Download Symmetry and Group Theory in Chemistry - Lecture Slides | CHEM 634 and more Study notes Chemistry in PDF only on Docsity! CHEMISTRY 673 Symmetry and Group Theory in Chemistry Tim Hughbanks B.S. in Chemistry, U. of Washington, 1977 Ph.D., Cornell, 1983 Faculty member at TAMU since 1987 Office: Chemistry Building, Room 330 Office phone: 845-0215 Office Hrs: Tues. 2:00 - 4:00 PM Other times are OK too! e-mail: trh@mail.chem.tamu.edu 1 2 CHEMISTRY 673 This course is for 3 credits. Lecture: 2 × 75 min/week; TTh 11:10 - 12:25, Room 2122 Web site: http://www.chem.tamu.edu/rgroup/ hughbanks CHEMISTRY 673 Grades will be based on the homework (roughly 50%), midterm and final exams Class web site: http://www.chem.tamu.edu/ rgroup/hughbanks/courses/673/ chem673.html 3 4 What is Group Theory? A fairly “recent” branch of mathematics. Early principles were developed by Évariste Galois (killed in a duel in 1832 at age 21), and Niels Abel (died in 1829 at age 26 of TB). First formal definition of a group was given by Cayley in 1854. Fedorov pioneered the application of group theory to crystallography. Group Theory is the closest many chemists get to truly “modern” mathematics. Properties of Groups Closure: “product” of any two group elements (operations) is a group element (operation), including squares One element, the identity, commutes with all others Associative property holds (commutative property does not necessarily hold) Every element (operation) has an inverse — which is also a group element (operation) 9 10 Simple Examples The integers, under the operation of addition? The integers, under the operation of multiplication? Relevant Example: A simple symmetry group, C2v – what are the elements (operations)? – how do we define a product? Another Relevant Example: Rotation Matrices ‣ Claim: The set of all 2×2 matrices of the form forms an continuous, infinite-order group, where the product is assumed to be defined by the usual definition of matrix multiplication. ‣ Proof ? cosθ − sinθ sinθ cosθ ⎡ ⎣⎢ ⎤ ⎦⎥ 11 12 Symmetry Elements vs. Operations Mathematically, the members of a group are called “elements” In symmetry groups these “elements” are called “operations” - the term “element” is reserved for something else: The term “symmetry element” refers to a geometrical entity (a point, a line or axis, or a plane) about which the operation is defined. The Symmetry Operations Reflection (in a plane) σ Inversion (through a point) i Rotation (about a proper axis) Cn - through an angle 2π/n Improper Rotation Sn (about an improper axis) Identity (do nothing) E 13 14
