Continuum function energy-normalized

Continuum wave functions are spatially extended and are not normalizable in the usual spatial sense. Instead an energy normalization is chosen.186... 

In practical applications, the continuum is often approximated by a discrete spectrum. To this end, one conveniently introduces a potential wall at long internuclear separations and solves for the artifically bound states.171,172 Alternatively, basis set expansion techniques can be employed.195,196 In either case, the density of states depends on external conditions, that is, the size of the box or the number of basis functions. This dependence on external conditions has to be accounted for by the energy normalization. Instead of employing a single continuum wave function with proper energy E in Eq. [240], one samples over the discrete levels with energy E -... 

The vibrational energy of the continuum state is not quantized consequently, the vibrational wave functions for energies above the dissociation limit are labeled by the good quantum number, E. These continuum functions are energy-normalized, rather than space-normalized as are the bound vibrational wave-functions. 

Xv,j xe,j)2 is a differential Franck-Condon factor because the continuum function Xe,j is energy-normalized. It has the dimensionality of E x, whereas the usual Franck-Condon factor is dimensionless. Contrary to the slow variation of He with energy, the differential Franck-Condon factor often varies rapidly and in an oscillatory manner with energy. 

Since cf>i is an energy-normalized continuum function, q is dimensionless, as is e. 

There are various ways to define the norm of the continuum functions In numerical calculations the radial wavefunctions of the discrete levels in a box can be normalized to 1. Those unity box-normalized functions are written (p °g(R), where the dependence on the size L of the box is explicitly written. At large distances they behave as sine functions. Alternatively, the energy-normalized radial wavefunctions (/ ) are related to the previous ones by the density of states in the box at the energy dn/d °, so that... 

For the case cited above, the ponderomotive energy is approximately 1 eV. For typical short pulse experiments today, this energy can easily be hundreds of electron volts. Therefore the wave function of a photoelectron in an intense laser field does not resemble that of the normal field-free Coulomb state, but is dressed by the field, becoming, in the absence of a binding potential, a Volkov state [5], This complex motion of the photoelectrons in the continuum is very difficult to reproduce in terms of the field-free atomic basis functions, so that we have chosen to define our electron wave functions on a finite difference grid. These numerical wave functions have the flexibility to represent both the bound and continuum states in the laser field accurately. 

Exact values of critical exponents are more difficult to obtain, because variational bounds do not give estimations of the exponents. Then the result presented by M. Hoffmann-Ostenhof et al. [64] for the two-electron atom in the infinite mass approximation is the only result we know for /V-body problems with N > 1. They proved that there exists a minimum (critical) charge where the ground state degenerates with the continuum, there is a normalized wave function at the critical charge, and the critical exponent of the energy is a = 1. 

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