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Keywords:

  • coupled-cluster;
  • density functional theory;
  • model chemistry;
  • barrier heights;
  • atmospheric chemistry

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Supporting Information

In this work, we report a theoretical investigation concerning the use of the popular coupled-cluster//Kohn-Sham density functional theory (CC//KS-DFT) model chemistry, here applied to study the entrance channel of the inline image reaction, namely by comparing CC//KS-DFT calculations with KS-DFT, MRPT2//CASSCF, and CC//CASSCF results from our previous investigations. This was done by performing single point energy calculations employing several coupled cluster methods and using KS-DFT geometries optimized with six different functionals, while conducting a detailed analysis of the barrier heights and topological features of the curves and surfaces here obtained. The quality of this model chemistry is critically discussed in the context of the title reaction and also in a wider context. © 2013 Wiley Periodicals, Inc.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Supporting Information

Coupled-cluster (CC)[1-5] theory has achieved an enormous success in quantum chemistry calculations of relatively small molecules, particularly through the popular CCSD(T) method[6] (CC approach with singles, doubles, and noniterative triples), with the noniterative correction due to triply excited clusters being added to the CCSD[7-9] energy. The CCSD(T) method is extremely appealing for several reasons. One stems from its single-reference nature which makes it a “black-box” and easy to use method for quantum chemists. Besides, CCSD(T) is known for being one of the most accurate methods of describing electron correlation effects for nondegenerate ground states of molecules, while remaining rigorously size extensive.[10] Computationally, the CPU time scaling of the noniterative triples correction scales approximately as N7, where N is a measure of the size of the molecule, typically related to the number of basis functions. In this case the scaling is calculated as inline image, where no and nu are the numbers of occupied and unoccupied orbitals, respectively, used in the CC calculation. Due to the above mentioned properties of CCSD(T), it is often referred to as the “gold standard” of quantum chemistry, although it suffers from some drawbacks. As a single-reference method typically based on the Hartree–Fock (HF) determinant, CCSD(T) may fail in situations where nondynamical correlation effects play an important role, making the HF determinant a poor reference for CC calculations. Typical cases are molecular geometries far from equilibrium (such as saddle points, often associated with bond-breaking/bond-forming processes), transition metal compounds, and biradicals.

With the purpose of seeking an improvement of the noniterative approach to approximate triple excitations without resorting to multireference methods of any kind (i.e., maintaining its “black-box” nature) and without losing the accuracy of CCSD(T) for nondegenerate ground states, several alternative CC perturbation methods have been developed in recent years, namely inline image,[11] a-CCSD(T),[12] inline image,[13, 14] and CR-CC(2,3).[15-18] However, only the latter (completely renormalized CC method) maintains the accuracy near the equilibrium geometries[17, 19-22] while improving the description of the already mentioned problematic cases of CC. CR-CC(2,3), which aims at obtaining some nondynamical correlation, has already been utilized in the context of our work.[23, 24] A brief survey of all these different theories can be found in Ref. [21] and references therein.

Many theoretical studies of reactions involve the calculation of the relevant stationary points at some level of theory, which precede the thermochemical or kinetics calculations. For the latter, accurate saddle point geometries and energies are crucial in order to determine the associated rate constants with chemical accuracy, while some methods additionally require minimum energy paths (MEPs). Apart from the inherent theoretical difficulties (above mentioned and already discussed in the context of our own work in Refs. [23-26]) in obtaining accurate energies in geometries close to saddle points, technical issues also arise if one wants to employ CC methods in the optimization procedure: the Hessian matrix, essential to determine the nature of the respective stationary point, is in some popular software packages calculated with a fully numerical procedure. This is computationally inconvenient as the number of energy evaluations needed to obtain the Hessian grows very rapidly with system size, making the use of CC for this purpose attractive (i.e., having a favorable accuracy/cost ratio) only for small molecular systems.

The problem of accuracy versus computer cost has always existed throughout theoretical investigations, leading to methodologies that incorporate the best of both worlds: optimization employing a method with favorable scaling and/or analytical (or semianalytical) Hessians followed by single point energy calculation using a methodology that is considered the most accurate to describe the electronic structure details. Such methodologies are based on the empirical observation that the geometry is often less sensitive to the theoretical level than relative energies. A few simple and common model chemistries can be listed as examples: MP2//HF, CCSD(T)//HF, CCSD(T)//KS-DFT, MRPT2//CASSCF, and MRCI//CASSCF with the acronyms having the usual meaning: MP2 for Møller–Plesset second-order perturbation theory, KS-DFT for Kohn–Sham density functional theory, MRPT2 for multireference second-order perturbation theory, MRCI for multireference configuration interaction, and CASSCF for complete active space self-consistent field. These model chemistries were here written in a simplified version of their typical representation: inline image/ inline image// inline image/ inline image, where inline image/ inline image is the level of theory used to perform the geometry optimization while inline image/ inline image represents the level of theory employed to calculate the respective single point energy. A discussion concerning the basis sets will be incorporated in the next section.

More elaborate approaches have been developed, namely the so called composite ab initio procedures, where energies obtained from calculations with different levels of theory are combined with the intent of producing energy differences with high accuracy. Examples of such methods are the Gaussian-2[27] (G2), multicoefficient G2[28] (MCG2), Gaussian-3[29] (G3), multicoefficient G3[30] (MCG3), G3-RAD,[31] Gaussian-4[32, 33] (G4), other multicoefficient correlation methods[34-37] (MCCMs) and Weizmann-n[38] (Wn).

We will turn here our attention to the ubiquitous CC//KS-DFT model chemistry (with a small fraction of examples found in the literature[39-52]), widely used in the mechanistic study of medium-sized molecular systems due not only to the already mentioned popularity of CCSD(T) but also to the success of KS-DFT[53, 54] and, in particular, of the B3LYP functional,[55] even with its known caveats.[56] In fact, specific reasons make this model chemistry very appealing: a large number of functionals can make accurate predictions of minima and saddle point geometries and KS-DFT possesses good computational characteristics to use in the optimization procedures, as it formally scales as N4 and analytical gradients and hessians are usually available. Although facing also the single-reference handicap, density functional theory (DFT) has recently emerged as one of the most promising avenues for electronic structure calculations in large molecular systems. Despite well-known pending issues relating to the determination of the DFT functional (see, e.g. Ref. [[57] and references therein), a prolific amount of work has indeed appeared with this methodology which may justify our analysis in the present work.

Our aim in this study is manifold. First, because we have recently questioned the foundation of model chemistries by analyzing in detail what is perhaps the most physically sound of such approaches: MRCI//CASSCF or CASDC,[58] thus the result of adding the dynamical correlation to the CASSCF energy (in straight analogy therefore with the popular HFD scheme where the dispersion energy is added to the Hartree–Fock energy,[59, 60] and also paralleled in exchange perturbation theory when adding the first-order exchange to the second-order dispersion energy[61]). Because this is clearly applicable only to fairly small molecules, we have turned here to the more popular theories cited above. Besides, with different CC methods at our disposal we will have four other goals: (1) compare the parallelity between the pure KS-DFT intrinsic reaction coordinate (IRC) curves obtained in Ref. [26] and the same curves calculated with the CC//KS-DFT model chemistry; (2) study the influence of the different CC methods on the shape of the curves; (3) study the effect of the different functionals on the final CC//KS-DFT IRC curves; (4) compare, to the possible extent, the relative energies of the prereactive minimum and saddle points obtained with the CC//KS-DFT approach with the ones obtained with pure KS-DFT calculations[26] and at the MRPT2//CASSCF and CC//CASSCF levels of theory.[24] By analyzing these different items, we will be able to make an evaluation of the behavior of the CC//KS-DFT model chemistry for this particular molecular system. We emphasize that two important benchmark methods will not enter our discussion: the accurate MRCI//CASSCF level of theory (see elsewhere, Refs. [62,63] and references therein) for being computationally unaffordable, and the pure CC results based on CCSD(T) optimizations, for being also prohibitively expensive. This is a constraint present in many investigations, which makes the title system a good candidate for the study of the CC//KS-DFT model chemistry. Note that the role of a water molecule on the atmospherically important inline image ozone depleting reaction [23, 25, 64] has already been examined by us in previous occasions, [24, 26, 65] where we present details of its chemistry, chemical structures of the intermediates, and also a reaction scheme. Hopefully, the details of the present study will allow us to gather useful information in order to assess and clarify the advantages and disadvantages of using such an approach in this reaction and also in a wider context. A recent study [66] on barrier heights and structures in the context of proton transfer reactions is a good example of a broader investigation concerning some of the items we have described in this paragraph. Detailed information on the computational methods is given in the next section, and the results and discussion subsequently presented. The conclusions are gathered in the last section.

Computational Methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Supporting Information

Four CC methodologies were utilized throughout the course of this work. One of them was the CCSD(T) method of Molpro,[67] where its two open-shell CCSD variants[68, 69] were used: spin unrestricted CCSD (RHF-UCCSD) and partially spin restricted CCSD (RHF-RCCSD) with both methods being based on a high-spin restricted HF (ROHF) reference wavefunction. Note that in RHF-UCCSD, the CC wavefunction is allowed to be spin-contaminated while in RHF-RCCSD the amplitudes are subject to restrictions, such that spin contamination becomes negligible.[68] Perturbative triples corrections were computed in both cases according to Ref. [70]. Other utilized approach was the UHF-CCSD(T) of ORCA,[71] where the CC energies are based on an unrestricted HF (UHF) reference, which can now be spin-contaminated. We have also performed calculations using the ROHF-based CR-CC(2,3) method available in GAMESS,[72] which should in principle improve CCSD(T) when nondynamical correlation effects become important. We recall that the open-shell CCSD part of the CR-CC(2,3) algorithm yields equivalent absolute energies to the RHF-UCCSD method.[24] This will be useful when discussing the results in the next section.

Besides the CC calculations, we also present KS-DFT results obtained (but not reported) in previous work,[26] namely the IRC curves calculated with six different fourth rung[73, 74] density functionals employing the aug-cc-pVDZ (AVDZ) basis set: BH&HLYP,[75-77] BLYP-VBV,[25] and B97-K[78] (hybrid GGAs); TPSS-VBV,[25] BMK,[78] and M06-HF [79] (hybrid meta-GGAs). A justification for using such functionals is given in Ref. [26], where the importance of high percentage of exact exchange in reducing the self-interaction error[80, 81] (SIE) in the context of barrier heights calculations[82, 83] is discussed. We note that all our KS-DFT geometries were obtained at the unrestricted level. Optimizations at the restricted KS-DFT level were also tested, but we have encountered numerical convergence problems or it was impossible to find a saddle point at the restricted level. This may be related to the fact that unrestricted KS-DFT is known to describe strong static correlation effects and, consequently, systems with multireference character,[84] which we know to be the case here[24] (see next section). Also, it has been argued that use of the unrestricted KS-DFT formalism should be the rule in open-shell systems.[85] As a consequence, our KS-DFT calculations are spin contaminated, which is taken to be less problematic than spin contamination in UHF calculations.[85-87] Correction of spin contamination via spin projection and annihilation techniques was not considered in this work as they are not available in any of the used electronic structure packages. Besides, there is a great deal of criticism concerning these methods.[86, 88-90] To avoid possible ambiguities due to basis set differences, we report the IRC single point CC energy curves (based on the KS-DFT geometries) also with the AVDZ basis set. Thus, all our two-dimensional plots are reported at the CC/AVDZ//KS-DFT/AVDZ level. Due to the heavier computational burden of CC calculations compared with KS-DFT and because our full IRC paths calculated at the KS-DFT/AVDZ level include a large number of steps, our two-dimensional curves are shown with only seven and eight points along the oxygen and hydrogen abstraction IRC paths, respectively. We should recall that these curves, which were interpolated with splines, start at the saddle point and end at the prereactive minimum.[26, 65] Final single point energy CC calculations were obtained at the CC/AVTZ//KS-DFT/AVDZ level, in order to allow a comparison with our previous results.[24, 26] However, due to computational and hardware limitations, we have only gathered these final CC results with the RHF-UCCSD(T) and RHF-RCCSD(T) methods.

Results and Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Supporting Information

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 inline image and the saddle points (SP) as inline image. This reaction has two entrance channels: they correspond to the oxygen and hydrogen abstraction mechanisms, which are characterized by two saddle points, inline image and inline image, which represent the barriers for oxygen and hydrogen abstraction, respectively. Each of these saddle points is preceded by equally stable prereactive complexes between the inline image complex and ozone: inline image is connected to inline image while inline image is connected to inline image. Both prereactive complexes are also connected to a common saddle point, inline image, 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 inline image complexes, inline image and inline image comprise both entrance channels. In inline image, an OO bond in ozone is broken and another is formed between the free oxygen and the hydroperoxyl radical, forming inline image. Instead, for inline image 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 inline image. 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 inline image one.[23, 25] The reaction then proceeds with the inline image fragment separating from the inline image radical until another minimum is reached, inline image. However, the path from inline image to inline image is not always direct like in the hydrogen abstraction mechanism. In turn, the M06-HF functional predicts two extra structures between inline image and inline image : a minimum with an open-chain structure ( inline image) and a saddle point that breaks an OO bond and connects inline image to inline image. This is similar to the inline image reaction, where M06-HF shows the same behavior,[25] somewhat mimicking previous CASSCF results.[23] The central OO bond in inline image is then broken to form the final products, inline image. 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.

image

Figure 1. Schematic diagram showing the energetic and geometric attributes of the stationary points for the inline image reaction. The energies, in inline image, are relative to reactants, and the optimized structures predicted at the BLYP-VBV/AVDZ and M06-HF/AVDZ level of theory. Following previous work, the black dashed-line connecting inline image to inline image is not shown.

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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 inline image 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 inline image, are relative to the prereactive minimum ( inline image) 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 inline image when going from BHHLYP to BLYP-VBV, for both saddle points.

image

Figure 2. Schematic diagrams representing the IRC path for the entrance channel of the oxygen abstraction mechanism of the inline image reaction as a function of the distance along the reaction path. Each of the six panels shows seven interpolated points along the optimized IRC path obtained with a specific density functional. Also, at each of these seven points, four different CC methodologies were utilized to calculate and interpolate the respective single point CC energies. All curves were obtained with the AVDZ basis set. The energies, in inline image, are relative to the prereactive minimum ( inline image) of a given curve.

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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 O[BOND]O 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.[79] Moreover, M06-HF may capture the nature of the medium- and long-range forces which are of key importance in describing the inline image minimum.[26] This minimum, which was found only at the M06-HF and CASSCF level, is linked to inline image and represents an intermediate step in the formation of the O[BOND]O bond in the inline image fragment. Assuming, as shown elsewhere[23, 24] for this reaction (and the simpler inline image 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 O[BOND]O 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.[91] 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 O[BOND]H bond instead of an O[BOND]O 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. [92], where model chemistries involving CCSD(T) single point energies are shown to accurately mimic the experimental barriers and enthalpies of 22 hydrogen abstraction reactions.

image

Figure 3. Schematic diagrams representing the IRC path for the entrance channel of the hydrogen abstraction mechanism of the inline image reaction as a function of the distance along the reaction path. Each of the six panels shows eight interpolated points along the optimized IRC path obtained with a specific density functional. Also, at each of these eight points, four different CC methodologies were utilized to calculate and interpolate the respective single point CC energies. All curves were obtained with the AVDZ basis set. The energies, in inline image, are relative to the prereactive minimum ( inline image) of a given curve. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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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 inline image, 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[92] 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 inline image Å and inline image 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]

image

Figure 4. Schematic diagrams representing the IRC path for the entrance channel of the oxygen and hydrogen abstraction mechanisms of the inline image reaction as a function of the distance along the reaction path. Each set of three panels gathers the previously shown (Figures 1 and 2) six single point CC//KS-DFT energy curves for three different CC methods (RHF-RCCSD(T) was left out, see text). All curves were obtained with the AVDZ basis set. The energies, in inline image, are relative to the prereactive minimum ( inline image) of a given curve. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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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 ( inline image grid) and hydrogen ( inline image 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.[24] 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 inline image BHHLYP saddle point indicates a minimum at the CC//BHHLYP level, while a ramp with a slight concavity can be seen in the inline image 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.

image

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 inline image, between the RHF-RCCSD(T) and RHF-UCCSD(T) ( inline image) PESs calculated at the BHHLYP geometries. It can be seen that inline image 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 inline image and inline image (i.e., inline image) 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 inline image Å to 0.1275 at inline image Å, meaning that we start with negligible spin contamination at inline image Å but end up with 17% above the ideal value of 0.75 for inline image 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 inline image Å to 0.0448 at inline image Å. This corresponds to an initial and final value of 2.56% and 5.97% above 0.75, respectively.

We note that the correct calculation of inline image for open-shell species[95] is, to our knowledge, not implemented in Molpro, which calculates inline image through eq. (42) of Ref. [68]. We cannot evaluate the differences between both methods, so we must admit the possibility that the latter approach yields erroneous values of inline image.

Barrier heights

In Table 1, we gather the relative energies of the three stationary points along the reactants channel of the inline image 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,[26] MRPT2(11,11)/AV TZ//CASSCF(11,11)/VTZ[24], 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[24] 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, inline image. The average values for the two sets of CC methods differ only by 0.09 inline image and, coincidentally, have the same low standard deviation value of 0.16 inline image. This agreement should not come as a surprise, as the value of inline image 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 inline image at the CC//KS-DFT is about 2 inline image 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 inline image for inline image, thus smaller than the one 0.21 inline image reported at the pure KS-DFT level.[26]

Table 1. Energies relative to reactants (in inline image) of the three stationary points along the reactants channel of the inline image reaction as calculated at the generic CC/AVTZ//CASSCF(11,11)/VTZ [24] and CC/AVTZ//KS-DFT/AVDZ (this work).
Method inline image inline image inline image inline image inline image (ZPE)
  1. In the last four lines, we present the average and standard deviation of the KS-DFT/AVTZ//KS-DFT/AVDZ, [26] MRPT2(11,11)/AVTZ//CASSCF(11,11)/VTZ [24], and CC/AVTZ//KS-DFT/AVDZ (this work) levels of theory. The last two columns show the difference between both saddle points, with and without ZPE corrections calculated with the AV DZ basis set. Active space and basis set information will be dropped for convenience.

RHF-UCCSD(T)//CASSCF−12.700.98−1.71−2.29−7.46
RHF-UCCSD(T)//BHHLYP−14.66−6.921.408.323.55
RHF-UCCSD(T)//BLYP-VBV−14.69−6.700.327.022.63
RHF-UCCSD(T)//B97-K−14.60−5.730.696.422.01
RHF-UCCSD(T)//TPSS-VBV−14.76−6.340.476.812.29
RHF-UCCSD(T)//BMK−15.01−5.25−0.644.610.94
RHF-UCCSD(T)//M06-HF−14.89−3.371.114.470.76
RHF-RCCSD(T)//CASSCF−12.613.790.37−3.42−8.19
RHF-RCCSD(T)//BHHLYP−14.58−6.411.587.993.22
RHF-RCCSD(T)//BLYP-VBV−14.61−6.090.436.532.14
RHF-RCCSD(T)//B97-K−14.52−4.860.865.721.31
RHF-RCCSD(T)//TPSS-VBV−14.68−5.730.606.331.81
RHF-RCCSD(T)//BMK−14.91−4.16−0.513.65−0.02
RHF-RCCSD(T)//M06-HF−14.79−1.751.353.10−0.62
MRPT2//CASSCF−12.04 ± 0.20−0.77 ± 0.683.72 ± 2.754.49 ± 2.12−0.12 ± 2.12
KS-DFT−12.68 ± 1.58−5.08 ± 2.470.22 ± 1.875.30 ± 2.821.14 ± 3.04
RHF-UCCSD(T)−14.79 ± 0.16−5.48 ± 1.310.39 ± 0.655.87 ± 1.231.72 ± 0.83
RHF-RCCSD(T)−14.70 ± 0.16−4.52 ± 1.720.55 ± 0.695.07 ± 1.580.92 ± 1.19

We now turn to the barrier heights, beginning with the oxygen abstraction saddle point, inline image. Both average values and error bars encompass the KS-DFT average value of −5.08 inline image. 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. [24]. 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 inline image, which describes the breaking and forming of the more complicated O[BOND]O 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 inline image and inline image rise when going from RHF-UCCSD(T) to RHF-RCCSD(T), with this increase being of 0.08 inline image for inline image and 0.61 inline image for inline image. 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.[24] In fact, the percentage of inline image in RHF-UCCSD that exceeds the ideal value is never above 0.2% for inline image, while for inline image it varies between 1.9% (TPSS-VBV) and 12.1% (M06-HF).

For inline image, 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 inline image. This decreasing differences should be related to the percentage of inline image 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 inline image and inline image, respectively.[24] 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 inline image) 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,[26] 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 inline image 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, inline image.[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,[102] 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 inline image, we have inline image while for inline image we obtain inline image. 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 inline image and inline image, 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,[101] a value of inline image which is much smaller than inline image “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 inline image can be used to evaluate the homogeneity of the electronic structure of the molecule. In our case, the inline image ratio yields inline image and inline image for inline image and inline image, respectively. These values are considerably lower than inline image, 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 inline image and inline image for inline image, and inline image and inline image for inline image. 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[24] that the most dominant excited configuration in the CASSCF wavefunction is the one corresponding to the inline image 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 inline image barrier heights, which involve the breaking and forming of the more complicated O[BOND]O bond, are visibly distinct from the other CC//KS-DFT calculations. Comparing the CC//M06-HF barrier heights with the pure KS-DFT ones[26] 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 inline image 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.

It should be noted that CR-CC(2,3) was shown to provide the best choice to describe bond-breaking in a study that analyzed eight radical reactions.[103] Our calculated CC//AV DZ results show that the CR-CC(2,3) barrier heights are higher than other CC//AVDZ barriers. For example, the average CR-CC(2,3) inline image barrier height is 2.30 inline image higher than the correspondent RHF-UCCSD(T) value. Similarly, this difference is of 3.01 inline image for the inline image barrier height. This means that inline image increases 0.71 inline image when comparing these two CC methods, implying not only that the rate constants are lower at the CR-CC(2,3) level, but also that the ratio between the hydrogen abstraction and the oxygen abstraction rate constants will be higher for the CR-CC(2,3) calculations.

Conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Supporting Information

The present work examined the use of the CC//KS-DFT model chemistries in the study of the inline image reaction, namely at its entrance channel, where a prereactive minimum and two saddle points exist. By carefully analyzing the results obtained here, we proposed to clarify four main points related to this reaction and tried to extrapolate our conclusions to other molecular systems. Our findings can be summarized as follows: (1) The CC//M06-HF single point IRC curves were found to bear the strongest resemblance to the pure KS-DFT IRC curves. This is particularly so for the oxygen abstraction mechanism. (2) Considering each functional separately, the different CC methodologies are not found to alter the shape of the single point IRC curves significantly. While the CC hydrogen abstraction curves almost overlap, the same general trend is observed for the oxygen abstraction curves, except at the region near the saddle point, where the distinct CC curves become easily distinguishable. (3) The effect of the different functionals on the final CC//KS-DFT IRC curves is very subtle for the hydrogen abstraction process, but quite visible in the oxygen abstraction one, with three distinct sets of results. One set contains the CC//M06-HF curve, another the CC curves using the B97-K and BMK functionals, and finally the last set involves the CC curves employing the BHHLYP, BLYP-VBV, and TPSS-VBV functionals. (4) The single-reference CC and KS-DFT barrier heights agree, on average, with each other, along with inline image and inline image (ZPE). Again, the CC//M06-HF results show a better agreement with the pure KS-DFT barrier heights, namely with the two best functionals, BLYP-VBV and TPSS-VBV.

If the objective is to base the calculations on a specific model chemistry, it appears that CC//M06-HF is among the best for the calculation of the CC barrier heights of the title reaction using KS-DFT optimized geometries. Recall that each saddle point represents a different type of atom transfer: a light atom (hydrogen), in which all functionals deal with the problem in a relatively consistent way and a heavy atom (oxygen) where M06-HF is believed to provide better geometries than the remaining functionals. Of course, as mentioned above, the use of CC//KS-DFT will always have an inherent uncertainty, which can only be clarified by performing extremely expensive (and often unaffordable) CC optimizations (leaving aside the multireference issue). The diversity of results concerning the oxygen abstraction mechanism reinforces our belief that the widespread use of this model chemistry, often justified with the argument that better relative energies are obtained, may in fact lead in some cases to a decrease in quality of the barrier heights (and consequently to poorer rate constants) when compared with the ones calculated with appropriate density functionals alone. This appears to be so particularly for saddle points involving heavy atom transfers and for many reaction studies where CC calculations use geometries calculated with B3LYP, a functional that is widely known as not being suited for saddle point calculations.

In fact, the results concerning the reaction here analyzed show that even if one is studying a molecular system for which a particular fine tuning of density functionals has been performed[23] (a methodology similar in spirit to the specific-reaction-parameter[104]), it seems advisable not to consider the results of just one functional, but work on an average sense for barrier heights (and eventually imaginary frequencies) using various functionals that are cost-effective for thermochemical kinetics. While such a procedure has been shown here to yield large standard deviations in comparison with the values obtained with CC//KS-DFT, the former recipe has the important merit of dealing with true saddle points, an assurance absent in CC//KS-DFT calculations for large molecular systems.

Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Supporting Information

Additional Supporting Information may be found in the online version of this article.

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