Bond-Length-Dependent Couplings

An improvement of the Hiickel model may be obtained if it is reahzed that the shorter the bond, the stronger is the coupling. Longuet-Higgins suggested the following equation  

the coupling (Po) is the same for all bonds. SetRo = 1.4anda = 0.3106 A. A calculation is started by assuming that all bond lengths are equal. Equations 3.25 and 3.26 are used to update the R v and p values of the different bonds. The iterative solution converges quite fast. 

The wave function must be invariant under rotation, disregarding a constant of absolute value equal to unity. We have reason to try the following expression  

For different values of zf, different values of 0 are obtained in Equation 3.28 and different orbitals in Equation 3.27. Other values of f do not lead to any new orbitals. 

It is practical to divide all matrix elements in Equation 3.17 by p / 0 and set (a - e)/p = X. The secular equation for the cyclic system of benzene may then be written as 

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One-dimensional crystals are subject to a distortion that leads to alternating short and long bonds. The half-filled valence band thereby develops a gap at the Fermi level. We have met this situation already for polyenes. The result is easily calculated using the Hiickel model with bond-length-dependent coupling p. 

The Peierls distortion is confirmed in accurate models. If Pol is given the value of 3 eV in a bond-length-dependent coupling model, the band gap has the value Ipl = 1.57 eV. 

The only tenn in this expression that we have not already seen is a, the vibration-rotation coupling constant. It accounts for the fact that as the molecule vibrates, its bond length changes which in turn changes the moment of inertia. Equation B1.2.2 can be simplified by combming the vibration-rotation constant with the rotational constant, yielding a vibrational-level-dependent rotational constant. 

Joachim C (1987) Ligand-length dependence of the intramolecular electron transfer through-bond coupling parameter. Chem Phys 116 339... 

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