Hamiltonian hydrogenlike atom

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. ... 

The unperturbed Hamiltonian for an electron in a hydrogenlike atom with nuclear charge + Ze is... 

In hydrogenlike atoms, the total electronic Hamiltonian now becomes... 

We now extend our discussion of hydrogenlike atoms to complex atoms with a total of p electrons. The nonrelativistic Hamiltonian operator for such atoms in the absence of external fields is... 

Suppose we take the interelectronic repulsions in the li atom as a perturbation on the remaining terms in the Hamiltonian. By the same steps used in the treatment of helium, the unperturbed wave functions are products of three hydrogenlike functions. For the ground state,... 

Notice that each individual one-electron hamiltonian (5-4) is just the hamiltonian for a hydrogenlike ion, so it has as eigenfunctions the Is, 2s, 2p, etc., functions of Chapter 4 with Z = 2. Such one-electron functions are referred to as atomic orbitals Representing them with the symbol 0/ (e.g., 0i = Is, 02 = 2s, 0s = 2px, 04= etc.) we have, then. 

The one-electron operators in the resulting approximate hamiltonian for an atom are hydrogenlike ion hamiltonians. Their eigenfunctions are called atomic orbitals. 

The details of a HF-SCF computation can now be examined in detail. In the HF-SCF approach, the Hamiltonian for an atom is written in terms of a summation of hydrogenlike terms plus the electron repulsion terms. 

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