Atomic orbitals atomization energies

The Bom-Oppenheimer approximation is not peculiar to the Huckel molecular orbital method. It is used in virtually all molecular orbital calculations and most atomic energy calculations. It is an excellent approximation in the sense that the approximated energies are very close to the energies we get in test cases on simple systems where the approximation is not made. 

The hydrogenie atom energy expression has no 1-dependenee the 2s and 2p orbitals have exaetly the same energy, as do the 3s, 3p, and 3d orbitals. This degree of degeneraey is only present in one-eleetron atoms and is the result of an additional symmetry (i.e., an additional operator that eommutes with the Hamiltonian) that is not present onee the atom eontains two or more eleetrons. This additional symmetry is diseussed on p. 77 of Atkins. 

Asimple example is the formation of the hydrogen molecule from two hydrogen atoms. Here the original atomic energy levels are degenerate (they have equal energy), but as the two atoms approach each other, they interact to form two non degenerate molecular orbitals, the lowest of which is doubly occupied. 

The neglect of electron-electron interactions in the Extended Hiickel model has several consequences. For example, the atomic orbital binding energies are fixed and do not depend on charge density. With the more accurate NDO semi-empirical treatments, these energies are appropriately sensitive to the surrounding molecular environment. 

The lowest energy molecular orbital of singlet methylene looks like a Is atomic orbital on carbon. The electrons occupying this orbital restrict their motion to the immediate region of the carbon nucleus and do not significantly affect bonding. Because of this restriction, and because the orbital s energy is very low (-11 au), this orbital is referred to as a core orbital and its electrons are referred to as core electrons. 

Spin-orbit coupling is a relativistic effect that is well reported in tables of atomic energy levels, and this gives a guide. Relativistic effects are generally thought to he negligible for first-row elements. 

Alternative methods are based on the pioneering work of Hylleraas ([1928], [1964]). In these cases orbitals do not form the starting point, not even in zero order. Instead, the troublesome inter-electronic terms appear explicitly in the expression for the atomic wavefunction. However the Hylleraas methods become mathematically very cumbersome as the number of electrons in the atom increases, and they have not been very successfully applied in atoms beyond beryllium, which has only four electrons. Interestingly, one recent survey of ab initio calculations on the beryllium atom showed that the Hylleraas method in fact produced the closest agreement with the experimentally determined ground state atomic energy (Froese-Fischer [1977]). 

Each atomic energy level Is associated with a specific three-dimensional atomic orbital. 

Elements beyond the second row of the periodic table can form bonds to more than four ligands and can be associated with more than an octet of electrons. These features are possible for two reasons. First, elements with > 2 have atomic radii that are large enough to bond to 5, 6, or even more ligands. Second, elements with > 2 have d orbitals whose energies are close to the energies of the valence p orbitals. An orbital overlap description of the bonding in these species relies on the participation of d orbitals of the inner atom. 

Using first-order perturbation theory, show that the spin-orbit interaction energy for a hydrogen atom is given by... 

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