Hydrogen hydrogenlike atomic orbitals

Concerning molecules, the wave function (molecular orbital) for a hydrogenlike molecule, for instance, is expanded in terms of hydrogen-like atomic orbitals Xaj(f) belonging to hydrogen-like atoms / = 1,2, respectively, as... 

We can use the quantum mechanical model of the atom to show how the electron arrangements in the hydrogenlike atomic orbitals of the various atoms account for the organization of the periodic table. Our main assumption here is that all atoms have the same type of orbitals as have been described for the hydrogen atom. As protons are added one by one to the nucleus to build up the elements, electrons are similarly added to these hydrogenlike orbitals. This is called the aufbau principle. 

Because of the factor exp (—0.5r ), the density distribution p is concentrated on the nucleus. The authors compare this density distribution with the corresponding Hartree-Fock density (appropriate for the potential used), and even with the density distribution related to the hydrogenlike atom (after neglecting l/ri2 in the Hamiltonian, the wave function becomes an antisymmetrized product of the two hydrogen-like orbitals). In the latter case, the electrons do not see each othei, and the eorresponding density distribution is too concentrated on the nucleus. As soon as the term l/ri2 is restored, the electrons immediately move apart, and p on the nucleus deereases by about 30%. The second result is also interesting the Hartree-Fock density is very elose to ideal-it is almost the same curve. ... 

As a result of the spherical symmetry of the potential energy, the energy of the atomic orbitals for hydrogen and hydrogenlike ions depends only upon the value of the principal quantum number (n), and is given by... 

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 Bohr model can be readily extended to hydrogenlike ions, systems in which a single electron orbits a nucleus of arbitrary atomic number Z. Thus, Z = 1 for hydrogen, Z = 2 for He+, Z = 3 for Li++, and so on. The Coulomb potential (7.5) generalizes to... 

This picture of polyelectronic atoms leads to hydrogenlike orbitals for these atoms. They have the same general shapes as the orbitals for hydrogen, but their sizes and energies are different. The differences occur because of the interpiay between nuclear attraction and the electron repulsions. 

We consider the electron in a hydrogen atom or hydrogenlike ion (He, Li +, ) orbiting around a nucleus of atomic number Z. The attractive Coulomb potential in atomic units e /ATt Q = 1) can be written as... 

The hydrogen-atom potential-energy function is even, and the hydrogenlike orbitals can be chosen to have definite parity (Problems 7.17 and 7.23). 

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