Accuracy of spin-component scaled ADC(2) excitation energies and potential energy surfaces.

In a recent paper of this Journal (Tajti and Szalay, JCTC 2019, 15, 5523) we have shown that failures of the CC2 method to describe Rydberg excited states, as well as potential energy surfaces of certain valence excited states can be cured by spin- component scaled (SCS) versions SCS-CC2 and SOS-CC2 by a large extent. In this paper, the related and popular Second Order Algebraic Diagrammatic Construction (ADC(2)) method and its SCS variants are inspected with the previously established methodology. The results reflect the similarity of the CC2 and ADC(2) models, showing identical problems in the case of the canonical form and the same improvement when spin-component-scaling is applied.


Introduction
In two recent papers 1,2 in this Journal, we have extended the large pool of benchmark calculations on vertical excitation energy 3-14 with characterisation of the excited state potential energy surfaces using Coupled-Cluster (CC) type methods. In the first paper we have found 1 that the popular CC2 method 15 fails sometimes badly, depending on the type of the excited state. The parent CCSD method (termed either EOM-CCSD 16-18 or CCSD-LR 19,20 ) gives usually reasonable results, but the inclusion of approximate triple excitations is necessary to achieve high accuracy. Since this is generally not possible for larger systems, improved second order approximations are requested. 12,13 Along this line, in our recent study 2 we also investigated spin-component scaled versions of CC2 21,22 and found that both the scaledopposite-spin (SOS-CC2) and spin-component-scaled (SCS-CC2) variants cure the problem of CC2 to a considerable extent and the resulting potential energy surfaces run quite close to those obtained with CCSD.
Besides CC2, another second order method often used in applications is the so called ADC(2) (2nd order Algebraic Diagrammatic Construction) method. 23,24 This method is closely related to CC2, 25 and it is a widespread experience that CC2 and ADC(2) results are of very similar quality. 7, [25][26][27] In this paper we augment the results of our previous paper 2 and present the results of the same type of calculations, now using the ADC(2) and its spin-component scaled variants SOS-ADC(2) and SCS-ADC(2). We believe that this addition is necessary to get a complete picture about the performance of the second order methods which can be used and are being used to describe excited states of larger molecules.

The ADC(2) method
The 2nd order Algebraic Diagrammatic Construction (ADC(2)) method has been introduced by Schirmer, 23,28 based on a diagrammatic perturbational expansion of the polarization propagator. 23 Later the ADC(2) method has been reintroduced by Trofimov and Schirmer. 24,29 The real advent of the method came when its relation to CC2 has been established and its implementation in program codes, e.g. in TURBOMOLE 30 pursued. The close relation of CC2 and ADC(2) has been discussed by Hättig 25 and recently in detail by Izsák,27 showing that ADC(2) can be considered as an approximation to CC2 by neglecting T 1 amplitudes (thus enforcing strictly second order) and symmetrising the Jacobian. As has been shown e.g. in ref. 12, neglect of T 1 amplitudes has a negligible effect on excitation energies. Symmetrisation, on the other hand, avoids technical issues related to the non-Hermitian formulation. 25 These two approximations also result in a slightly lower cost.

Spin-component scaled ADC(2)
The concept of spin-component scaled CC2, i.e. the introduction of spin scaling factors C OS and C SS into the formulae of the effective Hamiltonian has been first established by Hättig et al., 21 in the spirit of spin-component-scaled MP2 originally proposed by Grimme. 31,32 In case of the SCS (spin-component-scaled) variant the same-spin integrals are multiplied by a factor of C SS =1/3, while the opposite-spin integrals by C OS =6/5. In the SOS (scaledopposite-spin) version only the opposite-spin integrals are retained (C OS =1.3, C SS =0.0).
Using the above mentioned similarity of CC2 and ADC(2) theories, the same spin-scaled modifications can be introduced into ADC(2) in a straightforward manner, leading to the SCS-ADC(2) and SOS-ADC(2) models. The latter was first realized by Winter and Hättig,22 and later by Dreuw and co-workers 7,33 and, recently, by Mester and co-workers. 34

Computational details
The same methodology as in the parent paper 2 has been used. To benchmark the accuracy of vertical excitation energies, 64 valence and 56 Rydberg states of 23 molecules have been calculated using the aug-cc-pVDZ basis set, with the core electrons not correlated. The calculations were performed using the TURBOMOLE 30  Finally, the surface plots that follow the Frank-Condon forces in mass-weighted steps (for a detailed definition, see our previous paper 1 ) have also been evaluated in the cc-pVDZ basis by the MRCC program. The errors of ADC(2) lie close to those of the unscaled CC2 model. In particular, the severe underestimation of Rydberg excitation energies is very apparent, with an error of -0.42 eV being worse than even CC2 by 0.04 eV. Nevertheless, the valence-Rydberg misbalance, shown in the last column of table 1, is exactly as large as that of CC2. The two spin-component scaled ADC(2) variants show statistics very similar to their CC2 counterparts, the difference of the mean errors being no more than 0.03 eV, and even the standard deviations follow the same pattern. The above mentioned misbalance is reduced by spin-component scaling by as much as 38 and 55 percent for the SCS-ADC(2) and SOS-ADC(2) models, respectively, essentially identical to the SCS-CC2 and SOS-CC2 results.

Franck-Condon gradients
Measures on the gradients of the excited state energies at the ground state equilibrium geometry are presented in tables 2, 3, 4, and 5. The ADC(2) gradients, as evaluated by their length and their direction relative to the reference vectors, show an accuracy similar to that of CC2, with no clear indication of one being better than the other. Compared to CC2, in many cases a larger error of the gradient length is accompanied by a more accurate direction of the vector, or vice versa. However, in situations where the description is clearly problematic (e.g. in some n-π * states of nucleobases), both measures are inferior to the CC2 result. For such states, the effect of spin-component scaling is generally also similar, marginally inferior compared to the respective CC2 variant, i.e. both SCS-ADC(2) and SOS-ADC(2) provide significantly better gradients compared to the canonical model. The only exception seems to be the π-π * state of formamide, where no improvement is seen for the ADC(2) methods, while the CC2 result is clearly surpassed by the spin-component scaled versions. This can be, however, attributed to the fact that the CC2 and ADC(2) single excitation vectors of this state show a slightly different mixing of dominant π-π * contributions. For excited states where the CC2 gradient is found to be satisfactory, the spin-component scaling in ADC (2) shifts the canonical values in the very same direction as seen in the CC2 cases. Although this does not clearly lead to more accurate gradients, the differences remain negligible as the nominal errors remain low in all such cases.

Surface scans
The excited state potential energy surface scans shown on figures 1, 2, 3, and 4 generally seem to emphasize the above findings even further. The ADC(2) and CC2 curves run very close to each other, at least in the first half of the explored PES. That is, wherever ADC(2) and CC2 gradients were found to be exceptionally inaccurate (like in case of the first n-π * state of cytosine), the respective plots form a well recognizable, distinct pair. The spin-component scaled ADC(2) models show a pattern similar to the one found earlier for their CC2 counterparts: 2 their divergence from the reference curve, as compared to regular ADC(2), is reduced considerably, giving nearly CCSD quality results for most investigated states. SOS-ADC(2) performs slightly better than SCS-ADC(2) in this analysis, but the difference of the two variants remains minor compared to how much they both improve over spin-unscaled ADC(2).

Conclusions
The results presented above affirm earlier conclusions that ADC(2) and CC2 are closely related methods: not only statistics on vertical excitation energies for both valence and Rydberg states are close, but also the Frank-Condon gradients show very similar patterns.
In case of the surface scans one sees a failure of ADC (2)