Hydrogenlike

Normalized radial functions for a hydrogenlike atom are given in Table A 1.1 and plotted graphically in Fig. A 1.1 for the first ten combinations of n and /. It will be seen that the radial functions for Is, 2p, 3d, and 4f orbitals have no nodes and are everywhere of... 

The natural orbitals %2v and %3p are, in contrast to the hydrogenlike functions, localized within approximately the same region around the nucleus as the Is orbital. This means that the polarization caused by the long-range interaction is associated mainly with an angular deformation of the electronic cloud on each atom. If %2p and %3p are expanded in the standard hydrogen-like functions, an appreciable contribution will again come from the continuum. 

The energy levels of hydrogenlike one-electron ions of atomic number Z differ from those of hydrogen by a factor of Z2. Predict the wavelength of the 2s — Is transition in He+... 

Energy levels of a hydrogenlike atom of atomic number Z ... 

Evidence has been advanced8 that the neutral helium molecule which gives rise to the helium bands is formed from one normal and one excited helium atom. Excitation of one atom leaves an unpaired Is electron which can then interact with the pair of Is electrons of the other atom to form a three-electron bond. The outer electron will not contribute very much to the bond forces, and will occupy any one of a large number of approximately hydrogen-like states, giving rise to a roughly hydrogenlike spectrum. The small influence of the outer electron is shown by the variation of the equilibrium intemuclear distance within only the narrow limits 1.05-1.13 A. for all of the more than 25 known states of the helium molecule. 

In practice one will - in fact - not be interested in the accuracy of/(r) as a function of r, but rather in the error of matrix elements like that over a hydrogenlike Is function... 

The expectation values of various powers of the radial variable r for a hydrogenlike atom with quantum numbers n and I are given by equation (6.69)... 

The normalized radial wavefunctions for hydrogenlike atoms can be expressed by... 

This formula for hydrogenlike atoms with nuclear charge Z, is often formulated in terms of the dimensionless fine-structure constant a = 4Tre0hc/e2 as... 

The largest Lyapunov exponents for the relativistic hydrogenlike atom in a uniform magnetic field... 

Molecular Species (M = Cr, Mo, W) A Transition from Halogenlike to Hydrogenlike Chemical Behavior for Gold. 

EXO 0748-676, Cottam et al. (2002) have found absorption spectral line features, which they identify as signatures of Fe XXVI (25-time ionized hydrogenlike Fe) and Fe XXV from the n = 2 —> 3 atomic transition, and of O VIII (n = 1 —> 2 transition). All of these lines are redshifted, with a unique value of the redshift z = 0.35. Interpreting the measured redshift as due to the strong gravitational field at the surface of the compact star (thus neglecting general relativistic effects due to stellar rotation on the spectral lines (Oezel Psaltis 2003)), one obtains a relation for the stellar mass-to-radius ratio ... 

If kfi = In, then the functions Rni r) are just the familiar hydrogenlike radial functions, expressed in atomic units ... 

It is interesting to examine the momentum-space orthonormality relations of the Fourier transforms of hydrogenlike Sturmian basis sets. If we let... 

The momentum-space orthonormality relation for hydrogenlike Sturmian basis sets, equation) 17), can be shown to be closely related to the orthonormality relation for hyperspherical harmonics in a 4-dimensional space. This relationship follows from the results of Fock [5], who was able to solve the Schrodinger equation for the hydrogen atom in reciprocal space by projecting 3-dimensional p-space onto the surface of a 4-dimensional hypersphere with the mapping ... 

Basis sets of the type discussed in this paper can only be applied to bound-state problems. It is interesting to ask whether it might be possible to constmct many-electron Sturmian basis sets appropriate for problems in reactive scattering in an analogous way, using hydrogenlike continuum functions as building-blocks. We hope to explore this question in future publications. 

Since the hydrogenlike Sturmian basis functions form a complete set, the term %i,o,o(xy-R) can be represented as a single-center expansion in terms of functions localized at the origin ... 

The Fourier transforms of the hydrogenlike orbitals were shown by Fock [18] to be expressible in terms of 4-dimensional hyperspherical harmonics when momentum space is mapped onto the surface of a 4-dimensional unit hypersphere by the transformation ... 

With this transformation, as Fock was able to show, the Fourier-transformed hydrogenlike orbitals become ... 

In treating complex atoms and molecules, these hydrogenlike wavefunctions are still too complicated in most applications. In treating low-energy (up to a few eV) problems, the behavior of the wavefunctions near the nuclei of the atoms is not important. Therefore, in the hydrogen wavefunctions, only the term with the highest power of r is significant. Based on this observation, Zener (1930) and Slater (1930) proposed a simplified form of the atomic wavefunctions ... 

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