MO Models of Electronic Excitation

The vibrational structure may be explained as follows For each state of a molecule there is a wave function that depends on time, as well as on the internal space and spin coordinates of all electrons and all nuclei, assuming that the overall translational and rotational motions of the molecule have been separated from internal motion. A set of stationary states exists whose observable properties, such as energy, charge density, etc., do not change in time. These states may be described by the time-independent part of their wave functions alone. Their wave functions are the solutions of the time-independent Schrodinger equation and depend only on the internal coordinates q = q q-,. .. of all electrons and the internal coordinates Q = Qi, Qi,. . . of all nuclei. 

Within the Born-Oppenheimer approximation (cf. McWeeny, 1989 Section I. I) the total wave function of a stationary state is written as 

The vibrational wave functions xHQ) eigenfunctions of the vibrational Hamiltonian fIyJj,Q), which is defined for a particular electronic stated as an operator containing Ef, the electronic plus nuclear repulsion energy of state j, as the potential energy of the nuclear motions. For every electronic state j, there is a different potential energy and therefore a different vibrational Hamiltonian fiyJj.Q). 

Due to the product form of the total wave function in Equation (1.12) the energy of a stationary state can be written as 

Similarly, a rotational component and a translational component are obtained when all 3N displacement coordinates of the N nuclei are used rather than the internal coordinates, which are obtained by separating the motion of center-of-mass and the rotational motions. 

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During the six years since the previous NATO school on the Quantum Theory of Polymers (1) three important developments in this topic have occurred. First, reliable and affordable molecular-orbital (MO) models of the electronic excitation spectra of large organic molecules (2) and polymers (3,4) have been developed and applied for a variety of purposes, including the design of xerographic photoreceptors (2,5) and battery electrodes (3,6). Second, the role of disorder in localizing electronic excitations... 

In calculations and interaction diagrams, only the most simplistic MO models will be chosen to represent ground and excited states of reactants. An olefin then has a bond framework largely neglected in discussing the reactivity of the molecule. The bonding level will be characterized by a jr-electron wave function with no nodes between the two basis fi orbitals of the ir-bond. The first jr-antibonding level has one node in the wave function, and a first excited state has electron-occupancy of unity in each level. 

For many years, investigations on the electronic structure of organic radical cations in general, and of polyenes in particular, were dominated by PE spectroscopy which represented by far the most copious source of data on this subject. Consequently, attention was focussed mainly on those excited states of radical ions which can be formed by direct photoionization. However, promotion of electrons into virtual MOs of radical cations is also possible, but as the corresponding excited states cannot be attained by a one-photon process from the neutral molecule they do not manifest themselves in PE spectra. On the other hand, they can be reached by electronic excitation of the radical cations, provided that the corresponding transitions are allowed by electric-dipole selection rules. As will be shown in Section III.C, the description of such states requires an extension of the simple models used in Section n, but before going into this, we would like to discuss them in a qualitative way and give a brief account of experimental techniques used to study them. 

The HF term includes intemuclear repulsions, and the perturbation correction E(2) is a purely electronic term. E1 2 is a sum of terms each of which models the promotion of pairs of electrons. So-called double excitations from occupied to formally unoccupied MOs (virtual MOs) are required by Brillouin s theorem [89], which says, essentially, that a wavefunction based on the HF determinant Z> plus a determinant corresponding to exciting just one electron from Dl cannot improve the energy. 

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