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Where the summation is over all atomic orbitals on B. > of the core hamiltonian operator, B(~A) and is essentially an atomic quantity (the energy of cPIL in the bare field of the core of its own atom). U IL,. is obtained semiempirically from atomic data by methods to be discussed in the following sections. The remaining terms in Eq. 23) give the electrostatic interaction of an electron in cP" with the cores of other atoms B. We shall consider approximations for these terms shortly. Next we consider the off-diagonal core matrix elements H,,> between different atomic orbitals cP" and cP> on the same atom A.

80) follows similar lines to the closed-shell theory given in Sec. 2. Again, we may carry out linear transformations among the ""a orbitals to ensure that they are mutually orthbgonal, and similarly with the y,,11 set. The functions y"iaa and fl{3 will automatically be orthogonal by virtue of the spin functions. As in the restricted formalism, the total electronic energy can be conveniently calculated by separating the hamiltonian into a oneelectron part:Je 1 and a two-electron part:Je2 as in Eqs.

L'IXu) 1'" Now if the additional condition of Eq. 1) applies for the old basis set, the integral (~I'IXu) will vanish unless ~ = I' and X = u. If both these conditions are satisfied, the integral is 'Y AD and Eq. 17) I' using properties of an orthogonal matrix. 23), argument applies and we obtain finally Thus the same result would have been obtained if the approximations had been applied directly to the new set of orbitals. These approximations are therefore equivalent to the complete neglect of differential overlap for all sets of orthonormal atomic orbitals.

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Aproximate Molecular Orbital Theory by Pople J., Beveridge D.

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