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Unformatted text preview: Chem 120A The Born-Oppenheimer Approximation 04/07/06 Spring 2006 Lecture 30 READING: Engel (QCS): Ch. 12 and Ch. 15.4 A most important approximation: the Born-Oppenheimer Approximation The Born-Oppenheimer (BO) approximation is the central approximation used in describing the behav- ior of molecules. This approximation makes use of the disparity between nuclear and electron masses, M N ∼ 2000 m e . This disparity in the masses allows us to separate the electron and nuclear degrees of freedom in order to approximately solve the Schr¨odinger equation for a molecule. The total Hamiltonian for a molecular system is a sum of the kinetic energies of all the electrons and nuclei and all of the relevant two-body interaction terms. In position representation: ˆ H = − ∑ i − ¯ h 2 2 m e ∇ 2 r i + ∑ α − ¯ h 2 2 M α + 1 2 ∑ i 6 = j e 2 | r i − r j | − 1 2 ∑ α 6 = β Z α Z β e 2 | R α − R β | − ∑ α , i Z α e 2 | R α − r i | , (1) where α represents the α th nucleus of mass M α and charge Z α e , ~ R α is the position of the α th nucleus, and i represents the i th electron of mass m e with position ~ r i . We can write this Hamiltonian as a sum of separate terms, with one term for the kinetic energies of the nuclei and another term that contains all of the interactions and the kinetic energies of the electrons: ˆ H ≡ ˆ T N + ˆ H e (2) where ˆ T N = − ∑ α ¯ h 2 2 M N ∇ 2 R α ( nuclear kinetic energy ) (3) and ˆ H e = ˆ T e + V ee + V eN + V NN , (4) where ˆ T e = − ∑ i ¯ h 2 2 m e ∇ 2 r i ( electron kinetic energy ) , V ee = 1 2 ∑ i 6 = j e 2 | r i − r j | ( electron − electron interactions ) , V eN = − ∑ α , i Z α e 2 | R α − r i | ( electron − nucleus interactions ) , and V NN = 1 2 ∑ α 6 = β Z α Z β e 2 | R α − R β | ( nucleus − nucleus interactions ) ....
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