B1777 - ATOMI, MOLECOLE E SIMMETRIE

Learning outcomes

At the end of the course, the student is familiar with the basics of symmetry and group theory; they understand the fundamental principles of quantum mechanics and can apply group theory to the study of the electronic structure of atoms and molecules. The student will apply the main methods of group theory and quantum mechanics in exercises and laboratory sessions to study electronic properties, particularly the energy levels of atoms and molecules.

Course contents

The course will be structured according to the following macro-topics:

1 Symmetry. Introduction to symmetry as a geometric concept. Examples of symmetry in nature, with reference to systems in chemistry, physics and materials science: from molecules to crystalline solids. Simple symmetry operations (e.g. identity, rotation, reflection, roto-reflection, inversion). Formal introduction to symmetry elements and operations.

2 Symmetry operators. Matrices, determinants and their properties. Inverse of a matrix. Review of linear algebra and matrix representation of geometric transformations in the plane and in space. Introduction to operators, their definition, operator algebra and representations (*). Introduction to vector spaces (*). Characteristic equation of a matrix: eigenvalues and eigenvectors (*). Diagonalization of a matrix. Similarity transformations. Numerical exercises.

3 Groups. Definition of group. Examples of groups in relation to molecular systems and crystalline systems. Definition of point group and examples. Definition of space group and examples. Group properties, class definition and group order. Multiplication tables (group tables). Classification of point groups and determination of point groups in molecular systems. Examples and applications.

4 Representations. Matrix representations of symmetry operations: identity, rotation, reflection, roto-reflection, inversion. Matrix representations of point groups and group tables. Basis vectors, basis functions, concept of complete set of basis functions, linearly independent functions, orthogonality of basis functions and reminders of similarity transformations (*). Equivalent and reducible representations. Examples and applications.

5 Irreducible representations and character tables. Characters of representations and their properties. Characters and classes: examples. High orthogonality theorem and irreducible representations. Irreducibility criteria. Character tables and their construction: practical examples for point groups.
Reduction of representations, projection operators and symmetry adapted bases. Representation of direct product and direct product groups. Examples and exercises regarding molecular systems.

6 Symmetry, representations and quantum mechanics. Introduction to the Schrödinger equation (*). Hamiltonian operator: kinetic energy, potential energy operators and their transformations (*). Group of the Schrödinger equation. Wave functions as bases for irreducible representations. Non-degenerate and degenerate solutions of the Schrödinger equation. Normal and accidental degeneration. Symmetry of functions and evaluation of integrals using symmetry rules (vanishing integrals). Outline of matrix elements (Dirac notation), expectation value and selection rules (*).

7 Electronic structure and molecular orbital theory. Atomic orbitals, quantum numbers and atomic configurations (*). Linear combination of atomic orbitals (LCAO): cation of the hydrogen molecule (H2+) and hydrogen molecule (H2). Symmetry adapted linear combinations (SALC). SALC for homonuclear diatomic molecules. SALC for heteronuclear diatomite molecules. Hybrid orbitals. Polyatomic molecules: examples and applications relating to H2O, NH3, CH4, C2H4, C6H6.
Notes and examples on ligand field theory and examples (octahedral systems).

8. Molecular vibrations. Harmonic oscillator (*). Cartesian coordinates, internal coordinates and symmetry coordinates. Normal modes of vibration for simple polyatomic molecules. Applications of group theory to molecular vibrations. Critical discussion of the case of water. Notes on selection rules (IR and Raman spectra).

9. Outline of solid state symmetry. Translational symmetry, cyclic groups and Bloch's theorem. Wave vector group. Direct lattice and reciprocal lattice. Brillouin area.

(*) connections and references with the “Condensed Matter” module

Readings/Bibliography

Suggested books:

1. “Group Theory and Chemistry”, David M. Bishop.

2. “Symmetry and Spectroscopy”, Daniel C. Harris, Michael D. Bertolucci (Chapters 1, 3, 4).

3. “Molecular Quantum Mechanics”, fifth edition, Peter Atkins, Roland Friedman (Chapters 5, 8, 10).

(Note: there are no books fully covering the course topics. The lecturer will provide notes, scripts and suitable complementary indications).

Teaching methods

The course includes theoretical lectures and numerical exercises for the practical application of group theory on atomic, molecular and periodic systems.

During the lectures the theoretical concepts will always be accompanied by examples, trying to connect the new concepts as much as possible with those acquired in other courses such as chemistry, physics and quantum mechanics.

Assessment methods

The exam for the integrated course of Quantum Theory of Matter will focus on the topics covered in the modules of "Condensed Matter" and "Atoms, molecules and symmetries".

Concerning the module "Atoms, molecules and symmetries", the exam includes a written part, in which practical exercises related to group theory will be given, and an oral part, in which the written assignment will be discussed, and theoretical and applicative aspects of group theory will be assessed, with reference to the electronic and vibrational structure of simple systems (molecules).

There is no passing threshold for the written exam, all students must also take the oral exam.

The final grade of the integrated course will be calculated by the arithmetic mean of the grades obtained in the two modules. To pass the integrated course, it is necessary to exceed 18/30 in both modules.

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Teaching tools

1) PC + beamer, openboard, lecture handouts
2) Computer exercises
3) Lecture materials presented in the lab will be made available to the student through the platform VIRTUALE.

Office hours

See the website of Daniele Fazzi