Normalisation continuum states

In fig. 3. 1, wavefunctions for bound states and for the continuum state lying near threshold (e —> 0) are shown. In order to compare them, one has to normalise the continuum functions, which are not square-integrable. This difficulty is resolved by requiring... 

We can numerically solve for Xk > th continuum state at an energy that corresponds to the eigenvalue of a pseudostate. Recall toat Xk satisfies scattering boundary conditions, and Xn zero for large r, square-integrable and normalised. If however we project Xk O " sis subspace, P)(P, we find for a good choice of basis that... 

Note that continuum orbitals are normalised in the sense discussed in section 6.4. If two orbitals < ) are identical the determinant vanishes because it has two equal columns. No two electrons can be in the same orbital. The iV-electron state p) therefore obeys the Pauli exclusion principle. We call it a configuration. 

In this section we first summarise the meaning of the notation for the channel and collision states with box normalisation and in the continuum limit L —> 00. We then define notation for the limit 6 —> 0-1- and write the corresponding integral equations. 

With box normalisation the channel states d>,) are countable. For discrete target states the index i stands for the internal quantum numbers n,j,m,, v of the target and projectile and the box quantum numbers nix,nty,ni characterising the relative motion. When L —> 00 the box quantum number set is replaced by the momentum continuum k,. The limiting procedure is summarised as follows... 

For the [diotodissociation of a triatomic molecule, the asymptotic form of the final state, continuum ravefunction, correctly normalised on the energy sccile[42], may be written as[39] ... 

The XAFS spectrum, %(E), is defined as the normalised oscillatory structure of the X-ray absorption, e.g. %(E) = ( l(E)- Po(E)/ Jo(E)), where Po(E) is the smooth varying atomic-like background absorption. Essentially, the XAFS spectrum implicates the quantum-mechanical transition from an inner, atomic-like core orbital electron to an unoccupied, bound (pre-edge transition) or unbound, free-like continuum level. The oscillatory structure therefore reflects the unoccupied part of the electronic bands/structure of the system in the presence of a core-hole. Note that this differs from the initial, ground state by physical effects induced by the fact that the... 

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