For the Mobius geometry, the boundary conditions differ from the standard particle in a ring problem. Supposing to have a strip of length and , we can see that general Mobius boundary conditions for the wavefunction are: \n* \n* or using the spherical azimuthal angle : For an -carbons, the proposed ansatz linear combination of atomic orbitals (LCAO) is: where is the angle at each -th carbon atom and is the -th AO. Thus, for circular carbon rings, the general Mobius boundary condition can be rewritten as: Using this equation and the Euler rule we can find the right value satisfying previous boundary conditions: From the last equation we see that to fulfil the general boundary conditions, must be a half-integer number. The coefficients of the ansatz become: From figure above, it can also be seen that the overlap between two consecutive AOs is at a constant angle , and for this reason resonance integral it's considered as a constant into the Huckel matrix we will write later. It could be simply written as: where is the standard Huckel’s resonance integral value (the one with ).Nevertheless, the presence of a axis as the only symmetry element brings to a full phase change at the end of the ring, e.i. between the first and the -th carbon atoms. For this reason, in the Huckel matrix the resonance integral between carbon and is .For the generic carbons Mobius system, the Huckel matrix is: Eigenvalues equation can now be solved. Since is a matrix, we will have eigenvalues and MOs. Defining the variable we have: Hence we obtain a system of equations, in which the first one () and the last one () have a coefficient: All these equations can be easily solved using Euler's rule, leading to hence

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