5.61 Physical Chemistry 1 Lecture Notes MODERN ELECTRONIC...

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5.61 Physical Chemistry Lecture Notes 1 MODERN ELECTRONIC STRUCTURE THEORY: Electron Correlation In the previous lecture, we covered all the ingredients necessary to choose a good atomic orbital basis set. In the present lecture, we will discuss the other half of accurate electronic structure calculations: how we compute the energy. For simple MO theory, we used the non-interacting (NI) electron model for the energy: E NI = E μ μ = 1 N = ψ μ 1 ( ) ˆ H 0 μ = 1 N ψ μ 1 ( ) d τ Where, on the right hand side we have noted that we can write the NI energy as a sum of integrals involving the orbitals. We already know from looking at atoms that this isn’t going to be good enough to get us really accurate answers; the electron-electron interaction is just too important. In real calculations, one must choose a method for computing the energy from among several choices, and the accuracy of each method basically boils down to how accurately it treats electron correlation. Self Consistent Fields The Hartree Fock (HF) Approximation The Hartree-Fock method uses the IPM energy expression we’ve already encountered: E IPM = E μ μ = 1 N + J μ ν K μ ν μ < ν N E μ = ψ μ 1 ( ) ˆ H 0 ψ μ 1 ( ) d τ J μ ν ψ μ * 1 ( ) ψ ν * 2 ( ) 1 r 1 r 2 ψ μ 1 ( ) ψ ν 2 ( ) d r 1 d r 2 d σ 1 d σ 2 ∫∫ K μ ν ψ μ * 1 ( ) ψ ν * 2 ( ) 1 r 1 r 2 ψ μ 2 ( ) ψ ν 1 ( ) d r 1 d r 2 d σ 1 d σ 2 ∫∫ Since the energy contains the average repulsion, we expect our results will be more accurate. However, there is an ambiguity in this expression. The IPM energy above is correct for a determinant constructed out of any set of orbitals { } μ y and the energy will be different depending on the orbitals we choose. For example, we could have chosen a different set of orbitals, { } ' μ y , and gotten a different energy:
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5.61 Physical Chemistry Lecture Notes 2 E ' NI = E ' i μ = 1 N + J ' μ ν K ' μ ν μ ν N How do we choose the best set of orbitals then? Hartree-Fock uses the variational principle to determine the optimal orbitals. That is, in HF we find the set of orbitals that minimize the independent particle energy . These orbitals will be different from the non-interacting orbitals because they will take into account the average electron-electron repulsion terms in the Hamiltonian. Thus, effects like shielding that we have discussed qualitatively will be incorporated into the shapes of the orbitals. This will tend to lead to slightly more spread out orbitals and will also occasionally change the ordering of different orbitals (e.g. s might shift below p once interactions are included). Now, the molecular orbitals (and hence the energy) are determined by their coefficients. Finding the best orbitals is thus equivalent to finding the best coefficients. Mathematically, then, we want to find the coefficients that make the derivative of the IPM energy zero: E IPM c μ = c μ E μ μ = 1 N + J μ ν K μ ν μ ν N = 0 After some algebra, this condition can be re-written to look like an
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