When can a Computer Simulation act as Substitute for an Experiment? A Case-Study from Chemisty

Johannes Kästner and Eckhart Arnold

3.4.2 Modeling Techniques

The H-tunneling simulation makes use of a number of approximations and modeling techniques within the realm of quantum mechanics. To describe the atomic motion, it uses instanton theory (also called harmonic quantum transition state theory, HQTST) which is an approximation to quantum mechanics. Its approximations are mathematically well defined, and it has been applied for decades to calculate tunneling rates (Vai 82, Tak99). “This approach has been shown to be quite accurate in comparison to analytic solutions and results from quantum dynamics, especially low temperatures where other semi-classical tunneling approaches often underestimate transmission coefficients” (Goumans/Kaestner 2010, p. 7350). Instanton theory is based on (an approximation to) Feynman's path integral method.

The electronic motion in turn was described by density functional theory, a different formulation of quantum mechanics. The theory itself is exact (Hoh 64), but the functional involved is unknown and has to be approximated. Goumans/Kaestner (2010) performed reference calculations with the CCSD(T)/CBS ab-initio method, another approximation to the exact quantum mechanical result with well established accuracy. Then they compared different forms of functionals, all of which are frequently used and were empirically found to be credible in many cases in chemistry. They used the MPWB1K functional for the simulation since it provided a satisfactorily good match to the CCSD(T)/CBS reference data, and because it was at the same time the relatively best match in comparison to several other tested functionals.

In all quantum chemical simulations, the quantum mechanical wave function has to be expanded in a finite basis set. The error introduced by this expansion is already accounted for in the comparison of the functionals, because CCSD(T)/CBS does not show this error (CBS stands for complete basis set limit).

The computer programs used by Goumans/Kaestner (2010) (NWChem, ChemShell, Gaussian, Molpro) are well established and used by many researchers throughout the world. Significant errors would likely have been found prior to their study and can, thus, be regarded as very unlikely. The program implementing the instanton theory was partially written specifically for the one study presented here. Before the production calculations, it was tested extensively against cases with known results. This is in line with the established best practices in the field and will usually not even be mentioned in the scientific papers.

Additional approximations, like the truncation of a number series or convergence criteria, were tested by extending them and monitoring the change of the result. The used values are reported to allow the exact reproduction of the calculations by other scientists.

Summing it up, in order to realize the simulation, the authors made use of a number of modeling techniques, including several levels of approximations. While one could say that this inevitably introduces some degree “motleyness” and “autonomy” (Winsberg 2001), these characterizations are not very fitting in this particular example, because the simulation discussed here is diligently built to reflect the theory as closely as possible and does not draw on any independent phenomenological considerations. Also, many techniques are used to keep deviations from the theory in check, like the comparison of different functionals with reference calculations as well as various measures of testing and error checking. Ultimately, when interpreting the results it is taken into account that the simulation results may - due to the employed functional - deviate from the real values in a certain way: “Since the functional we used overestimates the classical barrier, the real rates [of $H_2$ formation] will be higher than our calculated ones.” (Goumans/Kaestner 2010, p. 7352)