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In the Schrödinger equation
for an atom or molecule the Hamiltonian and the wave function depend on the electrons and the nuclei. In the Hamiltonian for an atom we can distinguish four terms
with Tn corresponding to the kinetic energy of the nucleus, Te to the kinetic energy of the electrons, Vne to the Coulomb attraction between the nucleus and the electrons, and V ee to the Coulomb repulsion between the electrons. For a molecule there is a fifth term.
| Hmol = Tn+Te+Vnn+Vne+Vee. |
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Vnn is the Coulomb repulsion between the nuclei. In the last expression Tn and Vne contain contributions of all nuclei of the molecule.
We see that mathematically there is no distinction between electrons and nuclei. When we think of a molecule, on the other hand, we do distinguish between them. A molecule has a well-defined structure; i.e., we assume that the nuclei have well-defined positions with respect to each other, which we associate with the positions of the atoms that form the molecule. The electrons do not have such well-defined positions. The reason why this is an acceptable way to think about molecules is the very different mass of the nuclei and the electrons. Even the lightest nucleus (the proton of a hydrogen atom) has a mass that is almost 2000 times that of an electron. As the forces on the electrons are of about the same magnitude as the forces on the nuclei, electrons move much more rapidly than nuclei. Nuclei are so sluggish that we can think of them as fixed at certain positions. Electrons move in a blur around them.
We can use this difference to simplify the Schrödinger equation and the way we deal with atoms and molecules in quantum mechanics. We start by giving the nuclei fixed positions. The most obvious choice is to pick them so the a molecule has its normal structure. For example, in a CO molecule the atoms are 1.23 Å apart, so we put the oxygen nucleus at a distance of 1.23 Å from the carbon nucleus. Because the nuclei are fixed, they have no kinetic energy and we drop Tn from the Schrödinger equation.
Fixing the nuclei also means that we do not need to solve the Schrödinger equation for them. We only have a Schrödinger equation for the electrons. This looks as follows.
with
In the electronic Hamiltonian Hel the kinetic energy of the nuclei Tn is missing, because of the reason mentioned above. The repulsion between the nuclei Vnn is removed as well. This is because the variables of the Schrödinger equation for the electrons are only the coordinates of the electrons, and Vnn does not depend on them. We can also say that the repulsion between the nuclei does not affect the electrons. The arguments of the electronic wave function Yel are only the coordinates of the electrons.
Solving the Schrödinger equation for the electrons yields the electronic wave function Yel and the electronic energy Eel. This energy is not equal to the total energy of the molecule. There are two contributions missing, which correspond to the terms that are missing in the electronic Hamiltonian. The first is the kinetic energy of the nuclei. The nuclei hardly move and we can neglect this contribution. The second is the Coulomb repulsion between the nuclei is. This contribution is not small. It does not affect the electrons, but it has to be added to the electronic energy to get the total energy of the molecule. Because the nuclei are fixed and the Coulomb interaction is a simple expression depending only on the charges of the nuclei and their distances with respect to each other, this is quite easy to do.
So the whole procedure is as follows. We first fix the nuclei. We then solve the Schrödinger equation for the electrons to get the electronic energy. Finally we add the Coulomb energy for the interaction between the nuclei. This gives us the total energy of the molecule. This procedure is called the Born-Oppenheimer approximation . This approximation is almost always used when doing quantum mechanical calculations of molecules. There are only very few situations were the approximation is not a good one.
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