Ateneo - UPO

The course aims to describe some mathematical tools of general use in Chemistry for the solution of various problems, in particular for Spectroscopy and data analysis; students are expected to learn some theoretical bases and to exercise with specific chemical problems. The main contents involve Group Theory, function expansions in power series, and Fourier series and transforms.

Notes provided by the teacher. Mary Boas “Mathematical methods for the physical sciences”.

1) Provide a general knowledge of the Group Theory (GT), with applications to point symmetry groups; develop the skill to determine the molecular symmetry groups, and to obtain the relevan informations from GT; reach the competence to describe the symmetry species of molecular vibrations.
2) Obtain a deeper knowledge of the mathematical tools related to number and function series; develop the skill to obtain a power series expansion of a given function.
3) Illustrate the basic concepts and the techniques related to Fourier series and transforms; develop the kill to expand periodic signals in Fourier series and to compute the Fourier transform of various functions; get the competence of use Fourier transform in spectroscopic applications.

Communicative skills: obtain a suitable chemical lexicon, and the ability of use it in various applications related to the course arguments.

Basic knowledge of Calculus.

Lectures and exercises.

Classroom exercises to check the learning during the course.

Oral exam, with questions about the arguments treated during the lessons. In particular, students will be questioned about Group Theory, power series expansion of functions, Fourier series and transform.

The course provides an introduction to the formalism and the most important results of the Group Theory, almost exclusively using examples of point symmetry groups, of practical interest in Chemistry and Spectroscopy. Groups, classes and reducible and irreducible representations are defined. The Great Orthogonality Theorem is presented along with its corollaries, then the concept of character of representations is illustrated, and we show how to reduce generic representations in terms of irreducible representations. For symmetry groups more simple and common irreducible representations are obtained, describing the nomenclature and their character tables.
These concepts are applied, at the end of the first part of the course, to the definition of the species of symmetry of molecular vibrations, through the reduction of the normal representation for different molecules. We describe the importance of this technique in the analysis of IR spectra.
In the second part of the course the basic concepts on numerical series are recalled, along with the theorems ruling their convergence, with different exercises. Then we introduce function series and the concept of radius of convergence, to describe the development in power series of different functions.
In the last part of the course we define the Fourier series for the development of periodic functions (signals) and illustrate how to determine the coefficients of the series; Finally, the concept of Fourier transform is introduced for generic functions and we illustrate some examples, so that students develop the ability to calculate the Fourier transform at least for the most simple functions. The application of these concepts to Fouries transformed spectroscopy is analyzed in detail, with the description of an interferometer and of the related experiments.

Knowledge and understanding: knowledge of the principles and main theorems of the Group Theory, in particular related to point symmetry goups; knowledge of the main application sof the Group Theory to molecular symmetry; knowledge of the main theorems on numerical series, convergency tests, Taylor and McLaurin series; knowledge of the basic theorems on Fourier analysis; achievement of a suitable scientific language.

Applying knowledge and understanding: capacity to identify the symmetry group of a given molecule and to obtain its representations: capacity to apply group theory to molecular vibrations; capacity to develop in power series a given function and to calculate its convergency range; capacity to develop in Fourier series a given periodic function; skill to Fourier-transform a simple funtction.

Making judgements: skill to critically analyze the vibrational spectra of symmetric molecules; skill to interpret the operation of instruments based on the Fourier transform.

Communication skills: ability to report on scientific topics, in particular related to mathematical methods for chemistry, in a precise, concise and clear manner, both in written and oral form.

Learning skills: ability to use the teaching material for a critical and reasoned study, also for a subsequent autonomous acquisition of superior knowledge and for a continuous updating.