Hydrogen-like atom spectra

Analytic, exact solutions cannot be obtained except for the simplest systems, i.e. hydrogen-like atoms with just one electron and one nucleus. Good approximate solutions can be found by means of the self-consistent field (SCF) method, the details of which need not concern us. If all the electrons have been explicitly considered in the Hamiltonian, the wave functions V, will be many-electron functions V, will contain the coordinates of all the electrons, and a complete electron density map can be obtained by plotting Vf. The associated energies E, are the energy states of the molecule (see Section 2.6) the lowest will be the ground state , and the calculated energy differences En — El should match the spectroscopic transitions in the electronic spectrum. 

Spectrum of Dirac Hydrogen-like Atoms with Coulombic Potential... 

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. 

The oscillating part of the secondary electron spectrum fine structure in the expression obtained is determined by two interference terms resulting from scattering of secondary electrorrs of final and intermediate states (the latter are due to the second-order process only). Here intensities of oscillating terms are determined by the amplitudes and intensities of electron transitions in the atom ionized. In this section we make estimations of these values within the framework of the simple hydrogen like model using the atomic unit system as in the preceding section. This section s content is based on papers [20,22,29-31,33,35,37,45-47]. 

The quantum theory, however, is essentially a physical theory, developed to explain observations like the atomic spectrum of hydrogen. Chemical ideas do not emerge from it easily. The theory is also very mathematical. Most chemists have to accept the results of quantum-mechanical calculations on trust. My approach avoids these problems. While I bring in the quantum theory where it is helpful, my treatment is essentially chemical. This makes for an easier introduction to the subject, and leads, I believe, to a better understanding of the key ideas, and the chemical thinking behind them. 

What is the wavelength of the transition from n = 4 to n = 3 for Li " In what region of the spectrum does this emission occur Li " is a hydrogen-like ion. Such an ion has a nucleus of charge +Ze and a single electron outside this nucleus. The energy levels of the ion are —where Z is the atomic number. 

The X-ray excitation process frequently is analyzed in terms of an excitonic electron hole pair (e.g. Cauchois and Mott 1949). The excitonic approach to X-ray absorption spectra accounts for the fact that the excited state is a hydrogen-like bound state. The X-ray exciton is different from the well-known optical excitons. In the latter cases the ejected electron polarizes a macroscopic fraction of the crystal-fine volume because the lifetime of optical excitations is in the order of lO s. The lifetime of the excited deep core level state, however, is in the order of 10 — 10 s, much too short to p-obe more than the direct vicinity of excited atom. Following Haken and Schottky (1958) the distance r between the ejected electron and core hole of an excited atom for E = 1 turns out to be r oc [h/(2m 0))] Here m denotes the effective mass of the ejected electron, to is the phonon frequency and is the dielectric constant. A numerical estimate yields r 10 A. Thus the information obtainable in an L, spectrum of the solid is very local the measurement probes essentially the 5d state of the absorbing atom as modified from the atomic 5d states by its immediate neighbors only. It is not suited to give information about extended Bloch states. On the other hand it is well suited to extract information about local correlations within the 5d conduction electrons, whose proper treatment is at the heart of the difficulty of the theory of narrow band materials and about chemical binding effects. 

With the solution of the hydrogen atom, the list of analytically solvable systems to be considered here is complete. Planck s quantum theory of light described black-body radiation, and now the simplicity of the spectrum of the hydrogen (and hydrogen-like ions) is adequately explained by quantum mechanics. We will find in the next chapter that although an exact analytic understanding of the behavior... 

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