lecture28to29 - 5.61 Physical Chemistry Lecture#28 1...

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1 5.61 Physical Chemistry Lecture #28± MOLECULAR ORBITAL THEORY- PART I At this point, we have nearly completed our crash-course introduction to quantum mechanics and we’re finally ready to deal with molecules. Hooray! To begin with, we are going to treat what is absolutely the simplest molecule we can imagine: H + . This simple molecule will allow us to work 2 out the basic ideas of what will become molecular orbital (MO) theory. We set up our coordinate system as shown at right, with the electron positioned at r , and the two nuclei positioned at points R A and R B , at a distance R from one H A another. The Hamiltonian is R A easy to write down: H 2 H ˆ = 1 2 2 A 2 B 1 1 + 1 ˆ 2 r 2 M A 2 M B R ˆ r ˆ R r ˆ R ˆ R ˆ B A B A e- H B R B r r A r B R + Coordinates Electron H A H B e - -H A e - -H B H A -H B Kinetic Kinetic Kinetic Attraction Attraction Repulsion Energy Energy Energy Now, just as was the case for atoms, we would like a picture where we can separate the electronic motion from the nuclear motion. For helium, we did this by noting that the nucleus was much heavier than the electrons and so we could approximate the center of mass coordinates of the system by placing the nucleus at the origin. For molecules, we will make a similar approximation, called the Born-Oppenheimer approximation . Here, we note again that the nuclei are much heavier than the electrons. As a result, they will move much more slowly than the light electrons. Thus, from the point of view of the electrons, the nuclei are almost sitting still and so the moving electrons see a static field that arises from fixed nuclei . A useful analogy here is that of gnats flying around on the back of an elephant. The elephant may be moving, but from the gnats’ point of view, the elephant is always more or less sitting still. The electrons are like the gnats and the nucleus is like the elephant.
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2 5.61 Physical Chemistry Lecture #28± The result is that, if we are interested in the electrons, we can to a good approximation fix the nuclear positions R A and R B and just look at the motion of the electrons in a molecule. This is the B-Oppenheimer approximation, which is sometimes also called the clamped-nucleus approximation, for obvious reasons. Once the nuclei are clamped, we can make two simplifications of our Hamiltonian. First, we can neglect the kinetic energies of the nuclei because they are not moving. Second, because the nuclei are fixed, we can replace the operators R ˆ A and R ˆ B with the numbers R A and R B . Thus, our Hamiltonian reduces to ˆ r H ( R , R ) = − 2 1 1 1 + el A B 2 R r ˆ R r ˆ R R A B A B where the last term is now just a number the electrostatic repulsion between two protons at a fixed separation. The second and third terms depends only on the position of the electron, r , and not its momentum, so we immediately identify those as a potential and write: ˆ ( ) 2 r R A , R B ( ) 1 H R , R = − + V r ˆ + el A B 2 eff R R A B This Hamiltonian now only contains
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