# Mathematics around the periodic table | Science

The UN General Assembly has proclaimed 2019 as the International Year of the Periodic Table, to celebrate the 150 since the Russian chemist Dmitri Mendeléyev gave his first version of it. That ordering of the elements, according to its number of protons and its chemical affinities, it turned out to be a very valuable tool to explain known chemical processes and anticipate new discoveries.

The table raises many basic issues whose mathematical explanation, deductively and rigorously from the first principles, is a huge and fascinating intellectual challenge. An example is the very notion of periodicity in the ordering of the elements; another are the concepts of valence, orbital and ionization energy, but also that there are certain special numbers of electrons (2, 10, 18, 36, 54, …) that prevent noble gases from participating in any reaction.

The atomic model introduced by Max Born and Robert Oppenheimer is considered the most appropriate on which to build the mathematical theory of the table. According to this approximation, an atom consists of a nucleus (of charge Z), that we can suppose in the origin of coordinates, and of Z electrons quantized that are described by the wave function Ψ. Allowing us the license to disregard the "spin", this is a function whose value depends only on the positions of the electrons. The energy of the atom is given by the Hamiltonian, H, in whose expression appears the Planck constant, the mass of the electron, its charge and the positions, which are vectors in three-dimensional space. H is the sum of three different terms: one represents the electrostatic attraction proton-electron; another repulsion, also electrostatic, electron-electron; while the third captures the kinetic energy of the Z electrons.

The fundamental difference with the equations that describe the classical model is that the term corresponding to the kinetic energy (one half of the mass times the square of the velocity) is replaced by a second order differential operator in the quantum model. In addition, observable values such as position, kinetic moment, or energy become operators over the space of wave functions. These functions also have a probabilistic interpretation: the integral of the square of its absolute value gives the probability that the electron is in a given region of space.

But while in classical mechanics the possible values of energy are simply the values of the Hamiltonian, which in that theory is a function defined in the space of the phases, in quantum mechanics the values of the energy, obtained in the spectroscopic observations of the light emitted by the atoms are, precisely, the quantities E (called eigenvalues) for which the equation HΨ = EΨ has a non-zero solution.

In the case of Z = 1, which corresponds to the hydrogen atom, we know the E values (called Quantum numbers) and also the solutions of the equation HΨ = EΨ. Therefore, a very complete mathematical model is available to explain satisfactorily the chemical properties of this element.

However, the case Z> 1 is much more difficult and we lack explicit solutions. The space of the wave functions Ψ is much more complicated. One of its known properties is that they are square integrable and antisymmetric functions; that is, Ψ changes sign when exchanging the positions of two electrons. This property is fundamental in the theory and codifies the so-called Pauli's exclusion principle, which states that electrons are indistinguishable particles that obey the Fermi-Dirac statistics (associated with indistinguishable dice rolls).

The ground state, that is, the lowest possible energy state of the atom, is described by its wave function Ψ that minimizes the energy, and corresponds to the lowest eigenvalue of the Hamiltonian, HΨ = E (Z) Ψ, in the space described above. wave functions (which we know are antisymmetric and integrable square). Knowing precise expressions for the value of that energy E (Z) is a fundamental first step to mathematically understand the periodicity of the table.

We now know that E (Z) has a series development, whose first three terms are powers of the number of electrons Z with decreasing exponents (7/3, 2 and 5/3), multiplied by some well-determined coefficients. The first of the terms was conjectured by Enrico Fermi in 1927 and rigorously demonstrated by Elliott Lieb and Barry Simon in 1977; the second proposed by J.M.C Scott in 1950 and tested by Webster Hughes in 1990; while the third was conjectured by Julian Schwinger and Paul Dirac and finally demonstrated by Charles Fefferman and Luis Seco in the year 1991. These terms can be deduced with simple physical intuitions, but all require complicated mathematical arguments to be rigorously demonstrated.

The following term of E (Z) is of more complicated nature (it is not a power of Z) and has an oscillatory character (trigonometric sum almost periodic), which gives a particular interest when mathematically justifying the periodic table, and It could very well be the term of the energy contributed by the valence electrons. Some results have been published in this regard, among them, the one that the author signs together with Fefferman and Seco, where we established a connection of this function with a famous conjecture of the theory of numbers (the so-called problem of the circle and the reticle). So we could use the estimates of trigonometric sums so useful in the analytic theory of numbers to advance the objective, still distant, to glimpse the chemistry from the most rigorous mathematical reasoning. The challenge remains open.

**Antonio Córdoba** is a director of ICMAT and a professor at the Autonomous University of Madrid

** 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 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)