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See discussions, stats, and author profiles for this publication at: Lecture Notes: Computational ChemistryData· January 2016DOI: 10.13140/RG.2.1.1157.1282CITATIONS0READS9211 author:Some of the authors of this publication are also working on these related projects:Intramolecular hydrogen bonding and optical transitions for selected flavonoids.View projectLecture NotesView projectJens Spanget-LarsenRoskilde University237PUBLICATIONS2,141CITATIONSSEE PROFILEAll content following this page was uploaded by Jens Spanget-Larsen on 06 January 2016.The user has requested enhancement of the downloaded file.
Lecture Notes: Three Lectures in Computational Chemistry Jens Spanget-Larsen RUC 2012-16 Largely based on a chapter by Warren J. Hehre in the textbook by Thomas Engel: “Quantum Chemistry and Spectroscopy”, Pearson–Benjamin-Cummings, 2006.
1 Computational Chemistry Lecture Notes 1 (RUC, NSM, September 2014) Isolated molecule Born-Oppenheimer approximation Potential energy surface Nuclear eigenvalue problem Electronic eigenvalue problem Molecular Mechanics MO theory The LCAO-MO procedure
2 Isolated molecule At this point, the system under consideration is a collection of mutually interacting particles, i.e., nuclei and electrons, a structureless “plasma”. We seek solutions to the time-independent, non-relativistic Schrödinger equation,)()(ˆRr,Rr,enenenenEHΨ=Ψ, involving the molecular Hamilton operator (atomic units, au): kinetic attraction repulsion repulsion kinetic energy between between between energy of elec- electrons electrons nuclei of nuclei trons and nuclei The molecular wavefunction)(Rr,enΨis a dynamical function of the coordinates of all electrons (r) and all nuclei (R). The eigenvalue problem is a many-body problem and cannot be solved exactly. Born-Oppenheimer approximation The nuclei are much heavier than the electrons. Hence, the electrons move very much faster than the nuclei. In the Born-Oppenheimer approximation, the motion of the electrons is decoupled from that of the nuclei, and the molecular eigenvalue problem is divided into two separate problems: One involving the motion of the electrons, and another involving the motions of the nuclei. The molecular Hamilton operator is divided into two parts,neenHHHˆˆˆ+=: In the electronic eigenvalue problem, the nuclei are considered as classical point charges at fixed positions in space. The nuclear coordinates Rare input parameters to the formulation of the electronic eigenvalue problem(see later), involving the electronic Hamiltonian,eHˆ. The solutions Ψe(r;R) and Ee(R) are called the electronic wavefunctionand electronic energy, respectively. They depend parametrically on the nuclear input coordinates RSolution of the electronic eigenvalue problem for a particular set of input coordinates Ris called a single point calculation, providing a “single point” on the potential energy surface (see below). In general, there are numerous solutions, corresponding to
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