Lec6 (1) - 10.675 LECTURE 6 RICK RAJTER 1. Today...

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10.675 LECTURE 6 RICK RAJTER 1. Today Variational Principle Derive HF (Hartree-Fock) Equations Interpretation of Solutions to HF equations 2. Variational Principle Idea: The closer our guess of C i ’s of Ψ trial , the lower our energy. ˜ Φ trial function E [Φ] = < ˜ | ˜ ˜ Φ | H Φ > Functional De±nition : maps functions to numbers 3. Functional Variation Φ Φ + δ ˜ vary ˜ ˜ Φ substitute in and solve ˜ Φ] = < ˜ Φ H | Φ + δ ˜ E [Φ + δ ˜ Φ + δ ˜ | ˜ Φ > expand this via linear ±rst order term. E [Φ + δ ˜ ˜ Φ | H Φ > + < ˜ | Φ > + higher order terms. ˜ Φ] = E [Φ]+ < δ ˜ | ˜ Φ | H δ ˜ E [Φ + δ ˜ ˜ ˜ Φ] = E [Φ] + δE Trying to approach the true solution such that H | Φ > = E Φ > δE = 0 | Normally, E will always be a min, so we don’t have to worry about local/global max solutions. There exists and in±nite number of solutions. ˜ H | Φ α > = E α Φ α > | ˜ α = 0 , 1 , 2 ... E

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Lec6 (1) - 10.675 LECTURE 6 RICK RAJTER 1. Today...

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