Rydberg Hamiltonian, diagonalized

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

While direct diagonalization of the Hamiltonian matrix works well for situations in which there is a finite number of states, as in Fig. 9.1, it is clearly hopeless to try it in this case. A useful WKB approach was proposed by Edmonds12 and refined by Starace.13 Using the fact that azimuthal symmetry exists, Starace writes the wavefunction of the spinless Rydberg electron in cylindrical coordinates as13... 

The residual interaction between these same-n Rydberg levels, expressed in the case (c) basis, but intermediate between cases (c) and (a), can now be viewed as a consequence of the electronic part of the Hamiltonian, since the case (c) basis functions are exact eigenfunctions only when the energy difference between isoconfigurational triplet and singlet states is zero. The off-diagonal matrix element in the case (c) basis is... 

In the CESE procedure, the coordinates of the Hamiltonian (40) are left real. The total function space is formally divided info two multi-dimensional parts of N-elecfron wavefuncfions, say Q and P, nof necessarily orthonormal between them. The Q space is fixed by considering judiciously the electronic structure and spectrum of the system under study. The P space is not completely fixed. It contains parameterized configurations with real as well as complex orbitals representing contributions from fhe multichannel high Rydberg and scattering states. It is the variational optimization of this space via the diagonalization of fhe matrix... 

Lastly we note that the width of the well identified bound states is zero. But if we diagonalize a hamiltonian with a Coulombic tail, where sits an infinity of discrete but loosely bound Rydberg states, we will reproduce in our projected calculation one discrete state which averages these states, and is surely also an admixture of continuum states. Typically cross sections to these states are small. If one is really interested in cross sections to them an adjunct basis, P, may be used which connects them to P through the potential matrix elements. A t-matrix expression which has the desired state as the final entry, but P%, as the approximate state wave function, can be used. For example, if charge transfer is responsible for a small amount of flux loss from the target nucleus it is not necessary to use a two-centered basis the procedure described can be used, e.g.,... 

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