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Sunday, February 7, 2010

Molecules and Radiation I: Molecular Structure


This post is part of a primer on infrared spectroscopy and global warming. The previous post  starts the process of looking at the interaction between infrared radiation and matter and discusses black-bodies and relationship between temperature and infrared radiation.  This post goes further and looks at how gas phase molecules interact with infrared radiation.




For a molecule to absorb radiation, several conditions must hold.  First, energy must be conserved: if the molecule absorbs energy from a photon, the molecule must be able to store that energy in some manner. 
Recall from an earlier post on this topic that radiant energy is quantized.  It turns out that energy levels in molecules are also quantized; so there must be an energy match between the photon being absorbed and the molecule absorbing the photon.  Second, there must be some type of interaction between the photon and the molecule.  The gasoline in a car has enough potential energy to move the car down the street, but just having enough energy is not enough.  There must be a mechanism for that energy to transferred into motion.

In the earlier post referenced above, I state that for my purposes, electromagnetic radiation can be treated as electric dipole radiation.  Take a minute to refresh your memory of what that means. The molecule in question must somehow interact with this dipole radiation to absorb a photon.  A subsequent post discusses this issue further.

There are some other laws that must be followed for a molecule to absorb radiation such as conservation of angular momentum and some considerations regarding symmetry.  I am not going to delve into these considerations at this point; so I focus on conservation of energy and the ability to interact with electric dipole radiation.  Understanding this process requires understanding a little bit about the structure of molecules.  This post starts that discussion.

The Hydrogen Chloride Molecule

I am going to use hydrogen chloride (HCl) as an example of a molecule that absorbs infrared radiation.  The reader is no doubt familiar with hydrochloric acid.  When hydrogen chloride is in an aqueous solution, it is known as hydrochloric acid.  Here, I am considering a lone molecule  of HCl in the gas phase. 

HCl is composed of two atoms a hydrogen atom and a chlorine atom.  The most common isotope of hydrogen hydrogen has a proton and an electron. The most common isotope of chlorine has 17 protons, 18 neutrons, and 17 electrons. That's a total of 54 particles that one must consider.  Each electron has a negative charge, each proton has a positive charge and each neutron has no charge.

In three-dimensional space each particle can move in the x, y, or z direction (I am neglecting spin of the individual particles).  Each direction that a particle can move is considered a degree of freedom.  So, there are:

     54 x 3 = 162

degrees of freedom in a hydrogen chloride molecule.  Luckily, we can get a better handle on all of these degrees of freedom by some simplifying approximations.  First, all of the proton and neutrons are packed together into atomic nuclei. The energies required to affect nuclei are extremely high and correspond to photons of much higher frequency than infrared (or visible or UV) radiation. For the purpose discussed here, the nucleus of an atom can be treated as a single particle.

Now there is a positively charged hydrogen nucleus, a positively charged chlorine nucleus and 18 negatively charged electrons for a total of 20 particles and 60 degrees of freedom. Most of these degrees of freedom involved motion of the electrons and some involve motion of the electrons.  Just as atoms interest with radiation via excitation of electrons, molecules can do the same thing.  In general, an infrared photon is not energetic enough to excite electrons in a gas phase molecule (solid state matter, such as metals and semiconductors, behaves a little differently and is beyond the scope of this discussion). It would be nice to separate the motion of the nuclei from the motion of the electrons so that I do not have to consider absorption and emission that arise from electron excitation, but am I justified in making such a separation?

The Born-Oppenheimer Approximation

 Approximation is inherent to the scientific enterprise.  Earlier, I made the approximation of treating electromagnetic radiation as electric dipole radiation.  There is a misconception that scientific inquiry must always deal in exact quantities and absolute proof.  The world is a very complicated place, and it is necessary to make some simplifying instructions to understand it.

In the realm of pure mathematics and geometry, there are theorems that can be proved exactly without approximation or simplifying assumptions (if one grants certain axioms).  In physics and chemistry, such exact results are rare.  The more common situation is that a simplifying approximation is made, and when it is found to be inaccurate, higher order corrections to the assumption are made.  Often, it is possible to determine a result with arbitrary accuracy.  How good is good enough?  If the answer to that question is known, then one knows how accurately one needs to modify approximations to get an answer that is good enough.

There is no exact solution to a three-body problem.  Consider the sun-earth-moon system.  It is not possible to exactly predict the orbit of the moon.  It is necessary to make approximations, and then it is possible to calculate the orbit of the moon to an accuracy that is good enough for whatever purpose one has in mind. The following graphic shows the chaotic motion of three interacting bodies.


Electrons move much faster than nuclei do.   The Born-Oppenheimer approximation entails picking coordinates for the nuclei and calculating the the energy levels of the electrons for that separation of nuclei.  One can then mover the nuclei and repeat the exercise.  The result is that one maps out an energy surface for the nuclei based on the electronic energy levels at each position.

Formally, one takes the wavefunction of the molecule and treats it as the product of an electronic wavefunction, and a wavefunction for motion of the nuclei:

    Ψmolecule(r,R) = Ψe(r;R)ΨN(R)

The wavefunction of the entire molecule is a function of the electron coordinates (r), and the nuclear coordinates (R).  In the Born-Oppenheimer approximation, it is treated as the product of an electronic wavefunction that is a function of r, and uses R as a parameter, and a nuclear wavefunction that is only a function of R.  Corrections to this approximation can be built by adding successively complex interaction terms back into the equation.

Do not be confused by the use of the word "nuclear" here.  I am referring to the coordinates of the nuclei, not the energetics of what happens within a nucleus.  This discussion has nothing to do with nuclear energy, fission, fusion, or associated concepts.  I am using the word "nuclear" to refer to inter-nuclear motion rather than intra-nuclear motion.

What does all this stuff about the Born-Oppenheimer approximation mean?  It means that the degrees of freedom that I have to worry about have been greatly reduced.  Although the Born-Oppenheimer approximation does break down in places, for the most part, i need only worry about the nuclear degrees of freedom.  The electronic wavefunction can be excited by energies that are greater than IR photons, but here the focus is on IR spectroscopy; so the focus is on the nuclear wavefunction.

The HCl molecule has two nuclei, a hydrogen nucleus and a chlorine nucleus. There are 3N degrees of freedom, where N is number of nuclei.  For HCl, I have reduced 162 degrees of freedom to 6 that are relevant for our purposes.  It is possible to reduce this number further.

The next post discusses these degrees of freedom and how they relate to the energetics of molecules relevant for the absorption and emission of IR.

Sources

  • Struve, Walter S., Fundamentals of Molecular Spectroscopy, John Wiley & Sons, New York, 1989
  • Steinfeld, Jeffrey I, Molecules and Radiation, The MIT Press, Cambridge, MA, 2nd edition, 1985
  • Atkins, P. W. Physical Chemistry, New York: W. H. Freeman and Company, New York, 3rd edition, 1986
  • Wikipedia: "N-body_problem" 









2 comments:

Leslie Lim said...

Wow. Awesome article. Please do more articles like this in the future. Very informational and knowledgeable. I will expect more from you in the future. For now i will just bookmark your page and surely I'm gonna come back later to read more. Thank you to the writer!


Rica
www.imarksweb.org

Leslie Lim said...

Wow. Awesome article. Please do more articles like this in the future. Very informational and knowledgeable. I will expect more from you in the future. For now i will just bookmark your page and surely I'm gonna come back later to read more. Thank you to the writer!


Rica
www.imarksweb.org