Lecture #20, October 15, 1999
	CURMUDGEON GENERAL'S WARNING. These "slides" represent highlights from lecture and are neither complete nor meant to replace lecture. It is advised not to use these as a reliable means to replace missed lecture material. Do so at risk to healthy academic performance in 09-105.
Back to waves and orbitals!
A review of some terms, symbols, and displays of wave functions for an electron in an atom.
Solving the Schrodinger wave equation for a system in which there is more than one center of force (a single nucleus) is difficult. This year, John Pople, retired Professor of Chemistry at Carnegie Mellon, shared the 1998 Nobel Prize in Chemistry for devising mathematical tools amenable to computation that have greatly eased the problem if one resorts to using a computer. Approximate wave functions can be assembled using the ones we're already intimately familiar with: the hydrogen-like, one-electron atomic orbitals.
Among the unusual properties associated with the mathematics of waves is that of "interference". This occurs even for classical waves such as water waves and sound waves. Depending on whether combining waves have the same sign of their amplitudes or opposite signs, the waves interfere constructively or destructively, respectively.
If we add together a 1s atomic orbital on one hydrogen atom with an identical 1s atomic orbital centered on another nucleus, the resulting orbital (whether there are electrons there or not) closely resembles the more exact orbital one would calculate by highly sophisticated mathematical techniques.Recalling the significance of an orbital, this would then indicate where an electron would be expected to be found if we placed an electron in the system. It would also be used to calculate the energy such an electron would have. Energy and geometry are our recurring themes, n'est pas?
An electron in an orbital about two protons has a potential energy indicated by the heavy curve if constructive interference of the atomic orbitals represents the electron's orbital. There is a certain proton-proton distance at which the energy is minimum. This is close to the bond length in the system with one electron.
If destructive interference represents the orbital in which the electron happens to be found, the linear combination results in a depeletion of electron density between the positively charged protons. They repel each other under these circumstances.
A schematic of the orbitals on separated protons relative to "close" protons. The constructive combination of atomic orbitals gives rise to a system lower in energy than the isolated system and is a bonding molecular orbital.
Molecular orbital energy diagram. Each "box" represents an energy state (orbital) which can accommodate up to two electrons. The label of each such state also indicates what the geometry (shape) of the orbital is by referring to previous discussion on how molecular orbitals are constructed from combinations of atomic orbitals. Shown is the electron configuration for H2
Re-defining how one calculates bond order within the context of molecular orbital theory.