lecture 34 - Chem 120A The Born-Oppenheimer Approximation,...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chem 120A The Born-Oppenheimer Approximation, part I 04/18/07 Spring 2007 Lecture 34 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 ∇ 2 R α + 1 2 ∑ i negationslash = j e 2 | r i − r j | − 1 2 ∑ α negationslash = 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 , vector R α is the position of the α th nucleus, and i represents the i th electron of mass m e with position vector 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 negationslash = 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 negationslash = β Z α Z β e 2 | R α − R β | ( nucleus − nucleus interactions ) ....
View Full Document

This note was uploaded on 02/19/2010 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at Berkeley.

Page1 / 4

lecture 34 - Chem 120A The Born-Oppenheimer Approximation,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online