Topography of the PESs
We start by briefly revisiting the main features of the title reaction, which were previously addressed in detail.[24, 26, 65] As Figure 1 shows, the reaction profile (Cartesian coordinates for all structures are given in the Supporting Information) displays a strong parallelism with the one reported elsewhere,[23, 25] with both reactions having the same number and type of stationary points along the minimum energy path. The minima are identified as and the saddle points (SP) as . This reaction has two entrance channels: they correspond to the oxygen and hydrogen abstraction mechanisms, which are characterized by two saddle points, and , which represent the barriers for oxygen and hydrogen abstraction, respectively. Each of these saddle points is preceded by equally stable prereactive complexes between the complex and ozone: is connected to while is connected to . Both prereactive complexes are also connected to a common saddle point, , which lies slightly higher in energy. However, this saddle point could not be optimized with all functionals, probably due to the flat nature of the potential energy surfaces (PESs). The reactants, the complexes, and comprise both entrance channels. In , an OO bond in ozone is broken and another is formed between the free oxygen and the hydroperoxyl radical, forming . Instead, for the OH bond in the hydroperoxyl radical is broken, leaving the H atom free to form a new OH bond, now with ozone, thus yielding . Note that the water molecule acts as a catalyst, with no additional bonds being formed or broken involving this molecule, justifying the parallelism observed between the title reaction and the one.[23, 25] The reaction then proceeds with the fragment separating from the radical until another minimum is reached, . However, the path from to is not always direct like in the hydrogen abstraction mechanism. In turn, the M06-HF functional predicts two extra structures between and : a minimum with an open-chain structure ( ) and a saddle point that breaks an OO bond and connects to . This is similar to the reaction, where M06-HF shows the same behavior, somewhat mimicking previous CASSCF results. The central OO bond in is then broken to form the final products, . Because our interest lies in the saddle points for oxygen and hydrogen abstraction, we will now focus on both entrance channels of the title reaction.
We proceed by analyzing Figures 2 and 3, where we show schematic diagrams representing the IRC path for the entrance channel of the oxygen (Fig. 2) and hydrogen abstraction (Fig. 3) mechanisms of the reaction as a function of the distance along the reaction path. In each figure, the six panels show by interpolated-point curves the optimized IRC path obtained with a specific density functional. Also, at each of these points, which represent a specific geometry along this path, four different CC methodologies were utilized to calculate and interpolate the respective single point CC energies. The energies, in , are relative to the prereactive minimum ( ) of a given curve, which marks the final step of the IRC optimization. From both Figures 2 and 3, one can see that the six pure KS-DFT curves are similar, with their largest differences being near the saddle point region, which leads to a broad range of relative energies with respect to the prereactive minimum. This can be seen more easily in Supporting Information Figure S1, where we show all KS-DFT curves together. Note, however, that the largest relative energies are due to BHHLYP (oxygen abstraction) and BHHLYP and M06-HF (hydrogen abstraction). We should stress that none of these functionals was constructed with the sole purpose of calculating accurate barrier heights. We should also point out the similarities of the BHHLYP and BLYP-VBV functionals, as they only differ in the amount of exact exchange: 50% and 40.4%, respectively. As expected,[25, 26] this difference produces a large impact on the relative energies, dropping almost 3 when going from BHHLYP to BLYP-VBV, for both saddle points.
We focus now on Figure 2. A similar trend can be seen in all six panels, with the relative energies of the CC methods showing the following order: RHF-RCCSD(T) >UHF-CCSD(T) >RHF-UCCSD(T) >CR-CC(2,3), which is particularly visible in the region close to the saddle point. This clearly reflects the differences of the CC methods in dealing with the problem of breaking and forming the OO bond. Note that the RHF-UCCSD(T) and CR-CC(2,3) curves almost match, which implies that the triples corrections contribute similarly in both methods, as their CCSD energy is equal. This is particularly valid in the M06-HF case. One also observes significant differences when comparing the KS-DFT curves and the CC//KS-DFT ones. The panels based on the BHHLYP, BLYP-VBV, and TPSS-VBV functionals are a good example. Conversely, the CC//M06-HF curves nearly overlap the pure M06-HF path. Note that M06-HF has full (100%) exact exchange and is believed to eliminate the well-known SIE at long-range. Moreover, M06-HF may capture the nature of the medium- and long-range forces which are of key importance in describing the minimum. This minimum, which was found only at the M06-HF and CASSCF level, is linked to and represents an intermediate step in the formation of the OO bond in the fragment. Assuming, as shown elsewhere[23, 24] for this reaction (and the simpler one), that the qualitative description of the CASSCF calculations is correct, and that M06-HF is able to better describe the breaking and forming of the OO bonds, it seems reasonable to say that the resemblance between the CC//M06-HF curves and the pure M06-HF IRC curve is not accidental. In fact, the good performance of M06-HF in heavy atom transfer reactions has been recently reported. The situation changes in Figure 3. The four CC curves are seen to overlap almost perfectly as they have a strong resemblance with the KS-DFT curve, with the most representative case being CC//M06-HF. These two facts may reflect the simpler nature of breaking and forming an OH bond instead of an OO bond, in line with previous findings that show a good accuracy for composite methods while studying hydrogen abstraction reactions. An example of this can be found in Ref. , where model chemistries involving CCSD(T) single point energies are shown to accurately mimic the experimental barriers and enthalpies of 22 hydrogen abstraction reactions.
We now turn our attention to Figure 4, where a given CC curve for both abstraction mechanisms is utilized for the various KS-DFT IRC paths. For a question of visibility, we decided to exclude one of the two CC methods of Molpro, and the choice was RHF-RCCSD(T), as we wanted to keep RHF-UCCSD(T) in order to make a direct comparison on the perturbative triple corrections between RHF-UCCSD(T) and CR-CC(2,3). We start by comparing the RHF-UCCSD(T) and CR-CC(2,3) curves of both IRC paths. While the difference shown in both panels is quite subtle for hydrogen abstraction, the same is not true for the oxygen abstraction, where, with the exception of CR-CC(2,3)//M06-HF, the relative energies of CR-CC(2,3)//KS-DFT are somewhat smaller than RHF-UCCSD(T)//KS-DFT. The previously addressed subject of nonparallelity is also visible in this figure. However, as each panel consists here of a specific CC method, the rationalization is different. Considering each panel individually one may imply that the larger overlap between the hydrogen abstraction curves is due to a very similar mapping of the CC PES by each density functional along its IRC path. The biggest difference in each panel is around 2 , at the saddle point. However, for the oxygen abstraction CC curves, one can see that each panel has a set of very different curves, which may now imply a distinct mapping of the CC PES along the IRC path for the various functionals. This is particularly visible near s = 2, where the largest variations are observed. A simple way to test this is to calculate the variation of the perpendicular looseness between the prereactive minimum and the point with the largest variation. Due to the nature of the curves and as the perpendicular looseness is the sum of the breaking and forming bond distances, we believe that this simple approach will give us an insight for the above observation, as we expect a smaller standard deviation among the KS-DFT structures for the hydrogen abstraction relative to the oxygen abstraction mechanisms. Indeed the results show Å and Å. Moreover, this is consistent with the general trend that saddle point structures for hydrogen abstraction reactions are relatively insensitive to the level of theory.[93, 94]
Despite the analysis made thus far, one important question remains to be answered: what is the shape of the CC PES calculated in the vicinity of a KS-DFT optimized structure? Stating differently, if grid points map a saddle point at the KS-DFT level, what will be the corresponding shape of the CC PES when mapping the same grid? To answer this question we show in Figure 5 three-dimensional diagrams representing a neighboring region of the saddle point for oxygen ( grid) and hydrogen ( grid) abstractions. The location of each saddle point, which was obtained at the BHHLYP/AVDZ level, is given by a black dot. Each set of three panels shows the saddle point region at the level of theory used for the optimization (left panel) and at the single point CC/AVDZ level (RHF-RCCSD(T) and RHF-UCCSD(T): middle and right panels, respectively). The choice for using only these two CC methods was based on convenience. Converging the UHF equations near both saddle points is tricky due to issues of stability of the UHF wavefunction. They are usually attributed to the failure of the single Slater determinant description of the HF method and growing importance of nondynamical correlation, which we believe to be the case here. Thus, it requires point-by-point separate calculations, instead of performing an automatic loop, which is impractical with our grids of 120 points. We have, therefore, discarded using UHF-CCSD(T) for these calculations. Additionally, we have not considered CR-CC(2,3) due to the extra computational time required. We should stress that discarding both CC methods for mapping these grids should not pose a problem, as we have no reason to believe (due to the similarities between the CC curves in the first panel of Figs. 2 and 3) that the resulting topologies would be dramatically different from the other calculations. The left panels of Figure 5 show what one could expect: a saddle point. However, as shown, the situation is distinct for the middle and right panels, with the resulting PESs having completely different topologies. For example, the grid used to map the BHHLYP saddle point indicates a minimum at the CC//BHHLYP level, while a ramp with a slight concavity can be seen in the CC//BHHLYP case. One can go further, stating that, most probably, the shape of the CC//KS-DFT PESs do not resemble at all a saddle point. Thus, it can only come as an enormous coincidence that the true CC saddle points (true in the sense that they result from a CC optimization) share the same geometry as the initial KS-DFT saddle point, irrespective of the density functional. The probability of occurring such a coincidence should in principle diminish with the number of atoms in the molecular system. By looking at Figure 5, one concludes that blindly performing CC//KS-DFT calculations involves an error of unknown magnitude, as there is no practical way of knowing what are the topological features of the CC PES that are being mapped. In fact, the middle and right panels of Figure 5 make us wonder whether the true CC saddle points even occur in a nearby region of the CC PES. Also, and perhaps more important, one wonders if only by accident can the energy differences between the true CC stationary point and the CC PES single point energy calculation based on a KS-DFT geometry be approximately the same and hence bear no implication in determining reaction barrier heights. Indeed, the above findings are replicated when using the CC//M06-HF model chemistry.
Figure 5. Three-dimensional diagrams representing a neighboring region of the saddle point for oxygen and hydrogen abstraction. The location of each saddle point (obtained at the BHHLYP/AVDZ level) is given by a black dot. Each set of three panels shows the saddle point region at the level of theory used for the optimization (left panel) and at the single point CC/AVDZ level (RHF-RCCSD(T) and RHF-UCCSD(T), middle and right panel, respectively, see text). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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Using the same grid as in Figure 5, Supporting Information Figure S2 shows the difference, in , between the RHF-RCCSD(T) and RHF-UCCSD(T) ( ) PESs calculated at the BHHLYP geometries. It can be seen that increases with the value of the reaction coordinate, R1. This should be related to the spin contamination of the RHF-UCCSD wavefunction, as the difference between the values of and (i.e., ) increases with R1, in both cases, affecting the calculation of the energies and the parallelity between both CC PESs. For example, considering the left panel of Supporting Information Figure S2, this difference along R1 (for the lowest value of R2) goes from 0.0027 at Å to 0.1275 at Å, meaning that we start with negligible spin contamination at Å but end up with 17% above the ideal value of 0.75 for Å. A similar trend is observed in the right panel, where we report a difference along R1 (for the lowest value of R2) that goes from 0.0192 at Å to 0.0448 at Å. This corresponds to an initial and final value of 2.56% and 5.97% above 0.75, respectively.
In Table 1, we gather the relative energies of the three stationary points along the reactants channel of the reaction calculated in this work at the CC/AVTZ//KS-DFT/AV DZ level and from previous work.[24, 26] In the last four lines of this table, we present the values of the average and standard deviation of KS-DFT/AVTZ//KS-DFT/AVDZ, MRPT2(11,11)/AV TZ//CASSCF(11,11)/VTZ, and finally of the newly calculated CC/ AVTZ//KS-DFT/AVDZ levels of theory. Note that the numbers presented in the last three lines of this table now include the M06-HF functional, which was initially excluded (along with BHHLYP) from recent numerical analysis due to the fact that M06-HF was not specifically recommended to yield accurate barrier heights. However, as its use has led to promising results in this work (as presented in the previous subsection), we decided to include it in our analysis. The last two columns show the energetic difference between both saddle points, with and without zero-poni-energy (ZPE) corrections calculated with the AVDZ basis set. We start by examining the results for the prereactive minimum, . The average values for the two sets of CC methods differ only by 0.09 and, coincidentally, have the same low standard deviation value of 0.16 . This agreement should not come as a surprise, as the value of obtained in the RHF-UCCSD calculations for the reactants and the prereactive minimum is very low, around 0.15% higher than the ideal value of 0.75. However, one should note that the stability of at the CC//KS-DFT is about 2 lower than at the remaining levels of theory. Another interesting source of information is the influence of the different geometries in the relative energies caused by altering the percentage of exact exchange in the functionals. This is visible by analyzing the differences between the CC results using BHHLYP and BLYP-VBV, which have 50% and 40.4% of exact exchange, respectively. Such a difference is only of 0.03 for , thus smaller than the one 0.21 reported at the pure KS-DFT level.
We now turn to the barrier heights, beginning with the oxygen abstraction saddle point, . Both average values and error bars encompass the KS-DFT average value of −5.08 . Also visible is the large difference between the CC//CASSCF and CC//KS-DFT barrier heights, which can be explained by the different geometries mapped at both levels of theory in the saddle point and possibly at the reactants. Recall that geometries resulting from the CASSCF calculations suffer from being performed with a reduced (smaller than full-valence CAS) active space and lack of dynamical correlation, thus justifying these discrepancies in geometric parameters, which can be consulted in Table 1 of Ref. . This effect caused by different geometries resulting from the KS-DFT calculations can also be seen in the large standard deviations of the CC saddle point results, particularly of , which describes the breaking and forming of the more complicated OO bond. There is now also a larger difference between the corresponding levels of theory than the one observed for the prereactive minimum. For example, looking at the TPSS-VBV results, one finds that both relative energies of and rise when going from RHF-UCCSD(T) to RHF-RCCSD(T), with this increase being of 0.08 for and 0.61 for . A similar trend is observed for the other cases, with this behavior being related to the nature of the regions of the PES that are being mapped, as saddle points are usually more prone to spin contamination problems, which in this case is also an indication of the importance of nondynamical correlation to describe both saddle points of the title reaction. In fact, the percentage of in RHF-UCCSD that exceeds the ideal value is never above 0.2% for , while for it varies between 1.9% (TPSS-VBV) and 12.1% (M06-HF).
For , similar trends are observed. Note the similarity between the KS-DFT and CC results and also the smaller differences between both CC methods when looking at correspondent levels of theory than when comparing with . This decreasing differences should be related to the percentage of in RHF-UCCSD that exceeds 0.75, which is, on average, 2.4% lower than the oxygen abstraction saddle point average of 5.2%. Note also that these deviations from 0.75 are much lower than the ones observed when using the CASSCF geometries, where we obtain deviations of 39.6% and 124% for and , respectively. These values also explain the different barrier heights calculated with the CC//CASSCF and CC//KS-DFT methods, which were discussed in the previous paragraph. Again, it is interesting to look at the relative energies of the CC calculations when considering the BHHLYP and BLYP-VBV geometries. The largest differences (more than 1 ) are seen for the hydrogen abstraction saddle point. Assuming that the geometries of the reactants at the KS-DFT level map a flat (or nearly flat) region of the CC PES, the reason for these differences will have to be attributed to the saddle points. First, one should recall that the KS-DFT hydrogen abstraction saddle point is characterized by an imaginary frequency more than twice larger than the one calculated for the oxygen abstraction, which implies a narrower and therefore steeper barrier. Second, if one assumes that the CC PES maintains these characteristics and also that the two KS-DFT geometries are relatively close to the CC saddle points, it seems likely that the larger differences between the CC//BHHLYP and CC//BLYP-VBV saddle points are caused by the steeper topography of the CC PES. Another way to make an assessment on the quality and difficulties inherent to CC calculations is to observe the T1 and D1 diagnostics, as well as their ratio, .[96-101] The T1 diagnostic is often used as a measure of the reliability of a CC wavefunction. In some cases, the larger the T1 value is, the more likely the wavefunction is multireference and hence the less reliable is the single-reference CC method expected to be. It is usually accepted that the upper limit is 0.02 for a closed-shell system. However, for open-shell systems it has been suggested that the upper limit for the T1 diagnostic should be 0.045, rather than 0.02. The values of the T1 diagnostic of both saddle points for RHF-RCCSD and RHF-UCCSD are very similar, so we have grouped them together. For , we have while for we obtain . On the other hand, the UHF-CCSD T1 values are considerably lower, with both saddle points having the same average of 0.017 and a standard deviation of 0.004 and 0.001 for and , respectively. This may suggest that the UHF-CCSD wavefunction is slightly more reliable than the one obtained after RHF-(R/U)CCSD calculations. Also, according to Lee, a value of which is much smaller than “indicates that there is a large variation in orbital rotation parameters in the coupled-cluster wave function, or in other words, there are problem areas in the molecule and other areas where the coupled-cluster approach is performing better.” Therefore, the ratio can be used to evaluate the homogeneity of the electronic structure of the molecule. In our case, the ratio yields and for and , respectively. These values are considerably lower than , a number that indicates a perfectly homogeneous system. Large values of t1 and t2 amplitudes can also indicate strong multireference character of the CCSD wavefunction. This is also observed in our calculations. For example, for RHF-UCCSD//M06-HF we obtain and for , and and for . A similar behavior is observed for all other CCSD calculations.
The results presented in Table 1 seem to indicate that there is, on average, a good agreement between the KS-DFT and CC//KS-DFT results in all columns of the table. However, the good results of the CC//M06-HF model chemistries reported in the previous subsection might be an indication that CC//M06-HF is the closest to the pure CC results. To investigate such a possibility, we have looked at the largest cluster amplitudes, particularly t2, as we know from our previous work that the most dominant excited configuration in the CASSCF wavefunction is the one corresponding to the double excitation. In fact, only the CC saddle point calculations using M06-HF geometries show this particular feature of having their largest t2 amplitudes associated with the above mentioned excitation. The previous discussion may give an indication of how different the M06-HF geometries are when compared to the ones based on the remaining functionals, and how their use in the mapping of the CC PES mimics certain expected characteristics that we know from previous calculations. Table 1 also shows that the CC//M06-HF barrier heights, which involve the breaking and forming of the more complicated OO bond, are visibly distinct from the other CC//KS-DFT calculations. Comparing the CC//M06-HF barrier heights with the pure KS-DFT ones obtained with the best functionals, BLYP-VBV and TPSS-VBV (best in the sense that they were optimized to match the CASSCF imaginary frequency of the oxygen abstraction saddle point of the identical reaction), one finds that they are in fact closer in energy than all other CC//KS-DFT levels of theory. This might be another indication that M06-HF geometries are more suitable to map the CC PES.