
Anionπ interaction refers to an attractive force between an anion and an electrondeficient π system [13]. Although it is now widely recognized as an important nonbonded interaction in supramolecular and biological chemistry [47], there is few direct spectroscopic determination of its intrinsic bonding strength, while its binding nature is still elusive [812].
Recently, Anstöter and coworkers have filled this void to provide a photoelectron spectroscopic determination of the binding strength for the iodidehexafluorobenzene (I^{}·C_{6}F_{6}) complex [13]. In combination with highlevel electronic structure calculations, the anionπ bond strength (D_{e}) in I^{}·C_{6}F_{6} was found to be 0.53 eV with an uncertainty less than ±0.03 eV. The interaction was claimed to arise for a large part from correlation forces (41%), with electrostatic interaction (23%) and polarization (28%) making up most of the remainder. In their experiment, I^{} was condensed onto C_{6}F_{6} using a molecular beam source. The I^{}·C_{6}F_{6} complexes were formed by the supersonic expansion, massselected by timeofflight, and intersected with light. The resultant photoelectrons were then analyzed. Theoretically, they have calculated the I^{}·C_{6}F_{6} complex with an equationofmotion coupled cluster theory, namely EOMIPCCSD(dT) [14], using the augccpVDZ (aVDZ for short) basis set [15, 16] at the frozencore approximation. A rigid potential energy scan was performed by changing the distance (R) between I^{} and the center of the C_{6}F_{6} ring. To explore the binding nature, an energy decomposition analysis (EDA) based on absolutely localized molecular orbitals (ALMO [17]) was carried out on the selfconsistent field (SCF) energy of the complex plus the correlation contribution. The QChem 5.0 computational package [18] was employed. Quantitatively, the aVDZ basis set used in their work was usually too small to be close to the basis set completeness, although it was a choice limited by the system size. Conceptually, it is generally accepted [810] that an anionπ interaction is principally due to the electrostatic forces including the ioninduced polarization, while the correlation forces are usually put in a respectively minor position in the anionπ interactions.
In the present work, we performed the CCSD(T) (i.e. coupledcluster with single and double and perturbative triple excitations) calculations on the potential energy curve of the I^{}·C_{6}F_{6} complex. The same aVDZ basis set [15] was employed for C and F atoms but without invoking the frozencore approximation. The same effective core potential and the respective basis set (augccpVDZpp [16]) were applied to I^{}. We have used the BoysBernardi counterpoise technique [19] to explore the basis set superposition error (BSSE). We have also performed the CCSD(T)F12 [20]/aVDZ calculations to further assess the basis set incompleteness. In an attempt to find an accurate and efficient alternative to the expensive CCSD(T) method, we have benchmarked the performances of XYG3 [21] and the longrangecorrected (lrc) XYG3 [22], two doubly hybrid (DH) functionals developed in our groups, using the same aVDZ basis set. To explore the binding nature, we have applied the local molecular orbital energy decomposition analysis (LMOEDA [23]) method to the CCSD(T) and XYG3 binding energies. The F12 [20] calculations were performed by using the Molpro package [24], the LMOEDA calculations were performed by using the GAMESS US package [25], and all other calculations were carried out by using a local developmentversion of QChem 5.0 computational package [18]. We found that CCSD(T)/aVDZ can have an uncertainty up to 0.113 eV due to the basis set incompleteness. We found that the electrostatic contribution is over 128% in the I^{}·C_{6}F_{6} complex, which is counterbalanced by the Pauli repulsion. The electrostatic contribution alone clearly outweighs the correlation contribution, although the latter is also important, making a 53% contribution at the equilibrium binding distance (R_{e}).
Our calculation results are presented in Tables Ⅰ and Ⅱ and FIG. 1. Table Ⅰ summarizes the results for the potential energy curve scan of the I^{}·C_{6}F_{6} complex, where the CC and the CF bond lengths in C_{6}F_{6} were fixed at 1.3948 Å and 1.3394 Å, respectively. The BSSE effects are indeed strong for the correlated methods. The CCSD(T)/aVDZ method predicts a binding energy of 0.565 eV, which is reduced to 0.452 eV after the full portion of the BoysBernardi counterpoise correction [19]. The equilibrium distance is found at R_{e} = 3.52 Å at the level of CCSD(T)/aVDZ, which is extended to 3.70 Å after the BSSE correction. The well of the potential energy curve is rather flat, where the energy difference from the distance R = 3.50 Å to 3.60 Å is only 0.003 eV (Table Ⅰ) at the level of CCSD(T)/aVDZ.
Figure 1. Energy decomposition analysis [22] applied to the I^{}·C_{6}F_{6} complex at the level of CCSD(T). Code: electrostatic contribution (OEE [42]), correlation contribution (COR), polarization contribution (POL), exchangerepulsion (EXREP = EX+REP [41]), the frozen density term (FRZ = OEE+EXREP [18]). Note that the electrostatic interaction (OEE) is always attractive, which clearly outweighs in magnitude the contribution from the correlation interaction (COR)
Table I. The potential energy (in eV) scan of the I^{}·C_{6}F_{6} complex.
The CCSD(T)F12/aVDZ method may provide a better way to account for the basis set incompleteness than the BoysBernardi counterpoise correction. Table Ⅰ summarizes the results from F12a and F12b. We may infer that F12a yields better results than F12b at the aVDZ basis set [24, 25]. As seen from Table Ⅰ, F12a predicts D_{e} = 0.531 eV at R_{e} = 3.54 Å. These compare very well with the fitting data of D_{e} = 0.53 eV at R_{e} = 3.56 Å by Anstöter and coworkers [13]. The F12b results are also satisfactory, which yields D_{e} = 0.519 eV at R_{e} = 3.55 Å.
It is well known that the BoysBernardi counterpoise technique will overcorrect the BSSE, as the monomer cannot take full advantage of the basis sets of the ghost atoms when they are actually occupied by electrons [26, 27]. Kim and coworkers have suggested to use 50% BSSE correction for smaller basis sets [26]. Table Ⅰ also gives the CCSD(T)/aVDZ results with 50% BSSE correction. If it is about to mimic the CCSD(T)F12a/aVDZ results, a 25% BSSE correction is recommended.
In the wavefunction theory, the HartreeFock (HF) theory, by definition, contains no correlation contribution. Still the bond strength is predicted to be 0.326 eV by HF, indicating the importance of electrostatic and polarization interactions. The equilibrium distance is found to be 3.90 Å at the level of HF/aVDZ. BSSE is not significant when the method does not take the excitations into the virtual orbitals into account. After a full portion of BSSE correction, D_{e} is reduced to 0.313 eV for HF/aVDZ. Hence, BSSE with HF/aVDZ is about one tenth of that with CCSD(T)/aVDZ.
Table Ⅰ also summarizes the results from B3LYP [2830]/aVDZ, which leads to D_{e} = 0.341 eV at R_{e} = 3.80 Å. The B3LYP results are found to be quite similar to those of HF, although the former shows a slightly stronger binding tendency than the latter. The BSSE effect for B3LYP/aVDZ shall be very similar to that for HF/aVDZ. The close similarity between B3LYP and HF indicates that B3LYP is insufficient to catch up with the correlation effects in the I^{}·C_{6}F_{6} complex with the LYP [29] correlation functional.
XYG3 [21] and lrcXYG3 [22] belong to the fifth rung functional of the Jocob's ladder [31], which not only hybridizes the local exchange (e.g., B88 [28]) with the HFlike exchange as it does in B3LYP, but also hybridizes the local correlation (e.g., LYP [29]) with the secondorder perturbation theory (PT2 [32]). The XYG3 type of doubly hybrid (xDH) functionals have been shown to be remarkably accurate in describing the noncovalent interactions of the main group elements [21, 22, 3336]. The generally good performance of xDH comes naturally from the right physics of the truly nonlocal correlation in the PT2 term.
For the I^{}·C_{6}F_{6} complex, the XYG3 method predicts the equilibrium distance for R_{e} = 3.54 Å with the binding energy being D_{e} = 0.509 eV. These results are very much similar to the CCSD(T)/aVDZ results with 50% BSSE correction (see Table Ⅰ). Our results highlight the usefulness of XYG3 as a costeffective substitute of the expensive CCSD(T) for complex realworld systems, indicating the superiority of the xDH functionals to the popular B3LYP functional in describing the correlation effects for the anionπ interactions. On the other hand, it is known that XYG3 does not recover 100% PT2 term at the long range for correlation/dispersion. Hence, it may still fall short of some correlation/dispersion interactions. The lrcXYG3 method [22] amends XYG3 by adding a distancedependent PT2 correlation, which leads to D_{e} = 0.555 eV at R_{e} = 3.53 Å. The BSSE effect is generally smaller [37], as the PT2 term is scaled by 0.3211. Hence, no BSSE corrections have been applied to all our DFT calculations as commonly done in the literature [21, 22, 3840], which further simplifies the calculations when applied to complex systems.
As for the binding nature is concerned, the most popular point of view so far is that the anionπ binding interaction is governed by the electrostatic contributions and the ioninduced polarizations together [812]. The electrostatic force requires the π system to possess a positive quadrupole moment, thus demanding the π ring to be substituted with electronwithdrawing substituents [112]. The results of Anstöter and coworkers [13], however, challenge this popular view that the electrostatic force is determinant in the anionπ interaction. They decomposed the interaction energy in the I^{}·C_{6}F_{6} complex at 3.70 Å and claimed that the relative ratio of electrostatic:polarization:correlation is 23:28:41. Hence, while the electrostatic force was found to be an important driver for the anionπ interaction in the I^{}·C_{6}F_{6} complex, they concluded that it was the correlation energy that was the dominant contributor [13].
We performed the energy decomposition on the CCSD(T) binding energies using the LMOEDA approach [23]. The analysis was applied to the HF energy, while the correlation/dispersion energy is obtained from the energy difference between CCSD(T) and HF. The results are summarized in Table Ⅱ, where, in addition to the correlation contribution (COR), and the polarization contribution (POL), there are also electrostatic contribution (OEE), exchange (EX) and Pauli repulsion (REP). The sum of the exchange and Pauli repulsion terms is generally recognized as the exchangerepulsion (EXREP) term [41], as it represents a quantum effect due to the properly antisymmetrized manyelectron wavefunction of the complex constructed from the unrelaxed nonorthogonal occupied orbitals of the monomers. The electrostatic term is also welldefined, which refers to the Coulomb interaction between two unrelaxed monomer electron density distributions on orbitals. As the electron clouds in an anion are always diffusive, it is important to emphasize its orbital nature in describing electrostatic interactions involving anions. To emphasize its difference from other widely used models [812], Q_{zz} (ESP), and quadrupole moment (Q_{zz}) where anions are simplified as point charges, we have recently named this electrostatic term as the orbital electrostatic energy (OEE) [42] and found it sheds new light on some typical behaviors of ionπ interactions [43]. Here we emphasize that, in the original ALMOEDA scheme [17, 18], the sum of OEE and EXREP is referred as the frozen density term (FRZ), corresponding to bringing infinitely separated monomers into the complex geometry without any relaxation of the monomer orbitals.
Table Ⅱ. Energy decomposition analysis (in eV) applied to the I^{}·C_{6}F_{6} complex.
As shown in Table Ⅱ for CCSD(T)/aVDZ at R = 3.70 Å, we found that the relative ratio of FRZ:POL:COR is 22:34:44, which is very close in numbers to the ratio of electrostatic:polarization:correlation of 23:28:41 claimed by Anstöter and coworkers [13], with an important qualitative difference that the electrostatic term is replaced by the frozen density term. FIG. 1 depicts the decomposition results along the potential curve. The electrostatic contribution between an anion (e.g. I^{}) and a positive quadrupole moment (e.g., Q_{zz} = +9.50 B for C_{6}F_{6} [13]) is always attractive and has to increase in magnitude as the interacting distance decreases. On the other hand, we have the Pauli repulsion, which represents an increasing wall effect upon the decreasing interacting distance. Hence the electrostatic interaction along with other attractive forces is counterbalanced by the Pauli repulsion. As clearly seen from FIG. 1, the OEE term is dominant. For CCSD(T)/aVDZ at R = 3.70 Å, we calculated the relative ratio of OEE:POL:COR to be 102:34:44. At R_{e} = 3.52 Å, the relative ratio of OEE:POL:COR is 128:39:53, while the FRZ ratio is reduced to 8%. Hence, our calculations disclosed that the previous calculations [13] on the electrostatic contribution are concealed by the contributions from the exchange and Pauli repulsion. The correlation effect is important, whereas the electrostatic contribution is actually determinant, being more than double of the correlation contribution in the I^{}·C_{6}F_{6} complex at the equilibrium binding distance.
We also extended the LMOEDA approach [23] to the xDH functionals, decomposing the B3LYP energy while augmented by the additional exchangecorrelation contributions from XYG3. The results are also summarized in Table Ⅱ. For XYG3/aVDZ at R_{e} = 3.54 Å, the relative ratio of OEE:POL:COR is found to be 144:47:61. Hence, the XYG3 results are in good accordance with the CCSD(T) results.
In summary, we have performed the CCSD(T)/aVDZ calculations on the potential energy curve of the I^{}·C_{6}F_{6} complex. The basis set incompleteness has been assessed using the BoysBernardi counterpoise techniques and the F12 techniques. We found that CCSD(T)/aVDZ can have an uncertainty up to 0.113 eV due to the basis set incompleteness, and the results from 25% BSSE corrections or F12a agree well with the fitting data from the literature. We have benchmarked the performances of the xDH functionals of XYG3 and lrcXYG3. Their results are found to be generally satisfactory within the uncertainty. As the xDH functionals formally scale as O(N^{5}), where \begin{document$ N $\end{document is proportional to the system size, they show promise as an economic substitute of the expensive CCSD(T) with an O(\begin{document$ N^7 $\end{document) scaling. We have applied the LMOEDA method to the CCSD(T) and the XYG3 binding energies, and reaffirmed the importance of the electrostatic forces in the anionπ interactions.
Theoretical Analysis of an Anionπ Complex: I^{}dC_{6}F_{6}
doi: 10.1063/16740068/cjcp2005069
 Received Date: 20200512
 Accepted Date: 20200519
 Publish Date: 20200627

Key words:
 Anionπ interactions /
 I^{}·C_{6}F_{6} /
 Electrostatic interaction /
 Correlation /
 OEE /
 XYG3 /
 CCSD(T)
Citation:  Zhangyun Liu, Zheng Chen, Xin Xu. Theoretical Analysis of an Anionπ Complex: I^{}dC_{6}F_{6}[J]. Chinese Journal of Chemical Physics , 2020, 33(3): 285290. doi: 10.1063/16740068/cjcp2005069 