One important principle in quantum mechanics is the fact that particles are respresented by waves (called wavefunctions). The wavefunction (usually denoted as the greek letter psi, Ψ) is a purely imaginary concept, but the square of the wavefunction, |Ψ|2
, does have a physical meaning; it is the probability that one will find the particle in a particular area (for example, |Ψ(0)|2
represent the probablity of finding the particle at the origin). (See figure A for an example of what a wavefunction and the corresponding probability density look like. In this case, you can imagine the wavefunction corresponding to the pz
orbital where the horizontal axis represents the position along the z-axis).
Now, why do we talk about Ψ and not just |Ψ|2
? Since |Ψ|2
is always positive, adding |Ψ|2
values will always lead to larger |Ψ|2
values in the areas of overlap. However, we know that wavefunctions can add constructively (increasing the probability density in the region of overlap) or destructively (decreasing the probability density in the region of overlap). Therefore, in order to figure out what happens when orbitals overlap, we need to add Ψ, not |Ψ|2
For example, consider the sittuations in figures B and C. In each we have the wavefunctions of two atoms that are about to bond (shown in blue and red). The sum of the two wavefunctions depends on whether they are in phase (so that they add constructively as in case A) or out of phase (so that they add destructively as in case B). In the case of constructive interference (B), you see that the probability density has a large peak between the nuclei. This means that the electrons will spend most of their time between the nuclei (represented by the vertical arrows). This situation is favorable because the negatively-charged electrons are sitting between the two positively charged nuclei, preventing them from repelling each other. Because this arrangement of electrons helps stabilize the molecule, this molecular orbital is a bonding orbital.
In contrast, destructive interference (C), leads to the loss of electrons from the space between the nuclei. Without electrons between them, the two positively-charged nuclei will repel eachother, leading to repulsion. Hence, orbitals that are formed from such destructive interference are called antibonding.
As for why the gap between orbitals decreases as conjugation increases, the best way to think of it involves some math. In a highly simplified model, you can think of electrons in a conjugated pi system as particles in a 1D box. If you look at the energy levels of the particle in a 1D box, you see that the energy spacing decreases as the length of the system increases.
To think of this in a way that does not involve math, consider a guitar string. A guitar string will support certain resonant frequencies (http://en.wikipedia.org/wiki/Standing_wave
). The difference in frequencies between resonant frequencies is smaller in longer strings than in shorter strings. Since frequency is directly proportional to energy, the energy differences of these resonant modes are smaller in longer strings than in shorter strings.
I hope these explanations helped. Quantum mechanics is a tough subject to comprehend because many of the phenomena are not observed in our daily lives and sometimes are counterintuitive. However, this is also what makes learning about quantum fun. Atoms and molecules live in a completely different world than we do and physical chemistry allows us to visit their world and see how amazing it is!