Hydrogen-like ions wavefunctions

The exact wavefunction corresponds to a = b = s and a = (3 = Z. The variational ground state energy of Z = 90 hydrogen-like ion in the Dirac-Pauli... 

The above example and selection rules are also applicable to hydrogen-like ions, which have a single electron. However, such systems are in the vast minority of atomic species whose spectra need to be understood. Recall that one of the final failings of classical mechanics was the inability to explain spectra. Although quantum mechanics does not provide analytic solutions for wavefunctions of multielectron systems, it does provide tools for understanding it. 

Both the mass—velocity and Darwin terms show unphysical behavior in variational calculations. To demonstrate this, consider a hydrogen-like atom (ion) with a single electron and a nuclear charge Z as a prototype example With a trial wavefunction of the form,... 

The hydrogen molecule, like the He atom, poses a real problem the Hamiltonian operator contains a term representing the repulsion between the electrons, and the presence of this term makes an exact solution of the Schrodinger equation impossible. In the case of the helium atom we turned to the hydrogen atom for guidance in the choice of approximate wavefunctions. In the case of the hydrogen molecule we turn to the hJ ion and assume that the wavefunction may be approximated by the product of two molecular orbitals... 

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