Finite overlap

The state R in Equation 6.4 strictly keeps the HL wave function of the product P, and is hence a quasi/spectroscopic state that has a finite overlap with R. If one orthogonalizes the pair of states R and R, by, for example, a Graham—Schmidt procedure (see Exercise 6.3), the excited state becomes a pure spectroscopic state in which the A—Y is in a triplet state and is coupled to X to yield a doublet state. In such an event, one could simply use, instead of Equation 6.5, the spectroscopic gap Gs in Equation 6.6 that is simply the singlet—triplet energy gap of the A—Y bond ... 

A very important consideration is that the atomic wavefunctions for a single atom iJ/(r) must extend to r = infinity, where they must vanish 0.4 nm or so away from the nucleus the ij/(r) are small in amplitude, but not zero placing another nonbonded atom centered less than 0.3 nm away causes a problem with orthogonality there is a finite overlap of wavefunctions [which is also... 

Rigorous one-dimensionality is of course a myth in organic conductors. The Coulomb interaction is inherently long range. There is also an unavoidable finite overlap and thus hopping between chains. These lead to two- and three-dimensional couplings and to dimensionality crossovers. (a) Single-Particle Crossover... 

The first approach is called the multiple reference frame (MRF) or inner-outer approach (inner-outer approach in fact defines inner and outer zones with a finite overlap whereas in the MRF approach there is no overlap between inner and outer regions). In this approach, flow characteristics of the inner region are solved using a rotating framework. These results are used to provide boundary conditions for the outer region (after azimuthal averaging), flow in which is... 

If 3 r becomes much larger than Gg and G, the diagonal energies of matrix (10) represent a better approximation. The wave-fimctions are not localized in the minima anymore because the ene etic (near-)degeneracy of the vibronic triplet is lifted. The finite overlap of the nuclear wave-functions xi demands that the electronic parts of the functions [Eq. (IS)] have to be orthogonal. The new eigenfimctions are the following linear combinations of 2> x> =... 

The bond length within the dissociating molecule increases upon adsorption. This process costs energy. The orbitals of adsorbate and surface need to have finite overlap in order for a covalent surface bond to form. The covalent interaction with the metal surface lowers the energy cost of dissociation. 

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