# Accuracy benchmarks for atomic multipole moments

Thermochemical accuracy is regularly quoted as 1 kcal mol$^{-1}$, or 4.2 kJ mol$^{-1}$. I'm expressing some data and I want to use benchmarks like this to validate them, however, they are not all in units of energy. My analyses involve various ranks of a multipole expansion.

I have found a reference when benchmarks dipole moments$^{1}$, but I can't find anything in the literature which similarly benchmarks higher rank multipole moments. Any pointers to the literature or comments from experience would be greatly appreciated.

To the best of my knowledge, there is no definition of "chemical accuracy" for higher-order (electric) multipole moments, in part because they are difficult to measure experimentally, with relatively large uncertainties. I found one recent paper that has a small comparison, prefaced with:

There are several techniques to determine experimentally the dipole moments, [4,5] but it is still very difficult to obtain precise experimental values of higher multipole moments such as quadrupole or octupole moments, [1,6,7] independently of the experimental conditions. Theoretical calculations are therefore essential but challenging for quantum chemistry methods. The accurate calculation of these properties is highly dependent on the method employed, [8] either regarding approximate density functionals [9] or methods based on wavefunctions. [10,11] Consequently, calculating the multipole moments is a way to assess any electronic structure method.

The authors use this work as a method development showcase for the Piris natural orbital functional 6 (PNOF6), the details of which I will generally leave out, but they provide valuable data for Hartree-Fock (HF), coupled cluster with single and double excitations (CCSD), and multireference configuration interaction with single and double excitations (MR-CISD, they abbreviate as MRSD-CI after the old abbreviation SDCI). I will leave in the PNOF data, since it (to my understanding) is a natural orbital-based almost black-box correlated method when the number of occ-virt correlated pairs can be controlled. For the molecules studied (small, organic, closed-shell), CCSD in a triple-$\zeta$ basis set is most likely a reasonable choice.

Table III. $\Theta_{zz}$ component of the quadrupole moments, in atomic units, computed with the Sadlej-pVTZ basis set at the experimental equilibrium [42] geometries.

\begin{array}{lllll} \hline & \text{HF} & \text{CCSD} & \text{Expt.} & \text{PNOF6} \\ \hline \ce{H2} & 0.4381 & 0.3935 & 0.39 \pm 0.01 & 0.3935 \\ \ce{HF} & 1.7422 & 1.7156 & 1.75 \pm 0.02 & 1.6939 \\ \ce{BH}^{a} & 2.6772 & 2.3388 & 2.3293~\text{(FCI/aVTZ)}^{b} & 2.3706 \\ \ce{HCl} & 2.8572 & 2.7233 & 2.78 \pm 0.09 & 2.7753 \\ \ce{HCCF} & 3.3530 & 2.9335 & 2.94 \pm 0.10 & 3.2482 \\ \ce{CO} & 1.5366 & 1.4889 & 1.44 \pm 0.30 & 1.4562 \\ \ce{N2} & 0.9397 & 1.1712 & 1.09 \pm 0.07 & 1.0530 \\ \ce{NH3} & 2.1258 & 2.1661 & 2.45 \pm 0.30 & 2.1080 \\ \ce{PH3} & 1.7217 & 1.5695 & 1.56 \pm 0.70 & 1.6507 \\ \ce{ClF} & 0.9413 & 1.0514 & 1.14 \pm 0.05 & 1.1122 \\ \ce{CH3F} & 0.3482 & 0.3002 & 0.30 \pm 0.02 & 0.3269 \\ \ce{C2H2} & 5.3655 & 4.6850 & 4.71 \pm 0.14 & 5.1531 \\ \ce{C2H6} & 0.6329 & 0.6234 & 0.59 \pm 0.07 & 0.6275 \\ \ce{C6H6} & 5.5418 & 5.6653 & 6.30 \pm 0.27 & 6.3571 \\ \ce{CH3CCH} & 4.2913 & 3.6939 & 3.58 \pm 0.01 & 4.1146 \\ \ce{CO2} & 3.8087 & 3.1966 & 3.19 \pm 0.13 & 3.6012 \\ \hline \text{MAE} & 0.2646 & 0.0902 & & 0.1517 \\ \text{MAE}~(\mu_{z}) & 0.0843 & 0.0177 & & 0.0309 \\ \hline \end{array}

• a. Calculations performed with the aug-cc-pVTZ basis set.
• b. Full CI calculation reported by Halkier [29].

I included the mean absolute error (MAE) for the dipole results from Table II to give an idea of the relative difficulty compared to calculating quadrupole moments. The quadrupole MAEs are about 3, 5, and 5 times larger than the dipole MAEs for each method.

Plotted, it looks like

which is admittedly not very useful other than to show pictorially that CCSD is generally in at least qualitative agreement with the experiment. I personally would consider HF in qualitative agreement to the level that HF usually gets geometries qualitatively correct, though bond lengths are usually too short due to lack of electron correlation. Mean-field electrostatics are the dominant contributor.

For percent errors,

\begin{array}{lrrr} \hline & \delta_{\text{HF to CCSD}} & \delta_{\text{CCSD to Expt.}} & 100 * \text{err}/\text{Expt.} \\ \hline \ce{H2} & 11.33 & 0.90 & 2.56 \\ \ce{HF} & 1.55 & 1.97 & 1.14 \\ \ce{BH} & 14.47 & 0.41 & 0.00 \\ \ce{HCl} & 4.92 & 2.04 & 3.24 \\ \ce{HCCF} & 14.30 & 0.22 & 3.40 \\ \ce{CO} & 3.20 & 3.40 & 20.83 \\ \ce{N2} & 19.77 & 7.45 & 6.42 \\ \ce{NH3} & 1.86 & 11.59 & 12.24 \\ \ce{PH3} & 9.70 & 0.61 & 44.87 \\ \ce{ClF} & 10.47 & 7.77 & 4.39 \\ \ce{CH3F} & 15.99 & 0.07 & 6.67 \\ \ce{C2H2} & 14.53 & 0.53 & 2.97 \\ \ce{C2H6} & 1.52 & 5.66 & 11.86 \\ \ce{C6H6} & 2.18 & 10.07 & 4.29 \\ \ce{CH3CCH} & 16.17 & 3.18 & 0.28 \\ \ce{CO2} & 19.15 & 0.21 & 4.08 \\ \hline \end{array}

this is further confirmation that CCSD is probably a good qualitative benchmark.

Table IV. $\Theta_{zz}$ and $\Theta_{xx}$ components of molecular quadrupole moments, in atomic units, computed using the Sadlej-pVTZ basis set at the experimental equilibrium [42] geometries.

\begin{array}{lrrr} \hline & \text{HF} & \text{MR-CISD} & \text{Expt.} & \text{PNOF6} \\ \hline \ce{H2O}~(xx) & 1.7966 & 1.8050 & 1.86 \pm 0.02 & 1.7808 \\ \ce{H2O}~(zz) & 0.0981 & 0.0950 & 0.10 \pm 0.02 & 0.0869 \\ \ce{H2CO}~(xx) & 0.1019 & 0.1100 & 0.04 \pm 0.12 & 0.0516 \\ \ce{H2CO}~(zz) & 0.0921 & 0.2230 & 0.20 \pm 0.15 & 0.1255 \\ \ce{C2H4}~(xx) & 2.7819 & 2.3700 & 2.45 \pm 0.12 & 2.5892 \\ \ce{C2H4}~(zz) & 1.4942 & 1.1700 & 1.49 \pm 0.11 & 1.3266 \\ \ce{O3}~(xx) & 1.1175 & 1.2830 & 1.03 \pm 0.12 & 1.2426 \\ \ce{O3}~(zz) & -0.2387 & 0.1680 & 0.52 \pm 0.08 & 0.3606 \\ \hline \text{MAE} & 0.1772 & 0.1448 & & 0.1066 \\ \hline \end{array}

The ozone result for Hartree-Fock most likely has the wrong sign due to its multiconfigurational ground state, with the MRCI result still in relative error of ~67%.

For octupole moments, they only present results for methane ($\ce{CH4}$), for which the first non-vanishing multipole moment is the octupole moment, due to its $T_{d}$ symmetry.

\begin{array}{ll} \hline & \Omega_{xyz}~(\text{a.u.}) \\ \hline \text{CCSD} & 2.0595 \\ \text{PNOF6(14)} & 2.1142 \\ \text{Expt.} & 2.95 \pm 0.17 \\ \hline \end{array}

Considering the lower bound of the experiment, the PNOF6 result is in 23% error, and considering the upper bound of the experiment, the CCSD result is in 36% error.

I won't draw any more conclusions about the data other than to say that in general, there may be large errors in the experimental data that are hard to predict (non-systematic). In that case, if you were to perform a study looking at convergence of a result with respect to some other parameter (say, cluster size), I would first look at convergence of said result with respect to method (HF, MP2, CCSD) and basis (aVDZ, aVTZ, ...) at the largest cluster size, just to check that CCSD/aVTZ is a reasonable benchmark if you're after absolute accuracy. Since experimental error is large, it may be more reasonable to just compare against higher-level, full cluster calculations.