Orbits, atomic energy

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. 

To obtain qualitative information about a molecule, such as its moleculip orbitals, atomic charges or vibrational normal modes. In some cast semi-empirical methods may also be successfully used to predict energy-trends arising from alternate conformations or substituent effects in qualitative or semi-quantitative way (but care must be taken in this area). 

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]). 

In Fig. 1 there is indicated the division of the nine outer orbitals into these two classes. It is assumed that electrons occupying orbitals of the first class (weak interatomic interactions) in an atom tend to remain unpaired (Hund s rule of maximum multiplicity), and that electrons occupying orbitals of the second class pair with similar electrons of adjacent atoms. Let us call these orbitals atomic orbitals and bond orbitals, respectively. In copper all of the atomic orbitals are occupied by pairs. In nickel, with ou = 0.61, there are 0.61 unpaired electrons in atomic orbitals, and in cobalt 1.71. (The deviation from unity of the difference between the values for cobalt and nickel may be the result of experimental error in the cobalt value, which is uncertain because of the magnetic hardness of this element.) This indicates that the energy diagram of Fig. 1 does not change very much from metal to metal. Substantiation of this is provided by the values of cra for copper-nickel alloys,12 which decrease linearly with mole fraction of copper from mole fraction 0.6 of copper, and by the related values for zinc-nickel and other alloys.13 The value a a = 2.61 would accordingly be expected for iron, if there were 2.61 or more d orbitals in the atomic orbital class. We conclude from the observed value [Pg.347]

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

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