lecture32to33 - 5.61 Physical Chemistry Lecture#32 1 MODERN...

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1 5.61 Physical Chemistry Lecture #32± MODERN ELECTRONIC STRUCTURE THEORY At this point, we have more or less exhausted the list of electronic structure problems we can solve by hand. If we were limited to solving problems manually, there would be a lot of chemistry we wouldn’t be able to explain! Fortunately, the advent of fast personal computers allows chemists to routinely use more accurate models of molecular electronic structure. These types of calculations typically play a significant role in interpreting experimental results: calculations can be used to assign spectra, evaluate reaction mechanisms and predict structures of molecules. In this way computation is complementary to experiment: when the two agree we have confidence that our interpretation is correct. The basic idea of electronic structure theory is that, within the Born Oppenheimer approximation, we can fix the M nuclei in our molecule at some positions R I . Then, we are left with the Hamiltonian for the electrons moving in the effective field created by the nuclei: N N M N H ˆ ≡ − 1 2 i 2 - ∑∑ Z I + 1 Eq. 1 i = 1 i = 1 I = 1 r ˆ R i < j r ˆ r ˆ i I i j Where the first term is the kinetic energy of all N electrons, the second term is the attraction between the electrons and nuclei and the third is the pairwise repulsion between all the electrons. The central aim of electronic structure theory is to find all the eigenfunctions of this Hamiltonian . As we have seen, the eigenvalues we get will depend on our choice of the positions of the nuclei E el ( R 1 , R 2 , R 3 ,… R M ). As was the case with diatomics, these energies will tell us how stable the molecule is with a given configuration of the nuclei { R I } – if E el is very low, the molecule will R 1 be very stable, while if E el is high, the molecule will be unstable in that configuration. The energy E el ( R 1 , R 2 , R 3 ,… R M ) is called the potential energy surface , and it contains a wealth of information, as illustrated in the picture at above. We can determine the equilibrium configuration of the Equilibrium Conformation Unstable intermediate Reaction Barrier R 2 E el (R 1 ,R 2 )
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2 5.61 Physical Chemistry Lecture #32± molecule by looking for the minimum energy point on the potential energy surface. We can find metastable intermediate states by looking for local minima i.e. minima that are not the lowest possible energy states, but which are separated from all other minima by energy barriers. In both of these cases, we are interested in points where E el = 0 . Further, the potential surface can tell us about the activation energies between different minima and the pathways that are required to get from the “reactant” state to the “product” state. Solving the electronic Schrödinger also gives us the electronic wavefunctions Ψ el ( r 1 , r 2 , r 3 ,… r N ), which allow us to compute all kinds of electronic properties average positions, momenta, uncertainties, etc as we have already seen for atoms.
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