Wavefunction energy-normalized

Vo(R) is the vibrational average of the interaction potential, Eq. 6.29. Free-state wavefunctions may be energy normalized,... 

To do this we must change the i wavefunctions from normalization per unit energy to normalization per state, i.e. Eq. (20.20). Using the derivative dtT,/dr, = 1/iy5 we may convert the squared wavefunctions 0, 2 from energy to state normalization by multiplying by 1/vj3. Equivalently, a bound 0, wavefunction which is normalized per unit energy has a normalization integral of v2. Since the wavefunction T = is composed of bound wavefunctions normalized per... 

Fig. 8.9. Contour plot of the potential energy surface of H2O in the BlA state as a function of the H-OH dissociation bond Rh-oh and the HOH bending angle a the other O-H bond is frozen at the equilibrium value in the ground electronic state. The energy normalization is such that E = 0 corresponds to H(2S ) + OH(2E, re). This potential is based on the ab initio calculations of Theodorakopulos, Petsalakis, and Buenker (1985). The structures at short H-OH distances are artifacts of the fitting procedure. The cross marks the equilibrium in the ground state and the ellipse indicates the breadth of the ground-state wavefunction. The heavy arrow illustrates the main dissociation path and the dashed line represents an unstable periodic orbit with a total energy of 0.5 eV above the dissociation threshold.

Fig. 9.9. Contour plot of the potential energy surface of H2O in the AlB state the bending angle is fixed at 104°. Superimposed are the total stationary wavefunctions I tot( ) defined in (2.70). The total energies are —2.6 eV and -2.0 eV corresponding to wavelengths of A = 180 nm and 165 nm, respectively. Energy normalization is such that E = 0 corresponds to three ground-state atoms.

Fig. 13.7. Contour plot of the A-state PES for a bending angle ae = 104°. Energy normalization is such that E — 0 corresponds to H+O+H. Superimposed are contours of luAX Oil2 where hax is the X —> A transition dipole function and 04 is the bound-state wavefunction of HOD with four quanta of excitation in the O-H bond. The filled circle indicates the barrier and the two especially marked contours represent the energies for the two photolysis wavelengths A2 = 239.5 and 218.5 nm used in the experiment. Adapted from Vander Wal et al. (1991).

The best possible variationally determined wavefunction of this form is that in which both the spacial orbitals total wavefunction be normalized. For such a wave-function, the variational method leads to orbital equations similar to the Hartree-Fock equations (10). These are.-26... 

Meg (in atomic units, IDebye = 0.3935 a.u.) is the electronic transition matrix element between the e and g electronic states, assuming the dipole length approximation, (ve is the energy normalized nuclear continuum wavefunction, and fj) is the initial state bound vibrational wavefunction. The overlap integral has units of cm1/2 (see Section 7.5). Note that 10 18 cm2=lMb... 

In this case, the individual orbitals, < )y(r), can be determined by minimizing the total energy as per equation Al.3.3. with the constraint that the wavefunction be normalized. This minimization procedure results in the following Hartree equation ... 

The essentials of the SSEA for numerically compufed energy-normalized N-elecfron wavefunctions were published in 1994 by Mercouris ef al. [54], The firsf application was not only to the multiphoton ionization of H (whose specfrum is known exactly analytically), as a test case, but also to the multiphoton detachment of the four-electron Li negative ion, with two free channels, Li ls 2s S and ls 2p P°. Li (or Be) is the first system of fhe Periodic Table for which the proper description of the zero-order electronic structure requires a multiconfigurational Hartree-Fock (MCHF) description. In the context of the review of the SSEA, we also discuss briefly the formulation of the problem in terms of the full atom-EMF interaction,Vext(f), which is computationally convenient as well as necessary for certain problems involving, say, off-resonance coupling of Rydberg states, for which use of just the electric dipole term is inadequate [55-57]. 

Due to their simple structure, the core wavefunctions ls 2s and ls 2p P° were represented by HP wavefunctions. The basis set of bound orbitals and of the energy-normalized scattering orbitals, si and s t, were computed numerically from separate calculations via the term-dependent HE scheme. 

The energy-normalized scattering wavefunctions, Isel, were computed numerically by the fixed-core HF method, for energies up to 4.03 a.u. in steps of 0.004 a.u. and for angular momenta Z = 0,1,..., 15. 

Obviously, there is much room for further development of the basic concepts of the SSEA and for improvement of its methodology, as well as for additional applications to new and challenging TDMEPs. In all cases, the fundamental issue is how to identify and construct the wavefunctions that are considered relevant to each problem. For example, the possibility of treating correctly the contribution from two-electron continue is an open question. Even if two-electron products of energy-normalized scattering states are used as basis sets, the computational requirements of this (multichannel in general) problem are huge, and so its solution would require dedicated effort and powerful computers. 

For a given Hamiltonian operator there will be an infinite number of solutions to this equation, each indicated by a different value of the index . We wish to find the ground state wavefunction, y/j, which has an energy Normally, equation (8.1) cannot be solved analytically and the wavefunctions that satisfy the equation are unknown. Under these circumstance it is necessary to formulate a trial wavefunction, which is expected to be a good approximation to the true ground state wave-function. 

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... 

The symbol "dr stands for integration over space and spin coordinates of the electrons dr = dvdco. Since both space and spin parts of our wavefunctions are normalized [cf. Eqs. (5-25) and (5-26)], the denominator of Eq. (5-46) is unity and may he ignored. The energy thus is given by the expression... 

If the wavefunction is normalized, the denominator is unity and the expectation value of the energy is written as ... 

---