66913 - Physical Chemistry 1

Learning outcomes

The students learn: to master their mathematical knowledge in order to apply it to problems in Physical Chemistry; the fundamentals of molecular symmetry and of quantum mechanics for following studies in atomic and molecular structure; to apply the methods of quantum mechanics for studying the electronic properties, especially the energy levels, of atoms and simple molecules.

Course contents

Linear algebra

Vector spaces, Matrix algebra, Matrices and Linear transformations, Determinants, Invertible matrices, Orthogonal matrices; Complex matrices, The eigenvalue problem, Similarity trasformations and diagonalization, Hermitian matrices.

Molecular symmetry and group theory

Symmetry operations and elements; The algebra of simmetry operators; Groups: definition, the multiplication table, some properties and definitions; Point groups; Symmetry operations as linear transformations in the ordinary 3D space; Matrix representations of symmetry groups; Base functions to build matrix representations; Equivalent representations; Reducible and irreducible representations; Great orthogonality theorem; Character tables.

Symmetry and quantum mechanics

The postulates of quantum mechanics: states, operators and observables; The Schroedinger equation; The meaning of the wavefunction; Time evolution; The matrix formulation; The symmetry of the Hamiltonian; Symmetry and degeneration; Integrals and selection rules.

Differential equations

Separable differential equations of first order, Linear first-order differential equations, Second-order homogeneous linear equations with constant coefficients, Examples: the classical harmonic oscillator and the particle in a one-dimensional box and in a ring, Second-order inhomogeneous linear equations.

Separation of variables, Examples: the particle in a rectangular box and in a circular box.

The harmonic oscillator and the rigid rotor

Hooke's law; diatomic molecules, reduced mass, harmonic oscillator approximation; energy levels of the harmonic oscillator; harmonic oscillator model and vibrational spectra of diatomic molecules; Hermite's polinomials; the rigid rotor; molecular rotation of diatomic molecules.

Hydrogen and hydrogen-like atoms

Hamiltonian and wave functions of the H atom. Separability in three 1D wave functions; angular part and spherical harmonics, Y( q , f ) ; Legendre equations, Legendre polynomials and Legendre associated functions; Ylm( q , f ) as wave functions of L2; properties of the components of the angular momentum; commutation between L and its components; radial wave functions, R(r); Overall wave functions Y nlm (r, q , f ) ; meaning of Y nlm and orbitals; R(r), R(r)*R(r) e 4 p r 2 R(r) * R(r) ; p ± 1 e px py orbitals.

Variational principle and  perturbation theory

Definition of the variational principle. Simple examples. He atom. Linear combinations of know functions to set up a trial function. The secular determinant. 1º and 2º order perturbation theory. Application to the He atom.

Multi-electrons atoms

Electronic interaction term. Atomic units Hamiltonian. Slater functions. Hartree-Fock limit and correlation energy. Electronic spin. Spin-orbit coupling and fine structure. Spin wave functions. Overall wave functions and symmetry properties. 6° postulate. Representation of asymmetric wave functions  with Slater determinants. Atomic term symbols. Quantum numbers L, S, J. Electronic configurations Hund rules. Selection rules. Zeeman effect.

Diatomic Molecules.

H2+ and H2 Hamiltonians. Born-Hoppenheimer approximation. Molecular orbitals theory. Overlapping, Coulomb, and exchange integrals. Bonding and anti-bonding orbitals. LCAO-MO treatment of H2.  Energy classification of molecular orbitals. Simmetry of molecular orbitals of omonuclear diatomic molecules. Configurations of I and II groups omonuclear molecules. Eteronuclear diatomic molecules. SCF-LCAO-MO method. Molecular electronic states and molecular term symbols. Symmetry properties. Excited electronic states.

Polyatomic molecules.

Hybrid orbitals. Electronic configurations and structures of H2O and BeH2. Walsh correlation diagram. pelectrons. Hückel theory of molecular orbitals. Butadiene and delocalization energy. Electronic states of benzene and pyrazine.

Laboratory work.

Writing a Fortran computer program to calculate the level's energies for the hydrogen atom, and its applications. Recording and interpretation of an atomic spectrum.

Readings/Bibliography

• Lecture notes on Group theory and simmetry, L. DORE, Pitagora 2013, 2nd ed.
• The Chemistry Math Book, E. STEINER, Oxford, 1997.
• Chimica Fisica, D. A. McQUARRIE e J. D. SIMON, Zanichelli (Bologna), 2000.
• Chimica Fisica, G. K. VEMULAPALLI, EdiSES (Napoli), 1998.

Teaching methods

The course is organized in two learning modules: Mathematical Methods for Chemistry (5 credits)  and Atomic and Molecular Structure (5 credits). Classes of the first module are given in the first semester; classes of the second module are given in the second semester.

Classes are organized as lectures in the classroom,  in-class exercises and, for the second part only, laboratory exercises.

Assessment methods

Learning assessment is evaluated only by means of the final examination. This aims at verifying the student's knowledge and skills by means of one test for each learning module.

For the first module there is first a written examination with exercises, which lasts 3 hours.  A minimum grade of 16/30 is required for the admission to the oral exam, where, after a discussion of the written test, two questions concerning the course contents are asked to the student.

For the second module, the student may choose to take a written exam or an oral one, both involving three questions and the solution of one exercise. If the students takes the written exam and obtain a low grade, but no less than 18/30, he is allowed to take also the oral exam to improve his grade.

The final grade is the arithmetic mean of the grades obtained for each learning module.

Teaching tools

Video projector, notebook, blackboard.

Office hours

See the website of Luca Dore

See the website of Walther Caminati