# Mathematics to describe the vibrations of molecules | Science

Everyone knows that molecules are composed of atoms, and these by nuclei and electrons. However, it is much less known that all of them are in perpetual motion, even at the lowest possible temperatures. These complex molecular dances, similar to ballet choreography, are studied in a discipline called molecular dynamics. It is a branch of science close to celestial mechanics, that on the other hand analyzes the dances of the celestial bodies, to the compass of the gravitational forces. In fact, both disciplines share methods and theories.

Each combination of atoms is only able to execute one specific dance, with a rhythm that repeats periodically in time. The corresponding frequencies constitute an unambiguous fingerprint that serves, for example, to detect explosives or drugs that may be in airport baggage or packages. Describing the dances is key, therefore, to characterize each of the substances.

To study them it is important to take into account the scenario in which they take place. They are the so-called molecular landscapes, described by hypersurfaces of multiple dimensions that express the relative energy of each configuration of atoms in the molecule. They have a complex topography formed by valleys and hills, slopes and slopes.

The valleys correspond to stable molecular states. In them the nuclei act as if they were united by elastic springs and vibrate all in unison. The movement of the nuclei follows a very regular form (described by mathematical objects called bulls, equal to the donuts of our breakfast). As excitement increases, for example, by supplying heat or energy with a laser, the nuclei separate from the valleys, making their movements more complicated (anharmonic). This complication increases, until they become chaotic and unpredictable, just like the weather.

Mathematically, this transition from the order of bulls to chaos can be analyzed through theorems of the modern theory of dynamic systems, developed in the second half of the 20th century. If the excitement is even greater, some atoms will have enough energy to surpass the different collars of the molecular landscape and explore other valleys of it, thus producing chemical reactions. Chemical reactions happen in very short times, in the order of the femtosecond, which is equal to one billionth of a second.

It is possible to determine precisely the speed of the reaction, thanks to a geometric theory that uses mathematical techniques similar to the previous ones, also derived from the celestial mechanics. This theory is based on an object called NHIM (from English normally hyperbolic invariant manifold), which determines that in the state of intermediate transition, between reactants and products, there is at least one direction with irregular or chaotic movement that governs the reaction. The NHIM has important properties such as its robustness, which means that it is not destroyed (although it can be modified) by external agents (for example, a laser).

This last property allows the use of lasers for the precise and specific control of chemical reactions, making them pass in the desired direction, in the same way that a surgeon uses his scalpel in a surgical operation. In this situation, the frequency of the laser (related to its color) is incorporated as another element of the molecular dance, causing it to change. To develop and apply this tool (and others) in an effective way, the presented mathematical techniques are fundamental, which give an account of what is the dynamics in the molecules, how the energy transfer occurs between them or the characteristics of the transition state.

**Florentino Borondo** is a professor in the Department of Chemistry of the Autonomous University and member of the Institute of Mathematical Sciences (ICMAT)

** Coffee and Theorems** is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between the mathematics and other social and cultural expressions and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".

Editing and coordination:**Agate Rudder** (ICMAT).

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