66913 - Physical Chemistry 1

Course Unit Page

Academic Year 2022/2023

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 mechanicsfor studying the electronic properties, especially the energy levels,of atoms and simple molecules.

Course contents

  1. Vector spaces and linear transformations: Vector spaces, Matrix algebra, Matrices and Linear transformations, Determinants, Invertible matrices, Orthogonal matrices; Complex matrices, The eigenvalue problem, Similarity trasformations and diagonalization, Hermitian matrices. Function spaces.
  2. 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.
  3.  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.
  4.  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.
  5.  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.
  6.  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.
  7.  Variational principle and  perturbation theory: Definition of the variational principle. Simple examples. Linear combinations of know functions to set up a trial function. The secular determinant. 1º order perturbation theory.
  8.  Multi-electrons atoms: Electronic interaction term. Atomic units Hamiltonian. Electronic spin. Spin wave functions. Overall wave functions and symmetry properties. Atomic term symbols. Quantum numbers L, S, J. Electronic configurations Hund rules.
  9.  Laboratory work: Hints and tutorials for using MATLAB. Writing of MATLAB routine to solve the Schrödinger equation for the particle in the box with infinite and finite barriers. Application of the implemented calculation to practical problems such as:
    1. Energy of confinement of electrons in atoms, molecules, nanoparticles, viruses;
    2. Calculation of the absorption wavelength of conjugated molecules;
    3. Exercises on SPIN Operator with MATLAB


  • Lecture notes on Group theory and simmetry, L. DORE, Pitagora 2019, 4th ed.
  • The Chemistry Math Book, E. STEINER, Oxford, 2008, 2nd ed.
  • Molecular Quantum Mechanics, P.W. ATKINS and R.S. FRIEDMAN, Oxford, 2010.

Suggested readings:

  • Quantum Mechanics. The Theoretical Minimum, L. SUSSKIND and A. FRIEDMAN, Penguin Books, 2015.
  • Quantum Mechanics, G. AULETTA, M. FORTUNATO and G. PARISI, Cambridge University Press, 2009.

Teaching methods

The course is organized in three learning modules: Mathematical Methods for Chemistry (5 credits)  and Atomic and Molecular Structure (4 credits) with the additional module of laboratory work (1 credit). Classes of the first module are given in the first semester; classes of the second third  modules are given in the second semester.

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

All students must attend Module 1, 2 on Health and Safety, online.

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, there will be a written test where the student will have to solve some numerical problems followed by a viva examination. During the semester of the teaching, the student can opt for passing two written tests.

At the end of the lab experiences, students have to elaborate a laboratory report. The laboratory part will be evaluated on the basis of the lab report and will complement the grade of the second module.

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 Francesco Zerbetto

See the website of Stefania Rapino