Formulation without Addition Theorem

Using the substitution method we eliminate the unknown vector Z2 from the last two matrix equations in (2.88) and obtain 

Equation (2.89) and the first matrix equation in (2.88), written in compact matrix notation as 

This formalism will be used in Sect. 2.5 to derive a recurrence relation for the transition matrix of a multilayered particle. 

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Density functional theory would never have played the important role it has now assumed in molecular quantum chemistry without the proof of an additional theorem, provided by Kohn and Sham (KS). They proved that it is possible to construct an auxiliary reference system of noninteracting electrons that has exactly the same electron density p(r) as that of the real system of interacting electrons. According to the HK theorem, this implies that there must be an associated potential "0 r) in which the noninteracting electrons move, according to a wavefunction which yields the same electron density as that in the real system. Since F describes a system of noninteracting electrons, it can be formulated as a Slater determinant, the MOs in which may be described as linear combinations of atomic orbitals. 

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