lecture 35 - Chem 120A Spring 2007 Born-Oppenheimer...

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Chem 120A Born-Oppenheimer Approximation, part II 04/20/07 Spring 2007 Lecture 35 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 (1) = T N + H e ( q , Q ) (2) 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 in the molecule 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 ) (3) 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 ) (4) 4. Finally, we solve the nuclear Schr¨odinger equations to obtain the total energy E n ν and the nuclear wavefunction φ n ν (The total wavefunction of the molecule is the product of the electronic and nuclear wavefunctions Ψ n ν = ψ n φ n ν ). ( T
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lecture 35 - Chem 120A Spring 2007 Born-Oppenheimer...

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