sec12 - XII The Born-Oppenheimer Approximation The...

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XII. The Born-Oppenheimer Approximation The Born- Oppenheimer (BO) approximation is probably the most fundamental approximation in chemistry. From a practical point of view, it will allow us to treat the electronic structure of molecules very accurately without worrying too much about the nuclei. However, in a more fundamental way, it underpins the way that most chemists think about molecules. Any time you see a chemist draw a picture like the one at right, you are implicitly making use of the framework suggested by the Bon-Oppenheimer approximation. So we are going to spend some time talking about this approximation and when we do and do not expect it to be valid. a. The Adiabatic Approximation For any molecule, we can write down the Hamiltonian in atomic units ( 1 = = = e m e p ) as (defining β α αβ r r r - , etc.) : - - + + - - = electrons ij i I I i I ij electrons i nuclei IJ IJ J I i nuclei I I I Z r R Z Z M H , 2 1 2 1 2 2 1 2 2 1 1 1 ˆ R r R E Nuclear Kinetic Energy Electronic Kinetic Energy Nuclear- Nuclear Repulsion Electron- Electron Repulsion Electron- Nuclear Attraction
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The physical motivation behind the Born-Oppenheimer Approximation is that the nuclei are much heavier than the electrons (e.g. a proton is 1800 times as heavy as an electron). At any given instant, the electrons will “feel” a Hamiltonian that depends on the position of the nuclei at that instant: ( ) - - + + - = electrons ij i I I i I ij electrons i nuclei IJ IJ J I i el Z r R Z Z H , 2 1 2 1 2 2 1 1 ˆ R r R . Where R denotes the dependence of el H ˆ on all of the nuclear positions { } I R at once. In the limit that the nuclei are infinitely massive, they will never move and the positions I R in the above expression will be fixed; i.e. the molecule will be frozen in some particular configuration. In this case, the I R ’s can be considered as parameters (rather than operators) that define the effective Hamiltonian for the electrons. For any fixed configuration of the molecule, then, one is interested in solving a Schrödinger equation that involves only the electronic degrees of freedom: ( ) ( ) ( ) ( ) R R R R el el el el E H Ψ = Ψ ˆ where we have noted explicitly that the Hamiltonian, its eigenstates and eigenvalues depend on the particular nuclear configuration. This is the key element of the BO approximation; it allows one to compute the electronic structure of a molecule without saying anything about the quantum mechanics of the nuclei. Once we have solved the electronic Schrödinger equation, we can write down the effective Hamiltonian for the nuclei by simply adding back in the terms that were left out of el H ˆ : ( ) R el nuclei I I I N E M H + - = 2 2 1 1 ˆ Hence, the nuclei move on an effective potential surface that is defined by the electronic energy, and we can define wavefunctions for the nuclei alone that are eigenfunctions of this Hamiltonian: ( ) N N el nuclei I I I N N E E M H Ψ = Ψ
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sec12 - XII The Born-Oppenheimer Approximation The...

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