385lec32 - PHYS 385 Lecture 32 - Molecules I 32 - 1 ©2003...

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Unformatted text preview: PHYS 385 Lecture 32 - Molecules I 32 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 32 - Molecules I What's important : • hydrogen molecular ion • Born-Oppenheimer approximation Text : Gasiorowicz, Chap. 20 Out next task is to extend the Schrödinger equation for the Coulomb problem to molecules, where the attractive potential experienced by bound electrons is no longer a single point nucleus, but rather two or more separate nuclear charges. Further, the separation between the nuclei is itself determined by the presence of the electrons - it is not fixed a priori . We consider here only the simplest molecule possible, namely the hydrogen molecular ion, consisting of a single electron bound by two separate nuclei. The state H 2 + has an internuclear separation of 1.060 Å, but is weakly bound at 2.791 eV. By bound, we mean the energy difference E (H 2 + ) - [ E (H atom ) + E (p)], i.e. , the energy difference between the molecular state and its component atoms. The simplest molecule H 2 + The potential energy of this molecule involves only three position vectors r A , r B and R AB : Proceeding as with the helium atom, the Hamiltonian operator can be written H = -( h 2 /2 m A ) ∇ A 2- ( h 2 /2 m B ) ∇ A 2- ( h 2 /2 m e ) ∇ e 2- ke 2 / r A- ke 2 / r B + ke 2 / R AB . This system has a total of 9 degrees of translational freedom, three for each of the particles. The potential energy contributions depend only on the separation between protons and electrons, so the centre of mass variables can be extracted immediately. protons and electrons, so the centre of mass variables can be extracted immediately....
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This note was uploaded on 09/07/2009 for the course PHYS 385 taught by Professor Davidboal during the Spring '09 term at Simon Fraser.

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385lec32 - PHYS 385 Lecture 32 - Molecules I 32 - 1 ©2003...

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