Module description
Learning aims & outcomes
An increasing amount of science and technology nowadays concerns with processes at the nanometer scale, typically involving functionalised structures consisting of nano clusters and molecules. A time scale of a few picoseconds is the natural one to investigate the vibrational/conformational properties of these systems and the relevant steps of their synthesis/assembly mechanisms.
Such a high time/size resolution poses extremely demanding constraints to experimental techniques. Also, almost all experimental techniques are non-directional, i.e. their results require interpretation based on theoretical modelling. Therefore, detailed theoretical descriptions and the quantum-based numerical modelling have become indispensable tools in modern research on these kind of systems, either as a guide for interpreting the experimental observations, but also, and increasingly, as independent investigation tools capable of quantitative predictions that can be supported (or not) by posterior experiments. At these scales quantum mechanical approaches must be used to provide accurate potential energy surfaces and structural/configurational properties that are at the heart of a wide range of techniques within the material modelling toolbox, a rapidly growing area of atomistic-based theoretical modelling in nano-science, that have increasingly being used to study realistic materials, both in academia and industry.
This course provides an introduction to the most widely used material modelling techniques. The course introduces the physics of many electron and phonon systems with a particular focus on the practical techniques that are being used by scientists nowadays. While the main goal of the course is to provide a theoretical background on the structure and quantum behaviour of matter at the nano-scale, examples of applications given during the course involve modern concepts on the nano-scale behaviour of functional materials, and provide an accessible introduction to some of the main theoretical and computational techniques used to model processes involving surfaces, interfaces, clusters, and macromolecules.
On successfully completing this course, a student should:
- Be familiar with the fact that the physical properties of complex nano-systems can be described within a coherent quantum mechanical framework, in particular that the many-electron problem can be attacked by mean-field techniques of different levels of complexity and accuracy.
- Be familiar with a number of modern computational tools widely used in material modelling, such as Density Functional Theory, Hartree-Fock, molecular dynamics, Monte Carlo, kinetic Monte Carlo, Nudged Elastic Band, Transition State Theory, and so on.
- Understand how these theoretical tools can be used as a basis for accurate modelling of real materials and their properties to provide quantitative predictions at the nanometer/picosecond size- and time-scales.
Syllabus
1. Foundations: mean-field modelling of many electron systems.
The Hamiltonian operator. The many-body problem: the general Schroedinger equation problem. The particle exchange operator, symmetry of a two-body wave function with spin. Wave function classes constructed from spin orbitals.
Variational techniques. Fermions, anti-symmetry of the wave function. Pauli principle and Slater determinants; Hartree-Fock. Correlation energy. A brief introduction into quantum chemistry methods such as CI. Modern self-consistent approaches: elements of Density Functional Theory and its extension to the investigation of the dynamical electronic behaviour.
2. Potential energy surfaces and vibrations
Born-Oppenheimer approximation, non-adiabaticity. The Hellman-Feynman theorem and the concept of classical interatomic force-field. Introduction to machine learning: regression problem for classical force fields. Vibrations of molecules and crystals treated both classically and quantum-mechanically. Quasiharmonic approximation. Free energy and equilibrium crystal structure; configuration entropy.
3. Practical aspects of modern electronic structure methods
LCAO method. Non-orthogonal localised basis set vs. orthogonal plane wave basis. BSSE correction, Pulay forces. Some details of modern DFT codes.
4. Molecular dynamics
Statistical ensembles: Microcanonical, NVT, NPT and Grand canonical. Important thermodynamic averages. Correlation functions. Molecular dynamics (MD); ergodicity; numerical algorithms. First-Principles MD. Classical potentials, the problem of transferability. MD in microcanonical and NVT simulations. Nose and Langevin simulations. Static and dynamic properties via MD. Examples of application.
5. Monte Carlo
Random numbers and distributions. Importance sampling. Markov chain and Metropolis algorithm. NVT, NPT and GCMC simulations. Examples of application.
6. Modelling dynamics
Kinetic MC. Transition rates within the harmonic model. Classical Transition State theory. Nudged Elastic Band method for finding energy barriers. Examples of application.
Assessment details
Details of the module's assessment/s
Quizzes 10%
Literature Review/Computational Project/Literature Assessment 30%
May exam 60%
Please note: module assessment may be subject to change. If you have any questions, please contact ug-physics@kcl.ac.uk
Teaching pattern
Asynchronous recorded lectures (3 hours per week)