Separation and contraction of variables in molecular spectroscopy

Spectroscopy is an ancient technique (it can be traced back to Newton, who observed the spectrum of the Sun using a glass prism in 1666), which plays a key role in the detection and identification of molecular and atomic species, and enables us to probe the physical conditions in which these species evolve. It can be applied to a wide range of fields, from the detection of molecules in other galaxies to the analysis of ancient works of art, not forgetting atmospheric sciences (pollutant tracking, fire detection, radiation balance of a planetary atmosphere, etc.), the remote detection of toxic substances or "dirty" bombs by intervention forces, the study of the structure and transformations of living molecules...

However, an atomic or molecular spectroscopy experiment provides only incomplete information. For one thing, not all the quantum states accessible to a molecule couple to electromagnetic radiation with sufficient intensity to give an observable line. There are so-called "dark" states that cannot be observed directly using spectroscopic methods. On the other hand, spectroscopy only gives information about the transition from a quantum state A to a state B, but it doesn't tell us which states A and B are. Not knowing the identity of the starting state means we can't determine the population of molecules in that state, and hence the temperature dependence of the observed spectrum. However, satellites are regularly launched to record molecular spectra and deduce the chemical composition and physical parameters (pressure, temperature...) of the probed media. To derive maximum information from the spectroscopic data collected, we turn to quantum theory.

The first quantitative theoretical methods were developed as early as 1892 by Lord Rayleigh. Our theoretical understanding of molecular spectra then took a quantum leap forward around 1926 with the arrival of quantum mechanics. More recently, the advent of computer science has enabled increasingly accurate theoretical predictions, and computational spectroscopy has become an indispensable tool for the analysis and interpretation of spectroscopic data.

We will attempt to provide an overview of computational spectroscopy or "in silico" spectroscopy, a rapidly developing field that has made spectacular progress in recent decades, combining advances in computer science with those of quantum theoretical methods.

More precisely, after presenting the major approximations based on the separation of variables in a molecular system, which serve as the starting point for most theoretical approaches, we'll focus on introducing a method that enables us to "contract" wave functions corresponding to the various separated variables. This is because molecular quantum states are...