Orbital functional theory of the -matrix

In the case of a scattering resonance, bound-free correlation is modified by a transient bound state of fV+1 electrons. In a finite matrix representation, the projected (fV+l)-electron Hamiltonian H has positive energy eigenvalues, which define possible scattering resonances if they interact sufficiently weakly with the scattering continuum. In resonance theory [270], this transient discrete state is multiplied by an energy-dependent coefficient whose magnitude is determined by that of the channel orbital in the resonant channel. Thus the normalization of the channel orbital establishes the absolute amplitude of the transient discrete state, and arbitrary normalization of the channel orbital cannot lead to an inconsistency. 

For an fV-electron target state, Ec is a sum of pair-correlation energies. In a two-electron system, such as atomic He, a major contribution to the correlation energy 

The //-electron target wave function is coupled to a continuum orbital pK for which nK — 0. Vanishing nK implies that the continuum electron does not modify the effective Hamiltonian Q that acts on occupied target orbitals (nt = 1). Q also acts on d K because 0 cancels out of the functional derivatives in -%j. This implies that pK is orthogonal to the occupied target orbitals. The result is to augment standard static-exchange equations with a nonlocal correlation potential vc. 

From Eq. (5.7), and Janak s theorem [185], the contribution of correlation energy to the mean energy of the continuum orbital within the //-matrix boundary is 

8 Variational methods for continuum states Table 8.1. Partial wave phase shifts for He 

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