Density Functional Steric Analysis of Linear and Branched Alkanes

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Density Functional Steric Analysis of Linear and Branched Alkanes Daniel H. Ess,*,† Shubin Liu,*,‡ and Frank De Proft§ Department of Chemistry and Biochemistry, Brigham Young UniVersity, ProVo, Utah 84602, United States, Research Computing Center, UniVersity of North Carolina, Chapel Hill, North Carolina 27599-3420, United States, and Vrije UniVersiteit Brussels, Pleinlaan 2, B-1050 Brussels, Belgium ReceiVed: September 8, 2010; ReVised Manuscript ReceiVed: October 15, 2010

Branched alkane hydrocarbons are thermodynamically more stable than straight-chain linear alkanes. This thermodynamic stability is also manifest in alkane bond separation energies. To understand the physical differences between branched and linear alkanes, we have utilized a novel density functional theory (DFT) definition of steric energy based on the Weiza¨cker kinetic energy. Using the M06-2X functional, the total DFT energy was partitioned into a steric energy term (Es[F]), an electrostatic energy term (Ee[F]), and a fermionic quantum energy term (Eq[F]). This analysis revealed that branched alkanes have less (destabilizing) DFT steric energy than linear alkanes. The lower steric energy of branched alkanes is mitigated by an equal and opposite quantum energy term that contains the Pauli component of the kinetic energy and exchangecorrelation energy. Because the steric and quantum energy terms cancel, this leaves the electrostatic energy term that favors alkane branching. Electrostatic effects, combined with correlation energy, explains why branched alkanes are more stable than linear alkanes. 1. Introduction

SCHEME 1: Hydrocarbon Reactions Studied 1

Experimental measurements and highly accurate ab initio calculations both agree that branched alkane hydrocarbons are thermodynamically more stable than straight-chain linear alkanes. Schleyer, Houk, Mo, (SHM), and co-workers have recently proposed a provocative explanation for this phenomenon; intramolecular 1,3-alkyl-alkyl interactions stabilize hydrocarbons, which they term protobranching or prototypical branching.2,3 The archetypal example of a stabilizing protobranch is seen in the transformation of two ethane (C2H6) molecules into propane (C3H8) and methane (CH4) (reaction 1, Scheme 1), which is -2.8 kcal/mol exothermic.2 SHM define a protobranching as “the net stabilizing 1,3-alkyl-alkyl interactions existing in normal, branched, and most cycloalkanes, but not in methane and ethane”.2 Scheme 2 shows that by definition, ethane has zero protobranches, while propane, butane, and isobutane have one, two, and three protobranches, respectively. The impact of this protobranching definition alters the schemes in which virtual thermodynamic quantities are computed and compared, for example, ring strain, conjugation, hyperconjugation, and aromatic resonance energies.2 In addition, the protobranching paradigm suggests that branched alkanes are more stable than linear due to more stabilizing protobranching interactions. Gronert has severely criticized SHM’s proposal that 1,3alkyl-alkyl interactions are stabilizing.4 In a response to SHM’s protobranching proposal, he points out that the reasons reaction 1 can be exothermic is either from unusual instability in ethane, unusual stability in methane or propane, or a combination of all three. Gronert favors the explanation that methane is unusually stable and that the widened angle in propane (112 versus 109° for a perfect tetrahedron) illustrates that 1,3* To whom correspondence should be addressed. [email protected] (D.H.E.); [email protected] (S.L.). † Brigham Young University. ‡ University of North Carolina. § Vrije Universiteit Brussels.

SCHEME 2: Number of Protobranches in Simple Branched Alkanes

alkyl-alkyl interactions are repulsive.5 Gronert has also pointed out that the protobranching effect cannot be rationalized by a through-space model of dispersive London forces. It is important

10.1021/jp108577g  2010 American Chemical Society Published on Web 11/18/2010

Density Functional Steric Analysis of Alkanes to note that Gronert’s repulsive view of 1,3-alkyl-alkyl interactions and SHM’s protobranching both lead to additivity schemes that accurately reproduce heats of formation for hydrocarbons.5,6 Gronert’s repulsive model is based on C-H bond strengths for primary, secondary, and tertiary alkanes with a lack of evidence for radical stabilization through hyperconjugation.4 In addition, Gronert has argued that a simple localized orbital mixing of two geminal C-H bonding orbitals leads to geminal closed-shell orbital destabilization. Ziegler and co-workers have carried out an energy decomposition analysis (EDA) of C-C and C-H bonds in alkanes to confirm Gronert’s proposal.7 Their calculations showed that C-C bonds are actually intrinsically stronger than C-H bonds but that after steric repulsion (defined as the sum of electrostatics and Pauli repulsions) is considered, they become weaker.8 Their EDA calculations also showed that this steric energy is more destabilizing with increasing substitution on the carbon center, presumably from the destabilizing 1,3-interaction outlined by Gronert. An early explanation for the stability of branched alkanes was offered by Pitzer and Calano based on dispersion interactions.9 However, SMH have pointed out that this dispersion explanation does not hold because the Hartree-Fock (HF) method lacks dispersion and is able to predict the correct stability trends.2 Ma and Inagaki have offered an orbital phase rational for why branched alkanes are more stable than linear.10 In their treatment, they found that σ cross conjugation between two C-H bonds and one C-C bond in an antiperiplanar conformation in branched alkanes results in greater orbital stabilization than straight-chain σ orbital conjugation.11 In a recent contribution, Kemnitz and co-workers have applied Weinhold’s natural bond orbital (NBO) analysis12 and valence bond theory to the stability of branched alkanes.13 They concluded that branching stability is the result of increased geminal carbon-carbon σ to σ* second-order donor-acceptor orbital interactions (hyperconjugation). Their analysis also concluded that steric repulsions destabilize branched alkanes and that Gronert’s hydrocarbon additivity scheme is successful due to linearly dependent variables rather than a correct physical description of alkanes. Although geminal hyperconjugation undoubtedly favors alkane branching, HF and density functional theory (DFT) methods that fail to adequately reproduce relative alkane energies predict significant hyperconjugation stabilization in branched alkanes.14 This is a result of a perturbative type of energy analysis that is referenced to a hypothetical localized state that does not analyze/ partition the total energy of each molecule. Using HF and MP2 methodology, Wiberg compared the energetics of t-BuR and n-BuR species (R ) H, CH3, Li, F, OH, O-, and NH3+) and found that electropositive substituents favor the linear isomer, while electronegative substituents favor the branched isomer.15 For example, the energy difference between t-BuF and n-BuF is -10.7 kcal/mol at the MP2/631G(d,p) level, while for t-BuLi and n-BuLi, the energy difference is 3.0 kcal/mol. On the basis of this argument, the energy difference between isobutane/butane and neopentane/ pentane is due to more electron-withdrawing groups, in this case, CH3 groups, attached to a single carbon. Wiberg was also the first to show that electron correlation, at the MP2 level of theory, was important to give accurate energy differences between branched and linear alkanes. Later in 2000, Curtiss and co-workers showed that B3LYP performed very poorly in comparison to Gaussian 03 theory for the calculation of enthalpies of formation for large n-alkanes.16

J. Phys. Chem. A, Vol. 114, No. 49, 2010 12953 In 2006, Grimme showed that popular density functional methods, such as the hybrid B3LYP method, generally fail to predict that branched alkanes are more stable than linear alkanes.17 In comparing HF, DFT, and MP2 methods, Grimme argued that electron correlation effects rationalize the stability of branched alkanes. Using localized molecular orbital MP2 calculations, Grimme was able to partition the electron correlation effects into regions of space and show that the electron correlation effects that stabilize branched alkanes result from interpair correlations between orbitals of similar type, that is, C-C with C-C and C-H with C-H. In contrast, electron correlation between C-C and C-H bonds and long-range correlations favor linear alkanes. Since Grimme’s report, there has been a flurry of papers exploring the energetic differences of linear and branched alkanes and the performance of DFT methods.18 Most recent, Corminboeuf and co-workers have shown that although n- and singly methylated alkanes are most stable in their branched forms, too much branching can cause destabilization for permethylated alkanes with R(CH3)2C · · · C(CH3)2R motifs.19 In this contribution, we are concerned with understanding the physical differences between branched and linear n-alkanes. It is intuitive that the isomerization from a linear to a branched alkane involves differences in steric energy. Like many concepts in chemistry, there is no exact definition of sterics. Liu has recently proposed that the Weiza¨cker kinetic energy (see section 2 for details) can be used as a novel definition of sterics in DFT.20,21 This DFT definition of the steric effect differs from the typical wave function definition resulting only from orthogonality and exchange antisymmetry of the wave function (the Fermi hole).12 This DFT steric definition is very much in accord with the early definition by Weisskopf who attributed sterics to “kinetic energy pressure”.22 Here, we utilize this DFT definition of steric energy in an energy partition analysis to understand the total physical differences between branched and linear alkanes. We are not concerned with isolating a specific intramolecular interaction. 2. Theory and Methodology In DFT, the energy of a system (E[F]) typically results from three terms, Ts[F], Ee[F], and Exc[F] (eq 1).23 Ts[F] is the noninteracting, one-electron kinetic energy, Ee[F] is the electrostatic energy, and Exc[F] is the exchange-correlation energy term. The electrostatic term is a composite of nuclear to electron stabilization (Vne[F]), electron-electron Coulombic repulsion (J[F]), and nuclear-nuclear repulsion (Vnn[F]) (eq 2). In an effort to quantitatively define steric energy within a DFT framework, Liu has proposed that steric energy can be defined as the Weiza¨cker kinetic energy, Es[F] (eq 3).21 This is different from the wave function evaluation of steric energy from the antisymmetrization of the wave function as a result of the Pauli exclusion principle that only results from same-spin exchange. Qualitatively, this Weiza¨cker kinetic energy term shows the amount of kinetic energy compression that is applied to electrons in a given nuclear configuration as measured by the change in the electron density gradient normalized to the total density. As a result of this steric energy definition, a new energy decomposition scheme can be defined that includes this steric term (Es[F]), an electrostatic term (Ee[F]), and a fermionic quantum term (Eq[F]). The electrostatic term is exactly the same as that defined in conventional DFT partitioning. An important difference between the classic DFT energy partitioning (eq 1) and this new scheme (eq 4) is that the Pauli component of the kinetic energy is defined here to be Ts - Es or Ts - TW. The

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TABLE 1: M06-2X/6-311++G(2d,2p) Exchange (∆Ex), Correlation (∆Ec), Kinetic (∆Ts), Electrostatic (∆Ee), Steric (∆Es), and Quantum (∆Eq) Energies for Reactions 1-10 in Scheme 1a rxn

∆Ex

∆Ec

∆Exc

∆Ts

∆Ee

∆Es

∆Eq

∆ENBO

∆E

1 2 3 4 5 6 7 8 9 10

2.5 2.1 6.0 2.2 0.1 6.7 12.8 4.6 6.7 9.0

-1.5 -1.6 -4.9 -2.6 0.0 -4.7 -9.6 -3.1 -4.7 -5.7

1.0 0.5 1.1 -0.4 0.0 2.0 3.1 1.5 2.0 3.4

-0.1 0.5 -3.2 -1.5 -0.1 0.1 -3.8 -0.4 -0.6 -0.4

-2.9 -2.4 -1.5 1.0 0.0 -7.5 -8.7 -5.1 -7.2 -12.1

-23.5 -19.0 -54.5 -29.8 -0.9 -67.8 -127.8 -48.8 -73.2 -85.6

24.4 20.0 52.4 27.9 0.8 69.9 127.1 49.9 74.6 88.6

10.0 10.7 32.8 11.1 -0.2 30.9 63.4 20.2 30.6 55.6

-2.0 -1.4 -3.6 -0.9 -0.1 -5.3 -9.4 -3.9 -5.9 -9.1

a ∆ENBO is the difference in NBO steric energy. The classic DFT partition of total energy (∆E) is the sum of ∆Exc + ∆Ts + ∆Ee. The steric energy partition of total energy (∆E) is the sum of ∆Exc + ∆Eq + ∆Ee (kcal/mol).

quantum energy, Eq[F], contains this Pauli component of the kinetic energy and Exc[F] (eq 5).

E[F] ) Ts[F] + Ee[F] + Exc[F]

(1)

Ee[F] ) Vne[F] + J[F] + Vnn[F]

(2)

Es[F] ) TW[F] ) (1/8)

∫ [|∇F(r)|2/F(r)] dr

E[F] ) Es[F] + Ee[F] + Eq[F]

(3) (4)

Eq[F] ) Exc[F] + Ts[F] - Es[F] ) Exc[F] + Ts[F] - TW[F] (5) All hydrocarbon structures were optimized and verified as minima with B3LYP/6-31G(d,p) using Gaussian 03.24 Energy evaluations and steric energy decomposition calculations were performed using the M06-2X functional with the 6-311++G(2d,2p) basis set in a modified version of NWChem 5.025 or in Gaussian 09.26 The M06-2X functional gives isomerization energies that are very close to that predicted by CCSD(T) and experiment.2 Zeropoint energies were not included in our evaluation. The NBO steric analysis12 was carried out using a stand-alone NBO code 5.027 or as implemented in Gaussian 09.28 In this STERIC analysis, the steric exchange is considered by Weinhold to be the Weisskopf’s “kinetic energy pressure” resulting from enforcement of mutual orthogonality and exchange antisymmetry of the wave function.29 Here, we use the pairwise-additive estimate that is the result of pairs of natural localized molecular orbitals compared before and after orthogonalization. 3. Results and Discussion The isodesmic reaction 1 (Scheme 1) involves the transformation of two ethane molecules into propane and methane (2 C2H6 f C3H8 + CH4). The ∆E is -2.0 kcal/mol (Table 1) at the M06-2X/6-311++G(2d,2p) level of theory, which is very close to the CCSD(T)/6-311++G(d,p) and SCS-MP2/6311++G(d,p) values of -2.1 and -2.0 kcal/mol, respectively. Adding SHM’s zero-point and thermal correction of 0.5 kcal/mol gives the M06-2X predicted value of -2.5 kcal/mol, compared to -2.8 kcal/mol found experimentally. For a complete list of experimental bond separation enthalpies, see Table 1 in ref 2. It is important to note that SHM as well as Grimme found that HF and the hybrid DFT methods, such as B3LYP, underestimate this

branching stability but do predict the correct stability of propane and methane versus two ethane units. Reaction 2 involves isomerization of butane to isobutane. The M06-2X ∆E is -1.6 kcal/mol. This is in exact agreement with SHM’s CCSD(T) value. The linear to branched isomerization of n-pentane to neopentane (reaction 3) is exothermic by -3.6 kcal/mol, while the isomerization to isopentane is only exothermic by -0.9 kcal/mol (reaction 4). Reaction 5 is an isodesmic reaction that provides a nice example where energetic effects are nearly equal. The reaction energy for propane and n-pentane giving two molecules of n-butane is only -0.1 kcal/ mol exothermic. Reactions 6-10 are isodesmic reactions that start with ethane units and result in methane along with either isobutane, neopentane, n-butane, n-pentane, or cyclohexane. These reactions are all exothermic, especially reactions 7 and 10, and SHM have shown that these bond separation reaction energies are related to the number of protobranches. n-Butane, n-pentane, isobutane, neopentane, and cyclohexane have 2, 3, 3, 6, and 6, respectively, and, on average, between 2.7 and 2.8 kcal/mol of stabilization per protobranch. Table 1 gives the M06-2X exchange (∆Ex) and correlation (∆Ec) energies for reactions 1-10. The sum of these terms is given in the classic DFT energy partitioning as ∆Exc along with the combined kinetic and electrostatic energy terms (∆Ts and ∆Ee). It is clear that the exchange energy favors the left sides of reactions 1-10, whereas the correlation energy favors branching (right sides of reactions 1-10) species. However, if exchange and correlation energies are combined, the exchange term dominates, and this quantity favors less branching. Grimme has previously shown that branched alkanes contain more correlation energy stabilization than linear alkanes.17 To quantify this relationship, Figure 1 shows a plot of exchange-correlation energy versus reaction energy for reactions 1-10. As expected on the basis of Grimme’s work,17 there is a reasonable linear correlation with an R2 regression value of 0.95. If only the correlation energy is plotted against the reaction energy, the R2 value drops to 0.81, and the slope of the line is positive (see Supporting Information). There is also a good linear correlation between the reaction energies and the electrostatic energy (∆Ee[F]), with R2 ) 0.87 (Figure 2, Table 1). For isomerization reactions 2-4, the correlation stabilization and the ∆Ee term stabilization are nearly equal in magnitude. However, for the bond separation reactions, the electrostatic term is more stabilizing than the correlation energy term. On the basis of these linear correlations and the difficulty of directly interpreting the physical meaning of these DFT energies, we have utilized a DFT steric energy decomposition to investigate the origin of alkane branching stabilization.

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Figure 1. Plot of reaction energy versus (a) exchange-correlation energy and (b) electrostatic energy (kcal/mol).

Figure 2. Plot of the reaction energy versus the change in the DFT steric energy, quantum energy terms versus steric energy terms, and the reaction energy versus the NBO steric energy.

The clustering of carbon-carbon bonds in branched alkanes intuitively suggests that there is a steric energy difference between branched and linear alkanes. Although not unique, a novel and conceptually compelling definition of sterics within the context of density functional theory is to use the Weiza¨cker kinetic energy (eq 3, section 2). Table 1 gives the M06-2X change in the Weiza¨cker kinetic energy or steric energy (∆Es) along with electrostatic (∆Ee) and quantum energy terms (∆Eq) for reactions 1-10. The total change in energy (∆E) is the sum of ∆Exc + ∆Eq + ∆Ee. For each molecule, the steric energy is a positive absolute energy that is related to the change in electron density gradient over the space of the molecule relative to the total electron density. For reactions 1-10, the change in steric energy (branched - unbranched) is negative. A plot of steric energy versus reaction energy also produces a nice linear correlation with an R2 value of 0.91 (Figure 2). This indicates that linear alkanes are more sterically demanding in a “global” molecular sense compared to branched alkanes. However, branched alkanes are obviously more “locally” congested at the branching carbon center, and the electron density associated with the branched nuclear configuration is more compact. Table 1 also shows that the increased steric energy stabilization in branched alkanes is mitigated by a nearly equal and opposite in magnitude quantum term (∆Eq). A plot of ∆Es versus

∆Eq terms gives a near-perfect linear correlation with R2 ) 0.99 (Figure 2). The quantum term contains the Pauli component of the kinetic energy and Exc[F]. Because this term contains Pauli repulsion, the more compact branched alkane results in more same-spin exchange repulsion, which results in significant destabilization. As a comparison, the wave-function-based NBO steric energies are also given in Table 1. In accord with the ∆Eq from the DFT steric decomposition, these values are positive and indicate that there is more exchange (Pauli) repulsion in the branched alkanes than in the unbranched alkanes.13 There is also a good correlation between reaction energies and the NBO steric energy change (Figure 2). The third term in the energy decomposition, ∆Ee, is smaller but always favors branched alkanes, except for reaction 4. As noted earlier, there is a linear correlation between the electrostatic energies and the energies of reaction. This suggests that in addition to correlation effects,17 classic electrostatic interactions also significantly contribute to stabilizing branched alkanes relative to unbranched alkanes. The electrostatic stabilization between electrons and nuclei is easily envisioned in light of the more compact nuclear structure of branched alkanes (see below), which is brought about by an increase in both steric energy and quantum energy terms.

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In the same spirit of our analysis, Laidig previously compared stabilizing and repulsive forces for the isomerization reactions of butane to isobutane and pentane to neopentane/isopentane (reactions 2-4).30 Laidig defined attractive forces as being only the nuclear-electron potential term (Vne) and repulsive forces as being electron-electron (Vee) and nuclear-nuclear repulsions (Vnn). In Laidig’s analysis, kinetic energy was considered to be a minor contribution to the change in energy, and correlation energies were neglected. With the HF method, Laidig found that upon isomerization from linear to branched alkanes, there is an increase in both attractive and repulsive forces, with attractive forces being slightly larger. This is in accord with our results that show large, but opposite, increases in Es and Eq terms (Table 1). Laidig concluded, although without quantifying atomic distances, that branched alkane carbon atoms are on average closer to other carbon atoms and that hydrogen atoms are closer to other hydrogen atoms, which results in more stabilizing potential energies for branched alkanes. Stated another way, atoms in the interior of a molecular are more stable than atoms near the exterior. To quantify relative carbon and hydrogen distances, we have compared the average pairwise carbon and pairwise hydrogen atom distances between butane/ isobutane and pentane/neopentane. For pentane, the average carbon pairwise distance is 5.36 Å. This average distance drops to 4.69 Å in neopentane. In butane, the average pairwise carbon distance is 3.41 Å, and it is 3.11 Å in isobutane. The average pairwise hydrogen distance in butane is 13.99 Å, and in isobutane, it is 13.33 Å. This geometric analysis indicates that although the nuclei in branched alkanes are more compact, which leads to destabilizing exchange repulsion effects manifest in the Eq term, the electronic structure is more stable due to a mitigating stabilization in the steric energy term as well as more stabilizing electrostatic interactions. 4. Conclusions Alkane isomerization and bond separation reaction energies were computed with the M06-2X functional. In accord with the previous work of Grimme,17 there is a clear connection between the increase in correlation energy of branched alkanes and their stability. In addition, we have performed a DFT steric energy decomposition that has revealed that branched alkanes have less destabilizing steric energy than linear alkanes. However, branched alkanes have a larger destabilizing quantum energy term (Pauli kinetic energy and exchange-correlation energy) because branched alkanes are more compressed. These effects mitigate each other, leaving the electrostatic energy term, in conjunction with electron correlation,17 responsible for branched alkane stability. Acknowledgment. D.H.E. thanks BYU for financial support and access to the Fulton Supercomputing Lab (FSL). D.H.E. especially thanks Tom Raisor, Lloyd Brown, and Ryan Cox of FSL. S.B.L. thanks UNC for computer support and acknowledges partial support from the Chemical Sciences, Geosciences and Biosciences Division of the Office of Basic Energy Sciences, U.S. Department of Energy (Grant DE-FG0206ER15788) and UNC EFRC: Solar Fuels and Next Generation Photovoltaics, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences (Award No. DE-SC0001011). F.D.P. wishes to acknowledge the Research Foundation-Flanders (FWO) and the Free University of Brussels (VUB) for continuous support to his research group.

Ess et al. Supporting Information Available: XYZ coordinates, absolute energies, full refs 24 and 26, and B3LYP steric energy analysis results. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) (a) Ipatieff, V. N.; Grosse, A. V. Ind. Eng. Chem. 1936, 28, 461. (b) Olah, G. A.; Molnur, A. Hydrocarbon Chemistry; Wiley: New York, 1995. (c) Rossini, J. J. Res. Natl. Bur. Stand. (U.S.) 1934, 13, 21. (d) Afeefy, H. Y.; Liebman, J. F.; Stein, S. E. Neutral Thermochemical Data. In NIST Chemistry Webbook, NIST Standard Reference Database Number 69; National Institute of Standards and Technology: Gaithersburg, MD, 1996; http://webbook.nist.gov. (2) Wodrich, M. D.; Wannere, C. S.; Mo, Y.; Jarowski, P. D.; Houk, K. N.; Schleyer, P. v. R. Chem.sEur. J. 2007, 13, 7731. (3) (a) Allen, T. L. J. Chem. Phys. 1959, 29, 951. (b) T Allen, T. L. J. Chem. Phys. 1959, 31, 1039. (4) Gronert, S. Chem.sEur. J. 2009, 15, 5372. (5) Gronert, S. J. Org. Chem. 2006, 71, 1209. (6) Wodrich, M. D.; Schleyer, P. v. R. Org. Lett. 2006, 8, 2135. (7) Mitoraj, M.; Zhu, H.; Michalak, A.; Ziegler, T. J. Org. Chem. 2006, 71, 9208. (8) (a) Eyring, H. J. Am. Chem. Soc. 1932, 54, 3191. (b) Ru¨chardt, C. Angew. Chem., Int. Ed. Engl. 1970, 9, 830. (c) Ruu¨chardt, C.; Beckhaus, H.-D. Angew. Chem., Int. Ed. Engl. 1980, 19, 429. (d) Grimme, S. Angew. Chem., Int. Ed. 2006, 45, 1. (e) Zavitsas, A. A. J. Chem. Educ. 2001, 78, 417. (f) Matsunaga, N.; Rogers, D. W.; Zavitsas, A. A. J. Org. Chem. 2003, 68, 3158. (g) Coote, M. L.; Pross, A.; Radom, L. Org. Lett. 2003, 5, 4689. (h) Wodrich, M. D.; Corminboeuf, C.; Schleyer, P. v. R. Org. Lett. 2006, 8, 3631. (i) Gronert, S. J. Org. Chem. 2006, 71, 7045. (j) Fernandez, I.; Frenking, G. Chem.sEur. J. 2006, 12, 3617. (9) Pitzer, K. S.; Catalano, E. J. Am. Chem. Soc. 1956, 78, 4844. (10) Ma, J.; Inagaki, S. J. Am. Chem. Soc. 2001, 123, 1193. (11) Ito, K. J. Am. Chem. Soc. 1953, 75, 2430. (12) (a) Badenhoop, J. K.; Weinhold, F. J. Chem. Phys. 1997, 107, 5406. (b) Badenhoop, J. K.; Weinhold, F. Int. J. Quantum Chem. 1999, 72, 269. (13) Kemnitz, C. R.; Mackey, J. L.; Loewen, M. J.; Hargrove, J. L.; Lewis, J. L.; Hawkins, W. E.; Nielsen, A. F. Chem.sEur. J. 2010, 16, 6941. (14) The table below gives our results for the difference in NBO delocalization energies (∆Edeloc) between pentane and neopentane for HF and DFT methods with the 6-31G(d,p) basis set. The second-order perturbative delocalization energies are computed by comparison of the fully relaxed electronic structure (∆ESCF) with a localized Lewis bonding state (∆Elocal) using the NBOdel keyword in the NBO 3.1 program attached to Gaussian 09. Although only the M06-2X and ωB97XD DFT methods predict ∆ESCF values close to experiment, all methods predict very similar differences in delocalization energies (∆Edeloc). This indicates that hyperconjugation does stabilize branched alkanes but is unlikely responsible for the total energy difference between branched and linear alkanes. In addition, NBO delocalization energies are very basis set dependent. With the larger 6-311G++(d,p) basis set only HF predicts delocalization interactions to stabilize neopentane over pentane. The NBO delocalization energies for reactions 1, 2, and 7 also show a similar result for HF and M06-2X methods. Again, despite similar donor-acceptor second-order stabilizing interactions, HF and M06-2X give very different ∆ESCF values.

Density Functional Steric Analysis of Alkanes

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