Quantum Mechanics in Chemistry

Quantum Mechanics has many strange and wondrous properties, such as the inability to determine the position of a bound particle exactly. An obvious outgrowth of the wave nature of matter, this is sometimes called the Heisenberg Uncertainty Principle:
dx dp >= h/(4p)

This uncertainty relation occurs between any non-commuting observables. vis.

dE dt >= h/(4p)

What does non-commuting mean? It means that the order in which you observe succesively these two quantities affects the outcome of the measurement. Remember in algebra we say that multiplication is commutative in that A*B=B*A. In Quantum Mechanics the way your measure something depends on how you measure it, and in particular, the order in which you measure things. In fact, the act of observation must perturb the system when particular 'pairs' of measurements are made. Note that the quantity of uncertainty, i.e. the absolute measure of how much the observer must affect the system, is measured by Planck's constant. (Sometimes the numerical factor (4p) is ommitted in these inequalities; it is a kind of 'geometric' factor)

Quantum Mechanics rules in a land of the small, where small means that products of non-commuting variables approach the size of h. Strange things that happen when you live at the scale of Planck's constant; This is the land where the electrons in molecules live:

Zero-point Energy, The Energy of Confinement.

When you live like an electron, it costs energy for you to be confined in any small region of space. The smaller the region of space you are confined to, the higher the energy. The smallest amount of energy a bound electron can have by virtue of QM is called the Zero-Point Energy. Even electromagnetc fields have zero point energy because photons are quantal. The energy to confine a particle is the the real basis behind why the electrons in an atom do not collapse into the nucleus and why moecules form. Sometimes this concept is thought of in reverse: an Organic Chemist might say that the 'delocalization energy' of the electron is what drives 'resonance' and makes 'aromatic' molecules special.
Mass Diffusion
If you try to confine an electron with forces you incur a zero-point energy cost, a localization energy. But what if you just place it at rest somewhere and leave it alone (no forces). Won't it just stay there? The answer is yes and no. The center of the probability distribution that describes where the electron is will remain at rest, but the distribution itself will spontaneously spread out like an inkblot. Mass 'diffuses' in space all by itself. (This happens on my desk all the time)
Tunnelling
One of the more fascinating quantum mechanical phenomena is the apparent violation of conservation of energy that occurs when a confined particle 'escapes' by going through a barrier instead of going over it. Suppose it takes a certain amount of energ, Eo, for a particle to escape a trap that is made of a high thin wall, but the particle doesn't have this energy. Outside of QM, the particle would be trapped forever. In QM, the particle has a finite probability of going through the wall, if it is thin enough. Some of the particles wavefunction can 'leak' past the barrier, and the particle can actually exist a little bit inside the wall itself. The particle never really has energy Eo, but for the short time it needs to get past the wall, it can pretend it does due to the energy.time uncertainty relation. The probability of tunnelling decreases exponentially with the width of the barrier. Moreover the tunnelling probability decreases exponentially with the square root of the mass of the particle, so only small light things tunnel.
Alpha particle release by the nucleus of an unstable atom (radioactive decay via alpha emission) is an example of a tunnelling process
Proton tunnelling influence acid/base chemistry and the aqueous mobility of the H⁺ ion.
Phase and Coherence
- Spin Echos
  Whenever an experiment can directly access the individual particles within a molecule, it clearly shows the QM nature of our world. Electron Paramagnetic Resonance is used to probe molecules by chemists, but it can directly interact and change the spins of individual electrons. When yo can do this, you can control a lot about the electrons, even time. A spin echo works by time reversal, at least time reversal of the electron spin relaxation. Consider, if you will, a Radio frequency pulse that is beamed towards a sample. The sample absorbs the pulse and gets hot. A 'special' RF pulse, used to 'adjust' the Rabi frequency of the electron, can then be beamed at the hot sample, reversing time. After a delay in time the sample gets cold and spits out a RF burst of energy, bringing it back to where it started. This can be done because of the coherent nature of the electron as a wave. Science fiction? Not.
  NMR imaging takes advantage of the Quantum Mechanical phenomenon of nuclear spin echos, and uses echo decay rates to differentiate between the soft tissues in your living body (QM making your life better)
- Lasers
  Lasers are an interesting and unique interaction of light with matter. In a laser, the energy in a solid, liquid, or gas can be converted into light energy, coherently, meaning the light is monochromatic and all 'in phase'. The coherence is created from random 'thermal' motion, so in a sence this is a disorder->order transition (we like those). Lasers can be used to cool atoms to a few millionths of a degree above absolute zero. In fact, we can use coherent laser pulses to make matter have a negative temperature for a time, locally and with some restrictions. (Don't worry, thermodynamics and the concept of absolute zero still work)
- Coherent control of chemical reactions.
  Light/matter interaction has been proposed as a means of controlling the evolution of a chemical reaction. Specifically, it is proposed that bonds may be made or broken selectively and at will, regardless of the energetics or equilibrium constant for the transformation. As yet, this is but a dream...
Probability and Probability Amplitude (Fermions and Bosons)
We have discussed how the wave function of a particle tells us where the particle is in terms of its probability, but we have left out an important step. The wave function represents the probability amplitude of the particle, but it probability of existing at that spot depends on the wave function squared. That's OK most of the time because a p orbital and a p orbital squared have similar shapes. However, it leads to an interesting observation, changing the sign of the wavefunction leaves the probability (what we observe) unchanged. This separates all indistingushable particles into two classes: those whose wavefunctions change sign when identical particles are interchanges (Fermions) and those whose wavefunctions are unchanged by interchange of identical particles (Bosons). The fact that electroms are Fermions gives us the Pauli Exclusion Principle.
If we can trick electrons in a wire into being Bosons, resistance to flow drops to zero -- the reason we can make superconductors.

The ability to use the Quantum Mechanical nature of electrons in molecules to predict the properties (Thermochemistry, Structure, Reactivity, etc.) of molecules is called Quantum Chemistry. We have made tremendous strides toward this goal, in that most properties of light main group molecules can be calculated to within 10%, just with a computer and a Periodic Table! Other molecules (open shell, heavy, 'floppy', large) are not as well understood, yet! The algorythms and computer programs that perform these calculations are commercially available (Hyperchem, Gaussian) but still under development by research scientists, some here at UF (Bartlett, Ohrn).

The molecular orbitals of CO from quantum mchemical (DFT-B3YLP aug-cc-pVTZ) calculation:

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