J. Phys. Chem. 1996, 100, 16147-16154
16147
Ring Opening of Cyclopropylidene and Internal Rotation of Allene Holger F. Bettinger,1a Peter R. Schreiner,1a Paul v. R. Schleyer,*,1a and Henry F. Schaefer, III1b Institut fu¨ r Organische Chemie, Friedrich-Alexander UniVersita¨ t Erlangen-Nu¨ rnberg, Henkestr. 42, D-91054 Erlangen, Germany, and the Center for Computational Quantum Chemistry, The UniVersity of Georgia, Athens, Georgia 30602 ReceiVed: May 9, 1996X
Density functional theory and ab initio quantum mechanical techniques were employed to study the ring opening of singlet cyclopropylidene 1-S and the internal rotation of singlet allene 2-S. The B3LYP, CISD, and CASSCF(4,4) geometry optimizations used a TZP basis set. Single-point TZ2P energies were evaluated at CCSD(T) and multireference configuration interaction with single and double excitations (MR-CISD) levels. Employing the four most important configurations at the MR-CISD/TZ2P//UB3LYP/TZP level gave singlettriplet separations for 1 of 11.1 kcal mol-1 and for 2 48.9 kcal mol-1. Asymmetric transition structures for the ring opening of 1-S were obtained at B3LYP/TZP and CISD/TZP. The barriers are 4.2-4.5 kcal mol-1 at CCSD(T)/TZ2P with these geometries. In contrast, only one Cs symmetric TS was located at CASSCF(4,4), and the ring opening barrier is only 3.6 and 3.1 kcal mol-1 at CCSD(T)/TZ2P and MR-CISD/TZ2P, respectively. At CCSD(T)/6-31G* an asymmetric structure for the TS results confirming the predictions of the B3LYP and CISD methods. The B3LYP/6-31G* intrinsic reaction coordinate (IRC) delineated the complex modes of rotation of the methylene groups during the ring opening. Although the overall rotation in going from 1-S to 2-S must be conrotatory, the ring opening of 1-S starts with a disrotatory motion of both methylene groups. However, before the C1 TS is reached, the sense of rotation of one of the methylene groups changes. At the B3LYP and CASSCF(4,4) levels, the transition state for the internal rotation of allene is a planar bent (C2V) open-shell singlet (1A2). The D2h open-shell singlet is a higher lying second-order stationary point. All D2h and C2V closed-shell species are significantly higher in energy. MR-CISD single-point activation barriers obtained from the B3LYP and CASSCF(4,4) geometries were 44.6 and 45.5 kcal mol-1, respectively. The B3LYP/TZP ring opening (4.8 kcal mol-1) and allene internal rotation (44.6 kcal mol-1) barriers are in remarkably good agreement with the CCSD(T)/TZ2P//B3LYP/TZP value (4.5 kcal mol-1) on the barrier and with the MR-CISD/TZ2P//B3LYP/TZP barrier for the latter (44.6 kcal mol-1).
Introduction
SCHEME 1: Ring Opening of Cyclopropylidene
The simplest cyclic carbene, cyclopropylidene, 1, readily undergoes ring opening to allene 2 (Scheme 1). This thermal isomerization must be facile, since cyclopropylidene has never been isolated.2 The ring opening of cyclopropylidene (generated by photolysis at 77 K) gave allene3 spontaneously. Since cyclopropylidene precursors can be generated by several methods easily, the opening process is valuable for the synthesis of allenes.4 Knowledge of the mechanism of this prototype reaction provides fundamental insights. Earlier ab initio calculations, at increasingly sophisticated levels of theory, show convincingly that the ground state of cyclopropylidene is the 1A1 singlet state (1-S), and that the opening reaction proceeds on a singlet potential energy surface (PES).5-7a,c,8 A hypothetical chiral cyclopropylidene with deuterium and tritium atoms trans substituted could, in principle, either give a single enantiomeric chiral allene or a racemate (Scheme 1). The reaction proceeds stereospecifically for substituted cyclopropylidenes, but this is probably due to steric influences.9 However, the observed conservation of dissymmetry in the ring opening of various unsymmetrically para-substituted cis-diarylcyclopropylidenes cannot originate from steric effects.10 Jones and Krause10 argued that the cis-2-p-methylphenyl-3-p′-bromophenylcarbene should yield lower optical purity or even the X
Abstract published in AdVance ACS Abstracts, September 1, 1996.
S0022-3654(96)01343-3 CCC: $12.00
SCHEME 2: cis-2-p-Methylphenyl-3-p′-bromophenylcyclopropylidene (left) and cis-2-p-Methylphenyl3-phenylcyclopropylidene (right)
reversed allene configuration compared to the less bulky substituted cis-2-p-methylphenyl-3-phenylcarbene (Scheme 2). Thus, the observed higher optical purity and relative configuration of the allene must reflect electronic effects operating during the ring opening. This view was questioned by Ruedenberg who attributes the observed stereospecifity to “long-range” dipole-dipole interactions rather than “short-range” covalent effects.7d Based on experimental results and semiempirical calculations, conrotatory,9 monorotatory,10 and nonrotatory11 mechanisms have been proposed for the ring opening. The isoelectronic cyclopropyl cation is well-known to open disrotatorily to the © 1996 American Chemical Society
16148 J. Phys. Chem., Vol. 100, No. 40, 1996 allyl cation.12 Hence, at least in the initial stages, the disrotatory motion of the cyclopropylidene methylene groups preserving Cs symmetry should be allowed according to simple MO arguments, e.g., the Woodward-Hoffmann rules.13 Indeed, ab initio calculations of Pasto (RHF/STO-3G),14 Radom (RHF/631G*),15 and Yoshimine6 (RHF/DZP) identify this Cs symmetrypreserving reaction channel. Since only two electrons are “active”, the disrotatory rotation of the cyclopropyl cation gives planar allyl cation straightforwardly. In contrast, the ring opening of cyclopropylidene involves four electrons and the overall sense of rotation to give D2d allene must be conrotatory. Therefore, the Cs symmetry of the disrotatory motion has to be broken along the reaction path. A bifurcation along the reaction coordinate results in two isoenergetic enantiomeric reaction paths which are related by Cs symmetry. As Valtazanos and Ruedenberg pointed out,16 least-energy orthogonal trajectories are unsuited as reaction path models for the description of bifurcations. Therefore, they introduced the concept of the “valley-ridge-inflection” point (VRI), which should be near the bifurcation. Where does the reaction channel of the ring opening reaction of 1-S bifurcate? There are two possibilities, before or after the TS is reached. If before, two asymmetric mirror image TS’s must be involved. Alternatively, the bifurcation can occur after one single Cs symmetric TS. The minimal basis set calculations of Pasto14 gave a TS with Cs symmetry, but asymmetric transition states were obtained with larger basis sets.6,15 Ruedenberg et al. studied this reaction comprehensively.7a-d The full optimized reaction space (FORS) MCSCF method with four electrons in four reactiVe orbitals,17 which is similar to the CASSCF method,18 was used with the STO-3G basis set to scan the PES depending upon three internal coordinates. At this level of theory, the reaction path bifurcates before reaching the enantiomeric TS’s.19 The situation changes when larger basis sets [like a tightly contracted DZP (C 9s5p1d/3s2p1d; H 4s1p/ 2s1p)] and a larger active space was chosen. A small but very significant change in geometry resulted since only one Cs symmetric TS was obtained. The activation barrier for the ring opening of cyclopropylidene decreases with increasing theoretical sophistication: 18.0 kcal mol-1 at RHF/4-31G//RHF/STO-3G,15 10.2 kcal mol-1 at CISD+Q/DZP//RHF/DZP including zero-point vibrational energy (ZPVE),6 and 7.5 kcal mol-1 employing a FORS-MCSCF of eight electrons in eight reactive orbitals and the abovementioned more tightly contracted DZP basis set including ZPVE.7c To our knowledge, an experimental activation energy is not available. Ruedenberg et al. also characterized the TS for the internal rotation of allene. At his best level of theory, the barrier is 45.1 kcal mol-1,7a,c in agreement with the experimental barriers for 1,3-dimethyl- and 1,3-di-tert-butylallene (46.2 and 46.9 kcal mol-1, respectively).20 Somewhat higher activation barriers (which would be expected for the unsubstituted species) were obtained by Seeger21 (50.1 kcal mol-1) and by Rauk et al.15 (48.0 kcal mol-1, MP3/6-31G**//RHF/3-21G). As Roth20 suggested and Seeger21 et al. confirmed computationally, the TS is a planar bent C2V symmetric species corresponding to a 1A singlet biradical. 2 Both Ruedenberg’s FORS-MCSCF and the older RHF calculations did not include dynamic correlation in the optimization of the stationary points. Roos,18c Pulay,22 and others concur that MCSCF wave functions do not describe dynamic correlation well. Instead, we employed density functional theory (DFT)23a-d to explore the effects of dynamic correlation on the ring opening of cyclopropylidene and the internal rotation of allene. DFT
Bettinger et al. has now developed into a valuable alternative to traditional ab initio methods.23c,d Molecular properties obtained with DFT are in good agreement with experimental results, even for demanding compounds of the transition metals.24 Although density functionals have been developed which take dynamic correlation partially into account, good results for systems which require strong nondynamic correlation sometimes are obtained as well.23c,24c-j The performance of DFT in multiconfiguration cases like, e.g., ozone or the singlet-triplet separation in carbenes, can compete with lower level post-Hartree-Fock methods.23c,24c-j Since both single-determinant and multiconfiguration TS’s are involved in the ring opening of cyclopropylidene and the internal rotation of allene, we decided to compare DFT results with more conventional ab initio methods such as CASSCF, CISD, MR-CISD, and CCSD(T). Methods A standard split-valence basis set, 6-31G*,25 and two triple-ζ basis sets were employed. The latter two consisted of Huzinaga’s26 (10s6p) primitive set for C and (5s) primitive set for H contracted by Dunning27 to (5s3p) for C and (3s) and H, augmented with one and two sets of polarization functions with orbital exponents Rd(C) ) 0.75, Rp(H) ) 0.75, and Rd(C) ) 1.5, 0.375, Rp(H) ) 1.5, 0.375 to yield the TZP and TZ2P basis sets, respectively. The TZP basis set was used for geometry optimization, the TZ2P for MR-CISD and CCSD(T) single-point energy computations. The DFT computations were carried out with Becke’s threeparameter hybrid method (B3)28 and the exchange functional of Lee, Yang, and Parr29 (B3LYP) as implemented in the GAUSSIAN 92 program package.30 Spin-unrestricted theory (UB3LYP) was used for all open-shell species and for the TS for ring opening (see below). All stationary point geometries were fully optimized at B3LYP/6-31G* as well as B3LYP/TZP and characterized as minima or saddle points by analytically evaluating harmonic vibrational frequencies. Only the TZP results will be reported. Furthermore, the geometries of the critical points on the PES were fully optimized with the TZP basis set using the complete active space SCF (CASSCF) and configuration interaction (CI) methods. The CI wave functions included all single and double excitations of the HF wave function with the core orbitals frozen (CISD).31 Stationary structures were characterized by numerical second derivatives. The Davidson correction for unlinked quadruple excitations32 is also taken into account and the associated energies are denoted by CISD+Q. A four-electron four-orbital active space (4,4) was used for all species in the CASSCF computations. The active space for C2V 1-S consisted of two a1 bonding orbitals, one b1, and one b2 antibonding molecular orbital. One of the a1 orbitals corresponds essentially to the CC bond that is broken during the reaction. The antibonding b2 orbital was chosen to provide correlation for the CC bonding electron pair. The second a1 orbital is essentially centered on the carbene carbon and the b1 MO is employed to provide out-of-plane correlation for the carbene lone pair. The same active space was used for optimization of 3. For the D2d symmetric 2, the active space consisted of two sets of degenerate π/π* orbitals. Pulay’s22 method for the selection of the active space was used for optimizations of the D2h and C2V open-shell allenic species: we determined the natural orbitals33 for the stable symmetry-broken UHF wave function and included all fractionally occupied natural orbitals in the active space. Single-point energies were evaluated for all stationary points using multireference configuration interaction theory34 including
Ring Opening of Cyclopropylidene
J. Phys. Chem., Vol. 100, No. 40, 1996 16149
CHART 1
TABLE 1: Experimental and Theoretical Geometric Data for Allene (2-S) (Bond Lengths in Å, Angles in deg) r(C-C) r(C-H) Θ(CCH) c
EDa
MWb
IRc
B3LYP
CISD
CASSCF(4,4)
1.313 1.102 120.9
1.308 1.087 120.9
1.308 1.076 120.9
1.302 1.085 121.3
1.302 1.079 121.0
1.312 1.075 121.0
a Electron diffraction, ref 3a. b Microwave spectroscopy, ref 3e. Infrared spectroscopy, ref 3a.
TABLE 2: Experimental and Theoretical Vibrational Frequencies for Allene (2-S) (in cm-1)
all single and double excitations (MR-CISD) with respect to the most important reference configurations. As with CISD, the core molecular orbitals were kept frozen. We determined the most important configurations using CISD natural orbitals, which showed that three references were needed for all species, except for 1 and 3 which required four references. The use of a different number of configuration state functions (CSF) complicates the comparison of energies. But, as we are particularly interested in the barriers of the two reactions, we decided to describe the ring opening of 1 with four references and the internal rotation of 2 with three references. Electron correlation effects were also included for 1, 2, and 3 with the coupled cluster method including all single and double substitutions and perturbatively included triple excitations [CCSD(T)],35 with the core SCF orbitals frozen. The CCSD(T) as well as the MR-CISD single-point energies were obtained with the TZ2P basis set using B3LYP, CISD, and CASSCF(4,4) geometries. As a final confirmation of the point group of the transition structure for ring opening of cyclopropylidene we also optimized 3 at CCSD(T)/6-31G*. To test the predictive quality of the calculation levels we employed, ∆EST for methylene was determined at CCSD(T)/ TZ2P//B3LYP/TZP+ZPVE and MR-CISD/TZ2P//B3LYP/ TZP+ZPVE using four references (Table 6). The results -11.0 and -11.3 kcal mol-1, respectively, may be compared to the experimental value of -9.1 kcal mol-1.36 Thirty-six points were computed along the intrinsic ring opening reaction path with the B3LYP/6-31G* method. The concept of the intrinsic reaction coordinate was first suggested by Fukui37 and implemented into the GAUSSIAN program package by Gonzalez and Schlegel.38 Calculations were carried out using GAUSSIAN 92,30 except for the CISD, MR-CISD, and CCSD(T) single-point calculations, which were done with PSI2.0.8.39 The 15 internal coordinates chosen for the complete description of C3H4 and the labeling of atoms are shown in Chart 1. Two CC and four CH bond lengths, four HCC angles and the CCC angle (φ), and the four dihedral angles which define the angles between the planes CCX and CCH were used for the full optimizations. Deviations from Cs symmetry defined by the Cs plane perpendicular to the CCC plane can easily be designated by two additional parameters, δ1 and δ2:
δ1 ) ||dlp1| - |dlp3||
and
δ2 ) ||dlp2| - |dlp4||
In Cs symmetry δ1 and δ2 are 0°; in D2d symmetry, e.g., allene, δ1 and δ2 are 90°. Results and Discussion Allene 2-S (Figure 1). D2d allene has a singlet 1A1 ground state. The computed geometries and vibrational frequencies agree very well with experimental data (Tables 1 and 2).3 The lowest lying allene triplet (2-T) is planar and bent (C2V, 3A2).
Ramana E B1 E E A1 B2 A1 B2 B2 A1 E a
535 820 838
IRa
852 1031
1071 1389 1432 1956 2993 3061
1980 2960
B3LYP
CISDb
CASSCF(4,4)
372.1 880.8 871.6 1019.0 1112.4 1472.5 1484.1 2049.1 3118.8 3122.9 3193.6
355.7 867.4 845.6 999.3 1081.3 1411.9 1467.6 2006.3 3066.2 3068.0 3147.7
238.8 896.7 818.5 1064.9 1110.0 1552.4 1601.7 2057.1 3277.7 3281.3 3360.7
Reference 3c. b Scaled by 0.95.
Figure 1. B3LYP/TZP, CISD/TZP (in parentheses), and CASSCF(4,4)/TZP (in brackets) optimized structures of cyclopropylidene (1-S and 1-T) and allene (2-S and 2-T) (bond lengths in Å, angles in degrees).
The singlet-triplet separation at MR-CISD/TZ2P//B3LYP/TZP is 48.9 kcal mol-1 (Table 6). Cyclopropylidene 1-S (Figure 1). The 1A1 (1-S) and the 3B (1-T) states are the lowest lying singlet and triplet states. 1 The singlet-triplet separation (∆EST) of 1 (Tables 4 and 6) is 14.9 kcal mol-1 (CCSD(T)/TZ2P//UB3LYP/TZP) and 11.1 kcal mol-1 (MR-CISD/TZ2P//UB3LYP/TZP including the four most important CSF’s). The coupled cluster value is somewhat higher, perhaps due to the multireference character of the triplet 1-T, as indicated by a larger T1 diagnostic.40 At the MR-CISD/ TZ2P//UB3LYP/TZP level the error in ∆EST for methylene is 2.2 kcal mol-1. Therefore, we predict a corrected singlet-triplet separation for 1 of 8.9 kcal mol-1. Slightly higher ∆EST values were obtained by Honjou et al.6a (12.4 kcal mol-1 at CISD+Q/ DZP//HF/DZP) and Cramer et al.8 (13.8 kcal mol-1 using
16150 J. Phys. Chem., Vol. 100, No. 40, 1996
Bettinger et al.
Figure 2. B3LYP/TZP, CISD/TZP, CASSCF/TZP, and CCSD(T)/6-31G* optimized geometries of the TS (3) for the ring opening of cyclopropylidene (1-S) to allene (2-S) for left to right. For geometric data other than the length of the breaking bond see Table 3.
TABLE 3: Internal Coordinates for the Optimized Structure of the TS (3) for the Ring Opening of Cyclopropylidene at B3LYP/TZP, CISD/TZP, CASSCF(4,4)/TZP, and CCSD(T)/6-31G* (Bond Lengths in Å, Angles in deg) method
φ
cc1
cc2
hc1
hc2
hc3
hc4
hcc1
hcc2
hcc3
hcc4
dlp1
dlp2
dlp3
dlp4
B3LYP CISD CAS(4,4) CCSD(T)
90.7 92.7 84.1 89.5
1.388 1.381 1.411 1.402
1.400 1.394 1.411 1.419
1.097 1.090 1.079 1.100
1.092 1.086 1.076 1.097
1.099 1.092 1.079 1.103
1.091 1.085 1.076 1.096
126.6 125.6 117.9 127.2
119.1 119.7 126.8 118.8
127.2 119.4 117.9 128.1
118.7 113.9 126.8 117.9
41.6 41.1 39.4 38.7
39.1 39.6 36.9 37.0
-57.9 -64.7 -39.4 -60.7
-49.9 -56.1 -36.9 -50.2
TABLE 4: Absolute Energies (in au), Number of Imaginary Frequencies [in Brackets], and Zero-Point Vibrational Energies (ZPVE, in Parentheses, in kcal mol-1) for the Stationary Points 1-S, 1-T, 2-S, 2-T, and 3 B3LYP/TZP geometries B3LYP/TZP CCSD(T)/TZ2P MR-CISD/TZ2P CISD/TZP geometries CISD/TZP CISD+Q/TZP CCSD(T)/TZ2P MR-CISD/TZ2P CASSCF(4,4) geometries CAS(4,4)/TZP CCSD(T)/TZ2P MR-CISD/TZ2P
1-S
1-T
2-S
2-T
3
-116.590 08 [0], (33.3) -116.279 98 -116.216 93
-116.569 32 [0], (34.3) -116.257 82 -116.200 72
-116.702 42 [0], (34.5) -116.389 59 -116.332 37
-116.623 69 [0], (32.8) -116.306 36 -116.251 78
-116.582 05 [1], (33.1) -116.272 48 -116.207 45
-116.190 82 [0], (34.8) -116.236 95 -116.280 03 -116.217 59
-116.294 23 [0], (35.6) -116.342 08 -116.389 58 -116.332 62
-116.174 18 [1], (34.6) -116.223 24 -116.273 02 -116.210 22
-115.841 02 [0], (35.7) -116.279 98 -116.217 18
-115.952 99 [0], (35.4) -116.389 86 -116.332 79
-115.822 12 [1], (35.2) -116.273 50 -116.211 37
various DFT methods and CASSCF(8,8) with cc-pVTZ and ccpVQZ basis sets at CASSCF(8,8)/cc-pVDZ geometries). The S-T gaps are usually much smaller for small hydrocarbon carbenes, e.g., 0.8 kcal mol-1 (CCSD(T)/TZ2P+f) for dimethylcarbene41 (recall that methylene has a triplet ground state, ∆EST ) -9.1 kcal mol-1). In accord with Walsh diagrams,42 the CCC angle φ is larger for the triplet than for the singlet species. Hence, the smaller φ in 1-S favors the singlet over the triplet. The structures of cyclopropylidene optimized with B3LYP/TZP, CISD/TZP and CASSCF(4,4)/TZP are very similar (Figure 1) and more compact than the previously reported CASSCF(8,8) structures,7c,8 which are characterized by rather long C1-C2/3 bonds (around 1.54 Å). Inclusion of the C1-C2/3 bonding and antibonding orbitals in the active space might elongate the corresponding bonds as antibonding orbitals are explicitly occupied. Isomerization of Cyclopropylidene to Allene. We were able to localize asymmetric TS’s at the B3LYP and CISD levels (Figure 2, Tables 3 and 4). This disagrees with Ruedenberg’s report7c of a single Cs symmetric TS at CISD with the more tightly contracted DZd basis set. The difference cannot be attributed entirely to our larger basis set because an asymmetric TS resulted even with the small 6-31G* basis set. Our absolute energies with the 6-31G* basis set are much lower than Ruedenberg’s (e.g., for 2, E ) -116.222 80 au vs -115.953 47 au).7c However, only one Cs symmetric TS was obtained with
CASSCF(4,4), in agreement with Ruedenberg’s FORS-MCSCF investigations. All structures resemble rather early TS’s, in accord with the expectation for a strongly exothermic reaction. However, the question still remains whether the TS is well described with a single determinant. There are several criteria which indicate the multireference character of a wave function. The CISD results reveal that for all optimized TS’s the HF wave function is the dominant configuration state function for the CI wave function; the first CI coefficient is around 0.94 in all cases. The second and third coefficients are 0.08 and 0.04. An even more reliable measure is the T1 diagnostic for CC methods introduced by Lee and coworkers.40 For all TS’s the value of T1 is much lower than 0.02, which is the upper limit for a truly reliable single-reference correlation treatment. These observations show that a method based on the ground state CSF including dynamic correlation should be sufficient for the description of the TS. Thus, dynamic electron correlation becomes most important. A similar assumption was made by Yoshimine et al.6 who used a single configuration HF wave function for the description of the TS. The HF-based CASSCF multireference method may describe nondynamical correlation sufficiently, but it should not be used if dynamic correlation is important.18c,22 The CASSSCF method can give significant errors, e.g., in locating the strongly correlated azulene ground state43 and in the TS for the Cope rearrangement.44
Ring Opening of Cyclopropylidene
J. Phys. Chem., Vol. 100, No. 40, 1996 16151
TABLE 5: Activation Barriers and Reaction Enthalpies (in kcal mol-1) for the Ring Opening of Cyclopropylidene (1-S) to Allene (2-S) Including ZPVE method B3LYP/TZP geometries B3LYP/TZP CCSD(T)/TZ2P MR-CISD/TZ2P CISD/TZP geometries CISD/TZP CISD+Q/TZP CCSD(T)/TZ2P MR-CISD/TZ2P CASSCF(4,4) geometries CAS(4,4)/TZP CCSD(T)/TZ2P MR-CISD/TZ2P
activation energy
reaction enthalpy
4.8 4.5 5.7
-69.3 -67.6
10.2 8.4 4.2 4.4
-64.1 -65.2 -67.9
11.4 3.6 3.1
-70.6 -69.3
DFT is a way to deal with the many-electron problem which is rather different to wavefunctions based on traditional ab initio techniques. In the Kohn-Sham formulation of DFT,23a-c equations must be solved, which are analogous to those of Roothaan and Hall, and Pople and Nesbet for restricted and unrestricted conventional Hartree-Fock theory.45 The only difference is that the HF exchange matrix elements Kµν are replaced by the exchange-correlation parts of the Fock matrices FµνxcR and Fµνxcβ. While this introduces some degree of electron correlation into the Kohn-Sham based DFT, and DFT exchangecorrelation energy does not correspond to the sum of the exchange and correlation energies in HF theory. The close formal analogy between the DFT and HF working equations emphasizes the single determinant character of the DFT wave function constructed from Kohn-Sham orbitals.46 We employed unrestricted DFT and an unsymmetrical initial guess in the optimization of the TS to make sure that an unrestricted solution was obtained.47 Nonetheless, a non-symmetry-broken wave function resulted, and a subsequent stability analysis of the restricted wave function revealed no triplet instability.48 Symmetry breaking is common to UHF theory and usually indicates the need for a many-determinant description.22a Furthermore, no spin contaminantion for this wave function was observed. Baker et al. have shown that Kohn-Sham SCF equations in general have relatively little spin contamination,46 in contrast to UHF wave functions. The expectation value of S2 of the stable UHF wave function using the UB3LYP/TZP TS geometry, was small (〈S2〉
