A Mountaineering Strategy to Excited States: Revising Reference Values with EOM-CC4.

In the framework of the computational determination of highly accurate vertical excitation energies in small organic compounds, we explore the possibilities offered by the equation-of-motion formalism relying on the approximate fourth-order coupled-cluster (CC) method, CC4. We demonstrate, using an extended set of more than 200 reference values based on CC including up to quadruples excitations (CCSDTQ), that CC4 is an excellent approximation to CCSDTQ for excited states with a dominant contribution from single excitations with an average deviation as small as 0.003 eV. We next assess the accuracy of several additive basis set correction schemes, in which vertical excitation energies obtained with a compact basis set and a high-order CC method are corrected with lower-order CC calculations performed in a larger basis set. Such strategies are found to be overall very beneficial, though their accuracy depends significantly on the actual scheme. Finally, CC4 is employed to improve several theoretical best estimates of the QUEST database for molecules containing between four and six (nonhydrogen) atoms, for which previous estimates were computed at the CCSDT level.

Mountaineering Strategy to Excited States: Highly Accurate Oscillator Strengths and Dipole Moments of Small Molecules.

This work presents a series of highly accurate excited-state properties obtained using high-order coupled-cluster (CC) calculations performed with a series of diffuse containing basis sets, and extensive comparisons with experimental values. Indeed, we have computed the main ground-to-excited transition property, the oscillator strength, and the ground- and excited-state dipole moments, considering 13 small molecules (hydridoboron, hydrogen chloride, water, hydrogen sulfide, boron fluoride, carbon monoxide, dinitrogen, ethylene, formaldehyde, thioformaldehyde, nitroxyl, fluorocarbene, and silylidene). We systematically include corrections up to the quintuple (CCSDTQP) in the CC expansion and extrapolate to the complete basis set limit. When comparisons with experimental measurements are possible, that is, when a number of consistent experimental data can be found, theory typically provides values falling within the experimental error bar for the excited-state properties. Besides completing our previous studies focused on transition energies [J. Chem. Theory Comput. 14 (2018) 4360-4379, ibid. 15 (2019) 1939-1956, ibid. 16 (2020) 1711-1741, and ibid. 16 (2020) 3720-3736], this work also provides ultra-accurate dipoles and oscillator strengths that could be employed for future theoretical benchmarks.

A Mountaineering Strategy to Excited States: Highly Accurate Reference Energies and Benchmarks.

Striving to define very accurate vertical transition energies, we perform both high-level coupled cluster (CC) calculations (up to CCSDTQP) and selected configuration interaction (sCI) calculations (up to several millions of determinants) for 18 small compounds (water, hydrogen sulfide, ammonia, hydrogen chloride, dinitrogen, carbon monoxide, acetylene, ethylene, formaldehyde, methanimine, thioformaldehyde, acetaldehyde, cyclopropene, diazomethane, formamide, ketene, nitrosomethane, and the smallest streptocyanine). By systematically increasing the order of the CC expansion, the number of determinants in the CI expansion as well as the size of the one-electron basis set, we have been able to reach near full CI (FCI) quality transition energies. These calculations are carried out on CC3/ aug-cc-pVTZ geometries, using a series of increasingly large atomic basis sets systematically including diffuse functions. In this way, we define a list of 110 transition energies for states of various characters (valence, Rydberg, n → π*, π → π*, singlet, triplet, etc.) to be used as references for further calculations. Benchmark transition energies are provided at the aug-cc-pVTZ level as well as with additional basis set corrections, in order to obtain results close to the complete basis set limit. These reference data are used to benchmark a series of 12 excited-state wave function methods accounting for double and triple contributions, namely ADC(2), ADC(3), CIS(D), CIS(D∞), CC2, STEOM-CCSD, CCSD, CCSDR(3), CCSDT-3, CC3, CCSDT., and CCSDTQ. It turns out that CCSDTQ yields a negligible difference with the extrapolated CI values with a mean absolute error as small as 0.01 eV, whereas the coupled cluster approaches including iterative triples are also very accurate (mean absolute error of 0.03 eV). Consequently, CCSDT-3 and CC3 can be used to define reliable benchmarks. This observation does not hold for ADC(3) that delivers quite large errors for this set of small compounds, with a clear tendency to overcorrect its second-order version, ADC(2). Finally, we discuss the possibility to use basis set extrapolation approaches so as to tackle more easily larger compounds.

Benchmark of Bethe-Salpeter for Triplet Excited-States.

We have evaluated the accuracy of the Bethe-Salpeter singlet-triplet transition energies as well as singlet-triplet and triplet-triplet splittings for 20 organic molecules, using as reference the CC3 values determined by Thiel and co-workers with both the TZVP and aug-cc-pVTZ atomic basis sets. Our excitation energies are obtained on the basis of GW quasiparticle energy levels that are self-consistently converged with respect to the starting DFT eigenvalues. In its current form, BSE/GW is often unable to provide a balanced description of both singlet and triplet excited-states. While the singlet-singlet and triplet-triplet energy separations are obtained accurately, triplets are located too close in energy from the ground-state, by typically -0.55 eV when using standard functionals to generate the starting eigenstates. Applying the Tamm-Dancoff approximation upshifts the BSE triplet energies and allows reducing this error to ca. -0.40 eV, while using M06-HF eigenstates allows a further increase and hence a reduction of the error for triplet states, but at the cost of larger errors for the singlet excited-states. At this stage, the most accurate TD-DFT estimates therefore remain competitive for computing singlet-triplet transition energies. Indeed, with M06-2X, irrespective of the application or not of the Tamm-Dancoff approximation and of the selected atomic basis set, the deviations obtained with TD-DFT are rather small.

Assessment of the Accuracy of the Bethe-Salpeter (BSE/GW) Oscillator Strengths.

Aiming to assess the accuracy of the oscillator strengths determined at the BSE/GW level, we performed benchmark calculations using three complementary sets of molecules. In the first, we considered ∼80 states in Thiel's set of compounds and compared the BSE/GW oscillator strengths to recently determined ADC(3/2) and CC3 reference values. The second set includes the oscillator strengths of the low-lying states of 80 medium to large dyes for which we have determined CC2/aug-cc-pVTZ values. The third set contains 30 anthraquinones for which experimental oscillator strengths are available. We find that BSE/GW accurately reproduces the trends for all series with excellent correlation coefficients to the benchmark data and generally very small errors. Indeed, for Thiel's sets, the BSE/GW values are more accurate (using CC3 references) than both CC2 and ADC(3/2) values on both absolute and relative scales. For all three sets, BSE/GW errors also tend to be nicely spread with almost equal numbers of positive and negative deviations as compared to reference values.

Excited-State Geometries of Solvated Molecules: Going Beyond the Linear-Response Polarizable Continuum Model.

The theoretical determination of excited-state structures remains an active field of research, as these data are hardly accessible by experimental approaches. In this contribution, we investigate excited-state geometries obtained with Time-Dependent Density Functional Theory, using both linear-response and, for the first time, corrected linear-response approaches of the Polarizable Continuum Model. Several chromophores representative of key dye families are used. In most cases, the corrected linear-response approach provides bond distances in between the gas and linear-response data, the latter model providing larger medium-induced structural changes than the corrected linear-response model. However, in a few cases, the solvation effects predicted by the two continuum approaches present opposite directions compared to the gas phase reference.