Comparative Study of Ethane and Propane Cation Radicals by B3LYP

Sep 26, 1996 - A comparative computational study of the cation radicals of ethane and propane is made by using B3LYP density functional and high-level...
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15774

J. Phys. Chem. 1996, 100, 15774-15784

Comparative Study of Ethane and Propane Cation Radicals by B3LYP Density Functional and High-Level ab Initio Methods Han Zuilhof,* Joseph P. Dinnocenzo, A. Chandrasekhar Reddy,† and Sason Shaik† Department of Chemistry, UniVersity of Rochester, Rochester, New York 14627-0216 ReceiVed: April 11, 1996; In Final Form: June 18, 1996X

A comparative computational study of the cation radicals of ethane and propane is made by using B3LYP density functional and high-level ab initio methods [up to QCISD(T)/6-311++G(2df,p) and CBS-APNO model chemistry]. The properties investigated include the structures and energetics of the cation radical isomers, their isotropic hyperfine coupling constants, and their carbon-carbon bond dissociation energies, as well as the vertical and adiabatic ionization potentials of the parent molecules. The computational results are compared with experimental data where possible. In general, the B3LYP method exhibits good agreement with experiment. All of the methods show that the potential energy surfaces of the two cation radicals are very flat. The electronic origin of this phenomenon and the relationship between the cation radical isomers of a given species (so-called electromers) are discussed in terms of a valence-bond model.

Introduction Organic cation radicals form a fundamental class of compounds with a variety of intriguing properties and reactivity patterns. It would be desirable to identify an accurate but modestly demanding computational method for routinely investigating cation radicals and their reaction pathways. In practice, cation radicals pose a challenge to computational chemistry. For example, some hydrocarbon cation radicals are known to exhibit different isomeric forms that are often extremely close in energy and interconvertible via low-energy pathways.1 As such, the potential energy surfaces of these cation radicals tend to be very flat, rendering characterization of the various minima a difficult and often precarious task. Indeed, standard ab initio computational methods such as unrestricted Hartree-Fock (UHF), even with second-order Møller-Plesset theory corrections, have been found to yield erroneous results in certain cases.2 This makes it necessary to go to much higher computational levels that are intrinsically more demanding and currently prohibitive beyond a few heavy atoms. The goal of this paper was to test the newly developed density functional hybrid method B3LYP.3 This method was devised recently to account inter alia for the overbinding of the classical density functional theory (DFT) approach and has proven to be a reliable method for a variety of organometallic4a-d and organic problems.4e-k To test the B3LYP method, we have chosen to study two species: ethane and propane cation radicals. As discussed in the following, these cation radicals pose substantial computational challenges as well as unresolved experimental issues. The crucial feature of many cation radicals for our present discussion is the isomerism they often exhibit.1 This isomerism can be traced back to the availability of several closely spaced high-lying occupied orbitals in the corresponding neutral molecules. Ionization from each one of these orbitals followed by Jahn-Teller and pseudo-Jahn-Teller distortions,1a as well as state mixings and avoided crossings,1b,5 produces a soft potential energy surface with a variety of cation radical minima. We illustrate the isomerism problem by reviewing some of the relevant features of C2H6•+ and C3H8•+. The closely spaced, † Permanent address: Department of Organic Chemistry, The Hebrew University, 91904 Jerusalem, Israel. X Abstract published in AdVance ACS Abstracts, August 15, 1996.

S0022-3654(96)01092-1 CCC: $12.00

Figure 1. Highest occupied molecular orbitals for (a) ethane and (b) propane along with the state symmetries that correspond to the ionized states derived from the orbitals and the final (relaxed) states of the cation radical.

highest occupied orbitals of the two neutral species are depicted in Figure 1, along with the state symmetries that correspond to the ionized states derived from these orbitals. Ethane cation radical has a few low-energy isomers1b,6 that are nascent from the ionization of ethane into the 2Eg and 2A1g states. These isomers have been located by high-level ab initio calculations1b and are drawn schematically in Figure 2. One species has a diborane-type structure (DB), where the unpaired electron is delocalized over the C-C bond and the two bridging C-H bonds. Another isomer is a long-bond (LB) isomer in which the unpaired electron resides principally in the C-C bond. These species have been dubbed “electron-shift isomers”,1b a term that denotes different ground state isomers that preserve their atomic connectivity but differ in their electronic structures by the shift of one electron between different bonds. DB, which has C2h symmetry, is predicted to be the most stable C2H6•+ electron-shift isomer. This species appears to be the only one observed in ESR experiments, displaying a 1:2:1 three-line hyperfine splitting with a coupling constant of 152.5 G, which © 1996 American Chemical Society

Ethane and Propane Cation Radicals Study

J. Phys. Chem., Vol. 100, No. 39, 1996 15775

Figure 3. Isomers of propane cation radical: one-long-bond Cs structure (1-LB, left) and two-long-bond C2V structure (2-LB, right).

Figure 2. Isomers of ethane cation radical, diborane-type C2h structure (DB, top left), long-bond D3d structure (LB, top right), delocalized C2h structure (C2h, bottom left), and localized Cs structure (Cs(l), bottom right). Cs(l) has a mirror image isomer, Cs(r), which has θ1 > θ2.

is consistent with coupling to the two bridging hydrogens.7 The D3d symmetry LB isomer is predicted to be slightly higher in energy (1-2 kcal/mol), with very small hyperfine coupling constants for the hydrogens. It is worth noting that, in contrast with experiment, the LB isomer was predicted to be the most stable at UHF and UMP2 levels of theory that do not contain polarization functions on hydrogen in the basis set.1 Extensive ab initio calculations showed that UMP2 with at least a 6-31G** quality basis set is required to establish DB as the most stable isomer.1b,8 Even though this order survives at higher levels of geometry optimization (up to QCISD(T)/6-31G**), the geometric features of DB and the precise relative stability of DB and LB still show significant variation with computational level.1b Two other C2H6•+ isomers have been investigated in detail,1b which are characterized by Cs and C2h symmetries (see Cs and C2h in Figure 2). Both isomers are derived from the JahnTeller active 2Eg state shown in Figure 1. In C2h, the unpaired electron is delocalized over two CH2 moieties, while in Cs the electron is localized in one of them. As such, the Cs and C2h species can be viewed as localized and delocalized forms of the same electronic state.1b Both species are minima at the MP2(Full)/6-311G** level of theory, with C2h being more stable by 0.7 kcal/mol. However, single-point QCISD(T)/6-311G**/ /MP2/6-311G** computations suggest Cs to be more stable.1b From the recent high-level calculations of Ioffe and Shaik,1b the energetic ordering of the C2H6•+ isomers was predicted to be DB < LB < Cs (C2h), at levels of theory higher than UMP2. Early calculations by Radom et al.1a for propane cation radical predicted that the two most stable states are 2B2 and 2A′, both nascent from ionization of the b2 orbital and/or from mixing of b2 and a1 orbitals of the parent molecule (see Figure 1). Two additional isomers with C2V structures are the 2B1 and the 2A1 states, which are derived from ionization of the b1 and a1 orbitals, respectively. Both of these latter states are predicted1a to be significantly higher in energy and will only be discussed briefly. Figure 3 depicts the two low-lying isomers, which have one long C-C bond (1-LB, 2A′) and two long C-C bonds (2-LB, 2B ), respectively. In the C 1-LB structure, the unpaired 2 s electron is largely localized in the long C-C bond and the antiperiplanar C-H bond on the adjacent carbon atom. The C2V 2-LB structure has two equally lengthened carbon-carbon bonds, and the unpaired electron is delocalized into both bonds

as well as into the in-plane C-H bonds. As such, 1-LB and 2-LB are localized and delocalized isomers of the same cation radical state (Vide infra). Uncertainty surrounds the relative stability of the 1-LB and 2-LB C3H8•+ isomers. Initial semiempirical studies by Belville and Bauld concluded that 1-LB is more stable.6 This energetic order was later confirmed by Radom et al. using ab initio methods.1a Radom’s results also showed that the 2-LB structure was a saddle point for the electron-shift isomerism of the two 1-LB structures. This conclusion was not supported by subsequent work of Lunell, Feller, and Davidson,9 however, who used higher level ab initio calculations to argue that 2-LB is the more stable isomer and that it is a genuine minimum. Furthermore, these authors showed that the computed isotropic hyperfine coupling constants for 2-LB were in good agreement with experimental data for C3H8•+ in a SF6 matrix.7 No clear conclusion was presented with regard to the status of the 1-LB isomer. More recently, Toriyama and co-workers have published ESR data for propane cation radical in C2F6 and C3F8 matrices, which they interpret to originate from a 1-LB structure.10 This finding strongly suggests that the potential energy surface for propane cation radical is very soft, and, as such, the relative stability of the 1-LB and 2-LB C3H8•+ isomers remains as a challenging and unsettled computational problem. On the basis of the preceding discussion, it is apparent that the potential energy surfaces for both ethane and propane cation radicals are soft and therefore pose serious difficulties for any computational model. Accordingly, we felt that they could serve as critical tests for the B3LYP method. Strong motivations for benchmarking B3LYP for cation radicals are the disadvantages displayed by the standard ab initio methods for open-shell species, e.g., computational inefficiency and spin contamination.2a In comparison, B3LYP is significantly more computationally efficient than high-level ab initio methods.11 In addition, B3LYP exhibits only marginal spin-contamination problems (Vide infra). These provide attractive reasons to test the reliability of B3LYP as a tool for the study of cation radicals. The use of DFT methods for the investigation of cation radical chemistry has been rather limited up to now. Barone and Adamo have compared the results of a variety of DFT methods with those of MP2 and QCISD calculations in a study of formaldehyde cation radical12 and found that B3LYP performs significantly better than MP2 and previous DFT methods. Eriksson et al. and Huang et al. have demonstrated the applicability of DFT methods, in combination with large basis sets, for the computation of isotropic hyperfine coupling constants.13 The results of Eriksson et al. show, however, that geometry optimization is problematic for BLYP (e.g., in the case of ethane cation radical). Finally, various DFT methods have recently been used in a study of methane, ethane, and norbornane cation radicals by Shephard and Paddon-Row.14 These authors conclude that DFT methods with nonlocal gradient corrections yield satisfactory optimized geometries and good agreement with experimental isotropic hyperfine coupling constants. The present study complements previous DFT studies on open-shell species4g-h,12-14 and presents a systematic test of the

15776 J. Phys. Chem., Vol. 100, No. 39, 1996 B3LYP method for reproducing the properties of ethane and propane cation radicals. The focus of this investigation includes structures, relative isomer energies, isotropic hyperfine coupling constants, vertical and adiabatic ionization potentials, and carbon-carbon bond dissociation energies. To assess the reliability of the B3LYP results, they are compared with experimental data as well as with high-level ab initio levels, including UMP2, UMP4, QCISD, QCISD(T), and CBS-APNO, the latter being considered the best complete basis set method currently available for model chemistry.15,16 Theoretical Methods and Calculations All computations were performed with the Gaussian 94 series of programs17 on SGI workstations. Geometries were optimized with the Berny algorithm, using either internal or redundant coordinates. The optimized species were determined to be either minima or saddle points by frequency calculations. All notations are standard unless otherwise stated. The 〈S2〉 values for all B3LYP calculations on ethane and propane cation radicals were 0.753-0.754, while for the various ab initio levels 〈S2〉 falls in the range 0.758-0.770. CBS-APNO energies for some of the cation radicals required modification. For open-shell species, the standard CBS-APNO procedure15 starts with an optimization at the UHF/6-311G(d,p) level. The UHF level yields unsatisfactory results for both C2H6•+ and C3H8•+. For example, UHF does not locate the DB isomer of C2H6•+, and, as such, the standard CBS-APNO routine cannot be used to optimize the DB minimum. Another problem with the standard CBS-APNO procedure is that, to obtain an enthalpy at 0 K, the UHF frequencies are scaled and the corrected ZPE is then added to the potential energy of the QCISD/6-311G(d,p)-optimized geometry. This leads to meaningless results for both C2H6 and C3H8 cation radicals because the UHF/6-311G(d,p)-optimized geometries either significantly deviate from the QCISD/6-311G(d,p)-optimized geometries or, worse still, for some isomers the QCISD/6-311G(d,p)-optimized geometry is a minimum while it is a saddle point at the UHF/ 6-311G(d,p) level. To rectify these problems, it was necessary to circumvent the initial UHF procedure in the CBS-APNO routine. First, the geometry was optimized in an independent calculation at the QCISD/6-311G(d,p) or QCISD(T)/6-311G(d,p) level, and frequencies were calculated at the QCISD/6311G(d,p) level. The resulting geometry was then submitted in internal coordinates (Z-matrix) into the CBS-APNO procedure, leaving only a dummy variable to be optimized, i.e., a distance to a dummy atom [certain r(C-H) or angles can also be used if they do not vary with the level of calculation]. As a result of this trick, the CBS-APNO procedure ends up with the correct geometry and, more importantly, with the correct isomer identity, which allows for the inclusion of a meaningful QCISD-based ZPE correction. In the special case of C2H6•+, because the QCISD and QCISD(T) geometries for some isomers differ significantly,1b we have chosen QCISD(T)-optimized geometries for DB and LB as the initial input geometries for the CBS-APNO procedure. The modified procedure is designated in our tables as CBSAPNO(m). In this procedure the ZPE correction was determined from the unscaled QCISD/6-311G(d,p) frequencies (e.g., for the determination of carbon-carbon bond dissociation energies and adiabatic ionization potentials; Tables 6 and 11). However, when only comparisons of potential energies were needed, the ZPE term was subtracted from the CBS-APNO energy at 0 K, thereby providing a CBS-APNO(m) potential energy. This last procedure was used for comparison with potential energies obtained by other computational methods (see, for example, Tables 4 and 9).

Zuilhof et al. Calculated isotropic hyperfine coupling constants, a(H), were determined from the Fermi contact term F in the standard Gaussian 94 output by use of the following relation: a(H) (Gauss) ) 1595F (in atomic units).18 Vertical ionization potentials (IP) were obtained as the differences in absolute energies of the neutral species (C2H6 or C3H8) and their corresponding cation radicals, both calculated at the optimized geometries of the neutrals. Assignment of the symmetries of the vertical states in Gaussian 94 for ethane cation radical turned out not to be a trivial matter, and the final results could be obtained only by use of Z-matrix coordinates (rather than Cartesian). Initially the C2h point group was used with a slight change (0.000 01Å) in the bond length of the two antiperiplanar C-H bonds. At this point group, the 2Eg state separates into 2Ag and 2Bg states, which could be calculated individually, thereby yielding the vertical IPs associated with ionization to the 2Eg state. By working with the D3d point group and changing the guess orbitals as required, it was possible to obtain the second vertical IP corresponding to ionization to the 2A 1g state. Use of the STABLE option ascertains that this is not the lowest vertical state, and one obtains a lower solution that is virtually identical in energy to the 2Eg state but of unassigned symmetry. When the 6-31G* basis set was employed, we found that the only way to obtain symmetry assignments for the different vertical states was by taking five (pure) d-polarization functions instead of the default six. The symmetry of all the states (even when the program could not assign symmetry) could be ascertained by inspecting the spin and charge density distributions that are characteristic for the various states.1b We found this mode of assignment useful for bypassing deficiencies in the symmetry routines. Adiabatic ionization potentials were determined as the energy differences between the optimized neutral and the most stable isomer of the corresponding cation radical. These differences include electronic energies, zero-point energy corrections, thermal corrections (at 298.15 K) due to excitation of vibrational levels, and contributions from rotational and translational degrees of freedom. The carbon-carbon bond dissociation energy (BDE) for the cation radical of ethane (propane) was determined as the difference between the energies of methyl radical and methyl (ethyl) cation and that of the most stable cation radical isomer. This energy difference includes electronic energies, ZPE corrections, and thermal corrections (at 298.15 K). B3LYP yielded ethyl cation as a C2V structure with a symmetrical, bridging hydrogen atom between the two carbon atoms with all three basis sets used, in line with high-level ab initio calculations [e.g., our QCISD/6-311G(d,p) optimizations, necessary for obtaining the geometry in the CBS-APNO(m) procedure for the computation of the C-C BDE of propane cation radical; Table 11]. Results and Discussion A. Ethane Cation Radical. 1. Structures and Energetics. The B3LYP geometry optimization was carried out with three basis sets: 6-31G(d), 6-311G(d,p), and 6-311G(2d,p). The increased flexibility of the latter basis set due to the inclusion of two sets of d-functions on carbon was thought to be potentially advantageous for the description of elongated bonds.19 As benchmark levels we used two high-level ab initio methods to optimize all of the cation radical isomers: UMP4/ 6-311G(d,p) and QCISD(T)/6-311G(d,p) or QCISD/6-311G(d,p). Geometry optimizations were performed within three symmetries: C2h for DB, D3d for LB, and Cs for Cs/C2h. The resulting geometries for the isomers of ethane cation radical are given in Tables 1-3, together with previously published

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J. Phys. Chem., Vol. 100, No. 39, 1996 15777

TABLE 1: Geometrical Parameters for the Diborane (DB) Isomer of Ethane Cation Radicala

a

method

r(CC)

r(CH)

r(CH′)

∠(CCH)

∠(CCH′)

∠(H′CH′)

MP2(full)/6-31G(d) MP2(full)/6-311G(d,p) MP4/6-311G(d,p) QCISD(T)/6-311G(d,p)b B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p)

1.550 1.577 1.646 1.674 1.592 1.577 1.581

1.148 1.141 1.130 1.125 1.144 1.146 1.145

1.087 1.086 1.089 1.089 1.089 1.087 1.086

85.1 82.2 81.5 81.7 85.2 84.7 84.6

116.2 115.4 113.5 112.5 115.3 115.8 115.6

114.6 115.6 116.6 116.7 115.1 115.3 115.5

Point group, C2h; state symmetry, 2Ag. b Reference 1b.

TABLE 2: Geometrical Parameters for the Long-Bond (LB) Isomer of Ethane Cation Radicala

method

r(CC)

r(CH)

∠(CCH)

MP2(full)/6-31G(d) MP2(full)/6-311G(d,p) MP4/6-311G(d,p) QCISD(T)/6-311G(d,p)b B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p)

1.919 1.907 1.915 1.915 1.964 1.964 1.973

1.087 1.087 1.091 1.092 1.089 1.087 1.086

98.6 98.4 98.3 98.3 98.4 97.9 97.6

a

Point group, D3d; state symmetry, 2A1g. b Reference 1b.

data1b for comparison. Absolute and relative energies of these three isomers are presented in Table 4. The most important B3LYP computational result is that the diborane structure DB and the long-bond structure LB both turn out to be energy minima, in agreement with previous ab initio studies.1b,8 Furthermore, all methods predict DB to be more stable than LB by ca. 1 kcal/mol. Inspection of Tables 1 and 2 shows that the B3LYP structures of DB and LB are similar to those calculated by the high-level ab initio methods. The B3LYP diborane structure in Table 1 shows a very weak basis set dependence and has a C-C bond length of 1.577-1.592 Å. The weak basis set dependence is also apparent when comparison is made with the B3LYP/631G(d) and 6-31G(d,p) computations of Shephard and PaddonRow, which gave C-C bond lengths for the DB isomer of 1.596 and 1.586 Å, respectively.14 The B3LYP C-C distances in Table 1 are further seen to be comparable to the UMP2 results, but are somewhat shorter than the corresponding UMP4 and QCISD(T) distances, which are 1.65-1.67 Å. Despite this difference with the ab initio methods, B3LYP offers a significant improvement over other DFT methods.13,14,20 For example, Eriksson and co-workers20 obtained a C-C distance of 1.49 Å by use of Becke’s nonlocal gradient corrections for the exchange function21a and Perdew’s function for the correlation part,21b,c even with a large and flexible basis set. Related to the C-C bond length of the DB isomer are two critical features of the bridging C-H bonds. One is the bridging angle, ∠(CCH1), which is calculated to be ca. 85° with B3LYP in comparison with ca. 82° at UMP4 and QCISD(T). The second B3LYP feature is the in-plane C-H bond length, r(CH1), of 1.145 Å, which is in agreement with the UMP2 results but is slightly longer than the UMP4 and QCISD(T) values (1.125 and 1.130

Å). Thus, in comparison with UMP4 and QCISD(T), the B3LYP method yields a slightly shorter C-C distance and simultaneously a slightly longer bridging C-H distance. Inspection of Table 2 shows that for the long-bond structure, LB, the B3LYP deviations from the benchmark methods are in opposite directions to the deviations described for DB. For LB, the B3LYP method overestimates the C-C bond length (1.961.97 Å) in comparison with the ab initio results, which all give 1.91-1.92 Å. The values of the other two geometric variables appear to be nearly method-independent: r(CH) ) 1.09 Å and ∠(CCH) ) 98°. The geometry for the third isomer is detailed in Table 3. Here the structure of the resulting species is method-dependent, having either Cs or C2h symmetry. These species are mutually related as localized and delocalized forms of the same state.1b Results from the ab initio calculations are method-dependent, with a C2h structure at the UMP2/6-311G(d,p) level but a Cs structure at higher levels.1b These results, which have all been confirmed in this work by using frequency analysis where possible in Gaussian 94, resemble those of Ioffe and Shaik,1b which are based on single-point calculations along the C2hfCs path. In comparison, the B3LYP method gives a C2h structure irrespective of the basis set used. As pointed out by Ioffe and Shaik,1b the C2h structure represents the delocalized state for the interconversion of two mirror image Cs structures in which the unpaired electron is localized either on the right-hand or left-hand methylene groups. The ancestry of the C2h structure as the avoided crossing state of the two localized Cs electronshift isomers means that one expects either the C2h or the Cs isomer to be a true minimum, but not both.1b It is for this reason that the high-level ab initio methods locate one of the structures as a minimum and the other as a saddle point. It also suggests that the coexistence of C2h and Cs at the UMP2 level is an artifact of this method, which tends to create spurious minima.22 In contrast, B3LYP provides consistent results, although the identity of the true minimum may be questioned in view of the contrast with the highest ab initio levels. Inspection of Table 4 shows that the ab initio methods from MP2(full)/6-311G(d,p) onward predict the energetic ordering DB < LB < Cs in comparison with the B3LYP ordering, which is DB < C2h < LB. Despite this difference, B3LYP has a number of features in common with the benchmark ab initio methods. All of the methods compute DB to be most stable isomer, in line with experiment. In addition, both methods predict the potential energy surface to be very flat. Finally, they both predict the same energy difference between the DB and LB structures, ca. 1 kcal/mol. The performance of B3LYP is considerably better than its sister DFT methods BLYP and SVWN, which predict an energy difference between DB and LB of 2.5 and 18.7 kcal/mol, respectively, with a 6-31G(d,p)

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Zuilhof et al.

TABLE 3: Geometrical Parameters for the Cs and C2h Isomers of Ethane Cation Radicala,b

method MP4/6-311G(d,p) QCISD/6-311G(d,p) QCISD(T)/6-311G(d,p) B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p)

symmetry r(CC) r(CH1)/r(CH1′) r(CH2)/r(CH2′) ∠(CCH1)/∠(CCH1′) ∠(CCH2)/∠(CCH2′) ∠(H2CH2)/∠(H2′CH2′) Cs Cs Cs C2h C2h C2h

1.462 1.460 1.462 1.444 1.439 1.436

1.089/1.090 1.083/1.085 1.090/1.091 1.089 1.086 1.085

1.178/1.110 1.165/1.104 1.175/1.113 1.143 1.141 1.141

123.01/114.36 122.54/114.35 122.54/114.46 118.01 118.39 118.39

116.61/108.51 116.72/108.88 116.24/108.83 112.40 112.60 112.60

62.15/102.43 62.44/102.23 64.12/101.30 86.54 84.76 84.70

a In C symmetry atom pairs H and H ′ are identical; therefore, r(CH ) ) r(CH ′), and ∠(H CH ) ) ∠(H ′CH ′). b The state symmetries are 2B 2h 2 2 2 2 2 2 2 2 g for the C2h isomer and 2A′′ for the Cs isomer.

TABLE 4: Absolute Energy of the DB Isomer (in hartrees) and the Relative Energies (in kcal/mol) for the Other Minima of Ethane Cation Radical method

DB

LB

Cs/C2h

UMP2(full)/6-31G(d) UMP2(full)/6-311G(d,p) UMP4(SDTQ)/6-311G(d,p) QCISD(T)/6-311G(d,p) QCISD(T)/6-311++G(2df,p)//QCISD(T)/6-311G(d,p) CBS-APNO(m) - ZPEa B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p) B3LYP/6-311G(3df,3pd)//B3LYP/6-311G(d,p) B3LYP/6-311++G(3df,3pd)//B3LYP/6-311G(d,p)

-79.084 77 -79.184 07 -79.191 50 -79.194 14 -79.229 49

-0.02 1.57 0.28 0.40 0.78

not a minimum 4.82; Cs 6.01; Cs 5.71; Cs 4.55; Cs

-79.391 24b -79.410 50 -79.434 70 -79.436 51 -79.441 54

1.48b 0.42 1.50 1.20 1.34

4.18c 1.07;C2h 0.60;C2h 0.52;C2h 0.65

-79.441 62

1.31

0.65

aModified CBS-APNO procedure with ZPE correction deleted (see Theoretical Methods and Calculations section). b QCISD(T)/6-311G(d,p) geometry. c QCISD/6-311G(d,p) geometry.

basis set.14 The stability of the B3LYP results as a function of the basis set flexibility suggest that 6-311G(d,p), or even 6-31G(d), may be used as a routine basis set for the study of cation radicals. As mentioned earlier, the B3LYP and benchmark ab initio calculations make contrasting predictions regarding the relative stabilities of the C2h and Cs structures. This difference is maintained even for QCISD(T)/6-311G++(2df,p)//QCISD/6311G(d,p) and CBS-APNO calculations when compared with B3LYP single points using 6-311G(3df,3pd) or 6-311++G(3df,3pd) basis sets. These results imply that the computational differences originate in the treatment of correlation energy rather than in basis set effects. As discussed in the following, it is likely that the stability of the C2h structure is somewhat overestimated by B3LYP. Nonetheless, it is possible that B3LYP predicts the correct energetic ordering. This may be tested experimentally by examining ESR spectra in different matrices at a variety of temperatures. To the best of our knowledge, however, thus far only the DB isomer has far been detected by ESR.7 To benchmark our B3LYP results against experimental data, the isotropic proton hyperfine coupling constants and the C-C bond dissociation energies were computed for ethane cation radical, and the adiabatic and vertical ionization potentials were calculated for ethane. 2. Isotropic Hyperfine Coupling Constants. Since proton hyperfine coupling constants have been reported to be highly sensitive to slight changes in geometry,13a we report the computed values for this property at all three B3LYP-levels of geometry optimization. The effect of increased basis set

flexibility on the value of the hyperfine coupling constant was investigated at the B3LYP/6-311G(d,p)-optimized geometry with single-point calculations using the 6-311G(3df,3pd) and 6-311++G(3df,3pd) basis sets. Complementary ab initio computations at the UMP2, UMP4(SDQ), and QCISD levels of theory were performed to provide a comparison with other computational levels. The computed data are compiled in Table 5 along with the experimentally observed isotropic hyperfine coupling constant.7 The B3LYP results are seen to be in excellent agreement with the experimental data for the DB structure, even for the smallest basis set, 6-31G(d). Thus, the guidelines of Eriksson et al.13a and Chipman18 about the need for special, very flexible nonGaussian basis sets for the prediction of ESR data may not generally apply to B3LYP calculations.23 Neither the basis set nor the changes in geometry obtained with the three different basis sets seem to influence the B3LYP-derived hyperfine coupling constants to a significant degree. This remarkable stability of the B3LYP results is consistent with the conformity of calculated geometries and relative energies given in Tables 1-4. It is interesting to observe that B3LYP performs significantly better than all of the high-level ab initio methods in Table 5. In fact, the B3LYP results are closer to the experimental value than that derived by Lunell8 (131.3 G) using Davidson’s configuration selection SDCI method24 with a large flexible basis set. Carmichael and Chipman have noted that the conventional ab initio computation of hyperfine coupling constants requires the inclusion of triple excitations in the configuration interaction as well as a very flexible basis set.18,25 Our ab initio results, which were obtained with standard basis sets and do not include triple excitations, seem to be in line with their conclusion. In contrast, the B3LYP method nicely reproduces the experimental data and does not show any inconsistencies. This stability of the B3LYP method combined with its CPU efficiencysof at least 2 orders of magnitude in comparison with QCISD(T)smakes it an attractive alternative for obtaining hyperfine coupling constants of cation radicals. An interesting feature in Table 5 is the large B3LYP hyperfine coupling constant of the C2h structure of 138 G along with the two large ab initio hyperfine coupling constants for the Cs structure. If the C2h/Cs structure pair is observable as suggested by the B3LYP energetics in Table 4, then its ESR spectrum should be distinct from the spectrum of the DB isomer and give a series of five lines with spacing of ca. 138 G (or smaller for two rapidly interconverting Cs structures). This implies that ESR in different matrices might be a useful technique for probing the existence of such a structure. Such an investigation

Ethane and Propane Cation Radicals Study

J. Phys. Chem., Vol. 100, No. 39, 1996 15779

TABLE 5: Isotropic Hyperfine Coupling Constants (in Gauss) for the Isomers of Ethane Cation Radical structurea DB

a

LB

Cs/C2h

method

a(H1)

a(H2)

a(H)

a(H1/H1′)

a(H2/H2′)

MP4(SDQ)/6-311G(d,p)//MP4/6-311G(d,p) QCISD/6-311G(d,p) B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p) B3LYP/6-311G(3df,3pd)//B3LYP/6-311G(d,p) B3LYP/6-311++G(3df,3pd)//B3LYP/6-311G(d,p) experiment

95.1 93.4 147.6 152.4 151.3 152.7 152.7 152.5

0.1 -12.3 -8.4 -7.3 -7.5 -6.3 -6.3

7.8 -3.2 -1.8 -1.8 -2.3 -0.8 -0.8

-15.8/-2.0 -15.7/-1.9 -6.1 -5.6 -5.9 -5.2 -5.2

147.2/61.3 147.9/64.6 139.7 137.8 139.2 137.9 137.8

See Tables 1-3 for the atom numbering scheme.

TABLE 6: Computed and Experimental Carbon-Carbon Bond Dissociation Energy (BDE) of Ethane Cation Radical (in kcal/mol) and the Vertical and Adiabatic Ionization Potentials of Ethane (in eV) method

BDE

IP(vertical)a

IP(adiabatic)

B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p) CBS-APNO(m)b experimental

53.5 52.2 52.3 50.9 50.0d

12.18 12.25; 12.95 12.26 12.6,12.9c ≈12.0e

11.25 11.29 11.29 11.55 11.56 ( 0.02f

Lower value for ionization to 2Eg state; higher value for second IP to 2A1g state. b Includes standard thermal corrections to 298.15 K; see Theoretical Methods and Calculations section. c QCISD(T)/6-311G(d,p) data from ref 1b; for comments see ref 28. d Calculated from a thermochemical cycle using BDE(ethane) ) 89.7 kcal/mol, IP(ethane) ) 11.56 ( 0.02 eV, and IP(methyl radical) ) 9.84 ( 0.02 eV. Experimental data from ref 30. e Reference 26. See text for discussion. f Reference 30.

TABLE 7: Principal Geometrical Parameters for the 1-LB Isomer of Propane Cation Radicala

method

a

could serve as a critical test of whether the B3LYP method overestimates the stability of the C2h isomer. 3. Ionization Potentials and Bond Dissociation Energies. The B3LYP and CBS-APNO-computed ionization potentials and C-C bond dissociation energies (BDE) are presented in Table 6 along with experimental values. The experimental BDE derived from a thermodynamic cycle is 50.0 kcal/mol. This value is accurately reproduced by the CBS-APNO(m) method. The B3LYP data are all slightly greater than the experimental value by 2.2-3.5 kcal/mol. Given the inherent error of (2 kcal/mol in the experimental BDE, the B3LYP data suggest that the Becke three-parameter correction3 is useful for approximating BDEs. As seen in all previously described properties, the 6-31G(d) basis set performs well here too. An increase of the basis set to 6-311G(d,p) slightly improves the agreement with experiment, while the addition of an extra d-polarization function on carbon has no effect. The experimental adiabatic ionization potential of ethane (Table 6) is reported to be 11.56 eV, and this number is reproduced well by the CBS-APNO(m) method. The B3LYP data are lower by about 0.3 eV, in fair agreement with the experimental data. More problematic is the vertical ionization potential of ethane because the identity usually assumed between the peak maximum in the photoelectron spectrum (12.0 eV)26 and the vertical ionization potential is questionable for cases like ethane cation radical, which exhibit strong nonadiabatic effects upon ionization.5,27 As such, there is no consensus value for the vertical ionization potential, although the identity of the lowest vertical state is generally believed to be 2Eg (Figure 1). The ab initio values in Table 6 correspond to Ioffe and Shaik’s QCISD(T)/6-311G(d,p) and ST4CCD/6-311G(d,p) values for the vertical ionization potentials to the 2Eg and 2A1g states, respectively.1b,28 The B3LYP method gives the correct trend, with the lowest ionization potential being associated with the 2E state and the higher with the 2A state. Comparison of the g 1g

MP2/6-31G(d) MP2/6-31 G(d,p) MP4(SDQ)/6311G(d,p) QCISD/6311G(d,p) B3LYP/631G(d) B3LYP/6311G(d,p) B3LYP/6311G(2d,p) a

r(CRCβ) r(CβCγ) r(CRH1) r(CγH4) ∠(CRCβCγ) 1.901 1.893

1.466 1.468

1.088 1.088

1.116 1.118

105.60 105.54

1.911

1.476

1.089

1.115

106.16

1.897

1.477

1.090

1.117

105.78

1.896

1.467

1.090

1.126

105.59

1.906

1.461

1.087

1.125

106.50

1.913

1.458

1.086

1.124

106.26

Point group, Cs; state symmetry, 2A′.

B3LYP results with those of Ioffe and Shaik1b shows that the two sets match at the second vertical IP, but the B3LYP methods yield the first IP 0.3 eV lower than the highest ab initio levels. Since, as seen in Figure 1, the 2Eg state is the precursor state of the DB and C2h isomers (see ref 1b for an explanation), any lowering of the 2Eg state will carry over to the adiabatic DB and C2h isomers. This could explain why B3LYP estimates the C2h isomer to be relatively more stable than do the ab initio calculations. B. Propane Cation Radical. 1. Structures and Energetics. As discussed in the Introduction, optimization of propane cation radical leads to several isomeric structures. The two most stable ones are derived from ionization of the a1 and/or b2 orbitals shown in Figure 1. The first is a species of 2A′ state symmetry with a Cs skeletal structure. This structure, which possesses one long C-C bond of ca. 1.9 Å, is the 1-LB isomer described in Table 7. The electronic structure of 1-LB can be envisioned most simply as resulting from the ionic state, where an electron is removed from an orbital of mixed b2/a1 character. This amounts to taking an electron from the CR-Cβ σ orbital with some contribution from the in-plane σ(CγH4) bond. This mixing leads to significant delocalization of the charge and spin in this cation radical. For example, the group charges for CR, Cβ, and Cγ are 0.457, 0.240, and 0.303 using QCISD/6-311G(d,p) and 0.427, 0.260, and 0.313 at the B3LYP/6-311G(d,p) level of theory, respectively. This indicates that ≈30% of the positive charge is delocalized in the γ-CH3 moiety away from the long bond. The second species located in our study is the C2V structure with a 2B2 state symmetry. As seen from the geometric details

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TABLE 8: Geometric Parameters for the 2-LB Isomer of Propane Cation Radicala

method

r(C1C2)

r(C2H1)

r(C2H2)

r(C1H3)

∠(CCC)

∠(CCH1)

∠(H2CH2)

∠(H3CH3)

UMP2/6-31G(d) UMP2/6-311G(d,p) UMP4(SDQ)/6-311G(d,p) QCISD/6-311G(d,p) B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p)

1.579 1.580 1.595 1.600 1.592 1.590 1.589

1.121 1.121 1.118 1.117 1.123 1.121 1.121

1.088 1.087 1.088 1.088 1.089 1.086 1.085

1.089 1.088 1.089 1.090 1.090 1.087 1.086

94.15 93.73 94.32 94.50 95.56 95.55 95.40

91.66 90.66 90.82 91.47 92.58 92.10 92.07

113.43 113.81 114.07 114.00 113.25 113.44 113.49

112.19 112.84 112.45 112.80 111.94 112.01 112.07

a

Point group, C2V; state symmetry, 2B2.

TABLE 9: Absolute Energy (in hartrees) for the 1-LB Isomer of Propane Cation Radical and the Relative Energy Erel (in kcal/mol) for the 1-LB and 2-LB Isomers, together with the Frequency of the Lowest B2 Vibration in 2-LB method

E(1-LB)

Erel

UHF/6-31G(d) UMP2/6-31G(d) PUMP2/6-31G(d)//UMP2/6-31G(d) QCISD(T)/6-31G(d)//UMP2/6-31G(d) CCSD(T)/6-31G(d)//UMP2/6-31G(d) UMP2/6-311G(d,p) PUMP2/6-311G(d,p)//UMP2/6-311G(d,p) UMP4(SDQ)/6-311G(d,p) QCISD/6-311G(d,p) QCISD(T)/6-311++G(2df,p)//QCISD/6-311G(d,p) (CBS-APNO)(m) - ZPE B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p) B3LYP/6-311G(d,p);  ) 10.0b B3LYP/6-311G(3df,3pd)//B3LYP/6-311G(d,p) B3LYP/6-311++G(3df,3pd)//B3LYP/6-311G(d,p)

-117.905 11 -118.265 40 -118.266 75 -118.326 22 -118.325 82 -118.362 51 -118.364 06 -118.414 29 -118.416 48 -118.485 60 -118.726 82 -118.753 14 -118.785 98 -118.789 02 -118.873 02 -118.795 97 -118.796 11

9.53 -0.45 -0.94 1.00 0.97 -1.00 -1.53 1.35 1.68 0.31 -0.60 0.70 0.94 1.05 0.58 0.96 0.97

freq B2 vibr 2-LBa 874i 469

494 256 284i 107i 151i 168i

a Frequency of the lowest vibration in 2-LB (in cm-1). b Single-point calculation on B3LYP/6-311G(d,p) geometry using the SCIPCM solvent model in Gaussian 94 (ref 17).

in Table 8, this species corresponds to the 2-LB isomer with two long C-C bonds. The calculated geometry of 2-LB is virtually identical for all of the methods used. Its key features are the lengthening of the two carbon-carbon bonds to ca. 1.6 Å, and the slight lengthening of the in-plane σ(CH1) bonds. The 2-LB and 1-LB isomers have been optimized at a wide variety of different B3LYP and ab initio levels. The data are summarized in Table 9. While the 1-LB structure is a minimum at all levels of theory, the nature of 2-LB depends on the level of calculation. At the UHF/6-31G(d) level, the 2-LB species is a saddle point, in agreement with the study of Radom and co-workers using UHF/3-21G.1a Perturbative treatments of the electron correlation at the UMP2/6-31G(d), UMP2/6-311G(d,p), and UMP4(SDQ)/6-311G(d,p) levels of theory, however, predict 2-LB to be a minimum. An explicit account of configuration interaction using the QCISD method inverts the situation again, showing that 2-LB is a saddle point with a B2-type imaginary frequency (frequency calculations beyond the QCISD level are impractical at present). If we adopt the reasonable view that quadratic CI is a better model than the perturbative treatment (MPn; n ) 2, 4), we conclude that the benchmark ab initio method characterizes 2-LB as a saddle point. The results of B3LYP with all three basis sets used agree with this latter result and suggest that the 2-LB isomer is a saddle point having a B2-type imaginary frequency. As seen from the last column in Table 9, the 1-LB isomer is calculated to be the most stable one at all correlated levels of theory apart from UMP2 and CBS-APNO. Thus, our results

contrast with those of Lunell, Feller, and Davidson, who assigned the 2-LB structure as the lowest energy isomer.9 We agree, however, with their conclusion that the potential energy surface around the 2-LB isomer is quite flat, as is also apparent from the small values for the lowest B2 frequency vibration (Table 9). Furthermore, the relative energy of the B3LYP structures is in line with the highest ab initio levels. This relative energy match, as well as the identical characterization of 2-LB as a saddle point by both B3LYP and QCISD/6-311G(d,p), suggests that the B3LYP assignment is correct. The comparison with experiment sheds some light on the 2-LB/1-LB problem. The ESR data for propane cation radical in a SF6 matrix were originally assigned to a structure with approximately C2V symmetry.7 More recently, the EPR spectrum was redetermined by Toriyama et al. in a perfluoropropane matrix.10 This matrix was chosen because of its structural similarity to propane, which presumably allows for structural relaxation after ionization. The EPR spectrum of propane cation radical found in C3F8 is consistent with a Cs (1-LB) structure. On the basis of these data, it was concluded that a Cs structure was the most stable.10 Toriyama also attributed the unusual temperature dependence of the hyperfine coupling constants observed in a SF6 matrix to a partially relaxed 1-LB structure; i.e., they propose that the SF6 matrix prevents complete relaxation of the cation radical. We therefore conclude that the B3LYP computational data reproduce both the extreme flatness of the potential energy surface for propane cation radical and

Ethane and Propane Cation Radicals Study

J. Phys. Chem., Vol. 100, No. 39, 1996 15781

TABLE 10: Hyperfine Coupling Constants (in Gauss) for the 1-LB and 2-LB Structures of Propane Cation Radicala structure 1-LB

2-LB

method

a(H1)

a(H2)

a(H3)

a(H4)

a(H5)

a(H1)

a(H2)

a(H3)

MP2(FC)/6-31G(d) MP2(FC)/6-311G(d,p) MP4(SDQ)/6-311G(d,p) QCISD/6-311G(d,p) B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p) B3LYP/6-311G(3df,3pd)//B3LYP/6-311G(d,p) B3LYP/6-311++G(3df,3pd)//B3LYP/6-311G(d,p) experimentalb

10.9 11.6 10.4 0 5.2 3.2 2.6 4.1 4.1 18

4.6 5.1 6.4 -6.0 -4.6 -4.1 -4.5 -3.1 -3.0 0

9.0 9.0 10.0 2.8 5.0 6.0 5.7 6.7 6.7 6

58.6 57.4 48.7 72.2 98.4 97.2 97.0 96.3 95.9 88

4.6 4.1 3.9 6.0 8.4 9.0 9.3 9.0 8.9 9

89.1 86.3 76.5 88.4 104.4 105.1 105.8 104.4 104.3 100.0

0.5 1.0 0.5 -7.9 -6.4 -6.1 -6.1 -5.6 -5.6 -3.0

-1.7 -1.0 -0.4 -9.3 -7.4 -7.0 -7.0 -6.5 -6.4 2-3

a See Tables 7 and 8 for the atom numbering scheme. b Experimental values for 1-LB (ref 10b) and 2-LB[a(H1) and a(H2), ref 36; a(H3), ref 10b).

the relative energies of the 1-LB and 2-LB isomers deduced by Toriyama et al.10 The two high-energy propane cation radical isomers with C2V symmetry, 2B1 and 2A1, arise from ionization of the b1 and a1 orbitals, respectively (Figure 1). The optimized 2B1 isomer has long C-H bonds in the central methylene group [1.163 Å, B3LYP/6-311G(d,p)] and a relatively wide C-C-C angle (124.2°). Its B3LYP/6-311G(d,p) energy is only 3.8 kcal/mol above that of the lowest isomer (1-LB), which suggests once again that B3LYP might somewhat overestimate the stability of cation radical isomers with long C-H bonds. The 2A1 isomer has two slightly elongated C-C bonds [1.545 Å, B3LYP/6311G(d,p)] and a wide C-C-C angle (129.2°), and its B3LYP/ 6-311G(d,p) relative energy (6.2 kcal/mol above that of the lowest isomer) is in agreement with previous ab initio calculations.9 Both the 2B1 and 2A1 isomers possess two imaginary frequencies and, consequently, will not be discussed further. 2. Isotropic Hyperfine Coupling Constants. The isotropic hyperfine coupling constants computed from the Fermi contact terms for both 1-LB and 2-LB are presented in Table 10 along with the experimental results.7,10 The MP2, MP4(SDQ), and QCISD data are in rather poor accord with the experimental values, in line with the conclusions of Chipman and Charmichael that triple excitations are potentially important for reproducing experimental hyperfine coupling constants.18b,25 In contrast, the correspondence of our B3LYP data with experiment is generally good, but the fit is not as good as in the case of ethane cation radical [especially off is the a(H1) value for 1-LB]. For 2-LB, the B3LYP hyperfine coupling constants match the experimental values quite well and better than any of the conventional ab initio methods (Table 10). The agreement is even better than for the CISD values of Lunell et al.,9 which are 88.6, -5.8, and -7.1 G for a(H1), a(H2), and a(H3), respectively. At this point it is important to emphasize some of the hazards of comparing computed hyperfine coupling constants to experimental quantities when the species lies on a flat potential energy surface. As we have seen, the surface that connects 2-LB to 1-LB via the B2 vibrational mode is very soft and becomes even softer when medium effects are taken into account (see the entry with  ) 10.0, Table 9). This softness is further augmented by the fact that the ZPE contribution of the C-C stretching mode of 1-LB is on the order of the barrier separating the mirror image 1-LB isomers via their 2-LB saddle point. As a result, one might expect that the 1-LB geometry will be different for the isolated species and for the species embedded in a matrix. The same argument can be made for the 2-LB isomer, which can easily distort from C2V along the B2 vibrational mode, thus creating a species with slightly unequal C-C bonds. Perhaps more significant, vibrational averaging

TABLE 11: Computed and Experimental Carbon-Carbon Bond Dissociation Energy of Propane Cation Radical (in kcal/mol) and the Vertical and Adiabatic Ionization Potentials of Propane (in eV) method B3LYP/6-31G(d) B3LYP/6-311G(d,p) B3LYP/6-311G(2d,p) UMP2/6-311G(d,p) CBS-APNO(m)c experimental

BDE 30.0 26.5 26.6 22.1 22.5e,f

IP(vertical) 11.45 11.54; 11.71; 11.80a 11.53 12.12; 12.14; 12.19b 12.02d 11.5g

IP(adiabatic) 10.50 10.59 10.58 11.03 10.95 ( 0.05f

Ionization to 2B1, 2A1, and 2B2 states, respectively. b Ionization to and 2A1 states, respectively. c Adapted by using QCISD/6311G(d,p) frequencies rather than UHF/6-311G(d,p) frequencies for the thermal correction to 298.15 K. See Theoretical Methods and Calculations section. d Calculated by using the standard CBS-APNO procedure. e Calculated from a thermochemical cycle, by using BDE(propane) ) 87.8 kcal/mol, IP(propane) ) 10.95 ( 0.05 eV, and IP(ethyl radical) ) 8.12 ( 0.01 eV. f Experimental data: ref 30. g Reference 26. a

2B , 2B , 2 1

effects18b,c,29 are expected to be important for a flat potential energy surface like that for the 1-LB and 2-LB isomers. Thus, the comparison of computed to experimental hyperfine coupling constants must be made with these caveats in mind. Within these limitations, the B3LYP method seems to be a very useful tool for computing ESR hyperfine coupling constants of cation radicals, while standard high-level ab initio methods evidently need nonstandard modifications.25 3. Ionization Potentials and Bond Dissociation Energies. Table 11 shows the C-C bond dissociation energy (BDE) of 1-LB and the vertical and adiabatic ionization potentials of propane calculated with B3LYP using three different basis sets [6-31G(d), 6-311G(d,p), and 6-311G(2d,p)]. These results are compared with those of corresponding UMP2/6-311G(d,p) and CBS-APNO(m) computations (see Theoretical Methods and Calculations section), and experimental data. B3LYP performs reasonably well with regard to computation of the BDE, even though some overbinding may be reflected when compared to the benchmark CBS-APNO(m) and experimental values. The adiabatic IP for propane is reproduced very well by our modified CBS-APNO(m) procedure. B3LYP underestimates the adiabatic IP by an amount similar to that found for ethane (cf. Table 6). The comparison of computed and experimental vertical ionization potentials for propane presents some problems. The most recently accepted experimental value is 11.50 eV.30 Nonetheless, values as high as 12.7 eV have been reported (see cited literature in ref 1a). B3LYP predicts the lowest vertical IP to be 11.54 eV, in excellent agreement with experiment. The CBS-APNO(m) and UMP2 values are significantly higher, however. Although the order of the vertical states has not been

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firmly established experimentally, the differences between peak maxima in the propane photoelectron spectrum suggest that the three, lowest lying vertical states are separated by ca. 0.6 eV.1a The predicted spacing from ab initio calculations is much smaller and in disagreement with experiment.1a,31 Our UMP2/ 6-311G(d,p) calculations predict three closely spaced states within 0.07 eV, while the B3LYP spacing is somewhat larger (0.26 eV) but still smaller than the experimental one. Furthermore, the ordering of the vertical states is highly methoddependent. For example, while our UMP2/6-311G(d,p) ordering (2B2 < 2B1 < 2A1) matches the results of Radom,1a those of Peyerimhoff and co-workers31 are different (2B1 < 2A1 < 2B2) and agree with our B3LYP results. Finally, it should be noted that Peyerimhoff and co-workers have argued that peak maxima in the photoelectron spectrum are not reliable measures of the vertical IPs due to nonadiabatic effects.31 Given these limitations, we simply note that all of the theoretical methods listed in Table 11 provide a consistent picture, namely, of three closely spaced vertical states for propane cation radical and that B3LYP reproduces the experimental value for the lowest vertical IP. 4. Electromerism: Localized/Delocalized and ElectronShift Isomers of C3H8•+. It is important to understand the relationship between the 2-LB and 1-LB propane cation radical isomers as part of a general discussion of cation radical electromerism. In the context of cation radical chemistry, we define electromerism as isomerism that results from ground state ion radical structures that have the same atomic connectivity but in which the unpaired electron is localized in different moieties of the molecules.32 Electron-shift isomers are electromers that differ by single-electron excitation promoted along a distortion coordinate that interconnects them (Vide infra). In contrast, delocalized isomers are electromers that are simply resonance hybrids of electron-shift isomers. To illustrate these distinctions, we derive the relationship between the 1-LB and 2-LB isomers by using a valence-bond curve-crossing model. The same type of analysis has been used before1b to outline the isomeric relationship for the LB, DB, and C2h/Cs structures of the ethane cation radical.

First we consider the symmetry properties of the left/rightlocalized 1-LB structures under the symmetry operations of the C2V point group.33 As depicted in 1, the C2 axis of rotation and the σV mirror plane interconvert the two 1-LB structures. Consequently, their positive combination, eq 1a, is the totally symmetric 2A1 state, while their negative combination, eq 1b, is the 2B2 state, which is antisymmetric to the two symmetry operations.

Ψ(2A1) ) 2-1/2[1-LB(r) + 1-LB(l)]

(1a)

Ψ(2B2) ) 2-1/2[1-LB(r) - 1-LB(l)]

(1b)

Next we consider interconversion of the 1-LB structures along the C-C-C asymmetric stretch mode labeled as Q(B2) to emphasize its B2 symmetry at the C2V point group structure. This mode, drawn in 2, is the same one found to have an imaginary frequency when the 2-LB isomer is characterized as a saddle point. It is seen that this mode involves the simultaneous lengthening of one C-C bond and shortening of the other. Figure 4 shows the energies of the 1-LB electronshift isomers along this mode. Each of the electron-shift isomers is an energy minimum at the extrema of the diagram, where

Figure 4. Valence-bond curve-crossing diagrams for the isomerization of 1-LB(l) and 1-LB(r). The diagrams for large and small vertical energy gaps (G) are shown in (a) and (b).

the one-electron bond is long and the two-electron bond is short. The reaction coordinate Q(B2) interchanges these bond lengths while maintaining the same electronic structure. Consequently, 1-LB(l) becomes a vertical excited state, 1-LB(l)*, of the 1-LB(r) ground state at one Cs extremum of the Q(B2) coordinate. The analogous behavior is shared by 1-LB(r), which becomes a vertical excited state, 1-LB(r)*, along the reverse Q(B2) motion. It follows, therefore, that the two 1-LB forms intersect along the Q(B2) coordinate. This curve crossing along a mode that promotes an electron shift between two structures is characteristic of electron-shift isomers.1b Also shown in Figure 4 is the avoided crossing of the two 1-LB electron-shift isomers, which creates two states that are resonance hybrids of the 1-LB electron-shift isomers. As described by eqs 1a and 1b, these states have 2A1 and 2B2 symmetries. The sign of the avoided crossing matrix element between the 1-LB forms is negative, and therefore the bonding combination will be 2B2 and the antibonding combination 2A1.34a This is shown in the blowup of the avoided crossing region in Figure 4. Consequently, the 2-LB isomer with 2B2 state symmetry is shown to be the delocalized isomer of the localized 1-LB electron-shift isomers. The energies of the state curves along Q(B2) depend, inter alia, on the vertical gap of the 1-LB forms at the Cs extrema. This is shown in Figure 4a,b of the two avoided crossing diagrams. In (a), we present the situation with a large vertical gap, G, which leads to a 2B2 state that is a saddle point for the interconversion of the 1-LB structures. In contrast, (b) shows the situation expected for a small gap, G, where the avoided crossing is likely to generate the 2B2 state as a minimum. In this latter case, the 1-LB forms will not be energy minima. Thus,

Ethane and Propane Cation Radicals Study the model predicts that the 1-LB and 2-LB isomers will generally not coexist as minima, i.e., we expect an either/or trend. This seems to be the situation for the B3LYP and QCISD levels of calculation for the propane cation radical (Table 9).34b Can the 1-LB and 2-LB isomers ever coexist? Within the curve-crossing model, the coexistence of the two isomer types as minima will be possible only if a third valence structure (e.g., one-electron states of C-H bonds) can mix effectively into either the 2-LB (2B2) or 1-LB (2A′) states.35 With this in mind, it is interesting to note that the UMP2 and UMP4 methods predict a 2-LB structure with a shorter C-C bond, but longer in-plane C-H bonds in comparison with the QCISD method. This could lead to stronger mixing of the C-H cation radical states into 2-LB (2B2) by the MPn methods, which, in turn, might explain why these methods predict that both 1-LB and 2-LB isomers are minima. The preceding analysis of the electromers of C3H8•+ provides a means to systematically define the isomeric relationships expected for cation radical electromers. Figure 4 shows the general relationship that will be maintained between electronshift isomers of any variety in terms of the crossing of two structures along a specific promoting mode. In general, two electromers will be mutually related as electron-shift isomers if single-electron excitation of one of them produces the second species as an excited state and if the resulting state-pair interchange their ground/excited state relationships along a distortion coordinate that links the nuclear geometries of the two isomers. By using this definition, the two 1-LB(l, r) species are electron-shift isomers and so are the LB, DB, and Cs (or C2h) isomers in the case of ethane cation radical. The other type of isomeric relationship is illustrated by the avoided crossing diagrams in Figure 4. It is seen that 2-LB is a resonance hybrid of the two 1-LB(l, r) forms at their crossing point and is analogous to benzene and its Kekule´ structures at D6h geometry. Thus, 2-LB and the relaxed 1-LB(l,r) species are mutually related as delocalized/localized electromers of the same electronic nature. The same relationship applies in the case of ethane cation radical to the C2h and the two Cs(l,r) species (Figure 2). Thus, the C2h isomer is the resonance hybrid of the two Cs(l,r) forms at their crossing point,1b and as such the C2h and the two relaxed Cs(l,r) species are mutually related as delocalized/localized electromers. In general, mirror image electron-shift isomers will interconVert Via delocalized isomers, which may or may not themselVes be energy minima. Conclusions Application of B3LYP and high-level ab initio methods to investigate the properties of ethane and propane cation radical isomers reveals both advantages and disadvantages of the various methods. It is apparent from our results and previous ab initio results that reliable structures for the isomers of these cation radicals, their nature, and their energies require computational levels well beyond UMP2. On average, the B3LYP method performs better than standard ab initio methods up to UMP4. B3LYP gives the principal cation radical isomers in the correct energetic order and the correct nature (minimum or saddle point) and leads to hyperfine coupling constants in excellent agreement with experiment. In contrast, neither QCISD nor UMP4(SDQ) gives reliable hyperfine coupling constants. The B3LYP method shows little sensitivity to basis set changes from 6-311G(d,p) onward. In cases where this basis set is too large to be practical, the 6-31G(d) basis set seems to provide a good alternative, apart from calculation of the carboncarbon bond dissociation energies, where use of the 6-311G(d,p) basis set is recommended. The B3LYP method satisfactorily reproduces carbon-carbon bond dissociation energies, although

J. Phys. Chem., Vol. 100, No. 39, 1996 15783 the calculated values appear to contain a small amount of overbinding (ca. 3 kcal/mol) when compared to experimental and benchmark computational values. The general performance of the B3LYP method along with its overwhelming CPU advantage11 over the highest ab initio levels, such as QCISD and beyond, renders B3LYP a viable tool for exploring the structures and reactivities of medium- to large-sized cation radicals. There are, however, some shortcomings in the B3LYP functionals in comparison with the highest level ab initio methods. For example, adiabatic and vertical ionization potentials are calculated with B3LYP/6-311G(d,p) to be too low, with error margins of ca. 0.3 eV. Since the vertical IP of ethane and the stability of its C2h cation radical isomer are related, the lower IP computed by using B3LYP explains the high stability of this isomer relative to that obtained by using QCISD(T). It is important to note that the application of the CBS-APNO model chemistry to cation radicals requires caution to avoid meaningless results. The problems are largely due to the UHF/ 6-311G(d,p) starting point of the procedure. At this level of theory, the nature of the optimized stationary point (minimum or TS) differs in some cases from that obtained at correlated ab initio levels, yielding meaningless zero-point energy corrections for certain isomers. Also, the isomer that the initial UHF/6-311G(d,p) optimization yields might not be the lowest isomer at higher levels, which necessitates a prior optimization of the species of interest at a higher level and circumvention of the UHF optimization. In addition, due to the initial UHF calculation, state misassignments can occur in the calculation of vertical ionization potentials (see Theoretical Methods and Calculations section for an explanation).28 Clearly, a more flexible CBS-APNO procedure would be desirable with variable starting levels before the QCISD optimization. Finally, the present computational study and virtually all its predecessors (e.g., refs 1, 5, and 9) indicate that the potential energy surface of cation radical electromers can be quite flat. In fact, the modes that interconvert the electron-shift isomers can carry ZPEs on the order of the barriers separating the isomers. From a computational point of view, it is apparent that the harmonic approximation for these vibrational modes will not be valid. From the point of view of their physical and chemical behavior, the isomerism of the cation radicals lies at the borderline of the validity of the Born-Oppenheimer adiabatic approximation. As such, it remains an interesting problem to relate potential energy minima on such flat energy surfaces to the typical view of static structures in BornOppenheimer energy wells. In addition, the relationship of measured properties of these species to calculations on static structures presents challenges to computational chemistry. Acknowledgment. The authors thank D. Chipman (Radiation Laboratory, University of Notre Dame) for advice regarding ESR hyperfine coupling constant calculations and D. Danovich (Hebrew University) for assistance with the symmetry problems related to the vertical ionizations of ethane. Research at the University of Rochester was supported by the National Science Foundation (CHE-9312460) and at the Hebrew University by the Volkswagen Stiftung. H.Z. thanks the Netherlands Organization for Scientific Research (NWO) for a postdoctoral Talent-stipendium. References and Notes (1) (a) Bouma, W. J.; Poppinger, D.; Radom, L. Isr. J. Chem. 1983, 23, 21. (b) Ioffe, A.; Shaik, S. J. Chem. Soc., Perkin Trans. 2 1993, 1461. (2) (a) Bally, T.; Jungwirth, P. J. Am. Chem. Soc. 1993, 115, 5783. (b) Bally, T.; Truttmann, L.; Dai, S.; Williams, F. J. Am. Chem. Soc. 1995,

15784 J. Phys. Chem., Vol. 100, No. 39, 1996 117, 7916. (c) Bally, T.; Truttmann, L.; Wang, J. T.; Williams, F. J. Am. Chem. Soc. 1995, 117, 7923. (3) (a) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (b) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623. (4) See for example: (a) Holthausen, M. C.; Heinemann, C.; Cornehl, H. H.; Koch, W.; Schwarz, H. J. Chem. Phys. 1995, 102, 4931. (b) Heinemann, C.; Wesendrup, R.; Schwarz, H. Chem. Phys. Lett. 1995, 236, 194. (c) Schwarz, J.; Heinemann, C.; Schro¨der, D.; Schwarz, H.; Hrusa´k, J. HelV. Chim. Acta 1996, 79, 1. (d) Kudo, H.; Hashimoto, M.; Yokoyama, K.; Wu, C. H.; Dorigo, A. E.; Bickelhaupt, F. M.; Schleyer, P. v. R. J. Phys. Chem. 1995, 99, 6477. (e) Matzinger, S.; Bally, T.; Patterson, E. V.; McMahon, R. J. J. Am. Chem. Soc. 1996, 118, 1535. (f) Schleyer, P. v. R.; Jiao, H.; Sulzbach, H. M.; Schaefer, H. F., III; J. Am. Chem. Soc. 1996, 118, 2093. (g) Jensen, G. M.; Goodin, D. B.; Bunte, S. W. J. Phys. Chem. 1996, 100, 954. (h) Fox, T.; Kollman, P. A. J. Phys. Chem. 1996, 100, 2950. (i) Liu, R.; Tate, D. R.; Clark, J. A.; Moody, P. R.; Van Buren, A. S.; Krauser, J. A. J. Phys. Chem. 1996, 100, 3430 (j) Han, W.-G.; Suhai, S. J. Phys. Chem. 1996, 100, 3942. (k) Dobbs, K. D.; Dixon, D. A. J. Phys. Chem. 1996, 100, 3965. (5) Ko¨ppel, H.; Cederbaum, L. S.; Domcke, W.; Shaik, S. S. Angew. Chem., Int. Ed. Engl. 1983, 22, 210. (6) Bellville, D. J.; Bauld, N. L. J. Am. Chem. Soc. 1982, 104, 5700. (7) (a) Iwasaki, M.; Toriyama, K.; Nunome, K. J. Am. Chem. Soc. 1981, 103, 3591. (b) Toriyama, K.; Nunome, K.; Iwasaki, M. J. Phys. Chem. 1981, 85, 2149. (c) Toriyama, K.; Nunome, K.; Iwasaki, M. J. Chem. Phys. 1982, 77, 5891. (8) Lunell, S.; Huang, M.-B. J. Chem. Soc., Chem. Commun. 1989, 1031. (9) Lunell, S.; Feller, D.; Davidson, E. R. Theor. Chim. Acta 1990, 77, 111. (10) (a) Toriyama, K. Chem. Phys. Lett. 1991, 177, 39. (b) Toriyama, K.; Okazaki, M.; Nunome, K. J. Chem. Phys. 1991, 95, 3955. (11) For example, QCISD/6-311G(d,p) optimization and then frequency calculations for the 2B2 state of C2V propane cation radical took 3 and 21 days, respectively, on a single-user SGI Power Challenge R8000 workstation with 256 MB of RAM and ample disk space. In comparison, corresponding B3LYP/6-311G(d,p) computations only took a total of 11 h for both jobs on a 96 MB dual-user R6000 workstation! (12) Barone, V.; Adamo, C. Chem. Phys. Lett. 1994, 224, 432. (13) (a) Eriksson, L. A.; Malkin, V. G.; Malkina, O. L.; Salahub, D. R. J. Chem. Phys. 1993, 99, 9756. (b) Eriksson, L. A.; Malkin, V. G.; Malkina, O. L.; Salahub, D. R. Int. J. Quant. Chem. 1994, 52, 879. (c) Huang, M. B.; Suter, H. U.; Engels, B.; Peyerimhoff, S. D.; Lunell, S. J. Phys. Chem. 1995, 99, 9724. (14) Shephard, M. J.; Paddon-Row, M. N. J. Phys. Chem. 1995, 99, 3101. (15) (a) Montgomery, J. A., Jr.; Ochterski, J. W.; Petersson, G. A. J. Chem. Phys. 1994, 101, 5900. (b) Ochterski, J. W.; Petersson, G. A.; Montgomery, J. A., Jr. J. Chem. Phys. 1996, 104, 2598. (16) Ochterski, J. W.; Petersson, G. A.; Wiberg, K. B. J. Am. Chem. Soc. 1995, 117, 11299. (17) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Cioslowski, J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; HeadGordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94, ReVision C.3; Gaussian, Inc.: Pittsburgh, PA, 1995. (18) (a) Chipman, D. M. J. Chem. Phys. 1983, 78, 3112. For general discussions of this technique, see: (b) Chipman, D. M. Theor. Chim. Acta 1992, 82, 93. (c) Chipman, D. M. In Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy; Langhoff, S. R., Ed.; Kluwer Academic: Dordrecht, The Netherlands, 1995; pp 109-138. (19) For a general discussion of the choice of basis sets, see: Feller, D.; Davidson, E. R. Basis Sets for Ab Initio Molecular Orbital Calculations and Intermolecular Interactions. In ReViews Compututational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH Publishers: 1990, 1, p 1. (20) Eriksson, L. A.; Lunell, S.; Boyd, R. J. J. Am. Chem. Soc. 1993, 113, 6896. (21) (a) Becke, A. D. Phys. ReV. A 1988, 38, 3098. (b) Perdew, J. P. Phys. ReV. B 1986, 33, 8822. (c) Perdew, J. P. Phys. ReV. A 1988, 34, 7406.

Zuilhof et al. (22) Fox, G. L.; Schlegel, H. B. J. Am. Chem. Soc. 1993, 115, 6870. (23) Successful applications of the B3LYP method for the prediction of ESR parameters with flexible basis sets that include extra, very tight s-functions are published; see, for example, ref 13c and (a) Adamo, C.; Barone, V.; Fortunelli, A. J. Chem. Phys. 1994, 98, 8648. (b) Adamo, C.; Barone, V.; Fortunelli, A. J. Chem. Phys. 1995, 102, 384. It should be pointed out, however, that the B3LYP/6-311G(d,p)-computed hyperfine coupling constants for methyl radical [a(C) ) 26.0 G; a(H) ) -23.0 G] are in excellent agreement with the benchmark experimental data cited in these papers [a(C) ) 27.0 G; a(H) ) -23.0 G]. (24) (a) Tanaka, K.; Davidson, E. R. J. Chem. Phys. 1979, 70, 2904. (b) Feller, D.; Davidson, E. R. J. Chem. Phys. 1981, 74, 3977. (25) Carmichael, I. J. Phys. Chem. 1991, 95, 108. (26) Levin, R. D.; Lias, S. G. Ionization Potential and Appearance Potential Measurements 1971-1981; Publication report NSRDS-NBS 71; National Bureau of Standards, Department of Commerce: Washington DC, 20234; Library of Congress Catalog Card Number 82-2095. (27) Richartz, A.; Buenker, R. J.; Bruna, P. J.; Peyerimhoff, S. Mol. Phys. 1977, 33, 1345. (28) Application of the standard CBS-APNO procedure (ref 15) incorrectly leads to vertical ionization to the 2A1g state. This is related to the fact that CBS-APNO starts with a UHF/6-311G(d,p) calculation, at which level the lowest vertical state is 2A1g. Evidently, the CBS-APNO calculations that follow the UHF starting point do not break the symmetry and remain with the 2A1g state, despite the fact that the lowest vertical state is 2Eg. (29) (a) Fessenden, R. W. J. Chim. Phys. 1964, 61, 1570. (b) Krusic, P. J.; Meaken, P.; Jesson, J. P. J. Phys. Chem. 1971, 75, 3438. (c) Nelsen, S. F.; Frigo, T. B.; Kim, Y. J. Am. Chem. Soc. 1989, 111, 5387. (d) Barone, V.; Adamo, C.; Grand, A.; Brunel, Y.; Fontecave, M.; Subra, R. J. Am. Chem. Soc. 1995, 117, 1083. (e) Barone, V.; Adamo, C.; Grand, A.; Subra, R. Chem. Phys. Lett. 1995, 242, 351. (30) IST Standard Reference Database 25: Structures and Properties; Lias, S. G., Liebman, J. F., Levin, R. D., Kafafi, S. A., Eds.; National Institute of Standards and Technology; Gaithersburg, MD 20899; version 2.02 by Stein, S. E., 1994. (31) Richartz, A.; Buenker, R. J.; Peyerimhoff, S. D. Chem. Phys. 1978, 31, 187. (32) Electromer was previously introduced by Fry as a shorthand term to describe electronic isomers in an early theory of chemical bonding.32a The term was recently reintroduced to the literature to describe several inorganic electronic isomers of a nature similar to that described here.32b,c (a) Fry, H. S. Z. Physik. Chem. 1911, 76, 398. (b) Fiedler, A.; Schro¨der, D.; Shaik, S.; Schwarz, H. J. Am. Chem. Soc. 1994, 116, 10734. (c) Shaik, S.; Danovich, D.; Fiedler, A.; Schro¨der, D.; Schwarz, H. HelV. Chem. Acta 1995, 78, 1393. (33) The pictures in 1 contain the unpaired electron only in the C-C bond. However, as shown by NBO analysis (Glendening, E. D.; Badenhoop, J. K.; Reed, A. E.; Carpenter, J. E.; Weinhold, F. NBO 4.0 Program; Theoretical Chemistry Institute, University of Wisconsin: Madison, WI, 1996), each left/right structure contains contributions from both the C-C and the in-plane C-H bonds. The symmetry arguments as well as the essential features contained in Figure 4 are not affected by use of the simplified pictures. j r| and Ψ(1-LB(r)) ) |σrσlσ j l|, the (34) (a) By using Ψ(1-LB(l)) ) |σlσrσ one-electron matrix element is -〈σr|H|σl〉. Since the matrix element has a negative sign, the lower state due to the 1-LB(l)/1-LB(r) mixing is their negative combination, 2B2 (eq 1b), and the higher state is the positive combination, 2A1 (eq 1a). See: Shaik, S. In New Concepts for Understanding Organic Reactions; Bertran, J., Csizmadia, I., Eds.; NATO ASI Series Vol. C267, Kluwer Academic: Dordrecht, 1989; p 165. (b) Due to the mixing of the cation radical state with the unpaired electron in the C-H bond of the central CH2 moiety into the 2A1 state, the latter state in fact has a B2 mode with imaginary frequency dominated by the CH2 rocking motion. (35) See, for example, (a) Shaik, S.; Hiberty, P. C. AdV. Quant. Chem. 1995, 26, 99. (b) Pross, A. Theoretical and Physical Principles of Organic ReactiVity; Wiley-Interscience: New York, 1995; pp 120. (36) Matsuura, K.; Nunome, K.; Toriyama, K.; Iwasaki, M. J. Phys. Chem. 1989, 93, 149.

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