Performance of PNOF6 for Hydrogen Abstraction Reactions

Article pubs.acs.org/JPCA

Performance of PNOF6 for Hydrogen Abstraction Reactions Xabier Lopez,*,† Mario Piris,†,§ Fernando Ruipérez,‡ and Jesus M. Ugalde† †

Kimika Fakultatea, Euskal Herriko Unibertsitatea UPV/EHU, and Donostia International Physics Center (DIPC), P.K. 1072, 20080 Donostia, Euskadi, Spain ‡ POLYMAT, University of the Basque Country UPV/EHU, Joxe Mari Korta Center, Avda. Tolosa 72, 20018 Donostia-San Sebastián, Spain § IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain ABSTRACT: Radical formation through homolytic X−H bond cleavage in LiH, BH, CH4, NH3, H2O, and HF is investigated using natural orbital functional theory in its recent PNOF6 implementation, which includes interelectron-pair correlation, and the results are compared to those of the PNOF5 level of theory, CASSCF wave function methods, and experimental data. It is observed that PNOF6 is able to improve the estimation of the corresponding dissociation energies (De) with respect to PNOF5. When PNOF6 is combined with a better description of the electron pair, through the use of an extended number of coupled orbitals, we obtain further improvements of these quantities. The convergence of the corresponding De values with the number of coupled orbitals is also discussed, finding that a proper convergence of the results is attained with three orbitals. Next, we apply PNOF6 and its improved version PNOF6(3) to describe the thermodynamics of C−H homolytic bond cleavage for a data set of 20 organic molecules in which the C−H bond is broken in the context of different chemical environments. Finally, the radical stabilization energies obtained for such a general data set are compared with the experimental data, demonstrating that the inclusion of interelectron-pair correlation in natural orbital functional theory as in PNOF6 gives a resonable description of radical stability, especially as electron pair description is improved.

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INTRODUCTION Homolysis of X−H covalent bonds is an important process in biological1,2 and organic chemistry.3 In particular, understanding the thermodynamic stability and formation of radicals by hydrogen abstraction is a fundamental step for explaining oxidation of hydrocarbons4−6 and side chains of proteins, lipidperoxidation,2 the formation of reactive oxygen species7 (ROS), Fenton chemistry,8 and DNA damage.9 The proper description of the X−H homolytic bond dissociation curves is a fundamental step for the accurate characterization of the electronic structure of these important species.10−13 However, this requires the appropriate treatment of strong correlation effects, which unfortunately is not without difficulty. In wave function methods, the use of a single-reference methods such as Hartree−Fock (HF) leads to incorrect results; therefore, there is a need to include several determinants in the corresponding wave function, leading to computationally demanding methods. Valence bond theory has also been used for this type of system.14 Other alternative methods, such as density functional theory (DFT), suffer from methodological problems for treating strong electron correlation or near-degeneracy effects.15−17,41 However, it should be noted that accurate and cost-effective bond dissociation energies can be obtained in the context of open shell DFT calculations with unrestricted methods and symmetry-breaking solutions.18,19 Natural orbital functional theory20 (NOFT) is being configured as an alternative formalism to both DFT and wave function methods by describing the electronic structure in © 2015 American Chemical Society

terms of the natural orbitals and their occupation numbers. Various functionals have been developed in the last years, and a comprehensive review can be found in refs 21−31. Within the family of Piris natural orbital functionals,21−25 those satisfying the known necessary conditions for some of the known Nrepresentability of the second-order reduced density matrix (2RDM)32 are the best-suited to describe strong correlation effects. Thus, the so-called PNOF424 functional can properly describe dissociation curves of a variety of molecules and the ethylene torsion potential33 and shows an outstanding performance in the treatment of small diradicals and diradicaloids.34 Later, PNOF5 was developed,25 which is an independent electron pair model, showing the correct dissociation limit for a variety of molecules in various chemical types of bonding and bond order, integer number of electrons in the dissociation limit,35 and a good performance for a variety of hydrogen abstraction reactions.36 This model was further improved by a better description of the electron pair in the socalled PNOF5e37 (extended PNOF5). However, PNOF5 and PNOF5e show limitations with respect to orbital delocalization in aromatic systems such as in benzene, a key aspect in radical stabilization. A new functional PNOF6 was developed recently,38 which includes interpair correlations, and was Received: February 16, 2015 Revised: June 11, 2015 Published: June 11, 2015 6981

DOI: 10.1021/acs.jpca.5b01585 J. Phys. Chem. A 2015, 119, 6981−6988

Article

The Journal of Physical Chemistry A

The first term of the energy in eq 3 draws the system as independent F electron pairs described by the following NOF of two-electron systems

demonstrated to be able to treat aromatic systems, leading to a proper delocalization of the π natural orbitals. In the present paper, we analyze the performance of the interpair correlated PNOF6 and its improved version, PNOF6(Nc = 2,3,4), with an electron pair description ameliorated by inclusion of a higher number of coupled orbitals, for the description of X−H bond dissociations. To do so, we have evaluated the dissociation energy (DXH e ) for X−H bonds in LiH, BH, CH4, NH3, H2O and HF. Results are compared to those of PNOF5, in which the interelectron pair correlation is absent, and experimental data. Because of the widespread interest39 in the thermodynamic stability of organic radicals, we then analyze the cleavage of C− H bond in a data set of 20 organic molecules. Bond dissociation energy has often been used in the literature11,39,40 to provide a measure of radical stability and to test the performance of various theoretical methods. In addition, on the basis of the dissociation energies, we have estimated the radical stabilization energy (RSE) for a variety of hydrogen abstraction reactions, namely XH + Y • → X• + YH

Eg =

(1)

Δqp

Πqp

Orbitals

e−2Shqhp

−e−S(hqhp)1/2

q ≤ F, p ≤ F

γqγp e−2Snqnp

q > F, p > F

(6)

⎧ e −S h , p ≤ F p ⎪ αp = ⎨ S − ⎪ e np , p > F ⎩ γ Π qp

1/2 1/2 ⎛ γqγp ⎞ ⎛ γqγp ⎞ ⎟⎟ ⎜⎜hqnp + ⎟⎟ = ⎜⎜nqhp + Sγ ⎠ ⎝ Sγ ⎠ ⎝ F

S=

∑ hq , q=1

F

Sα =

F

∑ αq , q=1

Sγ =

∑ γq q=1

(7)

43

The solution in NOF theory is established by optimizing the energy functional with respect to the occupation numbers and to the natural orbitals separately, for which the iterative diagonalization procedure proposed by Piris and Ugalde44 has been employed. This scheme requires computational times that scale as N4, with N being the number of basis functions, as in the Hartree−Fock (HF) approximation. However, our implementation in the molecular basis set requires also fourindex transformation of the electron repulsion integrals, which is the time-consuming step, though a parallel implementation of this part of the code has substantially improved the performance of our program. One should remark the more favorable scaling of PNOF6 over CASSCF (n × N6, with n window size). In this sense, PNOF6 appears as an alternative method for

F f ≠ g p ∈Ω f q ∈Ωg

e−S(nqnp)1/2

γp = nphp + αp2 − αpSα

where p denotes a spatial natural orbital and np its occupation number. This involves coupling each orbital, g, below the Fermi level (F = N/2) with one orbital above it, so Ωg ≡ (g, g̃) is the geminal subspace containing the orbital g (g ≤ F) and its coupled orbital g̃ (g̃ > F). In the case of independent pairs, it was demonstrated37 that the description of each pair can be further improved by inclusion of more orbitals in each geminal, which gave rise to PNOF5e. Following this recipe, in this work we will consider a more general functional PNOF6(Nc) by coupling Nc orbitals to each orbital g. Accordingly, each subspace Ωg contains now an orbital g (g ≤ F) and its Nc coupled orbitals with q > F. If Nc = 1, PNOF6(1) corresponds obviously to the simplest formulation PNOF6.38 The PNOF6(Nc) energy for a singlet state of an N-electron molecule can be cast as

g=1

q > F, p ≤ F

where hp denotes the hole 1−np in the spatial orbital p. The magnitudes γ and Πγ are given by

(2)

F

(5)

q ≤ F, p > F

γ −Π qp

Sγ

g = 1, F

∑ Eg + ∑ ∑ ∑

(4)

where 1pq = ⟨pq|pq⟩ and 2pq = ⟨pq|qp⟩ are the usual direct and exchange integrals, respectively. 3 pq = ⟨pp|qq⟩ is the exchange and time-inversion integral,42 which differs only in phases of the natural orbitals with respect to the exchange integrals; therefore, 3 pq = 2pq for real orbitals. The PNOF6 ansatz for the off-diagonal elements of Δ and Π matrices is

METHODS We briefly describe here the theoretical framework of our approach. A more detailed description of PNOF6 can be found in ref 38. Recall that PNOF6 is an orbital pairing approach, which is reflected in the sum rule for the occupation numbers, namely

E=

p , q ∈Ωg , p ≠ q

int Epq = (nqnp − Δqp)(21pq − 2pq) + Πqp3 pq

■

np = 1,

int Epq

∑

where /pp is the matrix element of the kinetic energy and nuclear attraction terms, whereas 1pp = ⟨pp|pp⟩ is the Coulomb interaction between two electrons with opposite spins at the spatial orbital p. It is worth noting that the last terms of eqs 3 and 4 contain the interactions between the electrons in different pairs and inside a pair, respectively. The interaction energy Eint pq is given by

RSEs are frequently used to analyze the stability of radicals caused by different chemical substituents and are equivalent to the difference in bond dissociation energies of XC−H and Y−H species. The comparison of 210 RSEs calculated at PNOF6 with experimental data reveals that the introduction of interelectron pair correlation in PNOF6 leads to a reasonable description of the stability of these important radicals and consequently puts forward explicit inclusion of short-range electron correlation effects as a key development for making natural orbital functional theory a robust method for the elucidation of the molecular electronic structure.

∑

np(2/pp + 1pp) +

p ∈Ωg

10,12,18,19

p ∈Ωg

∑

int Epq

(3) 6982


Article


Table 1. Formation of Radicals from X−H Bond Dissociation, in Kilocalories Per Mole, with X = Li, B, CH3, NH2, OH, Fa LiH →·Li + H·

BH →·B + H·

CH4 →·CH3 + H·

NH3 →·NH2 + H·

H2O →·OH + H·

FH →·F + H·

MAE

95.0 102.4 109.4 111.2 111.8 95.1 106.8

99.4 106.7 112.9 114.7 113.4 99.4 114.8

122.2 113.2 116.0 119.8 − 107.8 126.6

17.5 14.6 11.0 9.1 − 19.2 8.8

97.6 104.8 112.2 114.4 114.2 97.3 112.2 115.9

103.0 110.4 117.7 120.1 118.4 102.7 122.0 126.0

114.3 119.5 125.1 127.8 128.7 113.4 136.9 141.1

17.1 11.8 7.0 6.5 6.4 16.5 3.4

cc-pVDZ PNOF5 PNOF6 PNOF6(2) PNOF6(3) PNOF6(4) CASSCF(2,2) CASPT2(2,2)

42.8 42.8 45.7 48.5 51.1 42.8 49.2

73.0 78.4 88.4 89.4 89.5 78.2 78.7

98.1 104.6 110.6 112.9 113.7 97.3 106.6

PNOF5 PNOF6 PNOF6(2) PNOF6(3) PNOF6(4) CASSCF(2,2) CASPT2(2,2) exptl

44.1 44.1 47.8 51.3 54.3 44.0 53.4 58.0

75.3 80.7 84.0 92.6 92.4 81.1 81.7 81.5

98.6 105.2 112.0 113.3 114.8 98.0 109.6 113.0

cc-pVTZ

a

Dissociation energies in kilocalories per mole, calculated from single-point energies at various levels of PNOF theoryb; ZPVEs were addedc to the experimental dissociation energies.56 bCalculations of the energy at the dissociation limit were done at an X−H distance of 5 Å. cZPVEs were taken from the Computational Chemistry Comparison and Benchmark DataBase57 and correspond to CCSD(T)/cc-pVTZ values.

CAS(2,2) and CAS(2,2)PT2 results are provided for comparison. X−H Homolytic Bond Cleavage. Homolytic X−H bond cleavage energies were calculated considering the following reaction: XH →·X + H· with X = Li, B, CH3, NH2, OH, F. The results are presented in Table 1 and Figure 1. The different

treating problems such as hydrogen abstraction and homolytic bond cleavage for which wave function methods are prohibitive. However, it should be emphasized that for the thermodynamics of X−H bond cleavage itself one could get a good performance with DFT methods, when symmetry-breaking solutions are used and calculations are done at the separated radical species.18,19 Geometries were optimized at the M06-2X45 level of theory using the GAUSSIAN09 program package.46 The dissociation limit was calculated by considering a frozen X−H distance of 5 Å and optimizing the rest of internal coordinates. At these geometries, single-point energies were evaluated at the PNOF5 and PNOF6(Nc) levels of theory. All calculations have been carried out using the DoNOF program package [DoNOF] with the correlation-consistent valence double-ζ (cc-pVDZ) or triple-ζ (cc-pVTZ) basis sets developed by Dunning.47 The matrix element of the kinetic energy and nuclear attraction terms, as well as the electron repulsion integrals, are inputs to our computational code. In the current implementation, we have used the GAMESS program48,49 for this task. In addition, we have also performed complete active space self-consistent field calculations (CASSCF)50−52 where the active space is defined by the distribution of two electrons in two molecular orbitals. Dynamic correlation effects are included through complete active space second-order perturbation theory calculations (CASPT2).53,54 The MOLCAS 8.0 suite of programs55 was used in all the wave function-based calculations.

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Figure 1. PNOF De dissociation energies, in kilocalories per mole, for X−H bonds (X = Li, B, CH3, NH2, OH, F) versus experimental values. Calculations carried out with the cc-pVTZ basis set.

RESULTS AND DISCUSSION In this section, the results are presented and discussed. They are organized as follows. First, the X−H bond dissociation energies for LiH, BH, CH4, NH3, H2O, and FH are studied using PNOF5, PNOF6, and PNOF6(Nc = 2, 3, 4) levels of theory. Then, we analyze the performance of PNOF6 for describing C−H bond cleavage in a variety of 20 organic molecules. Finally, radical stabilization energies are calculated based on the calculated C−H bond dissociation energies to test the capacity of PNOF6 to describe radical stability. In addition,

hydrides considered expand a wide range of De’s, from 58.0 kcal/mol of LiH to 141.1 kcal/mol of FH. The ordering in dissociation energies is LiH < BH < CH4 < NH3 < H2O < FH. In general, all methods reproduce satisfactorily these trends (see 1), except for the CH4 and NH3 ordering. The difference in experimental dissociation energies for these two molecules is very small, only 2.9 kcal/mol with NH3 having a higher 6983


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The Journal of Physical Chemistry A Table 2. C−H Bond Dissociation Energies (De), in Kilocalories Per Mole, for a Data Set of 20 Organic Moleculesa CH4 CH3CH3 CH3CH2CH3 CH3CH2CH3 CH3F CF2H2 CF3H CH3OH CH3COH H2CO CH3OCH3 CH3COOH C2H4 C2H2 CH3CCH HCN CH3CN CH3NO2 CF3CH3 C6H6 C6H6CH3 MAE

a b

exptl

PNOF6

PNOF6(3)

PNOF6’(3)

CAS(2,2)

CAS(2,2)PT2

112.7 109.7 106.9 108.5 108.7 111.8 113.5 103.2 100.8 95.6 102.5 100.8 119.3 141.8 97.1 132.5 103.9 106.5 113.8 120.5 96.1

104.8 104.6 103.0 104.7 112.8 115.3 116.7 108.5 114.2 107.1 113.0 114.4 125.9 144.5 111.5 139.1 119.0 132.3 117.4 128.4 112.3 8.8

112.8 111.2 110.8 112.7 111.4 118.3 121.2 113.4 110.6 112.0 113.7 113.8 128.9 147.9 112.1 141.2 119.4 127.5 114.6 129.8 111.9 9.0

103.1 101.7 99.8 101.9 105.4 106.9 107.7 96.0 104.8 93.9 101.4 98.6 120.2 133.9 98.0 137.3 104.4 113.1 106.4 110.2 93.3 4.9

97.3 96.0 94.5 96.5 97.9 100.3 101.7 94.5 91.7 81.6 95.6 92.9 103.7 127.0 92.1 126.5 94.8 98.2 99.1 101.6 89.6 11.1

106.6 104.2 102.5 104.8 103.0 105.4 107.3 99.1 97.4 87.2 100.7 100.6 115.8 149.7 94.5 125.8 98.3 103.3 107.9 114.6 103.7 5.0

a Calculations were carried out with the cc-pVDZ basis set. PNOF6’(3) implies that all orbitals below the Nf − 1 level have double occupations, with Nf the Fermi level. Experimental D0 dissociation energies were taken from refs 39 and 58, and ZPVEs were added at the M062X/cc-pVTZ level of theory to estimate experimental De’s. In the case of CH4, this leads to a experimental De’s of 112.7 kcal/mol, 0.3 kcal/mol lower than the value estimated in Table 1.

mol with the cc-pVTZ basis set). Introduction of dynamical electron correlation, using the CAS(2,2)PT2 level, reduces the MAE to 3.4 kcal/mol. One should note that DFT methods would give very good results for dissociation energies when calculations are carried out for dissociated radicals using open shell unrestricted DFT and symmetry-breaking solutions.18,19 Hydrogen Abstraction in a Data Set of 20 Organic Molecules. Dissociation Energies. We have considered a data set of 20 organic molecules to evaluate the performance of PNOF6 and PNOF6(3) for the C−H bond dissociation energy in these molecules. Our data set provides a convenient set to show the sensitivity of the C−H bond to different chemical environments, namely, functional groups with different degrees of electron withdrawing and donating ability (−F, −OH, −NO2, −CN, −CH3, ...); aromaticity (−C6H5); variety of C−X bonds (HCN, H2CO, CH3NO2, CH3CF3, ...), different chain lengths (CH4, CH3CH3, CH3CH2CH3); and different C−C bond orders, single (as in CH3CH3), double (as in C2H4), and triple (as in C2H2). As a result of this diversity, C−H dissociation energies (DCH e ) span a wide range of values, from 95.6 kcal/mol (H2CO) to 141.8 kcal/mol (C2H2). To the best of our knowledge, this is the most extensive data set for C−H cleavage treated so far with any natural orbital functional theory. We have decided to use the cc-pVDZ basis set because of the large number of compounds to be treated. On the basis of the results presented in the previous section, we expect some loss of accuracy due to the use of this basis set, but with no qualitative change of the main conclusions of our results. The results can be found in Table 2 and Figure 2. We obtain a similar MAE for both PNOF6 and PNOF6(3), namely, 8.8 and 9.0 kcal/mol, respectively. As in the previous section, these MAEs are between those of CAS(2,2), 11.1 kcal/mol, and

dissociation energy. In general, all PNOF methods give similar dissociation energies for these two molecules, but with the reverse ordering. Only PNOF6(3)/cc-pVTZ gives the correct order in the dissociation energies between CH4 and NH3. It should be noted that CAS(2,2)/cc-pVTZ also gives the reverse order, whereas CAS(2,2)PT2/cc-pVTZ recovers the correct trend. In general, we can say that there is a good semiquantitative agreement between all PNOF6 methods and experimental values. However, there are also quantitative differences between the various methods. For the six reactions considered, a mean absolute error (MAE) of 17.1 kcal/mol is obtained at the PNOF5/cc-VTZ level of theory. The introduction of PNOF6 reduces significantly the value of MAE to 11.8 kcal/mol. Improvement of the intrapair description through the inclusion of a larger number of coupled orbitals leads to a further improvement in DXH e values, with a very fast convergence with the number of orbitals. Thus, the MAE is only 7.0 kcal/mol for PNOF6(2), 6.5 kcal/mol for PNOF6(3), and 6.4 kcal/mol for PNOF6(4) with the cc-pVTZ basis set. Inclusion of more orbitals (not shown for the sake of brevity) leads to very similar values for the MAEs, so that we can say that the choice of three orbitals is a good compromise for the characterization of the electron pair. In general, PNOF6 leads to De’s that are higher than those of PNOF5, and the introduction of more coupled orbitals for the description of the electron pair enhances this tendency. The PNOF5 underestimation of bond strengths has been previously noted,25,36 and it is related to a partial lack of dynamical electron correlation, which is more important at the equilibrium structures than at the dissociation limit. Finally, if we compare the performance of PNOF6 with wave function methods, it is clear that PNOF6 and PNOF6(2,3,4) show a better performance than CAS(2,2) (MAE of 16.5 kcal/ 6984


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introduction of electron pair interaction in PNOF6/PNOF6(3) is key for treating aromatic stabilization of radicals in the context of natural orbital functional theory. The chain length of the molecule is also a factor influencing the C−H bond strength. It is known that a larger chain stabilizes the resultant radical.4,6,40 Therefore, as the chain length increases, DCH e (exptl) is expected to be lower: CH4 (112.7) > CH3CH3 (109.7) > CH3CH2CH3 (108.5). However, PNOF6 exhibits a rather poor sensitivity of radical stability values for these three toward chain lengths, with similar DCH e molecules that are within 0.2 kcal/mol. The improvement in the intrapair description with PNOF6(3) allows for a partial recovery of the correct trend, namely, CH4 (112.8) > for CH3CH2CH3 (112.7) CH3CH3(111.2); however, the DCH e is still quite similar to that for CH4. If we consider the same alkane, like CH3CH2CH3, and we measure both possibilities for hydrogen abstraction, namely from the central −CH2− or from the terminal −CH3 group, both PNOF6s correctly reproduce the more favorable hydrogen abstraction from the central carbon by 1−2 kcal/mol. There is also a sizable effect in hydrogen abstraction upon the inclusion of electron-withdrawing groups. For instance, fluorination59 and oxidation39 of methane tend to alter the dissociation energy of the C−H bond. Regarding fluorination, a decrease of DCH e is observed upon the inclusion of a first fluor, from 112.7 kcal/mol (CH4) to 108.7 kcal/mol in (CH3F). However, upon higher degree of fluorination in the fluoromethane, DCH increases again: 111.8 kcal/mol in CF2H2 and e 113.5 kcal/mol in CF3H. PNOF6 yields a DCH e for CH3F, 112.8 kcal/mol, that is higher than that for CH4, 104.8 kcal/mol, whereas PNOF6(3) recovers the correct experimental trend and yields a higher DCH e for CH4, 112.8 versus 111.4 kcal/mol. Nevertheless, both PNOF6 and PNOF6(e) describe the proper with the degree of fluorination in trend in increasing DCH e fluoromethane, namely, CH3F (112.8/111.4) < CH2F2 (115.3/ 118.3) < CHF3 (116.7/121.2). With respect to the oxidation of a methyl group, the performance of PNOF6 is limited. For instance, in going from CH3OH to H2CO, there is an important reduction in C−H bond strength, from 103.2 to 95.6 kcal/mol; however, the reduction of DeCH is more moderate for PNOF6 (108.5 to 107.1 kcal/mol) and PNOF6(3) (113.4 to 112.0 kcal/mol). In general, we observed that PNOF6 and PNOF6(3) yield De’s that are higher than experimental values for this set of molecules. Inspecting the natural orbital occupancies, we observed a tendency by PNOF6 and PNOF6(3) to lose dynamical electron correlation at the dissociation limit. This leads to a lack of balance in the dynamical electron correlation considered at equilibrium structures and at the dissociation limit. Therefore, the methods exhibit the propensity to give De’s larger than experiment values, specially for those molecules that show a significant degree of dynamical electron correlation at equilibrium. To further check this point, we repeated the PNOF6(3) calculations forcing all orbitals below the Nf − 1 level (with Nf the Fermi level) to double occupation. The results are presented in Table 2 and Figure 2 as PNOF6’(3). Effectively, now PNOF6’(3) leads to lower De’s, and the MAE decreases to 4.9 kcal/mol, similar to the value of CAS(2,2)PT2. Radical Stabilization Energies. The radical stabilization energy is a convenient measure of the effect of the substituent on a radical. This is defined as the energy change in the isodesmic reaction for hydrogen abstraction18,19,60 of eq 1. The RSE is equivalent to the difference in bond dissociation

Figure 2. C−H bond dissociation energies, in kilocalories per mole, for the data set of 20 organic molecules in Table 2. All calculations were done with the cc-pVDZ basis set. PNOF6’(3) implies that all orbitals below the Nf − 1 level have double occupations, with Nf the Fermi level.

CAS(2,2)PT2 methods, 5.0 kcal/mol. We will now comment on some specific trends. The degree of the C−C bond highly influences the strength of C−H bonds. Thus, the experimental DCH increases in the e following order:39 CH3CH3 (109.7) < C2H4 (119.3) < C2H2 (141.8) . Both PNOF6 and PNOF6(3) are able to reproduce appropriately this trend, namely, CH3CH3 (104.6/111.2) < C2H4 (125.9/128.9) < C2H2 (144.5/147.9) for PNOF6/ PNOF6(3). The effect of aromaticity can also be inferred from the comparison of these dissociation energies with that of the phenyl C−H bond (C6H6). This molecule, with a formal 1.5 C−C bond order, shows a quite high DCH e , 120.5 kcal/mol, even slightly larger than that in C2H4, with a formal bond order of 2. This is a clear signature of aromaticity in C6H6, partially lost upon hydrogen abstraction and radical formation. PNOF6/ PNOF6(3) also yields values of DCH e ’s for benzene that are larger than those for ethene, with values of 128.4/129.8 kcal/ mol at PNOF6/PNOF6(3). However, in the case of the benzylic C−H bond (C6H6CH2−H), the effect of the aromaticity works in the opposite direction. In this case, the C−H cleavage does not break the aromaticity. Furthermore, the radical itself is stabilized by the aromatic character of the phenyl ring, and consequently, one obtains a DCH much lower than e that for C6H6, namely, 96.1 kcal/mol versus 120.5 kcal/mol. Both PNOF6 and PNOF6(3) correctly describe this effect, and the DCH for the benzylic C−H bond (112.3/111.9) is also e much lower than that for the phenyl C−H bond (128.4/129.8), by a magnitude very similar to the experimental value. It is quite remarkable that the correct description of aromatic radical stabilization is provided by PNOF6 theory. In independent electron pair models, such as PNOF5, there are limitations in describing the aromatic character of molecules such as benzene. This has been proven to be related to an excess of localization of the natural orbitals. In fact, PNOF5 calculations for C6H6 and C6H6CH3 exhibit only a moderate difference in DCH e , namely, 109.8 versus 103.7 kcal/mol, respectively. Hence, the 6985


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between PNOF6’s and experimental data is satisfactory, and our data point to the potential of PNOF6 and PNOF6(3) to capture the correct trends in radical stability.

energies of XH and YH species. Thus, the RSE for the X,Y pair is defined as RSEXY = DeXH − DeYH

■

(8)

CONCLUSIONS We have shown that PNOF6 can yield satisfactory results for hydrogen abstraction reactions. In the case of XH systems, the results are superior to electron pair-independent methods such as PNOF5. The consideration of several orbitals for the description of the electron pair improves the results, with a fast convergence with the number of orbitals. The application of PNOF6 and PNOF6(3) methods to the calculation of the C− H bond dissociation energy in a data set of 20 organic molecules and the estimation of the corresponding radical stabilization energies support the use of PNOF6 as a semiquantitative theory for the description of this important set of reactions. An especially remarkable aspect of our results is the good qualitative treatment of aromatic stabilization of radical stability, a fact linked to the introduction of interpair correlation in PNOF6. Our results also suggest that in order to improve on the description of hydrogen abstraction, further NOF developments should address the problem of the loss of dynamical electron correlation in the dissociation limit. Work in this direction is in progress in our group.

DCH e

For the data set of 21 dissociation energies of Table 2, there are 210 possible combinations of RSEs ((21 × 20)/2). The comparison of the performance of PNOF6 and PNOF6(3) to yield these 210 RSEs (see Figure 3) is therefore a suitable and consistent data set for the determination of its adequacy for estimating the effect of substituents on the radical stability in organic molecules.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Finantial support comes from Eusko Jaurlaritza and the Spanish Office for Scientific Research. The SGI/IZO-SGIker UPV/ EHU is gratefully acknowledged for generous allocation of computational resources. X.L. thanks the Spanish Ministerio de Ciencia e Innovación (CTQ2012-38496-C05-04) for funding.

Figure 3. Radical stabilization energies, in kilocalories per mole, based on the combination of De’s dissociation energies of Table 2. All calculations were done with the cc-pVDZ basis set. The mean absolute errors with respect to the experimental values are 9.4, 7.3, and 5.6 kcal/mol for PNOF6, PNOF6(3), and PNOF6’(3) levels of theory (the latter implies that all orbitals below the Nf − 1 level have double occupations, with Nf the Fermi level), and 4.4 and 4.1 kcal/mol for CAS(2,2) and CAS(2,2)PT2, respectively.

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In general, there is a reasonable agreement with experimental RSEs for both PNOF6 and PNOF6(3) levels of theory. The mean absolute errors of RSEs with respect to the experimental data is 9.4 kcal/mol for PNOF6 and 7.3 kcal/mol for PNOF6(3). This suggests an improvement of PNOF6(3) over PNOF6, which is confirmed if we look at the values of the corresponding linear fit with respect to the experimental data. Thus, PNOF6(3) shows an r correlation coefficient, 0.8908, that is better than that of PNOF6, 0.8211. CASSCF methods have smaller errors, although in the same order of magnitude, 4.4 kcal/mol for CAS(2,2) and 4.1 kcal/mol for CAS(2,2)/ PT2. The suitable cancellation of errors of CAS(2,2) in calculating RSEs is remarkable. This method shows the highest MAE for De’s, but for RSEs, it shows MAE and r correlation coefficient similar to that of CAS(2,2)PT2. For completeness, we also provide the values obtained with PNOF6’(3). As expected, the MAE decreases to a value of only 5.6 kcal/mol, and the quality of the linear fit improves with an r value of 0.9176. In summary, taking into account the large number of hydrogen abstraction reactions considered, the correlation 6986


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Performance of PNOF6 for Hydrogen Abstraction Reactions

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