the mathematics of chemistry

Researchers have succeeded in explaining the complex vibrational dynamics of molecular systems using geometric methods.

Despite how feared mathematics has always been, today it is, without a doubt, one of the fashionable careers. This current success has its origin, above all, in the boom that we have seen in recent years with the development of big data and artificial intelligence, although it is also due to the fact that its importance has become evident in many other disciplines. Thus, without mathematics we would not be able to design bridges, manufacture airplanes, create ecological models, analyze social networks… The list can be practically infinite and chemistry is not out of it. In fact, over the last few years, numerous researchers have begun to study the behavior of chemical systems using geometric methods of the so-called dynamic systems theory of mathematics.

In a recent study, researchers from the Polytechnic University of Madrid (UPM), the Autonomous University of Madrid (UAM) and the Institute of Mathematical Sciences (ICMAT), all of these entities in Spain, have applied the Lagrangian descriptors of mathematics to the potassium cyanide (KCN) molecule, and have managed to explain and reproduce its complex vibrational dynamics even at high energies, a limit where the most traditional methods are not valid.

Lagrangian descriptors are indicators originally developed for the study of ocean dynamics that allow the identification of geometric structures (called invariant varieties) and that determine how a system (molecular, in this case) changes over time. The interesting thing about this tool is that, to use it, it is basically only necessary to integrate the equations of motion, without the need to carry out complex studies of the dynamics close to the trajectories, as occurs with traditional chaos indicators, which makes it much easier the study. The results of the work will serve for the development of a selective chemistry in which the vibrational energy can contribute to facilitating reactions that otherwise would not take place due to the high energy that would be necessary.

In the study, which has included the participation of researchers from the Complex Systems Group of the UPM, it has been verified that the geometric structures (invariant varieties) associated with a single periodic orbit of potassium cyanide determine how the entire molecule changes configuration causing the atoms that form it to vibrate in a specific way. According to Fabio Revuelta, a researcher at the UPM participating in the study: “knowledge of these vibrations is very important for the development of selective chemistry, in which (vibrational) energy can contribute to facilitating reactions that otherwise would not occur.” would be carried out by the high energy that would be required. In addition, although at low energy there are other methods for the study of vibrations in molecular systems, for example, the analysis of normal modes, these methods are not valid at high energies such as those at which the study has been carried out, hence the interest. of our work”, adds the researcher.

Potassium cyanide molecular system studied, formed by a potassium atom (K), a carbon atom (C) and a nitrogen atom (N), and coordinates (R, θ, r) that describe the configuration of the molecule . (Image: UPM)

Finally, the researchers also observed that the invariant varieties studied behave much more complexly than expected, folding and coiling on themselves more abruptly than other molecules. These foldings are responsible for the vibrational dynamics of the potassium cyanide molecule being much more chaotic than in other molecules also made up of three atoms. This is because the interactions between its constituents are much more complex in this case, although ꟷconcludes Fabio Revueltaꟷ: “that is another story that requires, once again, more mathematics”.

The study is titled “Unraveling the highly nonlinear dynamics of KCN molecular system using Lagrangian descriptors”. And it has been published in the academic journal Communications in Nonlinear Science and Numerical Simulation. (Source: UPM)