Modeling Noncovalent Radical–Molecule Interactions Using

Jan 16, 2013 - Conventional density-functional theory (DFT) has the potential to overbind radical–molecule complexes because of erroneous charge tra...
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Modeling Noncovalent Radical−Molecule Interactions Using Conventional Density-Functional Theory: Beware Erroneous Charge Transfer Erin R. Johnson,† Michela Salamone,‡ Massimo Bietti,‡ and Gino A. DiLabio*,§,∥ †

Chemistry and Chemical Biology, School of Natural Sciences, University of California, Merced, 5200 North Lake Road, Merced, California 95343, United States ‡ Dipartimento di Scienze e Tecnologie Chimiche, Università “Tor Vergata”, Via della Ricerca Scientifica, 1 I-00133 Rome, Italy § National Institute for Nanotechnology, National Research Council of Canada, 11421 Saskatchewan Drive, Edmonton, Alberta, Canada T6G 2M9 ∥ Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1 S Supporting Information *

ABSTRACT: Conventional density-functional theory (DFT) has the potential to overbind radical−molecule complexes because of erroneous charge transfer. We examined this behavior by exploring the ability of various DFT approximations to predict fractional charge transfer and by quantifying the overbinding in a series of complexes. It is demonstrated that too much charge is transferred from molecules to radicals when the radical singly unoccupied molecular orbitals are predicted to be erroneously too low in energy relative to the molecule highest occupied molecular orbitals, leading to excessive Coulombic attraction. In this respect, DFT methods formulated with little or no Hartree−Fock exchange perform most poorly. The present results illustrate that the charge-transfer problem is much broader than may have been previously expected and is not limited to conventional (i.e., molecule−molecule) donor−acceptor complexes.

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INTRODUCTION

hexamethylphosphoric acid triamide, two commonly used, very strong hydrogen-bond-accepting solvents.6 One difficulty that we encountered in our studies involving BnO•−amine prereaction complexes was a significant dependence between the properties of the complex and the nature of the density-functional employed. It is well-known that approximate density-functional theory methods have the potential to overbind noncovalent molecular complexes in which charge-transfer interactions occur, as was demonstrated recently for the tetrathiafulvalene−tetracyanoquinodimethane dimer.7 However, comparatively little is known about radical− molecule complexes. The central problem with charge-transfer complexes in DFT is that many conventional DFT methods predict orbital energy levels that are incorrect, with highest occupied molecular orbital (HOMO) energies that are too high and lowest unoccupied molecular orbitals (LUMO) energies that are too low. This problem is often described in the context of the DFT “fractional-charge” error. The result of this error is that the HOMO of a donor species is erroneously aligned with the LUMO of an acceptor species, resulting in charge transfer and overly large, primarily electrostatic, binding energies. Clearly the fractional-charge error has important consequences in the

The suggestion that noncovalent interactions between radicals and molecules influence the rate and outcome of chemical reactions is a long-standing one.1 In a particularly well-known study, Russell demonstrated2 that the interaction between chlorine atom (Cl•) and benzene, in the form of a chargetransfer3 π-complex (“acid−base” interaction in Russell’s words), reduces the reactivity of the radical with 2,3dimethylbutane and results in highly selective chlorination of the alkane. Soon after, Thomas reported that the oxidation of cumene is inhibited through the formation of a cumylperoxyl radical−trialkylamine complex.4 We recently investigated a number of noncovalently interacting radical−molecule systems using experimental and theoretical methods. We demonstrated by time-resolved laserflash photolysis that hydrogen atom abstraction reaction rates from amines by the benzyloxyl radical (PhCH2O•, i.e., BnO•) decrease with the degree of amine substitution, viz., tertiary < second < primary, whereas the opposite trend was observed for the cumyloxyl radical (PhC(CH3)2O•, i.e., CumO•).5 Densityfunctional theory (DFT) modeling showed that the ability of BnO• to form an unconventional hydrogen bond (H-bond) between the radical α-C−H moiety and the amine nitrogen atom (as shown in Figure 2) resulted in different reaction outcomes compared to CumO•, a species that cannot engage in strong H-bonding. Related phenomena were recently observed in the reaction of these two radicals with dimethyl sulfoxide and © 2013 American Chemical Society

Received: August 24, 2012 Revised: January 9, 2013 Published: January 16, 2013 947

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RESULTS AND DISCUSSION The incorrect behavior of molecule/radical energies as a function of fractional charge, as determined by various DFT methods, is illustrated in Figure 1. Figure 1 shows results for

modeling of charge-transfer complexes and complexes in which erroneous charge transfer occurs.8−12 Noncovalently bound complexes in which one component is a radical species may be more susceptible to problems associated with erroneous charge transfer because radicals often have energetically low-lying singly unoccupied molecular orbitals (SUMO)13 that can accept electrons from a donor species. In previous work on model complexes involving molecules and radicals, we demonstrated that the small energy difference between the ammonia HOMO and the hydroxyl radical (•OH) SUMO resulted in binding energy errors of ca. 50−130% for the molecule donor−radical acceptor complex for some conventional DFT methods. This finding suggested to us that reactions involving hydrogen atom transfers in molecule/ radical couples could be susceptible to modeling problems. In this work, we explore some aspects of the errors associated with charge-transfer complexes in which one of the components is a radical species. We consider the relationship between the nature of the DFT method, the associated predicted orbital energy levels, and the degree of binding in complexes. The origin of these errors in the context of the fractional-charge error in DFT is discussed.

Figure 1. Relative energy as a function of fractional charge removed from DABCO (upper set of five curves) and fractional charge added to BnO• (lower set of five curves). A properly behaved DFT method should predict straight lines.

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COMPUTATIONAL DETAILS Calculations of fractional charges associated with BnO• and diazabicyclo[2.2.2]octane (DABCO) were carried out with a beta version of the program QM4D.14 The structures used for the fractional-charge calculations were optimized at the CAMB3LYP/6-31+G(d,p) level of theory and were obtained using the Gaussian 09 package.15 The fractional-charge calculations are carried out by defining the orbital occupations, ni, 1 for i < f, δ for i = f, and 0 for i > f, where f is the index for the frontier orbital and δ is the fractional occupation. This is a generalization of Janak’s theorem16 to fractional charge.17 Binding energies for several radical−molecule complexes were calculated using a variety of density-functionals and included dispersion using the exchange-hole dipole moment (XDM) dispersion model.18 XDM is a nonempirical model of dispersion based on second-order perturbation theory, in which the instantaneous dipole moments responsible for dispersion attraction are modeled using the dipole moment of the exchange hole. If the Becke−Roussel (BR) model19 of the exchange hole is used, XDM becomes a meta generalized gradient approximation (GGA) functional and the dispersion correction is a negligible addition to the total computational cost. Two atom-independent parameters are required to damp the dispersion energy at small internuclear separations. Damping parameters used for all functionals considered in this work, along with statistics for performance on the fit set, are given in the Supporting Information. The XDM calculations were performed using a combination of the Gaussian 09 package and the postG program.20 Geometry optimizations used the aug-cc-pVDZ basis set and were followed by singlepoint energy calculations using the larger aug-cc-pVTZ basis. Our calculations incorporate corrections for dispersion and fairly complete basis sets, so our results should present a good indication of the effects of erroneous charge transfer. In our preliminary calculations, we did not use dispersion-corrected methods and we used 6-31+G(d,p) basis sets. Nevertheless, the results of the preliminary calculations (see the Supporting Information) qualitatively agree with the data presented herein.

the examples of incrementally removing an electron from DABCO, together with incrementally adding an electron to BnO•. The deviation from linearity is a quantitative measure of fractional-charge error because the correct behavior is linear.21 Hartree−Fock has distinctly different behavior from the DFT methods in that the energy vs fractional charge lines are downward curving.22 The DFT methods display improved behavior with increasing Hartree−Fock exchange (HFX). For example, BLYP,23,24 which has zero HFX, displays the worst upward curvature in Figure 1. Inclusion of 20% HFX in the DFT method, as in the case of B3LYP,24,25 improves the behavior to some extent. Nearly linear plots are obtained with BHandHLYP,24,26 which has 50% HFX, and CAM-B3LYP,27 which is a range-separated functional22,28−33 with 19% shortrange and 65% long-range HFX. It is clear from Figure 1 that some amount of HFX roughly midway between 0% (e.g., BLYP) and 100% (Hartree−Fock), such as BHandHLYP and CAM-B3LYP, is required to obtain the correct behavior with respect to charge transfer.34 What is the impact of the behavior that is illustrated by the fractional charge plots? In Figure 1, the slopes of the lines at zero charge reflect the orbital energies (HOMO for DABCO and SUMO for BnO•). The exact density-functional would predict a DABCO HOMO equal to its ionization potential (IP) and a BnO• SUMO equal to its electron affinity (EA). The substantial curvature associated with the BLYP and B3LYP plots reveals that the predicted orbital energies are lower than they should be, which indicates that the addition of an electron to BnO• and the removal of an electron from DABCO are more favorable than they should be. This encourages erroneous, enhanced electron transfer between the two species, which results in overly high Coulomb attractions between the two species. Moreover, these exchange functionals overstabilize the fractional charges, resulting in greater stability of the complex. Table 1 summarizes the impacts associated with the erroneous charge transfer on the BnO•−DABCO complex, with the %HFX used as a guide to the degree of the fractional charge problem associated with a particular method. The 948

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Table 1. Results from Method-XDM/aug-cc-pVTZ//Method-XDM/aug-cc-pVDZa Calculations on the BnO•−DABCO Complexb,c method 36

PBE BLYP B3LYP B97137 PBE038 BHandHLYP M06-2X39 CAM-B3LYP

%HFX

R(C−H)d

R(H−N)d

BE

BnO• SUMO

DABCO HOMO

charge

0 0 20 21 25 50 54 19/65

1.20 1.17 1.17 1.18 1.18 1.11 1.15 1.15

1.71 1.87 1.79 1.78 1.73 2.27 1.92 1.87

13.4 13.7 9.2 8.7 8.5 6.6 6.9 7.4

−0.189 11 −0.186 76 −0.153 85 −0.149 13 −0.136 02 −0.072 23 −0.090 86 −0.094 78

−0.156 79 −0.153 14 −0.193 61 −0.191 80 −0.200 28 −0.243 57 −0.246 34 −0.249 23

0.261 0.240 0.196 0.200 0.201 0.039 0.104 0.135

a

The dispersion-correcting, XDM approach was used for all of the methods except M06-2X. bResults obtained for the nondispersion corrected method/6-31+G(d,p) are qualitatively similar to those shown above; see the Supporting Information. cValues are shown for percent Hartree−Fock exchange (%HFX), selected bond distances (R, Å), binding energies of the complex (BE, kcal/mol), the BnO• singly occupied molecular orbital (SUMO) and DABCO highest occupied molecular orbital (HOMO) energies (atomic units), and the Hirshfeld charge transferred from DABCO to BnO• (electrons). dBond lengths between the labeled atoms in Figure 2.

complex has a C−H···N hydrogen bond, with secondary interactions occurring between the BnO• O atom and two DABCO H atoms that are α to the nitrogen; see Figure 2.

The structure of the complex is also strongly dependent upon the functional, viz., BnO• C−H bond length variability of 0.09 Å and H(BnO•)−N(DABCO) variability of 0.56 Å. However, there are only moderate correlations between these bond lengths and the binding energies/percent HFX. The origin of the overbinding with low %HFX can be understood by considering the degree of charge transfer from DABCO to BnO• that results from the poorly predicted orbital energies. Table 1 shows that methods with low %HFX predict the radical SUMO level to be lower in energy than the molecule HOMO. This makes charge transfer from the molecule to the radical a favorable process and results in the overly large BEs. We note that this effect is largely independent of whether dispersion is incorporated in the calculations and of the size of the basis set employed; see the Supporting Information. Although the results from Figure 1 suggest that CAM-B3LYP is the most suitable method for the modeling of the BnO•− DABCO complex, confirmation of this through the use of a theoretical methodology that is capable of producing unequivocal charge-transfer interactions is required. In such cases, we usually resort to complete basis set extrapolated coupled-cluster theory with singles, doubles, and perturbative triples: CCSD(T)/CBS.40 However, the complex is too large for this approach. Instead, we used the H3CO•−N(CH3)3 complex, which suffers from problems similar to those of the BnO•−DABCO complex. Full details of the results from the calculations on the model system are given in the Supporting Information section and we offer a summary here: CAMB3LYP predicts a BE of 5.1 kcal/mol, a value that is in good agreement with the value of 4.8 kcal/mol obtained by CCSD(T)/CBS. We therefore confidently conclude that the overbinding that is obtained using 0% HFX DFT methods for

Figure 2. Perspective view of the BnO•−DABCO noncovalently bonded complex. Optimized distances between the labeled atoms are given in Table 1.

Several DFT methods, with HFX ranging from 0 to 65%, were used. Keeping in mind that different DFT methods will produce results that are dependent upon all aspects of their formulation (i.e., not only exchange but correlation components of the functional as well), some general trends with % HFX can be extracted from the data in Table 1. Most obvious is the dependence of the degree of binding within the complex on %HFX: functionals with low %HFX produce greater binding energies (BEs) than functionals with higher %HFX. The BEs range from very high (viz., for BLYP, HFX = 0%, BE = 13.7 kcal/mol) to rather weak (viz., for BHandHLYP, HFX = 50%, BE = 6.6 kcal/mol). For comparison, the BE for the hydrogenbonded water dimer is ca. 5 kcal/mol.35

Table 2. Calculated Cl• SUMO and Benzene HOMO Energies (Atomic Units, au), Degree of Charge Transfer (Electrons), and π-Complex Binding Energy (kcal/mol) Obtained Using Method-XDM/aug-cc-pVTZ//Method-XDM/aug-cc-pVDZa method

Cl• SUMOb

benzene HOMOc

charge transferred to Cl•

BE(Cl•−benzene)

BLYP B3LYP BHandHLYP CAM-B3LYP

−0.284 91 −0.248 94 −0.180 87 −0.187 90

−0.224 66 −0.259 88 −0.297 30 −0.313 00

0.233 0.218 0.182 0.197

15.0 11.4 6.6 8.5

a

In connection with the data in Figure 1, we note that the methods having higher %HFX predict radical/molecule orbital energies that are closer to the experimental EA/IP than do lower %HFX methods. The extent of charge transfer was determined using Hirshfeld charges. bFrom ref 41, the experimental EA of Cl• is 0.133 au. cFrom ref 41, the experimental IP of benzene is 0.340 au. 949

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Table 3. Orbital Energies (Atomic Units, au) and Binding Energies of Complexes (kcal/mol) Using Various Density-Functional Theory Methods method

SUMO(•OCH3)a

SUMO(•OCHF2)

HOMO(H3COCH3)b

BLYP B3LYP BHandHLYP CAM-B3LYP

−0.195 35 −0.150 96 −0.066 43 −0.089 18

−0.265 73 −0.223 31 −0.135 60 −0.161 44

−0.213 92 −0.266 17 −0.331 59 −0.328 21

BE(•OCH3−H3COCH3) c

6.4 4.2d 3.5e 4.0f

BE(•OCHF2−H3COCH3) 12.5g 6.9h 5.2i 6.0j

a

From ref 41, IP(H3COCH3) = 0.3686 au. bThe calculated EA of C2H5O is 0.0634 au.43 cCharge transferred (in electrons) to radical moiety in the complex: 0.074. dCharge transferred (in electrons) to radical moiety in the complex: 0.029. eCharge transferred (in electrons) to radical moiety in the complex: 0.009. fCharge transferred (in electrons) to radical moiety in the complex: 0.018. gCharge transferred (in electrons) to radical moiety in the complex: 0.237. hCharge transferred (in electrons) to radical moiety in the complex: 0.155. iCharge transferred (in electrons) to radical moiety in the complex: 0.064. jCharge transferred (in electrons) to radical moiety in the complex: 0.097.

the BnO•−DABCO complex is due to a significant extent to erroneous charge transfer. The BnO•−DABCO complex BEs that are listed in Table 1 may be contrasted with the calculated binding energies of the BnOH−DABCO complex given in the Supporting Information. The BnOH LUMO lies substantially above the DABCO HOMO, and all of the functionals considered give consistent binding energies that range from 11.0 kcal/mol (B971) to 12.7 kcal/mol (BLYP). Therefore, eliminating the relatively energetically low-lying SUMO associated with BnO• by reduction with hydrogen also eliminates the overbinding in the complex. While the amount of HFX in a DFT could point to potential problems in the treatment of radical−molecule complexes due to erroneous charge transfer, this is not certain and will depend on the particular radical and molecule in question. However, a calculation of the orbital energies of two interacting species should provide a direct indication of whether a method is capable of modeling a complex under investigation. To illustrate this point in more detail, we return to the classic example of the Cl•−benzene complex. In Table 2 we collect the relevant orbital energies of the radical and molecule as obtained using DFT methods with different degrees of HFX, along with the calculated binding energies. Croft has studied the Cl•−benzene complex with many theoretical methods and concludes that it is bound by 6.9 kcal/ mol.42 Using BLYP, we predict a BE that is 15.0 kcal/mol. The overbinding is consistent with the fact that BLYP predicts the Cl• SUMO to be lower in energy than the benzene HOMO, and as expected, excessive charge is transferred to Cl• from benzene. A critical point to emphasize at this stage of the discussion is that some degree of charge transfer can be expected to occur between certain species, even when the DFT method chosen to treat them is properly behaved. We are not arguing that any DFT method that predicts a SUMO of a radical to be lower in energy than that of the HOMO of a molecule in a radical− molecule pair will result in a poor description of charge transfer between the two species. Nor are we suggesting that charge transfer does not occur when, in a radical−molecule complex, a radical SUMO lies higher in energy than a molecule HOMO. The latter point is illustrated by the data in Table 2: CAMB3LYP predicts a proper orbital ordering with the Cl• SUMO being higher in energy than the benzene HOMO and yet ca. 0.2 electron is transferred from the molecule to the radical. This kind of behavior is expected in radical−molecule complexes because radicals are inherently electron deficient. It is clear from the results presented herein (see also Table 3 below) that the seemingly small errors in charge transfer that arise because

of poorly predicted ordering in the energies of donor−acceptor species are responsible for overbinding in these complexes. In order to further explore the relationship between orbital energies and radical−molecule binding, we considered the example of X2HCO•−H3COCH3. In this complex, a radical C− H is directed toward the dimethyl ether O and secondary interactions between the radical O and the dimethyl ether CH groups are present. We do not expect these interactions to result in strong binding between the constituents of the complexes. In the case of X = H, we expect most DFT methods to predict orbital energies such that erroneous charge transfer does not occur. However, we might induce orbital energy changes, with X = F, to pull the radical SUMO energy level below that of the molecule for certain DFT methods. Our results in this connection are summarized in Table 3. For the H3CO•−H3COCH3 complex, the radical SUMO is higher in energy than the HOMO of the molecule. We therefore expect no erroneous charge transfer and no related overbinding in the complex. In accordance with this expectation, BLYP, B3LYP, BHandHLYP, and CAM-B3LYP predict binding energies within 2.9 kcal/mol of each other. However, when the radical is substituted with two F atoms, the SUMO energies drop significantly. In the case of BLYP, the SUMO for F2HCO• is lower than that of H3COCH3, and this results in a very large calculated binding energy of 12.5 kcal/mol, consistent with the fact that more than 0.23 electron is transferred from the molecule to the radical. The remainder of the DFT methods predict SUMO energies that are above the HOMO energies of the molecule, and these methods predict BEs of ca. 6 kcal/mol and less electron transfer to the radical (ca. 0.06−0.16 electron). We calculated the orbital energies using BLYP, B3LYP, and BHandHLYP for a variety of radicals and molecules, with the notion that these data could be used as a predictor of when these DFT methods fail as a result of erroneous charge transfer. The data, which are provided in the Supporting Information, support the concept that radicals having SUMOs lower than a molecule with which they are complexed, as determined by a low-HFX DFT method, will overbind the complex as a result of excess charge transfer. Conversely, those same DFT methods that predict radicals with SUMOs higher in energy than the HOMO of a molecule with which they are complexed will not display overbinding due to charge-transfer problems.

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SUMMARY

DFT approaches utilizing no Hartree−Fock exchange (e.g., BLYP, PBE) tend to predict the SUMO energy of radicals and the HOMO energies of molecules to be significantly lower than their electron affinities and ionization potentials, respectively. 950

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The Journal of Physical Chemistry A This illustrates the so-called “fractional-charge” error that occurs in many such DFT methods. Increasing the amount of Hartree−Fock exchange incorporated into the functional (e.g., for B3LYP, 20%, and for BHandHLYP, 50%), improves the quality of the orbital energy predictions as a result of reduced fractional-charge error. One key outcome of the fractional-charge error is that too much charge is transferred from molecules to radicals in radical−molecule complexes when radical SUMOs are erroneously predicted to be lower than molecule HOMOs. As a result, calculations on complexes involving such species predict significant overbinding owing to excessive attractive Coulombic interactions. The present results illustrate that the charge-transfer problem is much broader than may have been previously expected and is not limited to conventional (i.e., molecule−molecule) donor−acceptor complexes. In practice, the difference in Kohn−Sham orbital eigenvalues between the HOMO of the electron donor and SUMO (or LUMO) of the electron acceptor, along with the degree of Hartree−Fock exchange incorporated in the functional, can indicate whether conventional functionals are adequate to describe the chemistry or significant charge-transfer errors will be present. If the HOMO of the donor is higher or equal in energy to the LUMO of the acceptor, conventional, low Hartree−Fock exchange functionals will overbind the intermolecular complex. Beyond radical−molecule complexes, the charge-transfer problem may also affect the structures and energetics of transition states associated with, for example, atom-transfer reactions involving these species. Furthermore, certain DFT methods may exhibit difficulties with complexes/reactions involving anions, which may have HOMO energies that are higher than the LUMOs of some molecules. For example, charge-transfer error may lead to overstabilization of intermolecular prereaction complexes for nucleophilic reactions involving anions. In closing, users of conventional DFT should beware of erroneous charge transfer when modeling radical−molecule interactions. For systems susceptible to these problems, we recommend the use of range-separated functionals, such as CAM-B3LYP, that incorporate long-range Hartree−Fock exchange.

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ACKNOWLEDGMENTS

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REFERENCES

We thank Xiangqian Hu and Degao Peng for assistance with the QM4D program and Alberto Otero de la Roza for parametrization of XDM. We are also grateful to WestGrid for access to computational facilities.

(1) Boozer, C. E..; Hammond, G. S. Molecular Complex Formation in Free Radical Reactions. J. Am. Chem. Soc. 1954, 76, 3861−3862. (2) Russell, G. A. Solvent Effects in the Reactions of Free Radicals and Atoms. 11. Effects of Solvents on the Position of Attack of Chlorine Atoms upon 2,3-Dimethylbutane, Isobutane and 2-Deuterio2-methylpropane. J. Am. Chem. Soc. 1958, 80, 4987−4996. (3) Russell used the terminology “acid-base type interaction” to describe this charge-transfer interaction. (4) Thomas, J. R. Oxidation Inhibition by Trialkylamines. J. Am. Chem. Soc. 1963, 85, 593−594. (5) (a) Salamone, M.; DiLabio, G. A.; Bietti, M. Hydrogen Atom Abstraction Selectivity in the Reactions of Alkylamines with the Benzyloxyl and Cumyloxyl Radicals. The Importance of Structure and of Substrate Radical Hydrogen Bonding. J. Am. Chem. Soc. 2011, 133, 16625−16634. (b) Salamone, M.; Anastasi, G.; Bietti, M.; DiLabio, G. A. Diffusion Controlled Hydrogen Atom Abstraction from Tertiary Amines by the Benzyloxyl Radical. The Importance of C−H/N Hydrogen Bonding. Org. Lett. 2011, 13, 260−263. (6) Salamone, M.; DiLabio, G. A.; Bietti, M. Reactions of the Cumyloxyl and Benzyloxyl Radicals with Strong Hydrogen Bond Acceptors. Large Enhancements in Hydrogen Abstraction Reactivity Determined by Substrate/Radical Hydrogen Bonding. J. Org. Chem. 2012, 77, 10479−10487. (7) Steinmann, S. N.; Piemontesi, C.; Delachat, A.; Corminboeuf, C. Why are the Interaction Energies of Charge-Transfer Complexes Challenging for DFT? J. Chem. Theory Comput. 2012, 8, 1629−1640. (8) Ruiz, E.; Salahub, D. R.; Vela, A. Charge-Transfer Complexes: Stringent Tests for Widely Used Density Functionals. J. Phys. Chem. 1996, 100, 12265−12276. (9) Zhang, Y.; Yang, W. A Challenge for Density Functionals: SelfInteraction Error Increases for Systems with a Noninteger Number of Electrons. J. Chem. Phys. 1998, 109, 2604−2608. (10) Ruzsinszky, A.; Perdew, J. P.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E. Spurious Fractional Charge on Dissociated Atoms: Pervasive and Resilient Self-Interaction Error of Common Density Functionals. J. Chem. Phys. 2006, 125, 194112. (11) Vydrov, O. A.; Scuseria, G. E. Assessment of a Long-Range Corrected Hybrid Functional. J. Chem. Phys. 2006, 125, 234109. (12) Sini, G.; Sears, J. S.; Bredas, J. L. Evaluating the Performance of DFT Functionals in Assessing the Interaction Energy and GroundState Charge Transfer of Donor/Acceptor Complexes: Tetrathiafulvalene−Tetracyanoquinodimethane (TTF−TCNQ) as a Model Case. J. Chem. Theory Comput. 2011, 7, 602−609. (13) In a spin-unrestricted computational treatment, orbitals associated with α-spin generally have an analogous and nearly degenerate analogue with β-spin. We use the term singly unoccupied molecular orbital (SUMO) to refer to the unoccupied β-orbital that is generally associated with the highest occupied α-orbital. (14) QM4D: a quantum mechanical (QM)/molecular mechanical (MM) molecular dynamics (MD) simulation package. www.qm4d. info. (15) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; et al. Gaussian 09, revision C.01; Gaussian, Inc.: Wallingford, CT, 2009. (16) Janak, J. F. Proof That ∂E/∂ni=ε in Density-Functional Theory. Phys. Rev. B 1978, 18, 7165. (17) Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Many-Electron SelfInteraction Error in Approximate Density Functionals. J. Chem. Phys. 2006, 125, 201102.

ASSOCIATED CONTENT

S Supporting Information *

Results of calculations on the H3CO• + N(CH3)3 system (a model for BnO•−DABCO), •OtBu + N,N-dimethylaniline system, the BnOH−DABCO system, and the systems given in Tables 1−3 using 6-31+G(d,p) basis sets. XDM parameters used for the calculations and statistics associated with their fit set, as well as optimized coordinates for the structures described in this work. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: (780) 641-1729. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 951

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dx.doi.org/10.1021/jp3084309 | J. Phys. Chem. A 2013, 117, 947−952