Overlap integral Mulliken equations

Mulliken derived equations for the variation of overlap integrals with distance between the participating nuclei. For homonuclear diatomic molecules the equations for the overlap of two 2p orbitals in o and n modes are respectively ... 

RS Mulliken suggested a widely used method for performing population analysis [Mulliken 1955], The starting point is Equation (2.215), which relates the total number of electrons to the density matrix and to the overlap integrals. In the Mulliken method, all of the electron density (P ) in an orbital is allocated to the atom on which is located. The remaining... 

The first three underbraced integrals are just unity because we have implicitly used normalized molecular and atomic orbitals, being able to hold just two electrons the last integral is the overlap integral S12 and the last term, 2C1C2S12, is called the overlap population according to Mulliken [83]. Mulliken also heuristically partitioned the above equation into three different parts, namely... 

It is shown that the LCAO molecular Hartree-Fock equations for a closed-shell configuration can be reduced to a form identical with that of the Hoffmann extended Hiickel approximation if (i) we accept the Mulliken approximation for overlap charge distributions and (ii) we assume a uniform charge distribution in calculating two-electron integrals over molecular orbitals. Numerical comparisons indicate that this approximation leads to results which, while unsuitable for high accuracy calculations, should be reasonably satisfactory for molecules that cannot at present be handled with facility by standard LCAO molecular Hartree-Fock methods. 

Even if no integral parametrizations are introduced and the HF equations are all correctly solved, the method eventually turns out to be theoretically incomplete. Despite the correct treatment of electronic exchange (X) within Hartree-Fock theory, electronic correlation (C) is totally missing. This is easily shown for the case of the H2 molecule in which we use the bonding solution of the H2 molecular ion ( + = cr from Equation (2.15)) to build up an antisymmetrized molecular wave function. This means that we put both electrons (ri and rz) of the H2 molecule into the same ip+ orbital, and Pauli s principle is obeyed by means of the ct/ spinors. Neglecting orbital overlap and any pre-factors, for simplicity, the so-called Hund-Mulliken [124] (another name... 

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