Group properties of atomic orbitals and crystals

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Presentation transcript:

Group properties of atomic orbitals and crystals PHY 745 Group Theory 11-11:50 AM MWF Olin 102 Plan for Lecture 13: Group properties of atomic orbitals and crystals Reading: Chapter 5 in DDJ Correction: Cambell-Baker-Hausdorff equation Group properties of atomic orbitals Atomic orbitals in crystal fields 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Correction to previous lecture notes 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Symmetry properties of electronic states in an atom 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Geometric forms of spherical harmonics (real forms) https://www.mathworks.com/matlabcentral/mlcdownloads/downloads/submissions/43856/versions/9/screenshot.jpg 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Geometric forms of spherical harmonics (real forms) l=2 https://www.mathworks.com/matlabcentral/mlcdownloads/downloads/submissions/43856/versions/9/screenshot.jpg 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

2/10/2017 PHY 745 Spring 2017 -- Lecture 13

2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Using character tables to analyze the “crystal field” effects on atomic symmetry 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Example for the group Oh (48 group members) 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Compatibility of continuous rotation group with the cubic group: 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

l=0 1 1 1 1 1 1 1 1 1 1 l=1 3 -1 1 -1 0 -3 1 -1 1 0 l=2 5 1 -1 1 -1 5 1 -1 1 -1 G1 G15 G12+G25’ 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Visualization of l=2 orbitals from http://chemwiki. ucdavis G12 G25’ Octahedral symmetry 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Octahedral symmetry effects on d-orbitals G12 =eg G25’=t2g Energy diagram 2/10/2017 PHY 745 Spring 2017 -- Lecture 13

Tetrahedral symmetry effects on d-orbitals G12 =eg G25’=t2g 2/10/2017 PHY 745 Spring 2017 -- Lecture 13