On the form of molecular orbitals in conjugated rings and chains

The Huckel-Hund model of conjugated molecules assumes a planar frame of Σ bonds radiating at 120‡ from each C atom with the 2p π-electrons moving through it. The looseness of thepπ-electrons is a result of the splitting action exerted on the atomic levels of a given C atom by the perturbing field (C2v) of the three neighbouring atoms and of the resonance between the states [s], [px], [py], giving rise to the three planar Σ-bonds starting from the given C atom. Experimental evidence of varipus kinds is available. Bloch’s theorem on the form of the oneelectron wave functions in systems having translational symmetry follows directly from the identity of the probability density in points equivalent translationally. Conjugated rings and conjugated chains can be viewed as systems having a one-dimensional periodicity; they differ in the type of boundary conditions to be imposed on the one-electron wave functions. The form of the molecular orbitals derived in this way coincides with the form given by E. Huckel. The utility of group theory in buildingLCAO molecular orbitals relies on two theorems due to E. Wigner. The corollary of the second theorem tells us that with the set ofn 2pπ orbitals of the carbon atoms of a conjugated ring CnHn we can build through Wigner’s theorem (10), bases for each irreducible representation of- the group Cn, for the m = 0 representation ofDn antisymmetric in S′,S′’ (n- 2r) or inS (n = 2r + 1) and for the corresponding representation ofDnh antisymmetric in Σh for the m = r representation ofD2r antisymmetric inS′ and symmetric in S′’ (the axesS′ pass through opposite corners of the 2r sided polygon) and for the corresponding representation ofDnh antisymmetric in Σh; and finally for all the doubly degenerate representations ofDn and the corresponding representations ofDnh, antisymmetric in Σh. The π-electron molecular orbitals built through Wigner’s theorem in the groupsDn andDnh are identical. The π-electron molecular orbitals built through Wigner’s theorem in the groupCn are Huckel’s orbitals and represent, perfectly satisfactory molecular orbitals from the standpoint of the lull symmetry of the ring. However only them = 0,m = r Huckel’s orbital for C2rH2r, and only the m = 0 orbital for C2r+1H2r+1 coincide with the orbitals built for the corresponding representations ofDn of interest. The two π-electron molecular orbitals built using Wigner’s theorem for the doubly degenerate representations of Dn arc sums of Huckel’s orbitals belonging to complex conjugate representations ofGn built using the transformed sets of two differentOf. One can use also as degenerate π-electrou molecular orbitals in rings CnHn pairs of Huckel’s orbitals belonging to complex conjugate representations ofCn or any pair of independent linear combinations (e.g. sum and difference) of them. The π electron molecular orbitals going with ¦m¦=s hass + 1 nodes. It follows that the ground state of a2r (r = 2z + 1) membered ring is a totally symmetrical singlet; for r = 2z the orbital ¦m ¦ =z is only half full so that Mulliken’s method must bo applied to obtain the symmetries of the resultant states. In the ground state of a 2r + 1(r = 2s) membered ring the prc-eloctrons fill the molecular orbitals up to ¦m¦ = z- l and occupy three of the four states ¦ m s|=z so that Mulliken’s method must bo applied. The doublet ground state of a2r + (r = 2z + 1) membered ring has the symmetry of the orbital ¦m ¦ = z + 1. Conjugated chains with an even number of C atoms have symmetry C2h, while those with an odd number of C atoms have symmetry C2v. Huckel’s orbitals for these molecules are independent linear combinations of the orbitals that one obtains through Wigner’s theorem using these symmetry groups. For2r membered chains Huckel’s orbitals with evenk are linear combinations of the r orbitals of simmetryAu, while those with oddk are linear combinations of ther orbitals of symmetryBg. For a 2r + 1 membered chain Huckel’s orbitals with evenk are linear combinations of ther + 1 orbitals of symmetryB2, while those with oddk are linear combinations of ther orbitals of symmetryA2. The ground state of an even membered chain is singlet and totally symmetrical. The doublet ground state of a 2r + 1 memberod chain has symmetryB2 ifr = 2z, symmetryA2 if r = 2z + 1. A few examples are discussed in detail.