5210_chap10 - Chapter 10: Slide 1 Homonuclear Diatomic...

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Unformatted text preview: Chapter 10: Slide 1 Homonuclear Diatomic Molecules Chapter 10: Slide 2 Outline Math Prelim.: Systems of Linear Equations Cramers Rule MO Treatment of the H 2 Molecule LCAO Treatment of H 2 + Homonuclear Diatomic Molecules H 2 + Energies H 2 + Wavefunctions Heteronuclear Diatomic Molecules Hydrogen Molecular Ion: Born-Oppenheimer Approximation Chapter 10: Slide 3 Hydrogen Molecular Ion: Born-Oppenheimer Approximation The simplest molecule is not H 2 . Rather, it is H 2 + , which has two hydrogen nuclei and one electron. a b e R ab r a r b The H 2 + Hamiltonian (in au) ab b a e b b a a R r r M M H 1 1 1 2 1 2 1 2 1 2 2 2 +-- - - - = KE Nuc a KE Nuc b KE Elect PE e-N Attr PE e-N Attr PE N-N Repuls Chapter 10: Slide 4 Electrons are thousands of times lighter than nuclei. Therefore, they move many times faster Born-Oppenheimer Approximation The Born-Oppenheimer Approximation states that since nuclei move so slowly, as the nuclei move, the electrons rearrange almost instantaneously. With this approximation, it can be shown that one can separate nuclear coordinates (R) and electronic coordinates (r), and get separate Schrdinger Equations for each type of motion. ) ( ) ( 1 2 1 2 1 2 2 ab nuc ab el ab b b a a R E R E R M M = + + - - Nuclear Equation E el is the effective potential energy exerted by the electron(s) on the nuclei as they whirl around (virtually instantaneously on the time scale of nuclear motion) Chapter 10: Slide 5 Electronic Equation E E r r elect b a = = -- - 1 1 2 1 2 Because H 2 + has only one electron, there are no electron-electron repulsion terms. In a multielectron molecule, one would have the following terms: 1. Kinetic energy of each electron. 1. Attractive Potential energy terms of each electron to each nucleus. 1. Repulsive Potential energy terms between each pair of electrons Chapter 10: Slide 6 Outline Math Prelim.: Systems of Linear Equations Cramers Rule MO Treatment of the H 2 Molecule LCAO Treatment of H 2 + Homonuclear Diatomic Molecules H 2 + Energies H 2 + Wavefunctions Heteronuclear Diatomic Molecules Hydrogen Molecular Ion: Born-Oppenheimer Approximation. Chapter 10: Slide 7 Systems of Linear Equations: Cramers Rule 2 2 22 1 21 1 2 12 1 11 c x a x a c x a x a = + = + In these equations, the a ij and c i are constants. We want to solve these two equations for the values of the variables, x 1 and x 2 Cramers Rule 22 21 12 11 22 2 12 1 1 a a a a a c a c x = ts coefficien of matrix original of Det matrix Augmented" " of Det = 22 21 12 11 2 12 1 11 2 a a a a c a c a x = ts coefficien of matrix original of Det matrix Augmented" " of Det = Chapter 10: Slide 8 2 2 22 1 21 1 2 12 1 11 c x a x a c x a x a = + = + A Numerical Example 22 21 12 11 22 2 12 1 1 a a a a a c a c x = 22 21 12 11 2 21 1 11 2 a a a a c a c a x = 11 2 3 1 5 2 2 1 2 1 =- = + x...
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This note was uploaded on 04/11/2011 for the course CHEM 5210 taught by Professor Staff during the Spring '08 term at North Texas.

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5210_chap10 - Chapter 10: Slide 1 Homonuclear Diatomic...

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