The IPEA dilemma in CASPT2

We show that the use of the IPEA correction in CASPT2 for excited state calculations of organic chromophores is not justified.


Electronic Supplementary Information
This document is structured as follows. Section S1 collects all data related to the literature survey conducted in section 3 of the main manuscript and explains the procedure we followed to select the publications. In section S2, we provide additional computational details and data related to the CASSCF/CASPT2 and FCI calculations for the di-and triatomic molecules, described in section 4 of the main manuscript. And finally, section S3 presents the computational details, data, and additional discussion related to the CASPT2 calculations on the Thiel's benchmark set, described in section 5 of the main manuscript.

S1 Excitation Energies from the Literature Survey
The excitation energies considered in our analysis are collected in Table S1. The publications that entered our analysis were selected in the following manner. A simple search for the topic "CASPT2" in the SciFinder database generates over 2000 hits. Out of these 2000 hits, 770 refer to papers published until 2004. From those, we selected only studies concerned with the calculation of vertical excitation energies based on their title, i.e., we discarded adiabatic excitation energies or comparison of full potential energy surfaces. Further, we eliminated all studies where no experimental data was given for comparison. In case where several computational studies were provided, we included only the results obtained using the largest active space and/or basis set. Likewise, when multiple experimental values for the excitation energies were reported in the computational studies, we used the average of all of them as the reference data with one exception: since most computational studies were performed in the gas phase, we neglected experimental reference data obtained in solution if experimental reference data in gas phase was available.
For the organic molecules, we differentiated between valence and Rydberg excited states, excited states calculated in gas phase or any other environment, and excited states calculated with the standard non-diagonal CASPT2 variant 1 or any other CASPT2 variant. Table S1: Calculated (V calc ) and experimental(V exp ) excitation energies and energy differences (∆V ) in eV.

Molecule
State V calc V exp ∆V Character Environment Method Ref.                         The general computational details for our FCI/CASPT2 benchmark calculations are given in section 4.1 in the main paper. Here, we only discuss a few additional issues. All states comprised in our benchmark set are given in Tables S4 and S5 for the 6-31G and 6-311G basis sets, respectively.
As may be seen by inspecting these tables, the number of low-lying electronic states considered varies from molecule to molecule. Each number was inspired by results obtained in previous FCI calculations [42][43][44][45][46][47][48][49] and the goal to cover the most probable low-lying excitations that could be expected on the basis of simple orbital considerations regarding only the valence orbitals in the active spaces.
Initially, we pondered to use the already published FCI results for our reference data.
However, most of these studies use different basis sets often augmented with diffuse function to also describe Rydberg states. Since our aim was only to investigate valence excited states using a consistent treatment for all the molecules, all the FCI calculations were repeated using the 6-31G and 6-311G basis sets.
A comment is important here regarding the effect of lowering the symmetry in the calculations of the linear diatomic molecules. Symmetry lowering was unavoidable since both MOLPRO and MOLCAS only support the usage of Abelian point groups with the highest symmetric point group being D 2h . In Abelian point groups, there exist only one-dimensional irreducible representations. Thus, states that are degenerate by symmetry in the linear point groups, e.g., Π or ∆ states, transform according to different irreducible representations. The correspondence of the irreducible representations is shown in Table S2.
A symmetry lowering does not affect the degeneracies in the FCI calculations, but it can have an effect on the degeneracies in the CASSCF/CASPT2 calculations. When computing Σ and ∆ states, one of the components of the ∆ state and the Σ state can fall into the same irreducible representation. Since we used state-averaged CASSCF, the CASSCF wave functions of both components of the ∆ state then will differ leading to splitting of the energies of both components. The average splitting of the CASSCF energies is 0.07 eV and it is reduced again in the multi-state CASPT2 treatment to 0.05 eV. Note that the Π states are not affected by S20 Table S2: Correspondence of irreducible representations of the linear point groups D ∞h and C ∞v with the Abelian point groups D 2h and C 2v used in the calculation, respectively. to the (1σ + ) 2 (2σ + ) 2 (1π1π) 2 electronic configuration of the HB molecule. In the lowered C 2v symmetry the 1 ∆ state splits up into a 1 A 1 and a 1 A 2 state. The σ + orbitals become a 1 orbitals while the 1π orbitals split up into the 1b 1 and 1b 2 orbitals. The wave function of the 1 A 1 component is given by the linear combination of the configuration where both electrons are in the same orbitals, i.e., while in the leading configuration of the wave function of the 1 A 2 component the electrons in the π orbitals are in the symmetry-distinct b 1 and b 2 orbitals: In the original C ∞v point group, both components should be characterized as open-shell, since the (1π1π) double shell is not fully occupied. In the lowered C 2v symmetry, NOS( 1 A 2 ) ≈ 2

S22
while NOS( 1 A 1 ) ≈ 0, making the latter state "closed-shell" by our definition. Thus, both components are affected differently when employing the IPEA-shifted CASPT2 and to capture this difference, we decided to include both components of each ∆ state when calculating the errors in Table 2.

S2.3 Ground State Geometries of Di-and Triatomic Molecules
The Cartesian coordinates of the ground-state geometries of the di-and triatomic molecules used in the CASSCF/CASPT2 vs. FCI benchmark are given in Table S3. The geometries of the diatomic molecules were taken from the NIST database 50 while for the triatomic molecules we used geometries employed in previous FCI studies. 43,45   S25 Table S4: Total energies E X in a.u. and vertical excitation energies V X in eV of di-and triatomic molecules (X = FCI, IPEA, NOIPEA, CASSCF) calculated using the 6-31G basis set.

Molecule State Irrep
Transition      3.11   Figure 3 in the main paper shows one data point, labeled ψ 1 , at NOS ≈ 0 and ∆NOS ≈ −2, respectively, with an error considerably larger than that of the rest of the states.
This state is the shell ground state belonging to the electronic configuration (1σ + ) 2 (2σ + ) 2 (3σ + ) 2 (1π1π) 2 , and the 2 1 Σ + state may be described by a (3σ + ) 2 → (1π1π) 2 double excitation. For this transition, ∆NOS ≈ −2, and the resulting state is described by the closed-shell electronic configuration 4 . This is a rather unique transition. There is no corresponding double excitation from a open-shell ground state yielding a closed-shell excited state in our test set, so we simply report this unusual large error but refrain from speculating about its origin.

S2.6 Potential Energy Curves of Diatomic Molecules
For the first-row hydrides (HLi, HBe, HB, HC, HN, HO, HF) and the homodiatomic molecules , the potential energy curves were computed using CASPT2 (NOIPEA, ε = 0) and different ANO-RCC basis sets (MB, VDZ, VDZP, VTZP, VQZP). The active spaces and correlated electrons were the same as described in section S2.1. In Table S6, the total energies of these molecules are reported for various interatomic distances.           π/π * as well as n orbitals is given as a comment in Table S7. As special case is tetrazine.
Tetrazine possesses 14 electrons in the valence π and n orbitals. Finally, we wish to address the characterization of the excited states of triazine. Thiel and co-workers report the lowest-lying 1 ππ * excited states to be the 1 1 A 2 , 2 1 A 1 , 1 1 E , and 2 1 E states, 51 in accordance with the classification given in an earlier CASPT2 study by the Roos group. 2 However, by calculating the excited states of triazine in C s symmetry, as was done by Thiel and co-workers, our results suggest a different characterization for some of the states. We agree that the first two 1 ππ * excited states are the 1 1 A 2 and 2 1 A 1 states, however, we question the characterization of the other two states as 1 E states. These two states were reported to lie at 7.49 and 8.99 eV. We, too, find each one 1 ππ * state at 7.50 and 8.95 eV, however, the second component to both 1 E states is missing. The next higher-excited state of A symmetry appears at 9.08 eV. It is not a described by a π → π * excitation but is rather given by a (nn) → (π * π * ) double excitation. Since our calculations did not exploit the full D 3h symmetry of the molecule, one might wonder whether either one of the components of the 1 E states may have been lost due to symmetry splitting. However, this is unlikely since using the very same approach, our calculations predicted both components of two 1 E states of nπ * character with an energetic splitting smaller than 0.01 eV. Thus, both 1 ππ * states may require a new classification, which is, however, not of importance for the rest of this work. Table S7: Comparison of vertical excitation energies reported in the benchmark study by Thiel and co-workers 51 and vertical excitation energies computed in this work. Differences larger than 0.05 eV are marked as "significant" in the comments. The order of the molecules and excited states is adopted from the Supporting Information of Ref. 51.

S3.2 Ground State Energies of all Molecules
In Table S8

S3.4.1 General Considerations
In   Thus, the ground state is rather unaffected by the presence of the two worse described states.

S61
The other 1 A excited states show non-negligible mixing and there energy changes by 0.2-0.6 eV, and we, thus, exclude all 1 A excited states from our data set. Yet, we keep the ground state (1 1 A ) in our data allowing us to also keep the other states ( 1 A , 3 A , and 3 A ). Adenine For adenine, we find two SS-CASPT2 states with low reference weights: 3 1 A (ω = 0.18) and 4 1 A (ω = 0.01). However, since these states do not mix with the other 1 A states [see eq.(S17)] and we are only interested in the first two roots, we do not need to exclude any states of interest from our data set.

S3.4.3 Intruder State Problems in IPEA-CASPT2 Calculations
In Table S9 we show a list of excited states that we exclude from our data set due to possible intruder state problems. These excited states were obtained from CASPT2 calculations using IPEA shifts of ε = 0.08, 0.1337, and 0.16 a.u. In the previous section, we have discussed in detail all intruder state problems that occur when the IPEA shift is set to zero. We note again, that our interest was only directed at states reported in the Supporting Information of Ref. 51 In some cases, the intruder state problem affected only states that did not lie in our range of interest. When there was only negligible coupling with these states at the multi-state CASPT2 level, we felt it safe to keep the rest of states computed in the same calculation in our data set.
However, when the coupling was larger or one of states under our consideration was directly affected with an intruder state problem, we excluded these states (Table S9) from our test set (for additional discussion how to handle intruder state problem see section 5.1 in the main paper). We have also performed calculations with a negative IPEA shift parameter, namely, ε = −0.12 a.u. Here, we could obtain meaningful data only for a very small number of molecules due to significant intruder state problems. The intruder state problem almost always affected also the ground state of the molecule making it necessary to exclude all excited states of most molecules. Thus, instead of listing all the states we had to exclude, we ask the reader to simply inspect the small number of states reported in Table S10 for ε = −0.12 a.u.

S3.5 CASPT2 Results for Thiel's Benchmark Set
In Table S10 we          problem present, e.g., the 1 A states of imidazole or the 1 1 B 1g state of pyrazine. These states did not suffer from intruder state problems at the SS-CASPT2 level but rather were mixed with affected states at the MS-CASTP2 level. We chose to exclude them from our data set with the concern that these energies might be corrupted by the coupling to the affected states.
However, the normally appearing trend of increasing energy with increasing IPEA shift value

S72
-without any spike as, e.g., seen for the 1 1 B 2 state of furan -might raise the question if we had been overly cautious in excluding these states from our statistical analysis. Unfortunately, we see no way of clarifying this question and, thus, decide to rather stay on the safe side. In

S3.6 Basis Set Effects
Table S12 collects the contraction scheme of different basis sets used in this study and discussed in section 5.4.1 of the main paper. In Table S23 we list the MSEE of the CASPT2 excitation energies compared to experiment for the Thiel benchmark set (see also Figure 9

S3.7 Comparison to other Computational Results
In Table S24 we (3)]. 58 In Table S25 we show the mean signed and unsigned errors of the excitation energies compared to experimental reference data (see also Figure 12 in the main paper). S120