Quantum Mechanics of Molecular Systems

Schrödinger equation and wavefunctions

particle in a box
harmonic oscillator
anharmonic corrections
rigid rotor

molecular states

Born-Oppenheimer approximation

Slater-determinants
electron structure calculations for molecular systems (LCAO-MO)

electron-vibration coupling

transitions between states,

semiclassical curve crossing

Landau-Zener model

time dependent perturbation theory

Fermi's golden rule
optical transitions

After participating, the students are able to apply simple quantum mechanical models to molecular systems to analyse molecular states and transitions.

They are able

-  to describe Pi-electron systems of molecules with conjugated double bonds within the free electron model

- to formulate the Hamiltonian of a harmonic oscillator with ladder operators and to solve the time independent Schrödinger equation

- describe anharmonic effects with perturbation theory

- determine localized wave packets which solve the time dependent Schrödinger equation for free particles and particles in a harmonic potential

- to formulate the Hamiltonian of a molecular system and to apply the Born-Oppenheimer approximation to separate the motion of electrons and nuclei

- to describe the ground state of a many electron system with a Slater determinant

- to describe modern electron structure methods

- to apply the quasiclassical approximation to molecular transitions and to derive the Landau Zener rate expression with perturbation theory

- to derive the rate expression for molecular transitions into a continuum of final states and to apply it to optical transitions

- to interpret optical spectra of larger molecules on the basis of electron-vibration coupling

basic quantum mechanics of Bachelor level

The module consists a lecture and exercises.

During the lecture the learning content is presented. Necessary mathematical methods are explained and important theoretical results are derived explicitly. Functional relationships are shown with graphics and computer examples. Theoretical results are compared with experimental data from the literature with the help of computer presentations. After the lecture there is time for discussion.

Numerous problem examples with solutions deepen the learning content in the exercises. Here the mathematical derivations are discussed in more detail and their application is exercised using selected problem examples and calculations. Thus the students are able to explain and apply the learned knowledge on their own.

A series of interactive applets are introduced in the lecture and serve for individual studies visualizing functional relationships and the  dependency  of the theoretical results on the relevant parameters

additional notes and literature references are provided for further deepening of the learning content

Blackboard, laptop/projector, lecture notes, exercises and examples, Java programs, extra material (additional notes)

• P.O.J. Scherer & S.F. Fischer: Theoretical Molecular Biophysics, Springer-Verlag, (2017)
• H. Haken & H. Wolf: Molekülphysik und Quantenchemie, Springer-Verlag, (2006)
• F. Schwabl: Quantenmechanik, Springer-Verlag, (2007)
• Lecture notes

There will be an oral exam of 25 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using comprehension questions and sample calculations.

For example an assignment in the exam might be:

• describe the Pi-electron system of a molecule with conjugated double bonds within the free electron model
• formulate the Hamiltonian of a harmonic oscillator with ladder operators and to solve the time independent Schrödinger equation
• describe anharmonic effects with perturbation theory
• determine localized wave packets which solve the time dependent Schrödinger equation for free particles and particles in a harmonic potential
• formulate the Hamiltonian of a molecular system and to apply the Born-Oppenheimer approximation
• describe the ground state wavefunction of a many electron system
• describe modern electron structure methods
• apply the semiclassical approximation to molecular transitions and derive the Landau Zener rate expression with perturbation theory
• derive the rate expression for molecular transitions into a continuum of final states and apply it to optical transitions
• interpret optical spectra of larger molecules on the basis of electron-vibration coupling