Radical cations of cis and trans isomers of 1,3-di-and 1,3,5-tri-methyIcyclohexane, stabilized in various y-irradiated solute-halocarbon matrices at 77 K and above, have been investigated b y means of EPR spectroscopy. The hfs constants have been compared w i t h those calculated using semiempirical methods, I NDO (spin density) and MN DO (geometry optimizations). The SOMOs of the methyl-substituted cyclohexane cations are all similar t o either of the orbitals, a, or b, , following a Jahn-Teller split of the e, orbital (HOMO) of cyclohexane. The selection of SOMO depends on matrix and isomer. The trans isomers have one axial methyl group and the cations were found to take an a,-like SOMO in all matrices used, with large spin density o n the equatorial hydrogen attached t o the axially substituted ring-carbon. Using CF,CCI, as matrix the EPR signal due to the trans-l,3,5-trimethylcyclohexane cation contains (in addition) a spectrum of the b,-like SOMO superimposed. The cation of cis-I ,3-dimethylcyclohexane shows contributions from both SOMOs in the matrices CF,-cC,F,,, CF,CICFCI, and CF3CCI3, but exclusively a b,-like SOMO in cC6F12. A b,-like SOMO is observed for the cis-l,3,5-trimethylcyclohexane cation in all matrices. The abundance ratios, [a,]/[b,],for the cases where the matrix-stabilized cation shows contribution from both b,and a,-like SOMOs simultaneously, are found t o be ca. 1/4 from line shape simulations. No prominent temperature dependence of this ratio is detected. 'A" ( C,) A'(C,) 'A" (C,) 'A'(C,) ' A' ( C J 'A'K,) ' A" (C,) 'A"(C,)36 (2 H); 29 (2 H) 93 (2 H); 12 (2 H ) 38 (2 H); 29 (2 H) 107 (1 H); 64 (1 H); 35 ( I H) * 1 cal = 4.184 J.

isomeric radical cations of 1,2,4-Me,-cyclohexane, r-1 ,t-2,t-4-Me,-c-C6+ and r-I ,t-2,c-4-Me3c-C6+, have been studied at cryogenic temperatures adopting the halocarbon matrix technique. Both isomer cations show the same EPR hyperfine ( h f ) pattern with a triplet of triplets with alH = 6 0 G ( 2 H) and azH = 32 G (2 H) at 4.2 K. The hf parameters coincide with those of the cis and trans isomers of the 1,2-Me,-c-C6 cation, implying a similar electronic ground state structure in which the main part of the unpaired electron resides in the C-1-C-2-0-bond. The EPR spectra show a reversible temperature change between ca. 77 and 180 K. The temperature-dependent spectra can be explained by a dynamic effect in which the methyl groups on C-1 and C-2 rotate. Adopting a three-site jump model the rate constants were evaluated t o be ca. 5.0 x l o 6 s-I (100 K) to ca. 5.0 x lo7 s-1 (180 K) with an activation energy of 1.1 kcal moI.-l This low activation energy is compared t o methyl rotation in radical cations of propane+ and butane+, and is discussed in terms of the unpaired electron distribution within the framework of carbon atoms. Furthermore, it is shown that secondary and tertiary alkyl radicals are predominantly formed by thermal reactions of r-1 ,t-2.t-4-Me,-c-C6+ and r-1 ,t-2,c-4-Me3-c-C6+, respectively, in the CF,-c-C,F,, matrix.

a b s t r a c tA method for computing the inverse of an (n × n) integer matrix A using p-adic approximation is given. The method is similar to Dixon's algorithm, but ours has a quadratic convergence rate. The complexity of this algorithm (without using FFT or fast matrix multiplication) is O(n 4 (log n) 2 ), the same as that of Dixon's algorithm. However, experiments show that our method is faster. This is because our methods decrease the number of matrix multiplications but increase the digits of the components of the matrix, which suits modern CPUs with fast integer multiplication instructions.

Radical cations of the 1,2,3-t¡ isomers stabilized in various •-irradiated solute/halocarbon matrices have been investigated by means of ESR in the temperature range 4--77 K. At 4 K the ESR spectra ate dominated by contributions from an asymmetrieally distorted structure with the unpaired electron localized to the C1--C2 bond. On increasing the temperature a reversible change occurs in the ESR line-shape of the cations of the two symmet¡ isomers. Using a two-site jump model to reproduce the temperature dependent line-shape, the phenomenon is explained in temas of an interconversion between two such distorted structures, each being the mirror image of the other. The An'henius plot associated with the process is markedly nonlinear towards the low temperature region. The experimental data are also modelled by postulating that another (different) electronic ground state, having higher symmetry, becomes populated with the increase of temperature. In this way, the speclxal changes can be simulated using a three-site jump model which couple the thermally activated two-site jump process (E, ca. 0.137 kcal/mol) with a dynamical equilibrium between the asyrnmetrical ground state and a symmetricai structure 0.058 kcal/mol higher in energy. The energy barrier to pass from the distorted to the syrnmetrical structure was evaluated to be 0.085 kcal/mol. IntroductŸThe method to study radical cations of saturated hydrocarbons for the purpose of ESR spectroscopy was introduced with the halocarbon matrix isolation technique in the end of the seventies [1][2][3][4][5]. Cyclic alkanes with degenerate HOMO (Highest Occupied Molecular Orbital) have attracted attention due to the possibility of having Jahn-Teller distorted ground states [6][7][8][9][10][11]. A degenerate HOMO can also be split by introducing substituents, which has encouraged us to investigate a variety of alkyl-substituted cyclohexane cations [12][13][14][15][16][17][18][19].

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