One-Electron MO Models

however, an excited state can be described by just one singly excited configuration as is frequently possible to a good approximation for 

If electron interaction is taken explicitly into account by writing the Hamiltonian in the form 

The ionization potential and electron affinity of naphthalene were determined experimentally as IP = 8.2 eV and EA 0.0 eV. According to Koopmans theorem it is possible to equate minus the orbital energies of the occupied or unoccupied MOs with molecular ionization potentials and electron affinities, respectively (IP, = - s, and EA = - ). Thus, in the simple one-electron model, the excitation energy of the HOMO- LUMO transition in naphthalene may be written according to Equation (1.22) as 

Experimental values for the singlet and triplet excitations corresponding to the HOMO- LUMO transition are 4.3 and 2.6 eV, respectively. If the value of C in the above expression is equated to the electron repulsion terms in Equation (1.23) and Equation (1.24), 

due to the existence of the electron repulsion term — 2K,i the singlet excitation energy is only about half the orbital energy difference, and the exchange interaction is about one third of the Coulomb interaction J.  

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Perhaps you are thinking that the metal atoms in Fe6C(CO)i6]2 are now using more than three orbitals for cluster bonding. How else can they interact effectively with the interstitial atom But notice that the interaction described in Figure 3.8 is no different in principle than the addition of four H atoms to a square pyramidal B5H5 cluster or the addition of 4 H+ to [B5H5I4 as in the one electron MO method the electrons are added last. For the iron cluster, then, the equivalent model would be the insertion of a C4+ ion into the center of a [Fe6(CO)i6]6- cluster. No additional metal orbitals are needed. 

Fig. 6.1. The electronic structure of ZnS (a) simplistic qualitative one-electron MO energy-level diagram for a ZnS4 (tetrahedral) cluster (b) simplistic one-electron band model for ZnS.

Particularly illuminating is the free-electron MO model (FEMO) based on the assumption that Jt electrons can move freely along a one-dimensional molecular framework. Stationary states are then characterized by standing waves, and using the de Broglie relationship... 

The calculated one-electron MO energies in ground state are shown in Fig. 2, where the energies of the highest occupied MOs are set at zero. The valence band consists mainly of 0-2p orbitals. In the model (3), the band structure appears, and the conduction band is described mainly by Be-2sp, M-ispd, and Si-3spd orbitals. In all the results, the levels mainly occupied by Cr-3d orbitals appear well above the 0-2p valence. These levels are taken as the impurity levels to form the Slater determinants. 

Fig. 2. Calculated one-electron MO energies for each model cluster, (1), (2), and (3).

The one-electron MO energy levels calculated for the various model clusters of ruby and emerald are shown in Fig 2. The valence band mainly consists of 0-2s,2p orbitals. The width of the valence band decrease as the symmetry of the eluster approaches to eubic. The decrease of the valence band width is more significant in ruby than in emerald. The conduction band is not reproduced in these caleulations, because the cations of the host crystals are not included in the present clusters. There are impurity states with... 

The parameters of the model matrix elements used here are adjusted to fit Hartree-Fock energies. Inner-shell electrons are well described by means of the independent particle model within which the Hartree-Fock method yields accurate one-electron (MO) energies. At low incident velocities, where the MO concept is applicable, the vacancy transfer probabilities are sensitively dependent on the one-electron energies involved. 

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. 

Although in many cases, particularly in PE spectroscopy, single configurations or Slater determinants 2d> (M+ ) were shown to yield heuristically useful descriptions of the corresponding spectroscopic states 2 f i(M+ ), this is not generally true because the independent particle approximation (which implies that a many-electron wavefunction can be approximated by a single product of one-electron wavefunctions, i.e. MOs 4>, as represented by a Slater determinant 2 j) may break down in some cases. As this becomes particularly evident in polyene radical cations, it seems appropriate to briefly elaborate on methods which allow one to overcome the limitations of single-determinant models. 

The minimal basis calculation on the hydrogen molecule is a well-worn but eminently suitable example for our purposes. It has a convenient symmetry element and orbital basis calculations can be carried through which are quantitatively acceptable and yet not prohibitively unwiedly to report. We give below variational calculations on the H2 molecule using the familiar simplest AO basis in the one-electron-group (MO) model and the electron-pair (VB) model. These calculations have been performed explicitly to investigate the effect of symmetry constraints . 

The oxo-transfer chemistry of molybdenum in sulfite oxidase is probably the best characterized, in terms of synthetic models, structural and mechanistic data, of all the elements we have described up till now. The reaction cycle (Figure 17.5) involves binding of sulfite to the oxidized MoVI, two-electron reduction of the Mo centre and release of sulfate. The Movl centre is restored by successive one-electron transfers from a cytochrome (bs in mammals). The primary oxo-transfer reaction ... 

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