J. Phys. Chem. 1993,97, 12215-12219
12215
Theoretical Study of Deuterated Ethane Cations Leif A. Eriksson' and Sten Lune11 Department of Quantum Chemistry, Box 518, Uppsala University, S- 751 20 Uppsala, Sweden Received: July 9, 1993'
Equilibrium and transition-state geometries, torsional barriers, and relative abundances a t 4.2 and 77 K of different rotational isomers of deuterated ethane cations have been calculated, using ab initio MP2/6-3 1G** and MP4(SDTQ)/6-31GS* theories. The relative abundances of rotationally related isomers, predicted from Boltzmann statistics based on differences in zero-point vibrational energies (ZPE), are in good agreement with low-temperature matrix isolation ESR data. The rotational barrier in the singly deuterated cation is calculated to be 0.1 1 kcal/mol higher than in the undeuterated ion, in reasonable agreement with the experimentally estimated difference of >0.25 kcal/mol.
1. Introduction
Partial deuteration has proven very successful as an analytic tool in connection with electron spin-resonance (ESR) spectroscopy. The main reason for this is the difference in hyperfine structures between the proton and the deuteron, which enables a distinction between chemically inequivalent protons in the system under study.' In practice, deuteration takes place before the radical is produced (e.g., by irradiation) and the ESR experiment is performed. In cases where originally equivalent protons become inequivalent in the ionized molecule, e.g., through Jahn-Teller distortions, it may happen that the result of the experiment is a superpositionof spectra correspondingto different substitutional isomers. In two previous papers2.3 dealing with partially deuterated n-butane and cyclopropane radical cations, we have shown that the superposition pattern (Le., the population of the various substitutional isomers) can be fully explained by applying Boltzmann statistics to the calculated zero-point vibrational energies (ZPE), computed at the UHF or MP2 1evels.zJ In the present paper, we have investigated the undeuterated, partially deuterated, and fully deuterated isomers of the ethane radical cation, with the above mentioned aspects in mind. The results are compared with results from low-temperature matrix isolation ESR experiments by Iwasaki et ~ 1and . ~Toriyama et al.5 In addition, the transition-state structure, obtained after an internal rotation around the carbon-carbon bond, has been studied, and the ZPE-corrected rotational barrier is computed for both the undeuterated and singly deuterated species. 2. Theory
If the coupling between different vibrations in the molecule is neglected, the zero-point vibrational energy of, e.g., a C-H stretching vibration is in a simple harmonic oscillator approximation proportional to (k/mH)Il2. Here, k is the force constant of the particular bond, and m H is the mass of the hydrogen nucleus. Upon deuteration the deuterons will, in order for the system to lower its ZPE, preferentially occupy sites with large force constants-i.e., in general the shorter bonds. At low enough temperatures this effect is sufficiently strong to single out a unique ground-stateconformationof the radical (cf. ref 2-5). If, however, the temperature is increased, other conformations start to be populated so that in, e.g., an ESR experiment a superposition of spectra is observed. At sufficiently high temperatures, a complete
* To whom correspondence should be addressed, at the Department of Physics, University of Stockholm, Box 6730, 113 85 Stockholm, Sweden. Abstract published in Aduance ACS Abstracts, October 15, 1993. 0022-3654/93/2097- 12215$04.00/0
averaging of the various related positions will occur and vibrational stabilization of preferred sites of deuteration is no longer detectable. Clearly, the simple uncoupled harmonic oscillator approximation is not accurateenough for a detailedstudyofthevibrational motion of a molecular system. Instead, the potential hypersurface must be calculated,within the Bom-Oppenheimer approximation, and the ZPE extracted from normal-mode analyses performed for each isomer of interest. It has, indeed, been found that in particular cases, such as the cyclopropane cation,a9the deuteration occurs preferentially at the longer C-H bonds. Calculations have shown that this somewhat unexpected result is caused by strong effects on thevibrations in the low frequency part of the spe~trum,P which overtake the expected contributions from the C-H/C-D stretching vibrations. In normal cases, however, the results are expected to agree qualitatively with the simpler model. This has been verified for, e.g., the radical cations of methane,- n-butane,2 methylamine,lO and methanol." In the last case, it was furthermore noted that the internal rotation of the singly deuterated methyl group is hindered relative to that of the undeuterated species, indicating a fairly large barrier in the case of the CHIDOH+ cation (see below). The unsubstituted ethane cation has previously been the subject of extensive studies,5J2-'6 It has been concluded that ionization from the legorbital of the neutral molecule (D3dsymmetry)results in a structure of C2h symmetry with a 2Agelectronic ground state. In the cation, two of the hydrogens (in trans positions) become tilted inward, as depicted in Figure 1, and the carbon-carbon distance is, at the MP2/6-31G** level, about 1.58 A.Iz.13 The ethane cation will thus have two sets of chemically inequivalent protons: one group consisting of H1 and H2 (cf. Figure la), both lying in the molecular mirror plane and carrying essentially all of the unpaired spin density, and another group formed by the four out-of-plane protons H3-H6. The shorter C-H bonds are those corresponding to the four out-of-plane protons. In the present study, the vibrational frequencies and total ZPE have been calculated for all substitutional isomers of the partially deuterated ethane radical cation, using the MP2/6-31G**optimized geometry determined in a previouscontribution.12This has been found to be the lowest computational level that predicts the correct ground-state geometry for C2Hl,as found by matrix isolation ESR spectroscopy.4~5 We have also optimized the rotational transition state at the same level of theory, as well as the ground state and the transition state at the MP4(SDTQ)/ 6-31G** level. All geometry optimization calculations were performed using the GAUSSIAN 90 code.17 In the MP2 calculations, numerical force constants obtained from finite differences of analytical first derivatives were utilized for the 0 1993 American Chemical Society
12216 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993
(a)
\-
HI'
98.50
1.084
H2'
(98.5')
H4'
/%"
(106.7 106.6 ')
1.967
1.084 \-H6' (1.081)
(b) Figure 1. (a) Geometryof the 2A,electronicgroundstate(CZhsymmetry)
of the ethane radical cation, optimized at the MP4(SDTQ)/6-31G** (this work) and, in parentheses, MP2/6-31G** l2 levels. (b) Geometry of the *AIeclipsed transition state (CZ,symmetry) obtained at the same levels as in a (this work). geometry optimizations and frequency calculations. The MP4(SDTQ) calculations, on the other hand, were performed using numerical derivatives only, and within the frozen core approximation. In the calculations of the ZPE, we included for the ground state all vibrational frequencies (i.e., also that corresponding to the "reaction coordinate", the torsional motion), whereas for the transition-state structure this, now imaginary, frequency was excluded. Throughout, weuse thesplit-valence6-31G** basisset.18 This has been shown to yield highly accurate geometries and rotational barriers for similar systems, when used in conjunction with MP2 or MP4 cal~ulations.~~-2~ We have in a previous investigation2 found that the analysis of deuteration effects can be equally well based on the total ZPE obtained at the UHF/6-3 1G** or MP2/ 6-3 1G** levels, provided that the scale factor of 90%, commonly applied to HF frequencies,I8 is not used in the U H F case. For the ethane cation, however, the U H F level of theory is, as already mentioned, known to yield an erroneous ground-stateequilibrium ~ t r u c t u r e and ~ ~ Jis~thus not suitable for the present study. The MP2/6-31G1* level was therefore used throughout. All calculations are performed within the spin-unrestricted formalism (UMP2, UMP4). The rationale and justification for this is the very low spin contamination encountered for the ethane radical cation. At the U H F level, ( S 2 )= 0.761 for the groundstate structure and 0.758 for the transition state. This should be compared with the ideal value 0.750 for a radical system with total spin S = 1/2. 3. Results and Discussion
3.1 Geometric Structure and Rotational Barriers. The MP4(SDTQ)/6-3 1G**-optimized staggered equilibrium structure (Figure la) strongly resembles that obtained at the MP2 level,l2 except for a more elongated C-C bond (1.66 A vs 1.58 A). One can note that the difference of 0.08 A between the MP2 and MP4 bond lengths is remarkably large, compared toother bond length# or the present rotational transition-state structure (Figure 1b). This can be seen as a manifestation of the basic problem in the determination of the ground-state structure of the ethane cation, namely the strong interaction between the le, ionized and 3al, ionized ~tates.12-I~The potential energy curve for the ground *A, state of the cation is therefore very different from a normal,
Eriksson and Lune11
TABLE I: Absolute Energies (au), ZPE (kcal/mol), Rotational Barriers (kcal/mol), and Vibrationally Corrected Rotational Barriers (kcal/mol) for the Normal (Undeuterated) and Singly Deuterated Species of the Ethane Radical Cation, Obtained at the MP2/6-31G** and MP4(SDTQ)/&-31C**Levels of Theory MP2/ MP4(SDTQ)/ 6-31G** 6-31G** Ground State E (a4 -19.131 29' -19.163 13 ZPE (kcal/mol)b normal 44.441 singly deut. 42.42O/42.17lc Transition State E (au) -19.121 69 -19.161 36 ZPE (kcal/mol)b normal 44.383 singly deut. 42.416/42.476c 1.111 uncorrected rotational 2.259 barrier (kcal/mol) ZPE-corrected rotational normal 2.201 1.053 barrieP (kcal/mol) singly deut. 2.3 15 1.167 a Reference 12. Only calculatedat the MP2 level. Thesecorrespond to the in-plane/out-of-plane deuterated cases, respectively. For the MP4(SDTQ) correction, we have used the ZPE difference obtained at the MP2 level. e.g., Morse-like, potential, being extremely flat over a large C-C range. This obviously makes the theoretically determined equilibrium bond length very sensitive to the computational method used, which may explain the large effect of going from MP2 to MP4. From the point of view of steric interactions, it may be expected that the calculated rotational barrier should be lower at the MP4 level than using second-order Morller-Plesset perturbation theory. This is confirmed by the results in Table I, which lists the absolute energies for the various states as well as the rotational barriers, with and without vibrational corrections. The experimental barrier of rotation, determined by Iwasaki et a1.,4is reported to be 0.25 kcal/mol for the undeuterated species and >0.50 kcal/mol for thesingly deuterated one. The calculated barriers are thus considerably higher, also a t the MP4/6-31G** level, even though the improvement from MP2 to MP4 is significant. No previous computational studies have been reported of the transition state and rotational barrier of C2H6+. However, some data are available for similar systems, obtained a t various levels of theory. For the methanol radical cation (CHsOH+), Knight et al.11 found the rotational barrier to be 1.88 kcal/mol at the MP2/6-3 1lG(2df,2pd) level and "
