Lecture31Notes-Part2 - Chem 120A Spring 2006 Approximation...

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Chem 120A Approximation Examples, part II 04/12/06 Spring 2006 Lecture 31 1 Review of the Born-Oppenheimer Approximation Here we will outline the basic steps in using the Born-Oppenheimer approximation. 1. We begin by separating out an electronic Hamiltonian, H e ( q , Q ) from the total Hamiltonian, H ( q , Q ) . H ( q , Q ) = T N + T e + V ee + V NN + V eN = T N + H e ( q , Q ) (1) where T N is the nuclear kinetic energy, T e is the electron kinetic energy, V ee are the electron-electron interactions, V NN are the nucleus-nucleus interactions, and V eN are the electron-nucleus interactions. Here q describes the electronic cooridinates and Q describes the nuclear coordinates. The nuclear coordinates act as parameters in the electronic Hamiltonian. 2. Next for each nuclear configuration (parametrized by Q) we can solve the electronic Schr¨odinger equation to obtain the electronic energies H e ( q , Q ) ψ n ( q , Q ) = ε n ( Q ) ψ n ( q , Q ) (2) 3. For each nuclear configuration we can write a different nuclear Hamiltonian which incorporates the electronic energy eigenvalue, ε n ( Q ) , as a potential energy term. Thus each electronic energy be- comes a different potential energy surface for the nuclei (which is sometimes referred to as a Born- Oppenheimer surface). H n ( Q ) = T N + ε n ( Q ) (3) 4. Finally we can solve the nuclear Schr¨odinger equations ( T N + ε n ( Q )) φ n / nu = E n ν φ n ν (4) When can we use the B.O. approximation? For B.O. to be valid, it must be true that the electronic motion is fast compared to the nuclear motion so that the nuclear kinetic energy can be neglected when determining the electronic wavefunction. Thus B.O. relies on the fact that m e
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Lecture31Notes-Part2 - Chem 120A Spring 2006 Approximation...

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