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Unformatted text preview: 1 5.61 Physical Chemistry Lecture #30 MOLECULAR ORBITAL THEORY- PART II For the simple case of the one-electron bond in H 2 + we have seen that using the LCAO principle together with the variational principle led to a recipe for computing some approximate orbitals for a system that would be very difficult to solve analytically. To generalize this to the more interesting case of many electrons, we take our direction from our experience with the independent particle model (IPM) applied to atoms and we build up antisymmetrized wavefunctions out of the molecular orbitals. This is the basic idea behind molecular orbital theory – there are many variations on the central theme, but the same steps are always applied. Rather than go step- by-step and deal with H 2 and then Li 2 and then LiH … we will instead begin by stating the general rules for applying MO theory to any system and then proceed to show some illustrations of how this works out in practice. 1) Define a basis of atomic orbitals For H 2 + our atomic orbital basis was simple: we used the 1s functions from both hydrogen atoms and wrote our molecular orbitals as linear combinations of our basis functions: ψ = c 1 s + c 1 s 1 A 2 B Note that the AO basis determines the dimension of our MO vector and also determines the quality of our result – if we had chosen the 3p orbitals instead of the 1s orbitals, our results for H 2 + would have been very wrong! For more complicated systems, we will require a more extensive AO basis. For example, in O 2 we might want to include all the 2s and 2p orbitals on both oxygens, in which case our MOs would take the form ψ = c 1 2 s A + c 2 2 p xA + c 3 2 p yA + c 4 2 p zA + c 5 2 s B + c 6 2 p xB + c 7 2 p yB + c 8 2 p zB Meanwhile, for methane we might want to include the 1s functions on all four hydrogens and the 2s and 2p functions on carbon: ψ = c 1 1 s 1 + c 2 1 s 2 + c 3 1 s 3 + c 4 1 s 4 + c 5 2 s + c 6 2 p x + c 7 2 p y + c 8 2 p z In the general case, we will write: N AO ψ = ∑ c i φ i i = 1 and represent our MOs by column vectors: 2 5.61 Physical Chemistry Lecture #30 ⎛ c 1 ⎞ ⎜ ⎟ ψ = ⎜ c 2 ⎟ ⎜ ... ⎟ ⎜ ⎟ ⎝ c N ⎠ We note that for the sake of accuracy it is never a bad idea to include more AO functions than you might think necessary – more AO functions will always lead to more accurate results. The price is that the more accurate computations also tend to be more complicated and time consuming. To illustrate, note that we could have chosen to write the H 2 + MOs as linear combinations of four functions – the 1s and 2s states on each atom: ψ = c 1 s + c 1 s + c 2 s + c 2 s 1 A 2 B 3 A 4 B Now, when we use the variational principle to get the coefficients of the lowest MO , c , we are guaranteed that there is no set of coefficients that will give us a lower energy. This is the foundation of the variational method. Note that one possible set of coefficients is c 3 =c 4 =0, in which case our 4-function expansion reduces...
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This note was uploaded on 09/24/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.
- Spring '08