554017 Advanced Computational Chemistry

Description

This course addresses the methods used in ab initio molecular electronic-structure theory. Its focus is at wave-function-based methods and their origin in fundamental quantum mechanics.

The course covers the following topics:

• Exact and approximate many-electron wave functions and their physical interpretation
• One-electron basis sets
• Molecular integral evaluation
• Second quantization formalism
• Solution of the Hartree-Fock equations
• Configuration interaction, coupled-cluster and perturbation theories
• Numerical benchmarking and calibration of the considered models

The course is targeted at graduate or advanced undergraduate students in chemistry or physics with preliminary knowledge of quantum mechanics and an interest in working in the fields of computational and theoretical chemistry. Participants who are already familiar with quantum chemistry will deepen their knowledge on the fundamental basis of the commonly used methods, while students of other disciplines can get hands-on experience on the methods for solving the Schrödinger equation in the case of many interacting particles.

• Credits: 4 ECTS. Successful examination or a project work as well as a couple of obligatory exercises are required for the credits.
• Lecturer: Dr. Pekka Manninen, docent (email: pekka.manninen(at)csc.fi), tel. +358-50-3819039.
• Language: English if necessary; otherwise Finnish.
• Lectures: Mondays 3.30 pm to 5 pm; 30 hours in total. First lecture Jan 12 2009. Room A120 (Chemicum building, Kumpula campus).
• Literature: Selected parts of T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-Structure Theory (Wiley, 2002).
• Prerequisities: Either of the courses Introduction to computational chemistry (Johdatus laskennalliseen kemiaan) or Computational chemistry (Laskennallinen kemia) is recommended, but one or more courses in basic quantum mechanics and/or quantum chemistry is sufficient.

Course material

Lecture notes (pdf) (1.1 MB) for the course. Last update Jan 11 2009.

Examples of project work topics

The following are suggestions for project works. Feel free to suggest modifications into them or a topic that interests you more and/or fits to your current research.

Correlation-consistent basis sets In this project, you will acquaint yourself with the correlation-consistent basis sets, which are the most "physical" Gaussian basis sets in the literature, and present

• a survey of the underlying ideas
• the basis-set convergence in Hartree--Fock calculations on some small molecules for the electronic energy and geometrical parameters using the correlation-consistent hierarchy; as well as the correlation energy obtained by the MP2 and/or CCSD level of theory
• address the core-polarization and augmentation of the basis set.

Configuration state functions As discussed in the lectures, the exact non-relativistic wave function is an eigenfunction of the total and projected spins, whereas the Slater determinants are eigenfunctions of the projected spin only. We can set up a basis of functions that are simultaneous eigenfunctions of the orbital occupation-number operators as well as the operators for the projected and total spins - such spin-adapted functions are called configuration state functions (CSF).

• Consider the concept and properties of CSFs
• Present the construction of CSFs using the sc. genealogical coupling scheme
• Consider the transformations between determinant and CSF bases.

Size-extensivity

• Discuss the concept and the meaning of the size-extensivity in many-electron wave functions
• Consider the size-extensivity in exact wave functions as well as in approximate wave functions constructed by linear and exponential parameterization
• Give numerical examples on the issue.

Basis-set superposition error: In this project, you will familiarize yourself with a feature of finite basis sets not discussed on the lectures: the basis-set superposition error.

• Present the general concepts connected to the BSSE
• Discuss the counterpoise correction
• Include numerical estimates and discussion on BSSE in some rare-gas dimer.

CC2 model

• Derive the explicit form of the CC2 amplitude equations for a closed-shell case
• Demonstrate the performance of the CC2 model in
  • Total electronic energies
  • Bond distances and bond angles
  • Molecular dipole moments
  of a few small molecules and compare them with experimental results as well as other standard models. Remember to address the basis-set convergence too.

Operation CCSD overload

• derive the explicit (i.e. "implementation-ready") forms for the closed-shell CCSD wave function using the T1-transformed Hamiltonian
• carry out a highly detailed examination of the molecular energy of some small molecule (e.g., water or hydrogen fluoride) addressing
  • the basis-set convergence
  • the composition of the total energy.

Treatment of O₃ with multi-configurational wave functions

• Present the theory behind the CASSCF and CASPT models
• Build up a hierarchy of CASSCF wave functions for the ozone molecule and compare the obtained molecular geometry with the experimental results. Address the basis-set issues as well.
• Apply the CASPT2 theory to refine the description of dynamical correlation.

Møller-Plesset perturbation theory

• Present the general theory behind the Møller-Plesset theory and
  • Give the explicit forms of the MP1 and MP2 wave functions and the MP2 and MP3 energy expressions
  • Discuss the size-extensivity of MPPT
• Specialize the treatment to the closed-shell case
• Discuss the convergence in perturbation theory.

A detailed and well-written report (in English or Finnish) is expected together with a short presentation at the end of the course. The projects may be carried out in groups of two or three, but of course doing them solely is possible too.