Hamiltonian for electronic systems

An electron-atom or electron-molecule collision system consists of particles that never annihilate and that interact each other only via Coulomb forces. 

A theoretically attractive choice of HQ would be to confine all electrons within the closed channels to prevent them from going out as far as to the asymptotic region. For the hydrogen negative ion H, for example, this is possible by projecting out all the hydrogenic states yrK m(r,) in the open channels by the projection operator Q, = 1 - fnim( i), so that 

The formal theory of resonances due to Feshbach begins with the decomposition of the Hamiltonian in terms of a projection operator Q [8]. He defines Q as the projection onto the closed-channel space, just like the example of H discussed around Eqs. (4) and (5). Then, QBSs described well by the eigenfunctions Q4 of Eq. (5) with his Q may be called Feshbach resonances. A simplified picture would be that eigenstates Q t are supported by some attractive effective potential approaching asymptotically the threshold energy of a closed channel. If this is the case, then the energies EQ of 

Resonances unassociated with eigenstates of Feshbach s QHQ are often associated with the shape of some effective potential in an open channel, normally a combination of short-range attractive and long-range repulsive potentials, forming a barrier, within which a large part of the wavefunction is kept. These resonances are called shape resonances or potential resonances. They occur at energies above and usually close to the threshold of that open channel. 

In actual resonance calculations, explicit decomposition of the Hamiltonian is not always needed. The whole Schrodinger equation (1) is often solved somehow without explicit use of HQ or H. General methods for analyzing results from such a treatment are the main subject of this article. A comprehensive review of computational methods or computational results is outside of its scope. 

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