Hierarchy of Relative Bond Dissociation Enthalpies and Their Use to

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Hierarchy of Relative Bond Dissociation Enthalpies and Their Use to Efficiently Compute Accurate Absolute Bond Dissociation Enthalpies for C−H, C−C, and C−F Bonds Bun Chan* and Leo Radom* School of Chemistry and ARC Center of Excellence for Free Radical Chemistry and Biotechnology, University of Sydney, NSW 2006 Australia S Supporting Information *

ABSTRACT: We have used the high-level W1X-2 and G4(MP2)-6X procedures to examine the performance of a variety of computationally less demanding quantum chemistry methods for the calculation of absolute bond dissociation enthalpies (BDEs) and a hierarchy of relative bond dissociation enthalpies. These include relative bond dissociation enthalpies (RBDEs), deviations from additivity of RBDEs (DARBDEs), and deviations from pairwise additivity of RBDEs (DPARBDEs). The absolute magnitudes of these quantities decrease in the order BDE > RBDE > DARBDE > DPARBDE, and overall, theoretical procedures are better able to describe these quantities in the same order. In general, the performance of the various types of procedures improves in the order pure DFT → hybrid DFT → double-hybrid DFT → composite procedures, as expected. Overall, we find M06-L to be the bestperforming pure DFT procedure and M06-2X to be the best among the hybrid DFT methods. A promising observation is that even many pure and hybrid DFT procedures give DARBDE and DPARBDE values that are reasonably accurate. This can be exploited by using reference BDEs calculated at a higher-level of theory, in combination with DARBDE or DPARBDE values obtained at a lower level, to produce BDEs and RBDEs with an accuracy that is close to the directly calculated higher-level values. Strongly π-electron-withdrawing or π-electron-donating groups, however, sometimes represent challenges to these approximation methods when the substrate contains several of these substituents.

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INTRODUCTION Bond strength, as measured by the bond dissociation enthalpy (BDE), is an important quantity in chemistry. It determines the enthalpy associated with the breaking of existing bonds and the formation of new bonds in a reaction, which is often critical to chemical reactivity.1 While many BDEs have been determined experimentally,2 computational quantum chemistry has in recent years become an increasingly viable alternative for the accurate determination of bond strengths and thermochemical properties more generally.3−5 The application of the most accurate computational chemistry procedures remains prohibitively expensive in computational terms for all but the smallest molecules. However, the development of methods that are adequately accurate but less demanding on computational resources has broadened the range of theoretically accessible but reasonably reliable BDEs. In this regard, methods such as the Gn composite protocols,6 double-hybrid density functional theory (DHDFT) procedures,7 and regular DFT methods8 are attractive. Indeed, using a comprehensive benchmark set of accurate theoretical BDEs (the BDE261 set),4e we found, for example, that the composite protocol G4(MP2)-6X,9 the DHDFT method DuT-D3,10 and the DFT procedure M06-2X11 give mean absolute deviations (MADs) from the benchmark values of ∼5−7 kJ mol−1. However, despite the overall good performance of all three procedures, we find that they differ © XXXX American Chemical Society

from one another in terms of the robustness. Thus, the largest deviations (LDs) for the BDE261 set are −11.4 (G4(MP2)6X), −13.1 (DuT-D3), and −23.4 (M06-2X) kJ mol−1. In the same study,4e we demonstrated that many computational chemistry procedures yield substantially better performance for relative BDEs (RBDEs), when compared with that for absolute BDEs. Thus, the MADs from benchmark values for RBDEs for G4(MP2)-6X, DuT-D3, and M06-2X are 1.2, 1.3, and 2.0 kJ mol−1, and the corresponding LDs are −5.2, 8.6, and −17.9 kJ mol−1, respectively. Furthermore, we introduced the quantity deviation f rom additivity of the RBDE (DARBDE) and found that, for this quantity, all methods examined, including all the DFT procedures, have an MAD that is less than 1.0 kJ mol−1, and almost all methods have an LD that is less than 10 kJ mol−1. These encouraging observations have opened up the possibility for the accurate evaluation of absolute BDEs in an indirect manner, via the calculation of relative quantities obtained (accurately) by economical procedures such as DFT methods.12 In our previous work,4e we examined RBDEs and DARBDEs for methyl and fluoro substituents. In the present study, we explore further the potential for the use of relative quantities derived from BDEs to obtain accurate absolute BDEs Received: February 4, 2013 Revised: April 5, 2013

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dx.doi.org/10.1021/jp401248r | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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BDEs and related relative quantities for mono-, di-, and trisubstituted methanes. This set contains methane and 5 mono-, 15 di-, and 35 trisubstituted methane substrates, giving a total of 56 substrates. The thermochemical quantities under investigation in the present study include bond dissociation enthalpies (BDEs), relative BDEs (RBDEs), deviations from additivity of the RBDEs (DARBDEs), and deviations from pairwise additivity of the RBDEs (DPARBDEs). They are closely related to analogous quantities defined for radical stabilization energies.5 We now briefly describe each of these quantities. The C−H BDE in XYZC−H is defined as the enthalpy change for the dissociation reaction:

in an economical manner and examine substrates with a larger variety of substituents.

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COMPUTATIONAL DETAILS Standard ab initio molecular orbital theory and DFT calculations8,13 were carried out with the Gaussian 0914 and Molpro 201015 programs. Geometries were optimized with the B3-LYP/cc-pVTZ+d procedure. This is the prescribed method for geometry optimization in the W1X-2 protocol,16 which is the highest level of theory used in the present study. W1X-2 is a variant of W117 that employs a series of calculations to approximate the all-electron relativistic CCSD(T) level at the complete-basis-set limit. Following each geometry optimization, harmonic frequency analysis was carried out to confirm the nature of the stationary point as an equilibrium structure. To obtain the zero-point vibrational energies (ZPVEs) and thermal corrections for enthalpies at 298 K (ΔH298), we used B3-LYP/ cc-pVTZ+d harmonic vibrational frequencies scaled by 0.985, in accordance with the W1X-2 protocol. All relative enthalpies correspond to 298 K values in kJ mol−1. To obtain the benchmark single-point energies for the BDEs, we initially attempted to employ the W1X-2 protocol. However, we found that W1X-2 calculations for some of the molecules in our test set are computationally too demanding. Therefore, we have assessed the performance of a number of more economical composite procedures against the available W1X-2 data. As we shall see, the G4(MP2)-6X procedure9 is found to show the best statistical performance for this set of data, and we have adopted this method for obtaining our general benchmark values in the present study. We have investigated the performance of a variety of DFT and composite methods. As in our previous study,4e instead of examining a very large set of procedures, we have included a representative collection of methods in the present investigation.18 For conventional DFT procedures, we have included the pure functionals B-LYP,19 PBE-PBE,20 B-B95,21 and M06-L,22 the global hybrid methods B3-LYP,23 PBE1-PBE,24 B98,25 and M06-2X,11 the dispersion-corrected hybrid functionals B3-LYPD3,26 PBE1-PBE-D3,26 and B98-D3,27 and the range-separated hybrid procedures CAM-B3-LYP,28 LC-ωPBE29 and ωB97X.30 The Becke−Johnson variant31 of Grimme’s D3 procedure26 was used for obtaining dispersion corrections for the conventional DFT methods. The 6-311+G(3df,2p) basis set was used in conjunction with all of the above DFT procedures. We have included DuT-D310 as a representative wellperforming DHDFT method, and G46c and CBS-QB332 as prototypical composite procedures. For DuT-D3, the aug′-ccpVTZ+d basis set was employed, and the chemical core was frozen in the evaluation of the perturbative correlation energy, as prescribed in ref 10.

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BDE(XYZC−H) = ΔH(XYZC−H → XYZC• + H•) (1)

where X, Y, and Z are the substituents H, CHO, BH2, CH3, NH2, or CF3. The RBDE is the BDE of a specific substrate (XYZC−H) relative to that for CH4 RBDE(XYZC−H) = BDE(XYZC−H) − BDE(CH3− H) (2)

For a di- or trisubstituted methane substrate, we can obtain an additivity estimate of its RBDE (ARBDE) based on the RBDEs for the component monosubstituted substrates ARBDE(XYZC−H) = RBDE(XCH 2−H) + RBDE(YCH 2−H) + RBDE(ZCH 2−H)

(3)

Accordingly, the deviation from additivity of the RBDE (DARBDE) is DARBDE(XYZC−H) = RBDE(XYZC−H)−ARBDE(XYZC−H)

(4)

For a trisubstituted substrate, a further extension of the concept of DARBDE is the deviation from pairwise additivity of the RBDE (DPARBDE). This is again related to the deviation from pairwise additivity of the radical stabilization energy (DPARSE) introduced recently.5 Thus, the pairwise additivity estimate of the RBDE (PARBDE) is: PARBDE(XYZC−H) = RBDE(XYCH−H) + RBDE(YZCH−H) + RBDE(XZCH−H)−RBDE(XCH 2−H) −RBDE(YCH 2−H)−RBDE(ZCH 2−H)

(5)

and the deviation from pairwise additivity of the RBDE (DPARBDE) is simply

RESULTS AND DISCUSSION

DPARBDE(XYZC−H)

Substrates and Hierarchy of Quantities Examined. In order to more generally evaluate the applicability of the use of simple levels of theory in the calculation of a hierarchy of quantities derived from BDEs, we have chosen a prototypical set of substituents for the substrates. These include H (the reference), CH(O), BH2, CH3, NH2, and CF3, which cover electron-withdrawing as well as electron-donating substituents, with the electronic effects delivered either through σ or π interactions (or both). We will initially focus on the C−H

= RBDE(XYZC−H)−PARBDE(XYZC−H)

(6)

The quantities BDE (1), RBDE (2), DARBDE (4), and DPARBDE (6) represent a hierarchy of relative energies whose absolute magnitudes are expected to decrease in the order given. We now proceed to examine the performance of the various quantum chemistry procedures for the calculation of these quantities. B



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Choice of the Benchmark Procedure. As outlined in the Computational Details section, we initially attempted to obtain benchmark BDEs using the high-level W1X-2 procedure. However, we found that W1X-2 calculations for several molecules in our test set are computationally too demanding. We have thus employed the W1X-2 data that we were able to obtain (46 out of 56 BDEs) for the evaluation of alternative high-level but more economical procedures, namely G4(MP2)6X,9 G4,6c and CBS-QB3.32 The results are shown in Table 1.

Table 2. Benchmark G4(MP2)-6X C−H BDEs, RBDEs, DARBDES, and DPARBDEs (298 K Enthalpies, kJ mol−1)

X

Table 1. Statistical Performancea (kJ mol−1) of G4(MP2)6X, G4, and CBS-QB3 for the 46 Benchmark W1X-2 Bond Dissociation Enthalpies G4(MP2)-6X G4 CBS-QB3

MAD

MD

SD

LD

3.2 5.1 5.8

−3.2 −5.1 5.8

1.3 2.2 2.4

−6.5 −12.5 10.5

Y

Z

H H H H H CHO H H BH2 H H CH3 H H NH2 H H CF3 H CHO CHO H CHO BH2 H CHO CH3 H CHO NH2 H CHO CF3 H BH2 BH2 H BH2 CH3 H BH2 NH2 H BH2 CF3 H CH3 CH3 H CH3 NH2 H CH3 CF3 H NH2 NH2 H NH2 CF3 H CF3 CF3 CHO CHO CHO CHO CHO BH2 CHO CHO CH3 CHO CHO NH2 CHO CHO CF3 CHO BH2 BH2 CHO BH2 CH3 CHO BH2 NH2 CHO BH2 CF3 CHO CH3 CH3 CHO CH3 NH2 CHO CH3 CF3 CHO NH2 NH2 CHO NH2 CF3 CHO CF3 CF3 BH2 BH2 BH2 BH2 BH2 CH3 BH2 BH2 NH2 BH2 BH2 CF3 BH2 CH3 CH3 BH2 CH3 NH2 BH2 CH3 CF3 BH2 NH2 NH2 BH2 NH2 CF3 BH2 CF3 CF3 CH3 CH3 CH3 CH3 CH3 NH2 CH3 CH3 CF3 CH3 NH2 NH2 CH3 NH2 CF3 CH3 CF3 CF3 NH2 NH2 NH2 NH2 NH2 CF3 NH2 CF3 CF3 CF3 CF3 CF3 mean absolute value

a

MAD is mean absolute deviation, MD is mean deviation, SD is standard deviation, and LD is largest deviation from the W1X-2 values.

The mean absolute deviations (MADs) for the three composite methods increase in the order G4(MP2)-6X → G4 → CBS-QB3. This is somewhat surprising, as the G4(MP2)-6X procedure is the least computationally demanding among the three. We can see that the mean absolute deviations (MADs) have the same magnitude as the corresponding mean deviations (MDs) for all three methods. This is indicative of a fairly systematic deviation from the benchmark values. Accordingly, the standard deviations (SDs), which can be considered as a probe for nonsystematic deviations, are fairly small. However, they do increase in the same order as that for the MADs, i.e., G4(MP2)-6X → G4 → CBS-QB3. In addition, we find that G4(MP2)-6X gives a largest deviation (LD) that has the smallest magnitude. Taking all of the above into account, we consider that, while the G4 and CBS-QB3 protocols are comparatively more rigorous in their formulation, G4(MP2)-6X appears to be somewhat more appropriate as the benchmark level for this specific set of systems. C−H BDEs, RBDEs, DARBDEs, and DPARBDEs Obtained with G4(MP2)-6X. The set of G4(MP2)-6X C−H BDEs and related relative quantities are shown in Table 2. The mean absolute values (MAVs) of these quantities decrease in the order BDE (385.0 kJ mol−1) > RBDE (54.9) > DARBDE (20.9) > DPARBDE (7.2), as anticipated. The BDEs of a number of substrates investigated in the present study have been systematically evaluated previously5 using the G3X(MP2)-RAD33 and ROB2-PLYP34 procedures. We note that the G4(MP2)-6X values obtained in the present study and the previous high-level G3X(MP2)-RAD values are in good agreement with one another, with an MAD of 1.6 kJ mol −1 between these two methods. The ROB2-PLYP procedure typically leads to an underestimation of the BDEs when compared with G4(MP2)-6X and G3X(MP2)-RAD (on average by ∼7−8 kJ mol−1). The chemical aspects of the effects of substituents on C−H BDEs (as indicated by RBDEs) have been previously discussed in detail.5 We therefore simply note here a few of the more significant observations. Almost all systems examined have negative RBDEs, i.e., the substituents weaken the adjacent C− H bond, largely through stabilization of the resultant radicals. The few exceptions are for the RBDEs of CF3CH2−H (+9.2 kJ C

BDE

RBDE

DARBDE

DPARBDE

437.6 400.4 397.2 424.3 390.1 446.9 376.1 381.7 381.6 317.6 419.6 389.4 372.7 333.5 413.2 415.7 387.6 432.4 387.4 402.0 453.0 356.9 380.0 355.0 277.3 394.0 459.8 359.4 328.1 399.2 366.2 310.4 393.9 299.4 331.3 427.2 397.8 373.2 355.7 408.5 352.3 329.8 387.9 307.2 352.7 419.5 411.7 387.9 424.6 386.0 396.3 438.5 392.6 390.0 397.0 455.1 385.0

−37.2 −40.4 −13.3 −47.5 +9.2 −61.6 −55.9 −56.1 −120.0 −18.1 −48.2 −64.9 −104.2 −24.4 −21.9 −50.0 −5.2 −50.2 −35.7 +15.4 −80.7 −57.6 −82.6 −160.3 −43.6 +22.2 −78.2 −109.6 −38.5 −71.5 −127.2 −43.8 −138.2 −106.3 −10.5 −39.8 −64.4 −81.9 −29.1 −85.4 −107.8 −49.7 −130.4 −84.9 −18.1 −25.9 −49.7 −13.1 −51.7 −41.3 +0.8 −45.0 −47.6 −40.7 +17.4 54.9

+12.9 +21.7 −5.5 −35.3 +9.9 +32.7 −11.2 −16.2 +6.8 +4.7 +10.8 −1.1 +44.8 +2.6 −3.1 +31.0 +57.3 +5.2 −38.3 +21.6 +140.3 +12.8 +15.6 +30.0 −7.5 −29.1 −2.4 −6.0 −30.8 +8.3 +81.5 +29.8 +46.4 +42.5 −18.2 −6.5 −5.2 +5.0 −6.2 +3.9 +14.1 +24.5 +4.4 +56.7 +10.3 −4.3 +97.5 +38.2 −11.6 −10.2 20.9

−7.7 +0.9 +3.3 +19.3 −11.2 +64.2 +7.7 +45.4 −8.5 −1.3 +0.8 −5.7 +19.8 −8.0 −8.5 −16.5 +19.4 +46.2 −3.7 −0.7 +10.0 +0.3 −7.3 +0.6 −6.7 −0.1 −1.9 +1.9 −9.8 −2.1 +1.0 −36.9 −11.8 −13.8 −1.1 7.2



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mol−1), (CF3)2CH−H (+15.4), (CHO)(BH2)2C−H (+22.2), CH3(CF3)2C−H (+0.8), and (CF3)3C−H (+17.4), which are indicative of the destabilizing effect of strongly electronwithdrawing groups for the radical products. For DARBDEs, we first note that their mean absolute value (20.9 kJ mol−1) is smaller than the MAV (54.9 kJ mol−1) for RBDEs. We see large and positive values in a number of cases, such as those for (CHO)(BH2)2C−H (+140.3 kJ mol−1), (BH 2 ) 3 C−H (+81.5), and (NH 2 ) 3 C−H (+97.5). For (BH2)3C−H and (NH2)3C−H, the large departure from additivity has been previously rationalized in terms of large stabilizing effects in the closed-shell substrates relative to those in the radicals, resulting in stronger bonds.5 We find that such a rationalization can also be applied to the large and positive DARBDE for (CHO)(BH2)2C−H. Negative DARBDE values are generally seen for systems containing π-donor and πacceptor substituents, in accord with the well-known captodative effect.35 Regarding DPARBDEs, we can see that their mean absolute value (7.2 kJ mol−1) is smaller than the MAV for DARBDEs (20.9 kJ mol−1). This is perhaps to be expected, as pairwise additivity, which makes use of both di- and monosubstituted reference systems, should generally be more capable of capturing substituent effects in the trisubstituted substrates than additivity based on monosubstituted references alone. As an example, we note that the DPARBDE for (CHO)(BH2)2C− H is +64.2 kJ mol−1, which is notably smaller than the corresponding DARBDE of +140.3 kJ mol−1. Performance of Theoretical Methods for the Evaluation of C−H BDEs, RBDEs, DARBDEs, and DPARBDEs. We now use the G4(MP2)-6X benchmark C−H BDEs and related relative quantities to assess the performance of the various procedures. The MADs, MDs, SDs, and LDs are shown in Tables 3−6. For the various classes of theoretical procedures,

we can see that the performance generally improves in the order: pure DFT → hybrid DFT → double-hybrid DFT → composite procedures. The difference in MADs for the various procedures is most apparent for BDEs, with MADs of ∼20−30 kJ mol−1 for pure DFT procedures, ∼ 5−20 kJ mol−1 for hybrid DFT methods, < 5 kJ mol−1 for double-hybrid DFT procedures, and < ∼ 3 kJ mol−1 for composite protocols. The variation in the performance is somewhat smaller for RBDEs, with MADs up to ∼20 kJ mol−1 for pure DFT methods. For the calculation of DARBDEs and DPARBDEs, almost all methods perform well, with MADs that are less than 8 kJ mol−1 throughout for DARBDEs, and less than 5 kJ mol−1 for DPARBDEs. Among the pure DFT procedures, M06-L yields the smallest MADs for all four quantities, namely BDE, RBDE, DARBDE and DPARBDE. For the hybrid DFT methods, M06-2X shows the best performance across the board, with MADs that are comparable to those for the higher-level DuT-D3 method. The good performance of M06-2X is in accord with our previous findings.4e The use of range separation, in general, leads to an improvement in performance compared with the global hybrid DFT methods on which they are based. It can be seen that the inclusion of dispersion corrections leads to somewhat smaller MADs compared with the corresponding uncorrected global hybrid DFT procedures. A comparison of the MADs (Table 3) and the MDs (Table 4) shows that, for BDEs and RBDEs, the deviations from Table 4. Mean Deviations (kJ mol−1) from Benchmark G4(MP2)-6X C−H BDEs, RBDEs, DARBDEs, and DPARBDEs, for the Various Theoretical Procedures BDE B-LYP PBE-PBE B-B95 M06-L

Table 3. Mean Absolute Deviations (kJ mol−1) from Benchmark G4(MP2)-6X C−H BDEs, RBDEs, DARBDEs, and DPARBDEs, for the Various Theoretical Procedures BDE B-LYP PBE-PBE B-B95 M06-L B3-LYP PBE1-PBE B98 M06-2X CAM-B3-LYP LC-ωPBE ωB97X B3-LYP-D3 PBE1-PBE-D3 B98-D3 DuT-D3 CBS-QB3 G4

RBDE

DARBDE

B3-LYP PBE1-PBE B98 M06-2X

DPARBDE

pure DFT 21.2 5.9 3.9 17.5 5.6 3.4 23.5 7.6 4.1 17.2 4.4 2.3 global hybrid DFT 19.3 14.0 4.0 2.6 19.9 8.3 3.3 1.7 17.6 13.5 3.4 2.1 4.6 3.5 3.7 1.1 range-separated hybrid DFT 12.8 9.2 2.9 2.1 15.2 4.9 3.1 1.6 9.5 7.6 3.7 1.7 dispersion-corrected global hybrid DFT 15.9 12.0 4.0 2.5 18.2 7.3 3.3 1.7 11.0 10.0 3.6 2.0 double-hybrid DFT and composite procedures 4.6 4.6 2.7 0.9 3.3 2.5 1.2 0.6 2.4 2.4 0.5 0.3 31.1 26.2 22.3 22.2

CAM-B3-LYP LC-ωPBE ωB97X B3-LYP-D3 PBE1-PBE-D3 B98-D3 DuT-D3 CBS-QB3 G4

RBDE

DARBDE

DPARBDE

pure DFT −31.1 −21.2 −0.3 −0.1 −26.2 −17.2 2.1 1.4 −21.8 −23.2 5.1 1.6 −22.2 −17.2 1.6 1.5 global hybrid DFT −19.3 −14.0 −0.4 −0.3 −19.9 −7.9 1.5 0.7 −17.6 −13.5 1.5 −0.6 −4.4 −2.7 2.8 −0.8 range-separated hybrid DFT −12.8 −9.2 0.6 −0.6 −15.1 −0.5 1.9 0.3 −9.5 −7.6 2.2 −1.5 dispersion-corrected global hybrid DFT −15.9 −12.0 −0.1 −0.6 −18.2 −6.9 1.7 0.6 −11.0 −10.0 2.1 −1.0 double-hybrid DFT and composite procedures 1.9 2.5 −1.8 −0.6 2.8 −0.4 0.1 −0.2 −2.4 −2.4 0.3 −0.3

benchmark values for the pure and hybrid DFT procedures are largely systematic. Thus, the MDs have very similar magnitudes to those for the corresponding MADs. For DARBDEs and DPARBDEs, the systematic deviations are much less prominent, with MD values that are much closer to zero. We now turn our attention to the SDs (Table 5), which can be considered as a measure of nonsystematic uncertainties that D



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Table 5. Standard Deviations (kJ mol−1) from Benchmark G4(MP2)-6X C−H BDEs, RBDEs, DARBDEs, and DPARBDEs, for the Various Theoretical Procedures BDE B-LYP PBE-PBE B-B95 M06-L B3-LYP PBE1-PBE B98 M06-2X CAM-B3-LYP LC-ωPBE ωB97X B3-LYP-D3 PBE1-PBE-D3 B98-D3 DuT-D3 CBS-QB3 G4

RBDE

DARBDE

Table 6. Largest Deviations (kJ mol−1) from Benchmark G4(MP2)-6X C−H BDEs, RBDEs, DARBDEs, and DPARBDEs, for the Various Theoretical Procedures

DPARBDE

BDE

pure DFT 10.1 10.1 9.1 11.4 8.9 8.9 7.6 9.2 11.0 11.0 10.1 10.8 7.4 7.4 5.9 5.8 global hybrid DFT 7.0 7.0 6.1 7.4 5.7 5.7 4.5 4.0 5.6 5.6 4.8 5.6 3.6 3.6 3.9 2.6 range-separated hybrid DFT 5.0 5.0 4.4 5.5 6.2 6.2 4.7 4.0 3.8 3.8 4.1 4.3 dispersion-corrected global hybrid DFT 6.8 6.8 6.0 7.3 5.6 5.6 4.5 4.1 5.1 5.1 4.5 5.5 double-hybrid DFT and composite procedures 6.3 6.3 3.8 3.0 3.3 3.3 1.5 1.2 2.0 2.0 0.5 0.6

B-LYP PBE-PBE B-B95 M06-L B3-LYP PBE1-PBE B98 M06-2X CAM-B3-LYP LC-ωPBE ωB97X B3-LYP-D3 PBE1-PBE-D3 B98-D3 DuT-D3 CBS-QB3 G4

RBDE

DARBDE

DPARBDE

pure DFT −61.0 −51.1 −29.1 −43.0 −48.8 −39.9 25.5 −32.9 −48.7 −50.0 35.8 33.9 −42.7 −37.7 19.0 −23.1 global hybrid DFT −43.5 −38.2 −24.3 −26.5 −36.7 −24.7 16.8 13.5 −33.2 −29.1 16.1 −18.0 −15.4 −13.8 15.8 −10.0 range-separated hybrid DFT −28.2 −24.6 −16.7 −18.3 −25.7 19.1 13.6 11.9 −20.6 −18.6 −13.2 −16.3 dispersion-corrected global hybrid DFT −39.9 −36.0 −23.8 −26.6 −35.1 −23.7 16.6 13.1 −24.9 −23.9 15.9 −17.9 double-hybrid DFT and composite procedures 20.8 21.4 −13.4 −14.3 12.5 9.3 −3.9 −2.9 −8.1 −8.1 −1.8 −1.6

examined in the present study, namely the pure DFT methods such as M06-L. In our previous study,4e we demonstrated the possibility of using DARBDEs obtained using an economical theoretical procedure, in conjunction with accurate reference RBDEs obtained at a higher level, to provide an accurate estimate of the RBDE of the system of interest. Once an accurate RBDE has been estimated, it can then be combined with a highly accurate reference BDE to obtain an absolute BDE value of similar accuracy for the target system. We now further explore this strategy more generally. Thus, we estimate RBDEs for the set of systems for which values are available at our highest W1X-2 level, using the following six protocols: A. We use W1X-2 values for the RBDEs of monosubstituted systems in conjunction with G4(MP2)-6X DARBDEs for di- and trisubstituted systems to obtain RBDEs for the di- and trisubstituted substrates (eqs 3 and 4). We use the notation W1X-2:G4(MP2)-6X(DA) to describe this approximation scheme, with DA representing the use of DARBDEs. B. This is the same scheme as A, except that M06-2X is used for the evaluation of DARBDEs, i.e., W1X-2:M062X(DA). C. This scheme is identical to A, but with DARBDEs obtained using B3-LYP, i.e., W1X-2:B3-LYP(DA). D. This is identical to A, but the DARBDEs are obtained with M06-L, i.e., W1X-2:M06-L(DA). E. This is again the same as A, but with reference RBDEs being obtained with M06-2X and DARBDEs calculated using M06-L, i.e., M06-2X:M06-L(DA). F. The RBDEs for mono- and disubstituted systems obtained at the M06-2X level are combined with M06L DPARBDEs for trisubstituted systems to estimate the RBDEs for the trisubstituted systems (eqs 5 and 6), i.e.,

are inherently associated with the theoretical procedure. In accordance with this argument, we see in general only a small reduction in the SD values along the sequence: BDE → RBDE → DARBDE → DPARBDE. Overall, we see that the largest SD values are for the pure DFT procedures, and that the smallest occur for composite procedures. For the pure DFT procedures, M06-L has the smallest SD values for the four quantities considered. Among the hybrid DFT methods, M06-2X gives the smallest SD of 2.6 kJ mol−1 for DPARBDEs, which is an indication of its reasonable robustness in the calculation of this quantity. While the small MADs achieved by most methods for DARBDEs and DPARBDEs are encouraging, we caution that substantial deviations are still observed in specific cases. Notably, the LDs (Table 6) for the DPARBDEs for all the pure DFT procedures examined still exceed 20 kJ mol−1. The M06-2X procedure and the range-separated hybrid DFT method LC-ωPBE give LD values for DARBDEs with magnitudes of ∼10 kJ mol−1. On the other hand, the DuTD3 double-hybrid procedure, even with an MAD of just 0.9 kJ mol−1 for the DPARBDEs (Table 3), still has an LD of −14.3 kJ mol−1. The composite procedures, namely CBS-QB3 and G4, give LDs with magnitudes of less than 5 kJ mol−1 for both DARBDEs and DPARBDEs. Use of DARBDEs and DPARBDEs to Obtain Accurate RBDEs and BDEs. We have seen in previous sections that the G4(MP2)-6X protocol gives good agreement with a set of highlevel W1X-2 C−H BDEs. We also see that the G4 and CBSQB3 composite procedures, and the computationally less demanding DuT-D3 and M06-2X methods, are capable of achieving overall good accuracy when assessed against the larger set of benchmark BDEs and related relative quantities available at the G4(MP2)-6X level. In addition, it is evident that DARBDEs and DPARBDEs can generally be calculated with reasonable accuracy even with the most economical procedures E



Article

Table 7. Statistical Performance (kJ mol−1) With Respect to Benchmark W1X-2 Values for RBDEs Calculated using Lower-Cost Procedures and Estimated by the Various Approximation Schemesa method RBDE

G4(MP2)-6X G4(MP2)-6X

DARBDE

A W1X-2

M06-2X b

M06-2X

G4(MP2)-6X

B W1X-2

B3-LYP b

B3-LYP

M06-2X

C W1X-2

M06-L b

M06-L

B3-LYP

D

E

W1X-2

b

M06-2X

M06-L

M06-L

4.2 1.0 6.2 20.9

7.4 −7.0 6.0 −22.8

DPARBDE MAD MD SD LD

F b

M06-2Xc M06-L

1.6 −1.6 1.3 −4.9

1.0 −0.1 1.3 3.1

5.1 −5.0 3.2 −14.5

4.1 2.6 5.0 17.7

15.9 −15.9 7.5 −40.3

3.8 −1.1 5.4 −22.4

18.8 −18.8 8.3 −38.4

6.1 −3.8 6.5 −18.0

a

See text for the detailed descriptions of the approximation schemes A−F. bUsed for the reference monosubstituted systems only. cUsed for the reference monosubstituted and disubstituted systems.

using the computationally more efficient M06-L procedure for the largest systems, while still giving an accuracy that is comparable to the much better performing M06-2X method. If we restrict ourselves to the trisubstituted systems, we note that the MAD for scheme E [M06-2X:M06-L(DA)] is 8.2 kJ mol−1 and the SD is 6.9 kJ mol−1. In comparison, the use of DPARBDEs in scheme F [M06-2X:M06-L(DPA)] leads to a further improvement, with an MAD of 6.1 kJ mol−1 and an SD of 6.5 kJ mol−1. However, because the DPARBDE scheme involves additional high-level calculations for disubstituted systems, it may not be generally cost-effective. Performance of Various Procedures for the Calculation of C−H, C−C, and C−F BDEs and Related Relative Quantities. In the previous section, we demonstrated the use of DARBDEs and DPARBDEs to estimate RBDEs (and BDEs) for C−H bonds with reasonable accuracy. We now further explore the generality of such approaches, and examine the performance for C−CH3 and C−F bonds. Together with C−H, these three bonds cover a range of polarity for the bond that is being broken (Cδ−−Hδ+, C−C, and Cδ+−Fδ−). We first examine the performance of M06-2X, B3-LYP, and M06-L for the calculation of BDEs and associated relative quantities against the complete set of benchmark G4(MP2)-6X values (Table 8), as the success of the various approximation schemes relies on accurate predictions of DARBDEs and DPARBDEs by lower-level procedures such as M06-2X and M06-L. It is apparent that, for all three types of bonds, while M06-L and B3-LYP perform poorly for absolute BDEs, the performance improves for the relative quantities, especially for

M06-2X:M06-L(DPA), with DPA representing the use of DPARBDEs. We note that, when the estimated RBDEs are combined with the W1X-2 BDE for methane, the deviations of the resultant estimates of the absolute BDEs from the W1X-2 values would be identical to the deviations of the estimated RBDEs from W1X-2 RBDEs. Thus, we only need to discuss the performance of the six strategies above for the calculation of RBDEs. The results are shown in Table 7, together with statistics for G4(MP2)-6X, M06-2X, B3-LYP, and M06-L. The approximation scheme A [W1X-2:G4(MP2)-6X(DA)] that employs high-level W1X-2 and G4(MP2)-6X data has an MAD of just 1.0 kJ mol−1 and an LD of just 3.1 kJ mol−1 compared with the W1X-2 benchmark. Thus, by the use of only a small number of W1X-2 RBDEs for reference systems of modest size (monosubstituted), this approximation scheme has further improved on the performance of G4(MP2)-6X (MAD = 1.6 kJ mol−1 and LD = −4.9 kJ mol−1). Importantly, this enables the estimation of RBDEs and BDEs for systems that are beyond the reach of explicit W1X-2 calculations, with an accuracy that is somewhat better than the use of G4(MP2)-6X alone. When we replace G4(MP2)-6X in scheme A with M06-2X, we see that the resulting scheme B [W1X-2:M06-2X(DA)] also represents a slight improvement on the corresponding standalone method, i.e., M06-2X. The approximation schemes C [W1X-2:B3-LYP(DA)] and D [W1X-2:M06-L(DA)], respectively, represent improved B3-LYP and M06-L RBDEs obtained using DARBDEs calculated at these levels and W1X-2 reference RBDEs. In these cases, we can see that they give substantially improved results when compared with those from the direct RBDE calculations. For example, the MAD with B3LYP is improved from 15.9 to 3.8 kJ mol−1, whereas that for M06-L is improved from 18.8 to 4.2 kJ mol−1. The fairly good accuracy of standalone M06-2X RBDEs and the capacity of scheme D [W1X-2:M06-L(DA)] to provide substantial improvement to M06-L have opened up the possibility of combining M06-2X and M06-L for the economical estimation of RBDEs with reasonable accuracy. Thus, we arrive at scheme E [M06-2X:M06-L(DA)]. The resulting estimated RBDEs are of an accuracy (MAD = 7.4 kJ mol−1) that is slightly worse than those obtained from standalone M06-2X calculations (MAD = 5.1 kJ mol−1), but substantially better than those from M06-L calculations (MAD = 18.8 kJ mol−1). We note that the current set of systems can be straightforwardly calculated using M06-2X. However, the approximation scheme E would enable the evaluation of RBDEs (and BDEs) for substrates with larger substituents

Table 8. Mean Absolute Deviations (kJ mol−1) Against Benchmark G4(MP2)-6X C−H, C−CH3, and C−F BDEs and Related Relative Quantities for M06-L, B3-LYP and M06-2X

F

BDE

RBDE

C−H C−CH3 C−F

22.2 31.5 18.7

17.2 22.9 10.8

C−H C−CH3 C−F

19.3 45.1 22.5

14.0 25.6 13.7

C−H C−CH3 C−F

4.6 4.7 5.1

3.5 5.3 3.4

DARBDE M06-L 4.4 5.3 4.8 B3-LYP 4.0 3.9 4.5 M06-2X 3.7 3.5 3.6

DPARBDE 2.3 2.4 2.1 2.6 2.3 4.5 1.1 1.0 1.5



Article

DARBDEs and DPARBDEs. We can also see that M06-2X gives MAD values that are quite small throughout (less than ∼5 kJ mol−1). We now evaluate the capability of the various approximation schemes for the evaluation of RBDEs. Using the G4(MP2)-6X benchmark RBDEs, we examine the performance of two approximation schemes. The B′ scheme is similar to B (Table 7) but employs G4(MP2)-6X reference RBDEs, i.e., [G4(MP2)-6X:M06-2X(DA)] instead of [W1X-2:M06-2X(DA)]. Thus, it represents the use of RBDEs obtained at the G4(MP2)-6X benchmark level in combination with DARBDEs calculated using the most accurate low-level method (M06-2X, Table 8). The approximation scheme E [M06-2X:M06-L(DA)] is identical to that defined earlier in the text and in Table 7, and it represents a very economical approach using a combination of pure (M06-L) and hybrid (M06-2X) DFT procedures. The results are shown in Table 9.

schemes, we note that, the use of G4(MP2)-6X RBDEs for monosubstituted reference systems in scheme B′ does not always lead to an improvement over standalone M06-2X calculations. Nonetheless, the performance of scheme B′ is at least comparable to that for M06-2X. In accord with the results shown in Table 7, we note that the use of M06-2X reference RBDEs together with M06-L DARBDEs yields target RBDEs that are significantly more accurate than those obtained from standalone M06-L calculations, with an accuracy approaching that of M06-2X. Further Examination of the Approximation Schemes. We now further probe the capability of the approximation schemes for the estimation of RBDEs. We start by briefly applying the schemes to the RBDEs for larger substrates. Thus, we use the schemes B′ and E to evaluate the RBDEs of (nBu)3C−H and Ph3C−H (Table 10). For the n-butyl-substituted methanes, we find that M06-2X yields RBDEs that are in good agreement with the benchmark G4(MP2)-6X values, but M06-L deviates significantly from G4(MP2)-6X, with the deviations becoming larger in the order (n-Bu)CH2−H → (n-Bu)2CH−H → (n-Bu)3C−H. For phenyl-substituted methanes, however, the RBDEs obtained with both M06-2X and M06-L show significant deviations from the benchmark values, with the magnitudes of the deviations becoming larger with the number of phenyl substituents. We now turn our attention to the performance of the approximation schemes. It can be seen that for n-butylsubstituted methanes, both scheme B′ [G4(MP2)-6X:M062X(DA)] and scheme E [M06-2X:M06-L(DA)] give RBDEs that are in close agreement with the G4(MP2)-6X values. We note that the calculation of the (n-Bu)3C• radical represents the most time-consuming component for evaluation of the RBDE for (n-Bu)3C−H. At the G4(MP2)-6X level, the calculation of the (n-Bu)3C• radical consumed 8.5 days of CPU time. In contrast, the M06-L calculation for (n-Bu)3C• and the M06-2X calculation for the (n-Bu)CH2• radical that are used in scheme E consumed a combined CPU time of less than 2 h. This represents a saving in computer time of more than 2 orders of magnitude, with little compromise on the accuracy. For the phenyl-substituted methanes, the agreement between the approximation schemes and the G4(MP2)-6X RBDEs is somewhat less good, but the estimated RBDEs still represent substantial improvements compared with the directly calculated M06-2X and M06-L values. Notably, the values obtained with the more approximate scheme among the two estimation methods, namely scheme E, are in significantly closer agreement than direct M06-L with those obtained with the directly calculated M06-2X RBDEs. The use of scheme B′

Table 9. Statistical Performance (kJ mol−1) With Respect to Benchmark G4(MP2)-6X Values for C−H, C−CH3, and C− F RBDEs Calculated using Lower-Cost Procedures and Estimated by the Various Approximation Schemes method

M06-2X

B′

M06-L

E

RBDE

M06-2X

G4(MP2)-6Xa

M06-L

M06-2Xa

DARBDE

a

M06-2X

C−H C−CH3 C−F

3.5 5.3 3.4

C−H C−CH3 C−F

−2.7 −4.7 −1.4

C−H C−CH3 C−F

3.6 4.5 4.4

C−H C−CH3 C−F

−13.8 −17.6 −14.5

M06-L

mean absolute deviation 3.7 17.2 3.5 22.9 3.6 10.8 mean deviation 2.8 −17.2 2.3 −22.9 2.1 −10.6 standard deviation 3.9 7.4 4.0 9.2 4.2 7.1 largest deviation 15.8 −37.7 15.6 −46.2 16.1 −28.5

5.5 6.2 5.2 −4.1 −4.2 −2.3 6.3 6.9 6.9 −22.0 −24.6 −22.6

Used for the reference monosubstituted systems only.

For all three types of bonds, we see M06-2X significantly outperforms M06-L for all statistical quantities considered (MAD, MD, SD and LD). Regarding the two approximation

Table 10. Relative BDEs (kJ mol−1) for n-Butyl- and Phenyl-Substituted Methanes Obtained with Direct G4(MP2)-6X, M06-2X, and M06-L Calculations and the Approximation Schemes B′ and E method

G4(MP2)-6X

M06-2X

B′

M06-L

E

RBDE

G4(MP2)-6X

M06-2X

G4(MP2)-6Xa

M06-L

M06-2Xa

DARBDE (n-Bu)CH2−H (n-Bu)2CH−H (n-Bu)3C−H PhCH2−H Ph2CH−H Ph3C−H a

M06-2X −15.7 −29.2 −42.5 −54.6 −73.3 −90.6

−16.2 −29.4 −43.8 −59.1 −86.6 −110.4

−28.4 −42.3 −77.7 −97.1

M06-L −22.4 −40.3 −60.6 −72.9 −112.5 −139.2

−27.9 −41.9 −84.9 −97.7

Used for the reference monosubstituted systems only. G



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Table 11. Largest Deviations (kJ mol−1) From G4(MP2)-6X C−H, C−CH3, and C−F RBDEs for Values Obtained Using the Approximation Schemes B′ and Ea scheme B′

scheme E

RBDE: G4(MP2)-6X

a

RBDE: M06-2Xa

DARBDE: M06−2X X

Y

Z

DARBDE: M06-L deviation

X

Y

Z

deviation

BH2 CHO H CHO CHO

BH2 NH2 BH2 CHO CH3

BH2 NH2 BH2 BH2 NH2

−22.0 −21.3 −14.8 −13.9 −12.6

CHO CHO CHO CHO BH2

NH2 CH3 CHO BH2 BH2

NH2 NH2 NH2 BH2 BH2

−24.6 −15.8 −14.7 14.3 −13.9

BH2 CHO CHO CHO BH2

BH2 NH2 CHO CHO BH2

BH2 NH2 CHO NH2 CH3

−22.6 −16.8 16.1 −13.2 −13.1

C−H CHO BH2 BH2 CHO CHO

BH2 BH2 BH2 BH2 NH2

BH2 NH2 BH2 NH2 NH2

15.8 9.9 9.5 9.2 −8.5

CHO CHO BH2 BH2 CHO

BH2 NH2 BH2 BH2 CHO

BH2 NH2 NH2 BH2 CHO

15.6 −9.0 8.8 8.5 8.3

CHO BH2 CHO BH2 CHO

BH2 BH2 NH2 BH2 BH2

BH2 NH2 NH2 BH2 NH2

16.1 10.1 −9.3 8.3 7.4

C−CH3

C−F

a

For reference monosubstituted systems only.

chemistry procedures for the calculation of bond dissociation enthalpies (BDEs) and related relative quantities, namely relative BDEs (RBDEs), deviations from additivity of the RBDEs (DARBDEs) and deviations from pairwise additivity of the RBDEs (DPARBDEs). The following major findings have emerged. 1. The magnitudes of the four quantities considered generally become smaller in the order: BDE > RBDE > DARBDE > DPARBDE. In parallel to this, the performance of theoretical procedures in the calculation of these quantities improves in the same order. 2. In general, the performance of the various types of procedures improves in the order pure DFT → hybrid DFT → double-hybrid DFT → composite procedures, as expected. Such a variation is most prominent for BDEs but becomes less so for RBDEs and especially DARBDEs and DPARBDEs, with even pure DFT procedures giving DARBDE and DPARBDE values that are reasonably accurate. 3. We find M06-L to be the best performing pure DFT procedure for the evaluation of the present set of BDEs and related relative quantities. The M06-2X method is found to perform the best among the hybrid DFT methods. 4. Using reference BDEs calculated at a higher level of theory in combination with DARBDEs or DPARBDEs obtained at a lower level can lead to reasonably accurate BDEs and RBDEs for the target system. We consider that the approximation schemes that make use of DARBDEs are more appealing due to their lower cost and because they generally achieve a similar level of accuracy to the DPARBDE-based scheme. 5. An approximation scheme that makes use of W1X-2 reference RBDEs, in conjunction with G4(MP2)-6X DARBDEs, namely [W1X-2:G4(MP2)-6X(DA)], has a

provides further improvement over scheme E. In terms of timing comparisons, for example, for the calculation of Ph3C•, scheme B′ took 1/60 of the CPU time of that for the direct G4(MP2)-6X calculation, and scheme E took only 1/200 of that for G4(MP2)-6X. While the use of approximation schemes with DARBDEs can, on average, lead to quite adequate BDEs, it is also apparent that relatively large deviations still occur in some situations (see the LD values in Tables 6 and 7). It is thus worthwhile to unveil the main systems where the large deviations are observed. Table 11 shows the five largest deviations against G4(MP2)-6X benchmarks for RBDE values obtained using the approximation schemes B′ and E for the systems in Table 2. For RBDEs obtained using scheme B′ where G4(MP2)-6X reference values are combined with DARBDEs calculated with M06-2X, the largest deviations typically involve multiple CHO and BH2 substituents for all three types of bonds. Thus, strong π acceptors appear to represent a major difficulty for the evaluation of RBDEs using the approximation scheme B′. In addition, the NH2 group, a strong π donor, can also lead to large deviations in the presence of strong π acceptors. Nonetheless, most of the “large” deviations are ∼7−10 kJ mol−1, with only a small number of deviations being greater than 10 kJ mol−1 in magnitude (for X = Y = CHO and Z = BH2). For scheme E, which is theoretically less rigorous than scheme B, we find that the combination of CHO, BH2 and NH2 groups again gives quite large deviations, with magnitudes of ∼10−25 kJ mol−1. It is therefore necessary to be cautious when using this approximation scheme on substrates with multiple strong π-acceptor and π-donor groups.

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CONCLUDING REMARKS In the present study, we have used benchmark values provided by the high-level W1X-2 and G4(MP2)-6X methods to examine the performance of a variety of computational H



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Carbon-Centered Free Radicals and Radical Cations: Structure, Reactivity and Dynamics; Forbes, M. D. E., Ed.; John Wiley & Sons: New York, 2010; pp 83−104. (c) Hioe, J.; Zipse, H. Radical Stability − Thermochemical Aspects. In Encyclopedia of Radicals in Chemistry, Biology and Materials; Chatgilialoglu, C.; Studer, A., Ed., John Wiley & Sons: Chichester, U.K., 2012, pp 449−476. (d) Coote, M. L.; Lin, C. Y.; Beckwith, A. L.; Zavitsas, A. A. A Comparison of Methods for Measuring Relative Radical Stabilities of Carbon-Centred Radicals. Phys. Chem. Chem. Phys. 2010, 12, 9597−9610. (e) Chan, B.; Radom, L. BDE261: A Comprehensive Set of High-Level Theoretical Bond Dissociation Enthalpies. J. Phys. Chem. A 2012, 116, 4975−4986 and references therein.. (5) Menon, A. S.; Henry, D. J.; Bally, T.; Radom, L. Effect of Substituents on the Stabilities of Multiply-Substituted CarbonCentered Radicals. Org. Biomol. Chem. 2011, 9, 3636−3657 and reference therein.. (6) (a) Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. Gaussian-2 Theory for Molecular Energies of First- and Second-Row Compounds. J. Chem. Phys. 1991, 94, 7221−7230. (b) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Rassolov, V.; Pople, J. A. Gaussian-3 (G3) Theory for Molecules Containing First and Second-Row Atoms. J. Chem. Phys. 1998, 109, 7764−7776. (c) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K. Gaussian-4 Theory. J. Chem. Phys. 2007, 126, 084108−1−12. (7) See for example: Schwabe, T.; Grimme, S. Theoretical Thermodynamics for Large Molecules: Walking the Thin Line between Accuracy and Computational Cost. Acc. Chem. Res. 2008, 41, 569−579. (8) For an overview on density functional theory see: Koch, W.; Holthausen, M. C. A Chemist’s Guide to Density Functional Theory, 2nd ed.; Wiley: New York, 2001. (9) Chan, B.; Deng, J.; Radom, L. G4(MP2)-6X: A Cost-Effective Improvement to G4(MP2). J. Chem. Theory Comput. 2011, 7, 112− 120. (10) Chan, B.; Radom, L. Obtaining Good Performance with Tripleζ-Type Basis Sets in Double-Hybrid Density Functional Theory Procedures. J. Chem. Theory Comput. 2011, 7, 2852−2863. (11) Zhao, Y.; Truhlar, D. G. The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Functionals. Theor. Chem. Acc. 2008, 120, 215−241. (12) A number of alternative approaches have been proposed for obtaining accurate relative energies using a combination of high-level and low-level procedures. See for example: (a) Dapprich, S.; Komáromi, I.; Byun, K. S.; Morokuma, K.; Frisch, M. J. A New ONIOM Implementation in Gaussian 98. Part 1. The Calculation of Energies, Gradients and Vibrational Frequencies and Electric Field Derivatives. J. Mol. Struct. THEOCHEM 1991, 462, 1−21. (b) Izgorodina, E. I.; Brittain, D. R. B.; Hodgson, J. L.; Krenske, E. H.; Lin, C. Y.; Namazian, M.; Coote, M. L. Should Contemporary Density Functional Theory Methods be Used to Study the Thermodynamics of Radical Reactions? J. Phys. Chem. A 2007, 111, 10754−10768. (c) Wheeler, S. E.; Houk, K. N.; Schleyer, P. v. R.; Allen, W. D. A Hierarchy of Homodesmotic Reactions for Thermochemistry. J. Am. Chem. Soc. 2009, 131, 2547−2560. (d) Wodrich, M. D.; Corminboeuf, C.; Wheeler, S. E. Accurate Thermochemistry of Hydrocarbon Radicals via an Extended Generalized Bond Separation Reaction Scheme. J. Phys. Chem. A 2012, 116, 3436−3447. (e) Ramabhadran, R. O.; Raghavachari, K. Theoretical Thermochemistry for Organic Molecules: Development of the Generalized Connectivity-Based Hierarchy. J. Chem. Theory Comput. 2011, 7, 2094−2103. (f) Ramabhadran, R. O.; Raghavachari, K. Connectivity-Based Hierarchy for Theoretical Thermochemistry: Assessment Using Wave Function-Based Methods. J. Phys. Chem. A 2012, 116, 7531−7537. (13) See for example: (a) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986.

performance that is better than G4(MP2)-6X with only a small additional computational cost. Likewise, the use of M06-2X reference RBDEs, together with DARBDEs calculated using the computationally very efficient M06-L procedure, i.e., [M06-2X:M06-L(DA)], gives target RBDEs that are comparable in accuracy with standalone M06-2X calculations. 6. While the approximation schemes generally perform well, particularly for hydrocarbon substituents, large deviations from benchmark values are observed in some cases when the substrates contain multiple strongly π-electronwithdrawing and/or π-electron-donating groups.

ASSOCIATED CONTENT

S Supporting Information *

B3-LYP/cc-pVTZ+d zero-point vibration energies and thermal correction to enthalpies, and electronic energies for all species calculated at the various levels of theory (Table S1), deviations from benchmark BDEs, RBDEs, DARBDEs and DPARBDEs (Table S2), a comparison of the statistical performance of M062X with several additional DFT procedures (Table S3), and full citations for refs 14 and 15. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (B.C.); radom@chem. usyd.edu.au (L.R.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS we gratefully acknowledge funding (to L.R.) from the Australian Research Council (ARC) and generous grants of computer time (to L.R.) from the National Computational Infrastructure (NCI) National Facility and Intersect Australia Ltd.

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REFERENCES

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Hierarchy of Relative Bond Dissociation Enthalpies and Their Use to

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