Figure 1. The shapes of the four lowest-energy unnormalised wavefunctions for the particle-in-a-box model  (Equation 1) (left) and the harmonic oscillator (Equation 2) (right). The shaded zones are the classically forbidden regions.
The shapes of some of these wavefunctions are shown in Figure 1.
The use of analytical functions for these wavefunctions (Equation 1 and Equation 2) does not yield any insight. Furthermore, they obscure the similarities: the lowest-energy wavefunctions each have no nodes and one lobe, and are zero at the boundaries of the domain; as energy increases, the wavefunctions decrease in wavelength and increase in the number of lobes and nodes: see Figure 1. Use of a spreadsheet simulation quantum_well.xls (Figure 2) allows students to continuously vary the energy of a trial wavefunction to see how the energy affects the wavelength and overall shape of the wavefunction [12 - 14]. By easily switching between different potential-energy curves, the students can see the similarities in the shapes of the wavefunctions, even though their analytical forms (Equation 1 and Equation 2) may be different, or even non-existent in the case of the triangular (Figure 3) and other potential types.
Figure 2. A screen capture of the spreadsheet document quantum_well.xls for numerical simulation of solutions to the Schrodinger equation. Users are able to choose among a number of different potential energy functions (cell C11) and adjust the energy (cell C5) of the trial wavefunction. The shaded zones are the classically forbidden regions.
Figure 3. The spreadsheet document quantum_well.xls is able to find solutions to the Schrodinger equation for potential energy functions, which do not have an analytical form for the wavefunctions. The shaded zones are the classically forbidden regions.
Figure 4. A screen capture of the spreadsheet document anharmonicity.xls, which shows the relationship between the harmonic oscillator and the anharmonic Morse oscillator. Users are able to adjust the harmonicity (cell C9) to continuous and smoothly change the Morse oscillator (Equation 3) into its harmonic approximation (Equation 4).
Figure 5. Part of the infrared spectrum for carbon dioxide. The main peaks are due to (12C)(16O)2. The smaller peaks are due to other isotopomers.
Use of a spreadsheet simulation (Figure 6) allows students to see how the spectrum depends on initial-state and final-state bond lengths, temperature and other parameters . For example, Figure 7 shows that as temperature is varied, the relative intensities of the individual rotation-vibrational spectral lines change, but the transition energies (Equation 5) do not.
Figure 6. A screen capture of the spreadsheet document Vibrot.xls, which simulates the rotational structure in a vibrational or vibronic spectrum. Users are able to continuously and smoothly adjust the initial-state and final-state bond lengths (cells C9 and G9) and other parameters.
Figure 7. Simulation of part of an infrared spectrum showing how the peaks change intensity, but not position, as temperature is varied.
Figure 8. Variation of boiling points with mass for two series of chemical substances. The simple trend of boiling point increases with mass has been obscured by other factors in the case of H2O.
The use of models has several advantages: models are simpler than the systems they mimic, thus removing the unwanted complicating factors. However, to achieve physically meaningful results, the parameters for most models are tied to reality. For example, while we may use spheres to model atoms, these atoms have masses that correspond to real atoms: 4 g mol-1 for helium, 20 g mol-1 for neon, et cetera. A more convincing argument would be to show how a property (eg, boiling point) changes as some variable (eg, mass) is continuously varied. This is possible using a simulation, which can use values that do not correspond to any physical system: a simulation is more complicated than the system being simulated . For example, in a simulation, we could explore the effect of varying mass, by simulating an "atom", with the size and other characteristics of argon, but with the mass of helium, one order of magnitude smaller than the true mass of argon [15, 16].
It has often been claimed, "chemistry is an experimental science". However, in the mathematical approach to chemistry, either there are solutions (eg, Equation 1, Equation 2, and Equation 5) to mathematical-chemical equations (eg, the Schrodinger equation) or there are no solutions: where is there the possibility of "experimentation"? Simulations enable the student (or scientist) to play "what if" scenarios: what would happen to "property X" if the masses of the atoms were increased smoothly and continuously by an order of magnitude [15, 16]? Exploration is an important part of discovery and learning . Henon and Heiles, in their landmark 1964 paper on chaos theory, refer to simulations as numerical experimentation . Simulations ensure that the sub-disciplines of physical, theoretical, and computational chemistry are truly experimental science.
The success of simulations is also well supported by models of learning. Gardner refers to "multiple intelligences", one of which is the logical-mathematical intelligence . Other authors have referred to learning styles [20, 21]. Different students will have different preferred approaches to learning. Some may favour the symbolism of mathematical equations; others will not. Simulations enable those students who are weak in the logical-mathematical intelligence, or who do not favour abstract thinking to gain an appreciation of the significance and meaning of mathematical equations similar to those shown above. Table 1 lists some free-text responses from students on the use of the spreadsheet quantum_well.xls discussed in Section 2.1. Simulations quickly demonstrate qualitative trends, by generating a large number of solutions, but without the tedium of working the mechanics of the mathematics [22, 23]. Learning is enhanced when students can engage with material on more than one level: simulations help to engage students on both visual and tactile-kinesthetic levels, through visualisation  and "doing".
Table 1. Some free-text responses from Deakin University students on the use of the spreadsheet quantum_well.xls discussed in Section 2.1. Students' grammatical and spelling mistakes have not been corrected.
|"... you could see how the graph changed when you changed the energy"|
|"... easy to understand computer program"|
|"By reading about the Schrodinger Eqn I gained a better understanding of it, and by seeing the solns [solutions] on the spreadsheet also helped."|
|"It helped me learn how to use Excel better."|
|"It was interesting in terms of the varying potential models and their corresponding wavefunctions"|
|"... it was useful to visually see how the Schrodinger [sic] equation could be solved"|
|"Helped me to understand how energy levels are quantised ..."|
There is no doubt that specialist mathematical software can prepare more sophisticated simulations for teaching and learning than spreadsheets [5 - 9]. Ehrmann has introduced the term "worldware" to describe software that was not designed for instruction . Spreadsheets, like other worldware, are commonly available, inexpensive, and relatively easy to learn, because of the large user base. Most students have learnt how to use worldware before entering university: see Table 2. Since spreadsheets are commonly used in the commercial workplace, the use of spreadsheets provides another link between the world of chemistry education and the common world. Students like using spreadsheets: see Table 3. Furthermore, the use of worldware facilitates learning. Galbraith and Pemberton have shown that the use of specialist mathematical software (eg. MAPLE) can create additional barriers to learning . (Of course, there will always be a need to use more specialised software in some upper-level classes.) Software that isn't designed for instruction can be good for learning . Despite their shortcomings, the advantages of worldware  have resulted in increasing usage of spreadsheets in university chemical education [26, 27].
Table 2. Percentage of students reporting that they have the skill to use a type of software.(a)
The destinations of most chemistry graduates are in non-research environments, most of which use spreadsheets [see, for example, Table I in Ref. ]. The examples in this paper illustrate that, in many cases, university educators can both use simulations and align their own usage of software with that of their graduates' workplaces, without compromising any scientific or educational attributes of their visual aids. Thus university educators can be in a better position to prepare their students for future employment. This paper advocates that, where practical, university teachers should use spreadsheets to prepare simulation teaching aids. Spreadsheet simulations and other resources can be found in Spreadsheets in Education <http://www.sie.bond.edu.au/>, the Journal of Chemical Education, the Journal of Chemical Education's JCE Webware collection <http://jchemed.chem.wisc.edu/JCEDLib/WebWare/>, and the (United States) National Science Digital Library (NSDL) <http://nsdl.org>.
Table 3. Some free-text responses from Deakin University students to the question "What did you like about this unit [subject]?"
|"Use of computers especially Excel ..."|
|"I found the calculations a little difficult, but the use of Excel helped a lot"|
|"Use of Excel to perform multiple simple calculations"|
|"... sessions in computer room"|
The spreadsheets, quantum_well.xls, anharmonicity.xls, and Vibrot.xls can be obtained from the Journal's website, <http://www.sccj.net/publications/JCCJ/v5n3/a02/appendix.html>, or from the author, Associate Professor Kieran F. LIM, e-mail: firstname.lastname@example.org.
This paper is the full version of a lecture  presented at the 2005 Pacifichem Conference. K.F.L. thanks: Ms Jeanne LEE () for collaborations on some aspects of this work, and encouraging and helpful discussions on the other aspects; Dr Amanda Kendle (University of Western Australia) and Professor William F. Coleman (Wellesley College, USA) for collaborations on some aspects of this work; Professor Masato M. ITO and the other members of the organising committee for an invitation to present this work to the 2005 Pacifichem Conference. The work presented in this paper has received ethics clearances (EC 264-2001, EC 29-2002, EC 239-2002) from the Deakin University Ethics Committee.