Improved Description of Charge Transfer Potential Energy Surfaces via Spin-Component Scaled CC2 and ADC(2) Methods

Charge Transfer The performance of spin-component scaled variants of the popular CC2 and ADC(2) methods is evaluated for CT states, following benchmark strategies of earlier studies. The results show the capability of spin-component scaled approaches to reduce the large errors of their regular counterparts to a significant extent. Abstract The molecular level understanding of electronic transport properties depends on the reliable theoretical description of charge transfer (CT) type electronic states. In this paper, the performance of spin-component scaled variants of the popular CC2 and ADC(2) methods is evaluated for CT states, following benchmark strategies of earlier studies that revealed a compromised accuracy of the unmodiﬁed models. In addition to statistics on the accuracy of vertical excitation energies at equilibrium and inﬁnite separation of bimolecular complexes, potential energy surfaces of the ammonia–ﬂuorine complex are also reported. The results show the capability of spin-component scaled approaches to reduce the large errors of their regular counterparts to a signiﬁcant extent, outperforming even the CCSD method in many cases. The cost-eﬀective scaled opposite-spin (SOS) variants are found to provide remarkably good agreement with the CCSDT-3 reference data, thereby being recommended methods of choice in the study of charge transfer states.

Introduction the CT states with a very compromised accuracy. Our investigations, based on a set of local and CT states of bimolecular complexes, 31 revealed a significant underestimation of the CT excitation energy at the equilibrium structure of these systems. In ref 32 it was also shown that many issues related to the inappropriate description of CT states are connected to the asymptotic behaviour of the methods, i.e. the description of the charge transfer at infinite separation of the source and destination fragments. At these points the charge transfer corresponds to a simultaneous ionization on the source and an electron attachment on the destination fragment, and the associated excitation energy of the CT state should precisely equal the sum of the ionization potential (IP) and electron affinity (EA) of the respective subsystems. This condition is, however, not fulfilled by methods that are not size extensive in the excited state, e.g. CCSD. 33,34 (With CC methods, excited states are described either within the Equation-of-Motion (EOM) 35,36 or the Linear Response 12, 33,37,38 (LR) framework. Since these two provide identical excited state energies -the sole quantity discussed in the present work -we do not distinguish them and omit their notation for brevity.) In addition, the different description of the electron correlation by the various methods influences the accuracy of the IP and EA values considerably, resulting in large errors of the asymptotic CT excitation energy for some models: both CC2 and ADC(2) were found to severely underestimate the reference values. 32 These flaws seen on CT states are related to the case of Rydberg states in the sense that they both can likely be attributed to the inappropriate description of the electron moved far from its ground state position.
A significantly different accuracy of a method in the asymptotic and equilibrium regions has a consequence that the interconnecting potential energy surface (PES) of the CT state, normally characterised by the 1/R attraction of the two ions, will also be inaccurate. Such surface scans revealed important problems with the reliability of the CC2 and ADC(2) CT surfaces and the description of their avoided crossings with local excitations. 32 The concept of spin-component scaling, originally proposed by Grimme 39,40 for the MP2 energy and later by Hättig and co-workers for CC2 and ADC(2), 41 The first experiences on the performance of spin-component scaled methods on excited states were rather mixed, 45,52-54 which limited their popularity in applications. Recently, however, we pursued a quest to identify the terms responsible for the sometimes seriously bad performance of CC2 on Rydberg states 21 and excited state gradients, 22 also evaluating spin-component scaled methods in this context. 55 It was found that errors of the regular CC2 model -as compared to either CCSD or a higher level reference -can be mitigated to a large extent by these approaches. 55 Later the analysis was extended to ADC(2) and its spin-component scaled variants with a similar outcome, 56 which did not come as surprise knowing the very close relationship of the CC2 and ADC(2) theories. 11,15 In the present work, we investigate the performance of spin-component scaled variants of CC2 and ADC(2) on CT states, encouraged by these promising results on Rydberg states and PESs. 55,56 Since in our earlier studies no other pair of the C SS and C OS parameters was found to clearly outperform the common SCS and SOS methods, 55 we only check these two approaches and refrain from the ambiguous, hardly justifiable tweaking of these parameters.
In our examination we follow the benchmarking procedures established in refs 31 and 32 by evaluating CT states of a set of two-component non-covalent molecular complexes in the energetically significant, low-lying area.

Methods
We investigate the test systems from refs 31 and 32, i.e. the ammonia-fluorine, acetonefluorine, pyrazine-fluorine, ammonia-oxygendifluoride, acetone-nitromethane, ammonia-pyrazine, The structures of the complexes were taken from that study 31 and, with the exception of the ethylene-tetrafluoroethylene system discussed in detail, 31 correspond to the equilibrium distance between the two fragments.
As reference, the high-level CCSDT-3 57 data from ref 31 was used, where the excellent accuracy of this method was also confirmed via comparisons to CCSDT results on selected systems.
In all calculations we used the cc-pVDZ basis set of Dunning and co-workers 58 which, despite its limited size and the lack of diffuse functions, was confirmed to provide reliable results for such comparisons. 31 Core electrons were excluded from the correlation treatment, except for the calculation of ω descriptors and PESs (see below). All CC2, ADC(2) and spin-component scaled calculations were performed with the TURBOMOLE 48,59 package, while CCSD and CCSDT-3 reference data were taken from ref 31.
To rely on a clear definition of the charge transfer nature of the states, as well as to free the analysis from ambiguities connected to the sometimes rather intricate mixing of dominant characters of different type in the wave functions (and their variation with the methods), the states are characterized using the ω descriptors of Plasser and co-workers. [60][61][62][63][64] In this framework, the CT character ω CT (ranging from 0.0 of completely local states to 1.0 for pure CT ones) is defined as the weight of configurations in the wave function with charges separated on different fragments. The average exciton position ω P OS ranges from 1 to 2 in a two-component system, with the limits corresponding to local excitations on the first and the second fragment, respectively, while ω P OS ≈ 1.5 to CT and completely delocalized (one-to-one mixed) Frenkel resonance excitations. The participation ratio ω P R is close to 1 for both clear CT and local excitations, and in bimolecular complexes ω P R ≈ 2 is typical in completely delocalized Frenkel-type and charge resonance states. The detailed introduction and demonstration of these quantities can be read in refs 31, 32, 60-63. The ω descriptors were evaluated with the TheoDORE program, 63 using results from calculations performed with TURBOMOLE. We should note that, unlike in the analysis of ref 31 where the full transition density matrix was used even at the CC2 level to calculate these numbers, this workflow approximates the respective transition matrix elements by the corresponding components of the one-particle solution vector of the excited state. Thus, in the present work, even CC2 descriptors can slightly differ from the ones obtained in the rigorous approach of ref 31. The effects of this approximation are, however, minor (no more than 0.01, 0.05 and 0.04 in the CC2 ω P OS , ω P R , and ω CT values, respectively) and thus do not influence the analysis significantly.
To check the problems associated with the asymptotic behaviour, we calculate the CT excitation energies, as well as the associated ionization potential (IP) and electron affinity (EA) values with the spin-component scaled methods, and compare them to the respective CCSDT-3 data and the results of the unscaled variants. The IP and EA calculations were done using the continuum orbital strategy described in ref 65, allowing for the evaluation of these quantities with any method able to predict excitation energies. This approach makes it possible to treat all ionized states on an equal footing in a closed-shell framework, potentially providing more accurate associated transition moments. 66 It was successfully employed previously by us 32,66 even at the TDDFT level, and does not formally require a computer code to specifically support it: by including a separate, molecule-centered function in the basis set with a negligibly low exponent, ionized states are obtained in the excitation energy calculation as excitations to or from this orbital. The asymptotic excitation energies (E exc ∞ ) were obtained at 10,000 bohrs of intermolecular separation, where the interaction energy of the two charges is already negligible (below 0.003 eV).
The effects on the CT potential energy surface are examined on a system where the various issues were presented and explained previously, 32 the lowest totally symmetric CT state of the ammonia-fluorine complex and its interaction with a totally symmetric local excited state.

Identification of the CT states
To evaluate a performance of a theoretical method by comparison to results obtained with higher level reference data, it is vital to ensure that the correct samples are compared in a statistical manner. While establishing a benchmark set for CT states in our previous work, 31 it was revealed that forming a set of states that can be considered of CT type in all situations is not at all trivial and, due to the frequent mixing of CT and other (usually local valence) contributions in the wave function, requires one to follow a clear numerical definition in the classification of the states. To this end, threshold criteria for the ω descriptors (see above) prove to be an effective strategy. 31 In that work, states with ω CT > 0.5 (with ω CT evaluated at the CCSD level) were considered as charge transfer, while all others as local. (To also reflect a 'mixed' nature of some states, the ω CT ≤ 0.1, 0.1 < ω CT ≤ 0.9, and 0.9 < ω CT categories were used for analysis as well.) Nevertheless, if different methods treat the various contributions in the wave function differently (as it is usually the case), the values of the ω descriptors can also be considerably different, potentially causing a state classified as CT by the CCSD measures to be a local excitation with that particular method. As including such states in a side-by-side comparison might provide misleading results, we first have to verify that an identical set of states is being used in the statistical analysis. Note that we do not aim to exclude 'mixed' states in general, but rather to avoid ones with a significantly

Equilibrium excitation energies
The errors of vertical excitation energies relative to the reference CCSDT-3 values for the CT states investigated in ref 31 are presented in Table 1 and shifting the values further in the same direction. However, unlike to the case of Rydberg states where this results in about the same relatively low mean absolute error for both the SCS and SOS models, for CT states only the SCS parametrization predicts the excitation energies with a good accuracy and the SOS one overestimates them considerably. Table 1: Relative error of the calculated excitation energies (∆ CCSDT −3 (E exc eq ), in eV) at equilibrium separation with respect to CCSDT-3 results for the investigated CT states

Asymptotic limits
Statistics on the IPs of the source fragments, the EAs of the destination fragments and the asymptotic CT excitation energies are given in Table 2 However, the perhaps most important property of the asymptotic behaviour of a theoretical method is how accurately the CT excitation energy agrees with the sum of the IP and EA values of the source and destination fragments, respectively. Statistics on the difference between these quantities are presented in Table 3. As it was already shown in ref 32, CCSDT-3 reproduces the theoretical asymptotic energy with a very low ( < 0.03 eV ) error, while CCSD, due to the lack of size extensivity, shows a remarkable overestimation with the mean error being as large as 0.33 eV. Unscaled CC2 and ADC(2) also have a posi- tive, yet significantly lower mean error of 0.19 eV. Compared to them, the spin-component scaled methods behave noticeably better: the mean error is reduced to 0.14 eV and 0.11-0.12 eV with the SCS and SOS variants, respectively. The fact that this is accompanied with low standard deviations (in fact, lower than those of the unscaled models) emphasizes the consistency of this improvement throughout the benchmark set.  Table 3: Statistics on the deviation (E ∞ exc − (IP + EA), in eV) of the vertical excitation energies of the investigated CT states at 10,000 bohr separation from the sum of the IP (source fragment) and EA (destination fragment) values.            4 State characters at the avoided crossing  download file view on ChemRxiv SupportingInformation.pdf (878.04 KiB)