Symmetry properties of molecular orbitals

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Presentation transcript:

Symmetry properties of molecular orbitals PHY 745 Group Theory 11-11:50 AM MWF Olin 102 Plan for Lecture 15: Symmetry properties of molecular orbitals Reading: Chapter 7 in DDJ Linear combinations of atomic orbitals Molecular bonds Examples 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

2/15/2017 PHY 745 Spring 2017 -- Lecture 15

2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Electronic structure of an atom For simplicity we will first consider a single electron system; a H-like ion with atomic charge Ze E100=-13. 60569253 Z2 eV a0 = 0. 52917721092 Å 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Example – 1s state of hydrogen-like atom 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Electronic structure of H-like molecular ion (within Born-Oppenheimer approximation) -e rB rA RAB ZAe ZBe 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Electronic structure of H-like molecular ion – continued Ref. Pauling and Wilson, Introduction to Quantum Mechanics (1935) (now published by Dover) 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Electronic structure of H-like molecular ion – continued antibonding Y- E- HAA HBB E+ bonding Y+ 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

LCAO methods -- continued – angular variation http://winter.group.shef.ac.uk/orbitron/ l=0 l=1 l=2 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

LCAO methods -- continued – angular variation While, for atoms the “z” axis is an arbitrary direction, for diatomic molecules and for describing bonds, it is convenient to take the “z” axis as the bond direction. Atom symbol l=0 m=0 s l=1 m=0 p m= ±1 p l=2 m=0 d m= ± 1 d m= ± 2 d Bond symbol l=0 l=0 s l=1 l=0 s l=1 p l=2 l=0 s l= 2 d 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

LCAO methods -- continued – bond types sss pps ppp ddp 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Extension of ideas to polyatomic molecules “atomic site” representation 1 Example: C3v 3 2 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Relationship of atomic site representation to irreducible representations for C3v atomic site character 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Diagram (and following) from DDJ textbook (earlier version) Atomic site symmetry together with internal degrees of freedom such as LCAO Diagram (and following) from DDJ textbook (earlier version) 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Character table 3 3 0 0 1 1 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Diagram from DDJ textbook (earlier version) Atomic site symmetry together with internal degrees of freedom such as LCAO Diagram from DDJ textbook (earlier version) 2/15/2017 PHY 745 Spring 2017 -- Lecture 15

Analysis of LCAO symmetries for this case 2/15/2017 PHY 745 Spring 2017 -- Lecture 15