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Nature doesn't solve equations, so why should we? Mathematically-lean simulations in chemistry


Hugh M. Cartwright
Physical and Theoretical Chemistry Laboratory, Oxford University
South Parks Road, Oxford, England OX1 3QZ

06/30/97 to 07/01/97

In any university, computers are crucial tools for the teaching of chemistry. There is widespread agreement that computers can perform a vital role in developing student understanding, but less universal agreement on what that role should be. At one extreme, computers may be used as fast calculators, or to run spreadsheet macros or Basic programs. At the other extreme, they may drive complex, interactive simulations, perhaps taking advantage of multimedia or the Internet.
The versatility of the computer permits a wide range of uses. For example, their speed makes them ideal tools in the acquisition and analysis of experimental data; their memory makes make them powerful bibliographic search engines. However, one can argue that in neither role does the computer actually do much to help students understand the underlying chemistry. Instead, it amplifies and illustrates whatever understanding the student already possesses. Spreadsheets may help in the assessment of chemical data - and this is undoubtedly a valuable role - but, when using them, students' data massaging skills may develop at a faster rate than their chemical understanding.
As the power of computers increases, the range and complexity of chemical systems that can be simulated increases in proportion. It is now possible to routinely simulate in real time the interactions among several hundred (small!) molecules in the gaseous or condensed phases. Such simulations intrigue and stimulate students, and, if carefully designed, are effective teaching tools also. The recent dramatic growth of Internet applications employing Java provides new opportunities for chemists to provide such simulations to their students. Those chemists who might previously have been reluctant to face the trials of working with X and Motif, can now develop professional-looking, effective simulations and make them available, through the net, to a wide
As a consequence both of the increase in computer power and of the development of Java, chemical simulations and graphical applications of all types are appearing at a considerable rate. However, allowing
students to simulate a chemical process does not guarantee enhanced student learning.

This paper argues that simulations in which a mathematical description of natural phenomena can be derived from the behaviour of a physical system have notable advantages over those in which the
mathematical description is the basis for the simulation itself. We shall discuss simulations of both the "mathematics-rich" and "mathematics-lean" variety, and consider the advantages each may offer.
Suitable on-line links are provided.


1. Introduction


  • During the last decade the style of university chemistry teaching has changed radically. Computers were once rarities, but are now commonplace, and their use enhances many courses. As universities compete to attract students from a limited pool, the skill with which chemistry departments can weave new technology into their courses is becoming an important consideration for many candidates.

    Computers are somewhat cheaper than most scientific instruments and considerably cheaper than a professor. It is not surprising then that they are in widespread use (Table 1).



    Table 1.

    Typical computer use in university chemistry departments.


    * Provision of access to, and maintenance of, databases of scientific data or student records


    * Automation of exam setting and marking


    * Provision of word processing and graphic design facilities for students and staff


    * Tuition and guidance of students in problem-solving


    * Numerical computation in areas such as molecular dynamics


    * Control of instruments and data acquisition


    * Analysis of data using spreadsheets or mathematical packages


    * Control and display of simulations



      The contribution which computers make varies from one department to another, depending on the machines available, local expertise, and the teaching philosophy of the department. Consequently, their impact on student learning is far from uniform.

      Where the computer is used primarily as an administrative tool, for example to store student records, there is little direct benefit to the student. Similarly, when students have ready access, but the computer is used just to word process laboratory reports or assignments, it does not contribute a great deal to scientific understanding. Students' keyboard and computer skills improve, but the computer is not helping them to understand "why things are the way they are".

      Even when employed scientifically, if computers are used to help students to balance equations or follow VSEPR rules, the software may just be a disguised "programmed learning" text. The application of rules to determine molecular shapes or balance equations is, in a sense, a mechanical operation which can be carried out without knowledge of the underlying science.

      The primary aim of a chemistry course must be to develop in students an appreciation of how the material world behaves, and this requires that the computer play an active, positive role in learning. The key to this is the simulation. Simulations have long been a part of chemistry, and although longevity is not in itself a persuasive argument for their use, they find a place in most courses. We argue in this paper that one of the most valuable uses of the computer in a chemistry course is to run simulations which develop understanding of the molecular world.

      Detailed simulations of scientific phenomena can provide an insight which transcends a mathematical description, and an understanding of how materials behave is at least as important as being able to express that behaviour in algebraic terms. By interacting with a computer model students can develop an understanding that - at its best - becomes almost intuitive.



    2. The place of simulation in the undergraduate course


    • At the most fundamental level, a simulation visualises the - possibly idealised - behaviour of individual atoms and molecules.

      The level of detail may be limited by the facilities available, (it is a common observation that, no matter how much computer power is available to a chemist, it is never enough). Nevertheless, sophisticated simulation is not always necessary; a detailed view might conceal important generalisations, and thus be counterproductive. Consequently, both molecular level simulations and their coarser counterparts, in which behaviour in the bulk is simulated, have a role to play in a chemistry course. It is important to realise, though, that they are not equivalent. Visualisation of equations, which is essentially what most bulk-level simulations involve, is one step removed from the "real" world of atoms and molecules; we shall argue that in some ways this lessens their value and their impact in teaching.

      Computer visualisation can be introduced almost anywhere in a chemistry course, but grapeshot firing of simulation into it is rarely desirable. A simulation must justify its place no less rigorously than a lecture or practical experiment. (Equally, lectures and experiments should be able to justify their existence against the competition provided by simulations). To do this, the simulation must teach concepts more efficiently, or give students a special insight into a topic, or perhaps link together disparate areas of the subject. It must not be a soft financial option whose raison d'etre is to allow administration to replace professors, or keep students quiet when there are not enough spectrometers to go around.

      Variety is beneficial in learning, so a mixed diet of simulation, lecture, experiment and discussion will do much to stimulate students' appetite for chemistry.




    3. Categories of Simulation


    • Simulation is a broad field, and much variety exists, limited primarily by the inventiveness of those who write the software. Most simulations can be placed into one of the following categories:


      3.1 The Black Box Simulation


      • In a Black Box Simulation the computer mimics some of the behaviour of a real or imaginary instrument (Fig 1). A complete virtual instrument may be available, or even a virtual laboratory [10] ; in other cases users may select discrete components from a Grey Box and bolt them together to make their own Black Box. In this way they might, for example, construct an electrical circuit from standard items such as resistors and capacitors.



    Fig. 1. The spectrum calculated by a "Black Box" NMR instrument. This display is created by a Java applet [1]. Full size version of this figure.


    • Various factors may trigger the development of such simulations. Departments often wish to give students experience with techniques such as FTNMR, or X-ray crystallography, but are prevented from doing so by financial, technical or safety considerations. In a simulated experiment, problems which disrupt a real experiment, such as dangerous radiation fluxes or unstable samples, can be avoided. Conditions which would be difficult to produce in the laboratory can be investigated. Experiments may be interrupted in the middle to change operating conditions, or to allow the user time to reflect on what has been seen so far. The computer is thus a proxy for the real instrument, having advantages in speed, flexibility and cost.

      Black Box simulations can also be used to provide remote Internet access to real equipment. The Black Box element then consists of software which allows the student to control equipment in a user-friendly fashion (Fig. 2).


    Fig. 2. Data from an on-line optical rig, gathered remotely and displayed using Java applets [2]. Full size version of this figure.



      3.2 Simulation of equations.

      Most of the earliest simulations in chemistry were built around the interpretation of equations describing bulk behaviour. For example, the P-V-T behaviour of an ideal gas can be understood by investigating how the volume of a gas trapped by a moveable piston changes as the pressure or temperature of the gas is adjusted. Such a simulation is independent of the behaviour of individual molecules. Indeed, even the presence of molecules is not acknowledged, since it is the behaviour of an equation which is simulated (in this case, PV=nRT ), rather than the behaviour of an aggregation of molecules.

      While this paper argues that simulation of equations is not always the most productive way to proceed, sometimes such an approach is unavoidable. Fig 3 illustrates a simulation of the Belousov reaction, in which calculation at the molecular level is very involved, so that bulk calculation is the only realistic way to proceed.


      Fig 3. The development of concentric rings of concentration minima in the Belousov reaction [3]. Full size version of this figure.


        3.3 Visualisation of idealised behaviour

        Simulations need not be correct to be useful. From the Bohr planetary model onwards, students meet chemical models which are idealisations or approximations. On occasion it is helpful to isolate for inspection one aspect of molecular behaviour which, in a real system, cannot actually be isolated.

        Molecular vibration provides an example. Even molecules of modest size have many modes of vibration (according to the 3n-5 rule for linear molecules, or 3n-6 for non-linear molecules, where n is the number of atoms in the molecule). The vibrational motion which results is complex, even in quite simple molecules (Fig. 4).


        Fig. 4. A vibrating molecule displayed by a Java applet. [4] Full size version of this figure.


        • In any molecule all vibrations are active (though not usually excited) at room temperature, because of zero-point energy. A scientifically accurate depiction of a vibrating molecule shows all atoms moving in what appears to be chaotic motion. The amplitudes of vibration are small (of the order a few percent of the equilibrium bond lengths). The combination of these small amplitudes with the many simultaneous vibrations makes it impossible to divorce the vibration of a single segment of the molecule from vibrations of the whole.

          When students meet normal modes and characteristic frequencies, it is helpful to discuss the vibrations of functional groups as though a group were capable of vibrating in isolation from the rest of the molecule. This can be illustrated using an idealised model in which only one mode is active. This qualitative picture - though unrealistic - both clarifies the concept of characteristic functional group frequencies, and can be used to show how one specific vibration affects the symmetry of the molecule, thus leading into selection rules. What is essentially a scientifically unjustifiable simulation can in this way be of value.



        3.4 Simulation at the molecular level


        • Simulations at the molecular level - that is, those in which each molecule in the simulation is treated individually - encapsulate all simulations which describe the behaviour of material in bulk, since bulk behaviour is the aggregation of the behaviour of the constituent particles.

          Such simulations are computationally demanding, and represent the practical limit of simulations at present, since a further dissection into the movement of electrons and nuclei leads to intractable calculations. As an example of such a simulation, Fig 5 shows the establishment of a concentration gradient for argon atoms above a solid surface under the influence of (very strong!) gravity.


          Fig. 5. Simulation of the movement of gas phase argon atoms above a solid surface. [5] Full size version of this figure.


          In this simulation every atom (both in gas phase and solid phase) is treated individually, and no "bulk" equations are used.

          There are various advantages to such a simulation, which we shall shortly consider. We can note immediately that minimal assumptions are made in the calculation: we assume only that


          * collisions are elastic and that
          * energy is a function of vertical position.

          Complex behaviour (such as the Maxwell-Boltzmann distribution of molecular speeds, or the Boltzmann variation of density with height) emerges readily from the simulation, and it is the way in which detailed physical behaviour can be shown to be the result of simple physical interactions that represents one of the most potent features of molecular-level simulation



        4. Advantages of simulations.



        Fig. 6. The radiation field around a group of moving point charges [6]. Full size version of this figure.



          Fig. 7. The potential energy well for reaction between a hydrogen molecule and a fluorine atom [11]. Full size version of this figure.

            Interactivity is also a powerful argument in favour of simulation. Users quickly become bored if they are just passive consumers of information. They learn better when they need to respond frequently to, and have control over computer software.

            It should be appreciated, though, that interactivity carries with it some dangers. A conventional program which channels all users along a pre-defined track depends on that track being chosen with care, and on it being suitable for almost every user. Interactive software by contrast presents the user with many facilities and numerous choices. This flexibility may remove from the user any sense of direction, so that they become lost within the software package. The academic objectives of the work may become unclear, and the user may wander inefficiently through attractive but meaningless exercises. Interactive software must therefore be sufficiently structured that users do not lose sight of the educational target.

            The use of html wrappers and embedded Java applets can provide this essential structure, though there are of course other equally effective ways of ensuring that users are not submerged in a complex, multi-faceted simulation.



            4.2 Flexibility in topic ordering


            In any set of computer-mediated exercises the instructor must decide upon the most suitable order in which topics are to be presented. In mathematically-based or rule-based exercises a progression from mathematical simplicity to sophistication may be used.

            Exercises might progress from point masses to masses of finite volume and eventually to molecules. This may be a justifiable course, but there is no reason why an order determined by an increase in mathematical complexity should coincide with the order which is academically most desirable. It might instead be desirable to present a comprehensive simulation initially, and gradually uncover its mathematical basis as the underlying science is studied.



              4.3 Development of scientific intuition



              5. Advantages of mathematically-lean simulations


              Much science depends upon the interpretation and use of equations. Simulations of equations have a role to play, but the understanding which students gain from them is different from that which they derive from molecular simulations.

              Of course, all simulations make use of equations and rules. Without a prescription to determine how objects appear on the computer screen and interact with each other, there could be no simulation. In a mathematically-lean simulation, these equations and rules are kept to a minimum.

              For example, returning to PV=nRT , a molecular simulation of gases requires us only to assume that gas molecules move, that they do not interact except during collisions, and that collisions are elastic (no energy loss). If we wish to introduce non-ideality, we can allow the molecules to have non-zero size, or interact with a Morse or a 6-12 potential. Notice that our assumptions are physically based. That is, we are assuming properties of the components which make up the physical system, not assuming equations which describe observed behaviour,such as the van der Waals equation. This is the key idea behind mathematically-lean simulations.

              We have noted in section 3.4 that, from simple assumptions, complex behaviour can be derived. We now consider some further advantages of molecular-level simulation.


              5.1 Visualisation of behaviour "hidden" in equations



              Fig. 8. A frictionless hinged beam falling under gravity. [6] Full size version of this figure.



              5.2 Freedom in variable adjustment

              Fig. 10. The development of fractal growth at electrodes arising from the movement of copper ions in solution [7]. Full size version of this figure.


                5.3 Avoidance of mathematical complexity


                Fig 11. A display of a cold gas about to condense to a liquid; pairs of molecules have already begun to form [8]. Full size version of this figure.

                  The Clausius-Clapeyron equation expresses the dependence of vapour pressure of a volatile solid or liquid on temperature. It contains differentials (or logs in the integrated form). It is an important equation in thermodynamics, but though it is not particularly complex, it still presents students with difficulties, and they may not appreciate that vapour pressure rises approximately exponentially with temperature. A simulation of a collection of molecules, assuming a simple interaction potential, will allow students quickly to investigate the dependence of vapour pressure on temperature.

                  The experimental behaviour is not "built-into" the simulation - indeed thermodynamic properties are defined only for bulk systems, and the simulation is of individual molecules. Nevertheless, with a sufficiently large number of molecules, the user can measure vapour pressure and investigate P/T relationships.

                  In a two phase (gas/solid) system, one can illustrate the principles of Langmuir or BET behaviour using again simple interaction potentials. (Fig 12).


                  Fig. 12 Simulation of argon above a charcoal surface. The movement of the argon atoms is illustrated by the tracks. [5]. Full size version of this figure.


                  This leads us to a further point of crucial importance:- molecular simulations also simulate what we do not see.

                  This sounds like nonsense. What can it mean? Although the primary role of the simulation shown in Fig 12 might be to allow students to study adsorption on solids, further physical phenomena may be investigated using the same model. For example, the molecular tracks in Fig 12 hint that the molecules adsorbed on the surface are not actually stationary, but move around a little. This behaviour is not "what we are looking for" if we are using the simulation to study Langmuir adsorption; indeed, this movement across the surface could not be observed if the simulation was based solely on the Langmuir equation. In this sense then, the molecular level simulation may include within it behaviour we are not looking for (and may not even notice). The sudden discovery by a student of quite unexpected phenomena, such as movement of adsorbed molecules, is both a stimulant to learning, and often a source of delight to the student.


                  5.4 Insight into the physical origin of equations


                  Let us return to the ideal gas law. A simulation whose function was to allow students to understand the behaviour of ideal gases could be visualised either at the molecular level, or through direct interpretation of the ideal gas equation.

                  Most students need to memorise and be able to regurgitate equations in an exam. But they need also to appreciate the physical origin of gas pressure, that is, the bombardment of the walls of a container by molecules. There is more fundamental understanding to be gained by realising that, if the volume is restricted the number of collisions of molecules with the walls per second must increase, than by simply learning that P depends inversely on V.



                  Fig. 13 A simple simulation of gas molecules confined to a small container. [9]. Full size version of this figure.



                    5.5 Revelation of complex behaviour from simple equations

                    Fig. 14. Fractal growth resulting from a small centro-symmetric field applied to an electrode in a dilute solution of copper ions [7]. Full size version of this figure.


                    Fig. 15. Fractal growth by deposition of ions released from a point source. Full size version of this figure.

                    • Fig 15 shows the fractal arising when the solution contains only a small concentration of copper ions, and fresh copper to replace that deposited at the electrode is available only at a single point on the northwest side of the ring. The figure shows the path of a typical copper ion as it moves towards the growing fractal. It is easy to appreciate that, the stronger the electric field produced by the electrode, the more linear and less fractal in appearance the copper deposit becomes.

                      Fig 16 shows a fractal that results when we assume that the central electrode provides a potential concentrated along the Cartesian axes. The resulting 4-fold symmetry is striking.

                    Fig. 16. A fractal growth generated by an electrode which provides a field aligned along the x-y axes. Full size version of this figure.



                    6 Comment


                    • It would be unreasonable to expect simulation to sweep away the need for students to be able to understand and use equations. However, an appreciation of the physical world, promoted by interactive simulations, can lay the basis for a deeper knowledge of the raft of equations which chemists must know.

                      A chemist who understands only equations is really a mathematician. A chemist who has a feel for the way in which molecules behave, and who thinks that he or she knows what moleculeswant to do is a scientist.



                    Appendix A: The CoLoS group


                    • Several of the screen shots in this paper are from work by members of CoLoS. CoLoS brings together scientists from many disciplines in a variety of different countries, who share an interest in the effective use of the computer in understanding science.

                      Readers who share this interest may like to know that CoLoS is offering prizes for innovative scientific software available through the Internet. First prize is one thousand dollars, travel from anywhere in the world to a CoLoS conference, and day-to-day expenses for attendance at the conference.

                      The closing date for applications is expected to be February 1st 1998.


                    • Further details of the prizes.


                    • Further details of CoLoS.



                    References and background information.

                    1. Taken from an experiment in NMR spectroscopy under development in the Physical and Theoretical Chemistry Laboratory at Oxford University.

                    2. Taken from an on-line experiment in error analysis under development in the Physical and Theoretical Chemistry Laboratory at Oxford University. Data are generated by connecting to an optical rig through the Internet.


                    3. Data from Computer simulation of the Belousov-Zhabotinsky reaction, Chi Ho Lam, Chemistry Part II thesis, Oxford, 1996

                    4. A java applet showing a vibrating molecule:

                    5. Screen shot from a simulation written using X, C and Motif. Pete Bennett, Mathematical modelling and computer simulation of aspects of surface science, Chemistry Part II thesis, Oxford University, date.

                    6. An experiment under the control of the CoLoS program xyzet, developed at Kiel University, Germany.

                    7. Screen shot from an experimental CoLoS program on fractal growth, developed at Oxford University.

                    8. A screen shot from a CoLoS demonstration program on the thermodynamics of simple liquids. Andy Armstrong, Computational modelling and simulation of molecular phase dynamics. Chemistry Part II thesis, Oxford University, June 1995.

                    9. A screen shot for a simple simulation of gas-phase molecules.

                    10. The virtual lab in Oxford is at

                    11. An interactive computer simulation of collisional potential surfaces, Russell Strevens, Chemistry Part II thesis, Oxford University, England, 1995