Mathematics to describe the vibrations of molecules

Everyone knows that molecules are composed of atoms, and these are by nuclei and electrons. However, it is much less known that all of them are in perpetual movement, even at the lowest possible temperatures. These complex molecular dances, similar to ballet choreographies, are studied in a discipline called molecular dynamics. It Is A branch of science close to the celestial mechanics, which 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 capable of performing a specific dance, with a rhythm that is repeated periodically over time. The corresponding frequencies are an unequivocal fingerprint that serves, for example, to detect explosives or drugs that may be in airport baggage or parcels. To Describe the dances is key, therefore, to characterize each one 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 joined 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 the excitement increases, for example, by providing heat or energy with a laser, the nuclei separate from the valleys, becominging their more complicated movements (Anarmónicos). This complication increases, until they become chaotic and unpredictable, just like atmospheric weather.

Una técnico observa al microscopio confocal, una muestra de un corte raíz de Convallaria majalis en el Servicio de Microscopía Óptica y Confocal. Este servicio proporciona asistencia y formación en aplicaciones de Microscopía Óptica a los grupos de investigación pertenecientes al Centro de Biología Molecular Severo Ochoa.A technician observes the confocal microscope, a sample of a majalis boundary cut in the Optical and confocal Microscopy Service. This Service provides assistance and training in Optical Microscopy applications to research groups belonging to the Severo Ochoa Molecular Biology Center. Uly Martín

Mathematically, this transition from the order of the Bulls to chaos can be analyzed through theorems of the modern theory of dynamic systems, developed in the second half of the TWENTIETH century. If the excitement is even greater, some atoms will have enough energy to surpass the different hills of the molecular landscape and explore other valleys of the same, thus producing chemical reactions. The chemical reactions happen in very short times, the order of the femtosecond, which is equal to the Milbillonésima part of a second.

It Is possible to precisely determine 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 (normally hyperbolic invariant manifold), which determines that in the intermediate transition state, 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 robustness, which means that it is not destroyed (although it can be modified) by outside agents (for example, a laser).

This Last property allows the use of lasers for precise and specific control of chemical reactions, making them run 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 an element of the molecular dance making it change. In Order To develop and apply this tool (and others) effectively, the mathematical techniques presented are fundamental, which give an account of the dynamics of the molecules, how the energy transfer occurs between them or the characteristics of the state of transition.