# Models, Approximations and Reality

David Young
Cytoclonal Pharmaceutics Inc.

At about the senior year of their college career, chemistry students start to find that the information being taught seems to conflict with what was taught in introductory courses. The course instructors have not tried to intentionally deceive their students. However most individuals can not grasp the full depth and detail of any chemical concept the first time that it is presented to them. It has been found that most people learn effectively by first being given a basic description of the concepts then developing their detailed understanding over time. Depsite the best efforts of educators, a few misconceptions are at times introduced by attempting to avoid a detailed description in introductory courses. The part of this process, which seems to be most confusing to many students is that texts and instructors often do not point out the nature of the simplification being presented.

The scientific method is taught starting in elementary school. The first step in the scientific method is to form a hypothesis. A hypothesis is just an educated guess or logical conclusion from known facts. The hypothesis is then compared against all available data and the details developed. If the hypothesis is found to be consistent with known facts it is called a theory and usually published. Most theories have in common that they explain observed phenomena, predict the results of future experiments and can be presented in mathematical form. When a theory is found to be always correct for many years, it is evenually referred to as a scientific law. This process is very useful, however we often use constructs, which do not fit in the scheme of the scientific method.

One of the most commonly used constructs is a model. A model is a simple way of describing and predicting scientific results, which is known to be an incorrect or incomplete description. Models might be simple mathematical descriptions or completely non-mathematical. Models are very useful because they allow us to predict and understand phenomena without the work of performing the complex mathematical manipulations dictated by a rigorous theory. Experienced researchers continue to use models that were taught in high school and freshmen chemistry, however they realize that there will always be exceptions to the rules of these models.

A very useful model is the Lewis dot structure description of chemical bonding. It is not a complete description of the molecule since it does not contain the kinetic energies of the particles or coulombic interactions between the electrons and nuclei. The theory of quantum mechanics, which accounts correctly for these factors, does predict that only two electrons can have the same spacial distribution (one of alpha spin and one of beta spin). The Lewis model accounts for this pairing and for the number of energy levels available to the electrons under normal temperature and pressure. This results in the Lewis model being able to predict chemical bonding patterns and give a little indication of the strength of the bonds (single bonds, double bonds, etc.). However, none of the quantum mechanics equations are used in applying this technique.

Approximations are another construct that is often seen. Even though a theory may give a rigorous mathematical description of chemical phenomena, the mathematical difficulties might be so great that it is just not feasible to solve a problem exactly. If a quantitative result is desired, the best technique is often to do only part of the work. One approximation is to completely leave out part of the calculation. Another approximation is to use an average rather than an exact mathematical description. Some other common approximation methods are variations, perturbations, simplified functions and fitting parameters to reproduce experimental results.

Quantum mechanics gives a mathematical description of the behavior of electrons which has never been found to be wrong. However, the quantum mechanical equations have never been solved exactly for any chemical system other than the hydrogen atom. Thus the entire field of computational chemistry is built around approximate solutions. Some of these solutions are very crude and others are expected to be more accurate than any experiment that has yet been designed. There are several implications of this situration. First, computational chemists require a knowledge of each approximation being used and how accurate the results can be expected to be. Second, to get very accurate results requires extremely powerful computers. Third, if the equations could be solved exactly much of the work now done on super-computers could be done faster and more accurately on a PC.

This discussion may well leave one wondering where reality went. Some things are known exactly. For example, the quantum mechanical description of the hydrogen atom matches the observed spectrum as accuratly as any experiment ever done. If an approximation is used, one must ask how accurate an answer must be. Computations of energetics of molecules and reactions often attempt to attain what is called "chemical accuracy" meaning an error less than about 1 kcal/mol since this is sufficent to describe van der Waals interactions, the weakest interaction considered to effect chemistry. Most chemists have no use for answers more accurate than this.

A student of chemistry, or any science, must realize that theories, models and approximations are powerful tools for understanding and achieving research goals. The price of having such powerful tools is that not all of them are perfect. This may not be an ideal situation, but it is the best that the scientific community has to offer. Each individual must try to attain an understanding of the nature of our descriptions of the physical world and what results can be trusted to any given degree of accuracy.

## Further information

For more discussion of the application of the scientific method to chemistry see
L. Pauling, "General Chemistry", p. 13-19 Dover (1970)

An expanded version of this article will be published in "Computational Chemistry: A Practical Guide for Applying Techniques to Real World Problems" by David Young, which will be available from John Wiley & Sons in the spring of 2001.