Simulation Models

In scientific computing, we often are looking to model physical processes through computer simulation. Though generally computational scientists (physical scientists that is) are performing this work, often this simulation and modelling is not done in a very systematic or scientific way. This has been an issue for computational science since the inception of the computer.

This challenge was recognized many decades ago and was the driving force in John Pople's computational quantum chemistry research. Though Pople's chosen field was quantum chemistry, we would do well to learn from his approach to modelling for our chosen fields of research and study.

---

Here in general terms and for an outline of development and use of a model (from Pople.)

A theoretical model for any complex process is an approximate but well-defined mathematical procedure of simulation.

......

Five stages may be distinguished in the development and use of such a model:

1) Target
2) Formulation
3) Implementation
4) Verification
5) Prediction

---

Now an explanations of these stages from Pople's quantum chemical models:

1) Target
A target accuracy must be selected. A model is not likely to be of much value unless it is able to provide clear distinction between possible different modes of molecular behavior. As the model becomes quantitative, the target should be that data is reproduced and predicted within experimental accuracy. For energies, such as heat of formation or ionization potentials, a global accuracy of 1 kcal/mole would be appropriate.

2) Formulation
The approximate mathematical procedure must be precisely formulated. This should be general and continuous as far as possible. Thus, particular procedures for particular molecules or particular symmetries should be avoided. If this can be done, the procedure becomes a full theoretical model chemistry, which can be explored in detail as far a available resources permit.

3) Implementation
The formulated method has to be implemented in a form which permits its application in reasonable times and at reasonable cost. In recent times, this stage involves the development of efficient and easily used computer programs. It is closely comparable to the stage of building equipment in an experimental investigation.

4) Verification
The next step is to test the model against known chemical facts to determine whether the target has been achieved. If quantitative accuracy is being sought, this can be done by various statistical criteria such as the root-mean-square difference between the results of the theoretical model and experimental data. In selecting such a dataset, it is important to make it as broad as possible, while limiting it to experimental facts known to be of high quality. If the results of such a comparison do meet the target requirements, the model may be said to
be validated.

5) Prediction
Finally, if the model has been properly validated according to some such criterion, it may be applied to chemical problems to which the answer is unknown or in dispute. If the experimental dataset is sufficiently broad, there is a reasonable expectation that the results will be accurate to something like the target accuracy. This stage, of course, is the one of
most interest to the larger chemical community.

From - Nobel Lecture: Quantum chemical models by John A. Pople.
Reviews of Modern Physics, Vol. 71, No. 5, October 1999, page 1287.

---

As a computational scientist, all the stages should be of interest. As a scientific software developer then stages 3 and 4 are of the most interest though often all stages cross our path. Particularly stage 3 where often we as scientists are building our experimental apparatus (computer software) for the experiment ( computer simulation.) Interestingly enough as scientists we often do not build our software to the same standards we build our experimental equipment. Why? I am still trying to formulate the reasons we often seem to lose our scientific training when it comes to scientific software and computation.

Pople On Dirac's Famous Remark

The Schrodinger equation is easily solved for the hydrogen atom and found to give results identical to the earlier treatment of Bohr. With inclusion of the relativistic corrections via the Dirac equation, almost perfect agreement was found with experimental spectroscopic data. However, exact solutions for any other system was not found possible, leading to a famous remark by Dirac in 1929:

"The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved."

This was a cry both of triumph and of despair. It marked the end of the process of fundamental discovery in chemistry but left a colossal mathematical task of implementation. In retrospect, the implied finality of the claim seems excessively bold.

In 1929, there had only been one preliminary approximate quantum-mechanical calculation on the hydrogen molecule by Heitler and London, leading to a value of the bond energy of only about 70% of the experimental value. Nevertheless, the physicists were highly confident and most moved on to study the internal structure of the nucleus during the 1930s. In fact, their boldness was apparently justified, for no significant failure of the full Schrodinger-Dirac treatment has ever been demonstrated.


Nobel Lecture: Quantum chemical models
John A. Pople
Reviews of Modern Physics, Vol. 71, No. 5, October 1999, page 1287.