Quantification of the Ionic Character of Multiconfigurational Wave Functions: The Qat Diagnostic

Abstract

The complete active space self-consistent field (CASSCF) method is a cornerstone in modern excited-state quantum chemistry providing the starting point for most common multireference computations. However, CASSCF, when used with a minimal active space, can produce significant errors (>2 eV) even for the excitation energies of simple hydrocarbons if the states of interest possess ionic character. After illustrating this problem in some detail, we present a diagnostic for ionic character, denoted as Q _a^t, that is readily computed from the transition density. A set of 11 molecules is considered to study errors in vertical excitation energies. State-averaged CASSCF obtains a mean absolute error (MAE) of 0.87 eV for the 34 singlet states considered. We highlight a strong correlation between the obtained errors and the Q _a^t diagnostic, illustrating its power to predict problematic cases. Conversely, using multireference configuration interaction with single and double excitations and Pople's size extensivity correction (MR-CISD+P), excellent results are obtained with an MAE of 0.11 eV. Furthermore, correlations with the Q _a^t diagnostic disappear. In summary, we hope that the presented diagnostic will facilitate reliable and user-friendly multireference computations on conjugated organic molecules.

Figure 1

Structures of the molecules considered in this work.

Figure 2

Example showing the covalent triplet and ionic singlet states of ethene using delocalized canonical MOs (left) and localized MOs (right).

Figure 3

Transition densities between the ground state and the lowest two singlet states of naphthalene computed at SA-CASSCF (a,b) and MRCISD (c,d) levels of theory (isovalue: 0.001 au).

Figure 4

Overall errors of vertical excitation energies for SA-CASSCF and various MR-CI variants (using the aug-cc-pVDZ basis set) computed for the 11 molecules considered in this work. Results are reported as mean absolute and signed errors (MAE, MSE) determined separately for singlet and triplet states.

Figure 5

Errors of computed vertical singlet excitation energies plotted against the Q _a^t diagnostic measuring ionic character using (a) SA-CASSCF, (b) MR-CIS, (c) MR-CISD, and (d) MR-CISD + P, all using the aug-cc-pVDZ basis set. States are grouped according to type: ππ* (circles) and nπ* (squares) states; the out-of-plane ππ* state of cyanoformaldehyde is shown as an empty circle.

Figure 6

Comparison of the transition densities of the covalent and ionic ππ* states for three selected molecules computed at the SA-CASSCF/aug-cc-pVDZ level. Q _a^t descriptors and errors of vertical excitation energies are given below each plot.

Figure 7

Errors of computed vertical singlet excitation energies plotted against the Ω descriptor measuring single excitation character using (a) SA-CASSCF, (b) MR-CIS, (c) MR-CISD, and (d) MR-CISD + P, all using the aug-cc-pVDZ basis set. States are grouped according to type: ππ* (circles) and nπ* (squares) states; the out-of-plane ππ* state of cyanoformaldehyde is shown as an empty circle.