P$ is the one-particle density matrix, and the $I A O$ superscript on the density matrix indicates that this is the density matrix in the IAO basis. Additionally, the inclusion of $u_{A}$ , $v_{A}$ , and $r_{A}$ (the coordinates of atom A) ensures that the multipoles are origin-invariant.

The calculation of atomic dipoles and quadrupoles is controlled with the following $rem variable:

DO_ATOMIC_MULTIPOLES

DO_ATOMIC_MULTIPOLES
       Enables atomic multipole calculation
TYPE:
       BOOL
DEFAULT:
       FALSE
OPTIONS:
       FALSE Do not calculate IAO atomic multipoles TRUE Calculate IAO atomic multipoles
RECOMMENDATION:
       None

Example 10.5 Input for calculating atomic dipoles and quadrupoles via the IAO/IBO procedure.

$molecule
0 1
F         0.0000000000    0.0000000000    0.2314949765
H         0.0000000000    0.0000000000    1.1541050235
$end

$rem
jobtyp sp
method hf
basis aug-cc-pvtz
do_ibo true
do_atomic_multipoles true
$end

$loco
min_basis autosad
ibo_mem 1000
$end

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10.2.9 Atomic dipoles and quadrupoles