Undergraduate- and graduate-level physical chemistry courses are so mathematically intensive that they often (a) dissuade qualified students from pursuing chemistry as a major, (b) require more homework, testing, and grading time be devoted to tracking down hidden mathematical errors than is devoted to working with the subject matter, and (c) limit the complexity, and therefore real-world applicability, of assigned problems. In particular:
- The differential and integral calculus required for modern quantum mechanics and thermodynamics can be intensive.
- Only a small fraction of chemical mechanisms have differential rate equations that are analytically solvable.
- Real-world equilibrium systems often have many simultaneous, interdependent equilibria in play.
- Chemical systems are inherently three-dimensional, resulting in difficulties in visualization.
- Visualizing changes in parameters and their effects on complex calculations can often be difficult.
- Finding the equivalent of a carry-the-two error in a multi-page derivation is both frustrating and not pedagogically useful, especially when software packages are available that can perform the derivations flawlessly.
Extended example: Consider the v=1 state of the harmonic oscillator, whose probability density function is shown to the right.
A common question from physical chemistry students is, “If the system starts in the left-hand lobe, how does it ever get to the right-hand lobe?” And a common answer to this question given by many professors is that the system tunnels. However, that answer is incorrect. To understand this, you need to recognize that the question itself contains a contradiction. If the system starts in the left-hand lobe, then the system is not in the v=1 state.
We can model the state it is in by taking a linear combination of the various states and fitting the coefficients to the left-half of the v=1 wavefunction. But that resulting state is not an eigenfunction of the system, and thus it will evolve in time, governed by the time-dependent Schrodinger equation. To the right is an animation showing what happens to that wavepacket. And so the answer to the original question, how does a system that starts in the left-hand lobe move to the right-hand lobe is: It just moves.
Modern computational tools, however, can handle much of the mathematics drudgery for us. Chief among these tools is Mathematica™, which handles both symbolic manipulation and numerical calculations with ease. Unlike chemistry-specific software packages such as Gaussian™, Mathematica™ is not a black-box that hides the mathematical underpinnings of the calculations. Mathematica™ requires the students to work through the underlying mathematics enough to learn the material, but performs the calculations reliably and efficiently.
I have developed a set of Mathematica™ materials that span a full two-semester sequence of undergraduate physical chemistry, and am in the process of packaging these materials for publication. In the process of performing this packaging, I have further developed these routines to include the ability to modify Mathematica™’s native notational system, which can be opaque to non-programmers, into an approach that uses chemistry-standard notation. These modifications allow the chemist to use Mathematica™ in a way that is natural and easy to follow.