Infrared Spectroscopy of Symmetric and Spherical Spindles for Space Observation 1. Pierre-Richard Dahoo

      This book describes the theoretical methods developed in the framework of fundamental research for the interpretation of the spectra of ammonia molecules, which are characterized by a ternary axis of symmetry, from observed spectra in the IR range when these molecules are subjected to an environment in which the temperature and pressure modify their IR gas phase spectra or in nanocages. It describes the theoretical models for the study of ammonia and methane molecules in these media based on the theoretical models elaborated for the gas phase. The modification of the IR spectra of these molecules can also be interpreted, such as the shift of the band centers or the modification of the rovibrational spectrum in nanocages or on surfaces.

      This book is intended for students at master and doctoral levels, teaching academics and researchers, astronomers and astrophysicists who analyze the data derived from the interaction between electromagnetic radiation and matter in the IR range, in order to identify the chemical species and their environments.

      This book (Volume 3), which will be followed by a complementary book on applications (Volume 4), presents, to both beginners and experts in the field of spectroscopy, the methods developed through theoretical models using group theory for rigid and non-rigid molecules characterized by the tunneling effect phenomenon and large amplitude motions. Simulations of these models enable the analysis of IR data and the identification of molecules based on transitions and profiles not only in the gas phase from fundamental bands but also when they are forced to evolve in an environment that can be a nanocage or a surface.

      Ammonia and methane molecules are members of the class of the molecules with a ternary and quaternary axis of symmetry. The first part is organized into two chapters concerning the symmetry and the use of group theory for the study of symmetric and spherical tops, as well as a review of the line profile of molecules that evolve in nanocages or on surfaces. Then the theoretical models elaborated for the study of vibration–rotation spectra of polyatomic molecules in the gas phase are briefly recalled as developed in molecular physics. They allow the study of both cold and hot bands, as well as the combination bands, and have been extensively described in the books devoted to the spectroscopic study in the gas phase.

      In this volume, the theoretical description only covers the study of cold bands, whose spectra result from transitions from the fundamental vibration–rotation level. The IR spectroscopic study concerns the libration or hindered rotation movements that result from a strong coupling with the environment and the constrained rotational movements when the coupling with the environment is weak (rotation at low J).

The theoretical part resulted from molecular physics was partly inspired by the molecular physics lectures given in the 2nd year of master degree studies by G. Amat at the UPMC and those of J.M. Flaud and C. Camy-Peyret in the master degree of advanced studies, “Laser and Matter”, at the UPSUD, and the research works conducted at the CNRS laboratories (Paris, Orsay, Dijon, Grenoble and Reims), which are available particularly in the form of (PhD) theses on spectroscopic studies of symmetric or spherical top molecules in the IR range. The theoretical models specific to research work on the effects of an environment have been developed, particularly in the group of molecular physics in Besançon (L. Galatry, D. Robert, J. Bonamy, L. Bonamy, C. Girardet, A. Lahklifi, etc.), in order to analyze observations on molecules subjected to pressure or isolated in condensed phase media in collaboration with researchers from Paris and Orsay (L. Abouaf, H. Dubost, B. Gauthier, J.P. Boissel, P.R. Dahoo, etc.)

      These models are applied to study the molecules subjected to interactions in various media, whose effects manifest particularly at the nanometer scale, which modifies the profile of the IR spectra of these molecules. The theoretical inclusion model or the extended model proposed by Lakhlifi and Dahoo was explained in Volumes 1 and 2, and certain programs for the numerical calculation were described in these volumes, mainly in Volume 2. They are used to calculate the IR spectra of molecules in nanocages.

      Chapter 1 describes the main tools and methods developed in group theory and tensor algebra for their applications and IR spectroscopy. The finite and continuous point symmetry groups allow for classifying the energy levels associated with electronic, vibrational and rotational degrees of freedom by their symmetries connected to an irreducible representation of the group. The exchange of identical nuclei in a molecule, where tunneling inversion can be observed through experimentation, is studied as part of the permutation–inversion group.

      In Chapter 2, group theory methods are applied to determine the symmetry groups of symmetric top molecules with reference to the NH₃ molecule and the spherical top molecules with reference to the CH₄ molecule. The isotopologues (the hydrogen atom H is replaced by the deuterium atom D) of these molecules are also studied from the point of view of symmetry groups. The symmetry properties of NH₃ and CH₄ molecules and their isotopic varieties are presented in this chapter and their geometric symmetry groups are determined. It is shown that the loss of a symmetry element modifies the symmetry group from a more symmetric group to a less symmetric one, and consequently modifies the observable IR spectra. The theory of permutation–inversion groups is also applied to determine the corresponding CNPI groups.

In Chapter 3, the group theoretical methods are applied to determine the functions that generate irreducible representations of the symmetry group of symmetric top molecules and spherical top molecules by referring to NH₃ and CH₄ molecules and to their isotopes for various degrees of freedom, either electronic, vibrational, rotational and electron and nuclear spins. An approximate Hamiltonian is used, which makes it possible to express the total wave function as a product of wave functions by neglecting the vibration–rotation interaction. The statistical weights of the levels are calculated by applying the generic formula given by Landau, and the inversion phenomena is described in the molecular symmetry group following the approach of Bunker and Jensen. The various expressions used to calculate a line profile observable in a spectrum, as well as the method used to determine the selection rules in general, are recalled. By neglecting the couplings, the product form of the total wave function that describes the molecule in the zeroth-order approximation and its symmetry properties are taken into account. The objective is to identify the types of symmetry of the energy levels of each type of degree of freedom, and to determine the allowed transitions that give the selection rules in absorption, emission or diffusion spectroscopy. These transitions are the fingerprint of the molecule when it interacts with light in the studies of IR absorption, emission spectroscopy and Raman spectroscopy.

      In Chapter 4, the theoretical model developed for the study of IR spectra that result from transitions between the levels of vibration–rotation energy of symmetric top molecules is described with reference to the NH₃ molecule. The resolution of the Schrödinger equation gives eigenvalues of the molecular system that depend on the degrees of freedom of the nuclei and of the electrons in the molecule that do not have simple analytical solutions. An approximate Hamiltonian is built in the Born and Oppenheimer approximation in order to decouple the rapid motion of electrons from that of nuclei. For each electronic state, the Hamiltonian of nuclei can be used for the study of vibration–rotation motions of the molecule. The various steps in the calculation of the vibration–rotation energy levels are explained in the applications of quantum mechanics applied to molecules. The decoupling of the translation motion from the other degrees of freedom is achieved by studying the vibration–rotation motion in a reference frame attached to the equilibrium configuration of the molecule (mobile reference frame) whose origin coincides with its center of mass (Eckart–Sayvetz conditions). The vibration is then studied in the approximation of low amplitudes. A partial decoupling is achieved between the vibration and rotation motions of the molecule. The Van Vleck contact method is used for splitting the Hamiltonian into blocks, in order to study the IR spectra of the molecules. The transitions between the energy levels resulting from the interaction between the dipole moment or the induced dipole moment of the molecule and the IR electromagnetic radiation lead respectively to the absorption