Ab Initio and DFT Electronic Structure Methods

Abstract

Today's lecture: the structure of atoms, how atoms interact to form molecules, and how molecules interact with each other… all at the "first principles" level, i.e., no empirical constants or experimentally-derived information. Basics of quantum theory Neglecting relativistic effects, all matter is described at a fundamental level by quantum theory. The central feature of this theory is the multi-body wavefunction: Ψ , , … , Here, , , …, etc. are the positions of all fundamental particles in the system (electrons, protons, neutrons) and each is a vector The wavefunction evolves in time, . The wavefunction takes on complex values, of the form + For a single particle, such as a single electron, we have: ℘ ∝ Ψ * Quantum mechanics determines the wavefunction up to an arbitrary multiplicative constant. Therefore, we can normalize the wavefunction by demanding that the probabilities sum to one when integrated over the entire space of coordinates available to every particle. For a single particle, The wavefunction describes the evolution of probabilities. This is very different from Newtoni-an mechanics, in which each particle has an exact position at time , and not a distribution of probable positions. Quantum mechanics says that this distribution is the most we can possibly know about the system; we cannot predict the position of the particles to more accuracy. There is some inherent randomness in nature, and the best we can do is predict the probabili-ties of different possible outcomes. This may sound a bit strange, because we are not used to this kind of behavior at the macro-scopic scale. Indeed, for large objects, these probability distributions are very narrowly peaked relative to the object size, such that we can usually say, from a macroscopic scale of measure-ment, exactly where an object lies in space. For small objects, like atoms and electrons, these distributions become significant. The wavefunction is determined by the Schrodinger equation, the quantum-mechanical analo-gy of Newton's equations of motion. For a single particle traveling in a potential energy field, Schrodinger's equation reads: