Hydrogen-like Wave Functions

The parameter Z is an effective atomic number whose value is determined by the minimization of in equation (9.2). Since the hydrogen-like wave functions 01 and 02 are normalized, we have... 

Table 2.2 Complete Normalized Hydrogen-Like Wave Functions.

The unique properties of dilute metal-ammonia solutions depend not upon the nature of the metal species, but upon the solvated electron common to all these solutions. Thus, the electron-in-a-cavity model (17, 19, 21) seems best suited to describe the species present in these solutions since the model is independent of the type of cation present. Jortner and his associates (15, 16) have extended this model by assuming that the cavity arises from polarization of the medium by the electron. The energy levels of the bound electrons are obtained by using a potential function containing the static and optical dielectric constants of the bulk medium as parameters. Using one-parameter hydrogen-like wave functions for the first two bound states of the electron, the total energy of the ith state is expressed as... 

The plots for hydrogen-like wave functions of radial function R(r) versus r, the distance from the nucleus and the probability distribution function 4jrr2[R(r) 2 versus r are shown... 

In the variational method, the calculation principle is to define a N x N matrix whose elements are < H0 >, and to obtain its eigenvalues by proper diagonalization under minimization conditions with respect to variational parameters [17], The basis hydrogen-like wave functions used in this calculation are ... 

Fig. 3 shows a qualitative graphical representation of hydrogen-like wave functions for one-electron atoms which have to be replaced for many-electron atoms at least by Slater-type 107) analytical wave functions ifnlm (1) which are approximate as they contain no nodes in the radial part R ,. 

Whereas the Frenkel tight-binding description is valid for excitons localized to atomic dimensions, the Wannier approximation describes excitons delocalized over many lattice distances. The hydrogen-like wave functions associated with these states serve as envelope functions describing average probability in spatial regions of lattice-constant dimensions. 

The spatial wavefunction, P(1>2), for the helium atom can be written approximately as the product of two hydrogen-like wave-functions ... 

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