The Born - The Born-Oppenheimer Approximation We know that...

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The Born-Oppenheimer Approximation We know that if a Hamiltonian is separable into two or more terms, then the total eigenfunctions are products of the individual eigenfunctions of the separated Hamiltonian terms, and the total eigenvalues are sums of individual eigenvalues of the separated Hamiltonian terms. Consider, for example, a Hamiltonian which is separable into two terms, one involving coordinate and the other involving coordinate . (163) with the overall Schrödinger equation being (164) If we assume that the total wavefunction can be written in the form , where and are eigenfunctions of and with eigenvalues and , then (165) Thus the eigenfunctions of are products of the eigenfunctions of and , and the eigenvalues are the sums of eigenvalues of and . If we examine the nonrelativistic Hamiltonian ( 162 ), we see that the term
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(166) prevents us from cleanly separating the electronic and nuclear coordinates and writing the total wavefunction as , where represents the set of all electronic coordinates, and
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This note was uploaded on 11/22/2011 for the course CHEMISTRY CHM1025 taught by Professor Laurachoudry during the Fall '10 term at Broward College.

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The Born - The Born-Oppenheimer Approximation We know that...

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