(1)

In eq 1, V_M is the potential energy of a diatomic molecule with interatomic distance r and equilibrium bond length r_e, D_e is the well depth, and β is a parameter that governs the width of the Morse function well.¹ However, the second-order Taylor series (harmonic) approximation is discussed in most textbooks (1–7)

(2)

k is the force constant for the harmonic potential energy, V_H. This quadratic function permits exact solutions to the Schrödinger equation, shown by energy levels drawn inside the parabolic curve in Figure 1. Most textbooks show the Morse function and its quadratic approximation in diagrams, similar to Figure 1. Some students have difficulty understanding how the two curves differ only in the anharmonicity because the functional forms of the two equations are very dissimilar.

Figure 1. A graph from spreadsheet Anharmonicity.xls showing comparison of harmonic approximation to potential energy surface with the more realistic Morse curve description. (Here the anharmonicity factor = 100%.) The energy levels predicted by the solution of the Schrödinger Equation are also shown.

While the harmonic approximation is sufficient for understanding frequency-bond-order relationships, quantization, and zero-point energy, an anharmonic model such as the Morse potential is required for overtone spectroscopy (8–10) and the variation of bond length on vibrational excitation (11).

Figure 2. The harmonic potential is an excellent approximation to the Morse curve when the anharmonicity is very small. (Here the anharmonicity factor = 5%.) As anharmonicity is decreased, the dissociation energy increases, resulting in more bound energy levels (not all shown) for the Morse oscillator.

While North American chemistry majors are required to have a strong background in mathematics, this requirement disadvantages students who favor “intelligences” other than logical–mathematical intelligence in their learning (12, 13). Chemistry majors in other countries are not guaranteed to have the same mathematical background as North American students. In addition, it is desirable to make quantum–mechanical concepts associated with vibrational spectroscopy accessible to non-chemistry majors, who are more likely to be weaker in mathematical ability (14).

An advantage of simulations is that they can be used to demonstrate qualitative trends without requiring the user to work through the mechanics of obtaining a solution (15–18). Even if students have the mathematical ability to solve the equations, simulations enable such students to generate a large number of (usually) numerical solutions in a short time (13, 15). Thus simulations allow the user to perform “numerical experiments” (19) along the lines of “what if this parameter is varied?”

This WebWare paper describes how a Microsoft Excel spreadsheet simulation, Anharmonicity.xls, can be used to smoothly and continuously switch a plotted function between a Morse function (V_M(r) in eq 1) and its quadratic approximation (V_H(r) in eq 2). It can be used as a classroom demonstration or incorporated into a student-centered computer-laboratory exercise to examine the qualitative behavior of vibrational quantization. A description of the spreadsheet, instructions for its use, and a discussion of the concepts are included.

Acknowledgments

K. F. L. thanks Jeanne Lee () (Loyola College, Watsonia, Australia) for encouraging and helpful discussions.

Note

1. Some books and journal papers use α for the prefactor of (r – r_e), while others use β. β is also used for vibrational–rotation interaction constant (1–7). D_e is the well depth of the Morse function, but D (with no subscript) is a prefactor of one of the higher-order terms for the Morse energy.

Literature Cited

1. Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 2002.
2. Atkins, P. W.; de Paula, J. Physical Chemistry, 7th ed.; Oxford University Press: Oxford, 2001.
3. Silbey, R. J.; Alberty, R. A. Physical Chemistry, 3rd ed.; John Wiley and Sons: New York, 2000.
4. Hirst, D. M. A Computational Approach to Chemistry; Blackwells Scientific: Oxford, 1990.
5. Hollas, J. M. Modern Spectroscopy, 2nd ed.; Wiley: Chichester, 1992.
6. Banwell, C. N., Fundamentals of Molecular Spectroscopy, 3rd ed.; McGraw-Hill: London, 1983.
7. Bernath, P. F. Spectra of Atoms and Molecules; Oxford University Press: New York, 1995.
8. Bozlee, B. J.; Luther, J. H.; Buraczewski, M. J. Chem. Educ. 1992, 69, 370.
9. Mina-Camilde, N.; Manzanares, C.; Caballero, J. F. J. Chem. Educ. 1996, 73, 804.
10. Lim, K. F.; Collins, M. A. Aust. J. Educ. Chem. 2003, 61, 17.
11. Lim, K. F. J. Chem. Educ. 2005, 82, 145.
12. Gardner, H. Frames of Mind: The Theory of Multiple Intelligences, 2nd ed.; Fontana: London, 1993.
13. Preparing for a New Calculus; Solow, A. E., Ed. Mathematical Association of America: Washington, DC, 1994; Vol. 36.
14. Shiber, J. G. J. Chem. Educ. 1999, 76, 1615.
15. Day, R. Australian Educational Computing 1986, 1, 40.
16. Baker, J. E.; Sugden, S. Spreadsheets Educ. 2003, 1, 18 (accessed May 2005).
17. Lim, K. F. Using Comput. Chem. Educ. 2003, 2003 (Spring), paper 5 (accessed May 2005).
18. Lim, K. F. Using spreadsheets to teach quantum theory to students with weak calculus backgrounds, in Maths for Engineering and Science; C. Hirst, Ed., LTSN MathsTEAM: Edgbaston (UK), 2003, pp 24–25.
19. Hénon, M.; Heiles, C. Astron. J. 1964, 69, 73.
20. Microsoft Excel, Microsoft Corporation, Redmond, WA
21. Lim, K. F. J. Chem. Educ. 2002, 79, 135 [supplementary material: Journal of Chemical Education: Webware, paper WW003 (accessed May 2005)].