Molecular monoelectronics

The use of the exact Hamiltonian for calculating matrix elements between VB determinants leads, in the general case, to complicated expressions involving numerous bielectronic integrals, owing to the 1/r,-,- terms. Thus, for practical qualitative or semiquantitative applications, one uses an effective molecular Hamiltonian in which the nuclear repulsion and the 1/r,-, terms are only implicitly taken into account, in an averaged manner. Then, one defines a Hamiltonian made of a sum of independent monoelectronic Hamiltonians, much as in simple MO theory ... 

In the theoretical description of regular polymers, the monoelectronic levels (orbital energies in the molecular description) are represented as a multivalued function of a reciprocal wave number defined in the inverse space dimension. The set of all those branches (energy bands) plotted versus the reciprocal wave number (k-point) in a well defined region of the reciprocal space (first Brillouin zone) is the band structure of the polymers. In the usual terminology, we note the analogy between the occupied levels and the valence bands, the unoccupied levels and the conduction band. 

It is often useful to be able to determine the symmetry of a function that is a product of two (or more) functions whose symmetry is already known. This need arises, for example, when we consider a polyelectronic wave function that is written as a product of monoelectronic flmctions (atomic or molecular orbitals), or when we are interested in the overlap between two orbitals (see 6.5.1). 

The electronic criteria of aromaticity/antiaromaticity are based on the electronic structure of the molecules. Molecular orbitals (MOs) that are highly delocalized over the entire cyclic nuclear framework support cyclic electron delocalization which characterizes aromaticity. Based on the symmetry of the highly delocalized MOs we can distinguish between the orbital-types of aromaticity, e.g., a-, it-, 8-, and (p-aromaticity. Many planar cyclic clusters formulated as c-E and c-E L (E = element, L = H, C, O, NH, n = 3-6) exhibit the so-called multifold aromaticity, which arises from various combinations of the aforementioned orbital types. Molecular orbitals are multicenter monoelectronic wave functions constructed from the overlap of atomic orbitals (AOs) belonging to the nuclear centers. The o-aromaticity is supported by delocalized ct-MOs, which can result from the linear combination of S-, p-, d-, or f-AOs. Similarly jr-aromaticity is supported by delocalized tt-MOs... 

This theory endeavors to describe the molecule by a method intrinsically similar to that used for obtaining atomic orbitals but considering multicenter wave functions. Thus this approach consists of finding the best functions for describing the state of one electron in a field formed by the totality of the nuclei placed in their equilibrium positions. These monoelectronic molecular wave functions may be obtained according to the MO-theory by a linear combination of atomic orbitals (LCAO). 

On the oxidation side, the behavior of the folly desymmetrized catenanes 6 and 9 is particularly interesting from the viewpoint of molecular machines and it is the only one here discussed. Their electrochemical patterns are very similar and consist of three oxidative processes (for 6 " see Fig. 9.5a) the first two (Fig. 9.5c) are assigned to the two consecutive monoelectronic TTF oxidations [12], while the third one is ascribed to the oxidation of the DON unit. The first and second TTF oxidations exhibit the same features observed for a previously studied catenane [13, 14] and can be interpreted as follows after the TTF " oxidation, the electron donor ring circumrotates with respect to the electron accepting ring, delivering the DON unit into its cavity. 

As HF and DFT procedures are based on a variational principle, they can only obtain the lowest energy of the molecular system. To obtain the energies of excited electronic states (and so be able to study photochemical processes) it is necessary to go to a Cl calculation. The simple procedure is the Cl-singles (CIS) that just considers monoelectronic excitations [7]. A more precise technique is the complete active space (CAS) method that performs a full Cl over a selected (active) space of orbitals [8]. CAS methods are very powerful in the theoretical analysis of electronic spectra but are difficult to apply to reactivity as it is difficult to ascertain that the active space remains unchanged along all the reaction paths. Within the DFT formalism it is also possible to study excited electronic states using the time-dependent (TDDFT) formalism [9,10]. 

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