Theoretical Methods
Luke A. Burke

Introduction

Our conceptualization of chemistry is built on theory. The beginnings of the nineteenth century saw the development of Atomic Theory and only by the end of that century did we have the acceptance of the existance of atoms by all in the scientific community. The concept of the atom could explain so many physical and chemical facts. Besides, by that time, Thomsom had found a bit of the atom called an electron and Arrhenius' concept of anions and cations was finding wide-spread acceptance and went a long way in explaining concepts such as acid-base reactions and the electrical conductivity of salt solutions.

It wasn't until the beginning of the twentieth century that credible theories of the chemical bond began to appear. G. N. Lewis put forward his octet theory in 1912 and 'Lewis electron dot structures' were quickly used to explain the corresponance between bonding, structure, and the atom's position in the periodic chart. Up to that point, chemical theory was built on empiricism: from experimental facts one constructed a model (Kekule's hexagon for benzene, Fischer projections, Rutherford's nuclear atom, Bohr's orbits, etc.) that fit the facts and then could be used in turn to explain the properties of whole classes of substances.

Yet, there was very little of a rational foundation to chemistry. That is, starting with a collection of particles that we call nuclei and electrons, how could we use only the electrostatic forces of attraction and repulsion for charged particles to calculate any physical or chemical property? We could not, until 1927 and the beginnings of Quantum Theory.

This part of the course is designed to introduce students to the concepts in Quantum^M Chemistry which are needed to have a strong insight into the nature of the structure of molecules and the bonding of the atoms which combine to form these molecules. These concepts are mathematical in nature but the 'math' can be reduced to pictorial concepts which seem to be so useful to chemists involved with synthesis (Organic or Inorganic).
The student should come to this course equipped with general and organic chemistry, physical chemistry, at least one semester of calculus, and a computer account. We'll supply the rest. The first part of the course is an introduction to the concepts of Quantum Chemistry. It serves as an introduction to the concepts vital to a proper understanding of the new technique of molecular modeling used currently by molecular biologists and pharmacists, as well chemists and biophysicists. The gerund of the infinitive, to model, is spelled 'modeling' by many americans from the USA and modelling by those influenced by british english, e.g. canadians, irish, africans, and those east of the Khyber Pass. Since I tend to forget where I am so often, I will use the two orthographs throughout. (Although I did learn in a New York grammar school that one doubles the l to keep the antepenult o short in modelling. Otherwise, molecular modeling is akin to molecular yodeling.)

Tools of the Trade

Models from Quantum Mechanics

There are two competing models from Quantum Mechanics which are used extensively by chemists: the Valence Bond and Molecular Orbital models. These are derived from mathematical approximations that were needed to calculate the physical or chemical properties of molecules by means of the Schroedinger equation. Although the need for such mathemtical approximations is being eliminated by the ever-increasing speed and capacity of present day computers, the concepts derived from these two models are still rich enough to explain many of the mechanisms, rates, and energetics of organic reactions. It is therefore important to review at this point exactly what an 'orbital' is. An orbital is a mathematical function which describes the wave-like behavior of an electron. Most introductory chemistry textbooks define an orbital as a region in which an electron resides. Although one can use any definition one wishes, there are some definitions which are more useful then others. To descibe an orbital as a region is as scientifically sterile as describing an orbit as the path of one body around another. If one takes the definition of an orbit as the mathematical function which descibes the motion of two bodies rotates about each other, one has a much more predictive concept. An orbit can be used in Classical Mechanics to describe to motion of a particle around another. An orbital is used in Quantum Mechanics whereby the wave-like behavior of matter can be used in mathematical equations in order to calculate properties of that matter. Fortunately, the mathematical functions derived for small molecules can very often be 'pictorialized' as a blend of regions and functions (e.g. sigma and pi molecular orbitals) which can then be combined to represent the mathematical functions in larger molecules. This is often called 'Qualitative Molecular Orbital Theory'. If you have forgotten the mathematical concepts which form the basis of the chemical bond, please review a tutorial on the basis of Molecular Orbital Theory. The only mathematical functions that you will need to know for the tutorial are + - x / absolute value, and the e function (exp on your calculator). To do the excercises you will need a graphing calculator or a computer with a graphing program.
Models from Classical Mechanics

During an organic chemistry course, students should become familiar with molecular models, the real, plastic kind, in order to see how structure influences reactivity and chemical properties. One then acquires a three dimensional sense of the molecules. There are at least four shapes that one becomes familiar with: tetrahedral (sp3), trigonal planar (sp2), linear (sp), and bent (usually sp3). There are also the trigonal bipyramidal (dsp3) and octahedral (d2sp3) shapes more commonly found in inorganic compounds (although the SN2 transition state is roughly trigonal bipyramidal).

We have the idea that the atoms are bonded somehow to each other in molecules. From Rutherford's experiment, we have the idea of the nuclei as tiny spheres and the electrons are somewhere outside, flying around the nuclei and holding the whole thing together. We know from infrared spectroscopy that the nuclei are vibrating as if on springs and that only certain frequencies of ir light are absorbed because the bonds will only vibrate at certain frequencies. We also know that the higher the bond order (single, aromatic, double, triple bonds, i.e. 1, 1.5, 2, 3) the more energy it takes to vibrate a bond. We can rationalize this by saying that there are more electrons between the nuclei in a double bond than in a single bond and so the 'spring' is stiffer. We see this from the equation relating the energy imparted, E to the frequncy of the light absorbed, nu, where the the proportionality constant, h is called Planck's constant:

E = h * nu
or
E(oscillator) = h * nu

From X-ray diffraction studies we know that the nuclei have 'usual' bond lengths for typical types of bonds: C-C, 1.54A; C=C,1.32A; CC(aromatic),1.40A; C-H,1.08A; C-O,1.38A; C=O,1.24A; etc. An Angstrom (A) = 10**(-10)m. We also observe typical bond angles: 109.5deg for sp3, 120deg for sp2, 180deg for sp.

Molecular Mechanics is a branch of Computational Chemistry that deals with treating molecules as a collection of classical mechanical interactions as pictured below. It has the advantage of being able to treat molecules with thousands of atoms very quickly, as opposed to Quantum Mechanics which can only treat tens of atoms reliably and quickly enough. In Molecular Mechanics, each bond length and angle is assigned an optimum value and it costs energy to displace atoms from these values. The amount of energy needed depends on the stiffness of the spring and the amount of displacement. Typically for these molecular mechanics programs, one chooses an input 'geometry', or position of the nuclei and then the nuclei are moved in a way as to reduce the steric strain on each atom. At each step, the steric strain is reduced by eliminating the 'competition' between the atoms for remaining in a particular place. In other words, the 'stiffer' double bond atoms will 'push' the atoms in a single bond out of the way. One can go through thousands, even millions of positions seeking the least energy - or 'most stable conformation'. Sometimes the calculations leaves one in a higher enegy 'minimum' such as the gauche conformation rather than in a 'global minimum', such as the anti conformation, which is the most stable of all conformations.

We are know ready to make a model of molecules which resembles spheres held together by springs. From Hook's Law we know that the vibrational frequency is directly related to the strength of the spring, k or Hook's constant, and inversely to the mass of the spheres:

nu(oscillator) = (k/mass)**1/2
or
E(oscillator)/h = (k/mass)**1/2

It is also seen that the force needed to pull or compress the spring, E is equal to the square of the distance away from the rest position, d times k, Hook's constant:

roughly, E = k * d**2


Whereas distortions to optimal bond distances and angles can be measured experimentally, nonbonded interaction energies are more difficult contributions to evaluate. These energy contributions are presently the subject of much debate among developers of "Molecular Mechanics" programs.


Some of the names of these molecular mechanics programs are MM3, Amber, charmm. When one uses the "clean" command in the GausView (gv) program, the MM3 version is used.