Optimisation natural orbitals

We begin by using only the linear degrees of freedom contained in U to optimise our model and numerical approximations. We shall see that a careful consideration of these linear transformations suggests a natural generalisation of the usual AO basis in a way which enables us to use some of the conclusions of earlier sections where we discussed molecular symmetry. For a given set of non-orthogonal orbitals it is well-known from the work of Lowdin (14) that the simplest solution to equation (34),... 

For instance it allows the complete optimisation, orbitals and geometry, of benzene (D6h symmetry), which is described by two resonating structures and of the fictional molecule cyclohexatriene (D3h symmetry), whose wave function consist of just one of the structures. A comparison of the results gives a better insight in the nature and the persistence of resonance. 

The very special case of modelling a many-electron structvre by a single determinant is developed here. Since the only freedom within such a single-determinant wavefunction is in the forms of the spin-orbitals of which the determinant is composed, optimisation of this wavefunction naturally involves the optimisation of these orbitals. The equation which determines these optimum spin-orbitals is the Hartree-Fock equation. 

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