Extended Dynamically Weighted CASPT2: The Best of Two Worlds

We introduce a new variant of the complete active space second-order perturbation theory (CASPT2) method that performs similarly to multistate CASPT2 (MS-CASPT2) in regions of the potential energy surface where the electronic states are energetically well separated and is akin to extended MS-CASPT2 (XMS-CASPT2) in case the underlying zeroth-order references are near-degenerate. Our approach follows a recipe analogous to that of XMS-CASPT2 to ensure approximate invariance under unitary transformations of the model states and a dynamic weighting scheme to smoothly interpolate the Fock operator between state-specific and state-average regimes. The resulting extended dynamically weighted CASPT2 (XDW-CASPT2) methodology possesses the most desirable features of both MS-CASPT2 and XMS-CASPT2, that is, the ability to provide accurate transition energies and correctly describe avoided crossings and conical intersections. The reliability of XDW-CASPT2 is assessed on a number of molecular systems. First, we consider the dissociation of lithium fluoride, highlighting the distinctive characteristics of the new approach. Second, the invariance of the theory is investigated by studying the conical intersection of the distorted allene molecule. Finally, the relative accuracy in the calculation of vertical excitation energies is benchmarked on a set of 26 organic compounds. We found that XDW-CASPT2, albeit being only approximately invariant, produces smooth potential energy surfaces around conical intersections and avoided crossings, performing equally well to the strictly invariant XMS-CASPT2 method. The accuracy of vertical transition energies is almost identical to MS-CASPT2, with a mean absolute deviation of 0.01–0.02 eV, in contrast to 0.12 eV for XMS-CASPT2.


Excited States of Glycine
In order to investigate the dependence of vertical electronic excitation energies on the number of model states included in the perturbative calculation, we study the n N → 3s and the n O → π * transitions in the glycine molecule. The former is the transition to the first excited state of the totally symmetric irreducible representation (irrep) within the C s molecular point group, 2 1 A , while the latter is the excitation to the 1 1 A state. The glycine molecule has a very dense set of excited states below 10 eV of both valence and Rydberg type 1 and it is thus a suitable model to probe the effect of including various numbers of states of different character in the model space.
The computational details are as follows. The geometry was optimized at closed-shell MP2/cc-pVTZ level of theory 2,3 within the C s point group. This calculation was performed using the Gaussian program package, revision d01. 4 Then, state-average CASSCF calculations were performed for the two symmetry irreps. The active space was composed by 8 electrons in 5 a and 4 a molecular orbitals. The former were the n N and n O lone pairs, the 3s, 3p x and 3p y Rydberg orbitals, while the latter were the three bonding, non-bonding and anti-bonding π orbitals delocalized over the carboxylic acid group and the Rydberg 3p z molecular orbital. The basis set used was the aug-cc-pVTZ one. 5 Two separate calculations were carried out, one averaging over 11 1 A states, while the other one over 7 1 A states. Including this number of states was necessary in order to target n→ π * , π → π * and transitions to Rydberg states from both n and π orbitals. Using these wave functions, MS-CASPT2 and XMS-CASPT2 calculations were carried out, including different number of states in the model space starting with 3 states for the 1 A symmetry sector and 1 state for the 1 A one. To evaluate the excitation energy of 1 A states, the ground state energy was computed by a state-specific CASPT2 calculation using only the lowest CASSCF state (out of the 11 ones available). In the case of MS-CASPT2, calculations were carried out using the canonical method, as well as with a unique state-average Fock operator used for all states, in an analogous manner to the work by Kats and Werner 6 . We shall label results obtained with the state-average Fock operator as SA-CASPT2. Note that the use of a state-average Fock operator is completely analogous to XMS-CASPT2, the only difference between the methods being the use of rotated states in the latter one. All calculations were performed without the use of the IPEA shift, but including a 0.2 E h real denominator shift in order to avoid intruder state problems. These calculations were performed using a development branch of OpenMolcas, 7 version 19.11-88-g4984c848.
The vertical transition energies as a function of model space dimension are reported in here were selected because for all methods considered they were clearly identifiable through the transition density matrix and particle/hole transition natural orbitals. This allowed a true comparison between a method using a state-specific Fock operator and methods using a state-average one.

Avoided Crossing in LiF
In this section we show additional results for the dissociation of lithium fluoride. First, we report the potential energy curves in the case only two states are included in the calculation.
The computational details are the same as reported in the main text, with the only difference being the number of roots considered in both the SA-CASSCF and the various CASPT2 calculations. In Figure 3 we report the results of the reference calculation with MRCISD.
The results for XDW-CASPT2 for ζ = 50, 5000 and the limit ζ → ∞ are shown in
in Figures 11 and 12 for the 3-state calculation with ζ = 500.

Conical intersection of allene
In this section we report additional computational details of the calculation involving the distorted allene molecule that complement the description in the main text.
The total number of points in the two-dimensional scan is 6561, 81 for each variable.
The scan of the C-C-C-H torsion angle practically corresponds to a pyramidalization of the external carbon atoms. Note that in the original work, 10 the assignment of "variable 1" and "variable 2" in the surface plots were swapped as compared to the explanation in the text.
The IPEA shift was always set to zero for all methods. For some cases it was necessary to include a real denominator shift in order to converge the calculation for all points. In In Figure 13 we report the results for a model space with 6 states for XMS-CASPT2. As discussed in the main text, the PES remains virtually unchanged upon increase of the model space dimension.

Vertical excitation energies
In this section we report additional information regarding the computational details for the calculation of vertical excitation energies. Imposed molecular point group symmetry, number of active electrons, occupied and active orbitals and real denominator shifts are listed in Table 1. Note that if a shift was necessary for a method, it was also applied to the other ones in order to obtain comparable results.