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notes18 - 5.73 Lecture#18 18 1 Variational Method(See CTDL...

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18 - 1 5.73 Lecture #18 modified 10/9/02 10:21 AM Variational Method (See CTDL 1148-1155, [Variational Method] 252-263, 295-307[Density Matrices]) Last time: Quasi-Degeneracy Diagonalize a part of infinite H * sub-matrix : H (0) + H (1) * corrections for effects of out-of-block elements: H (2) (the Van Vleck transformation) *diagonalize H eff = H (0) + H (1) + H (2) coupled H-O’s 2 : 1 ( ω 1 2 ω 2 ) Fermi resonance example: polyads 1. Perturbation Theory vs. Variational Method 2. Variational Theorem 3. Stupid nonlinear variation 4. Linear Variation new kind of secular Equation 5. Linear combined with nonlinear variation 6. Strategies for criteria of goodness — various kinds of variational calculations 1. Perturbation Theory vs. Variational Method Perturbation Theory in effect uses basis set goals: parametrically parsimonious fit model, H eff fit parameters (molecular constants) parameters that define V(x) order - sorting H nk ( 1 ) E n ( 0 ) E k ( 0 ) < 1 — errors less than this “mixing angle” times the previous order non–zero correction term (n is in-block, k is out-of block) because diagonalization is order (within block). Variational Method best possible estimate for lowest few E n , ψ n (and properties derivable from these) using finite basis set and exact form of H .
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18 - 2 5.73 Lecture #18 modified 10/9/02 10:21 AM Vast majority of computer time in Chemistry is spent in variational calculations Goal is numbers. Insight is secondary. Ab Initio ” vs. “semi-empirical” or “fitting” [intentionally bad basis set: Hückel, tight binding – qualitative behavior obtained by a fit to a few microscopic– like control parameters] 2. Variational Theorem not necessarily normalized any observable α φ A φ φ φ a 0 If φ is approximation to eigenfunction of ˆ A belonging to lowest eigenvalue a 0 , then PROOF: eigenbasis (which we do not know – but know it must exist) A n = a n n φ = n n φ n φ A φ = φ n n, n n A n n φ = φ n 2 n a n φ φ = φ n n φ n = φ n 2 n completeness a n δ nn α φ A φ φ φ = a n n φ 2 n n φ 2 n subtract a 0 from both sides α a 0 = a n a 0 ( ) n φ 2 n n φ 2 n ≥ 0 expand φ in eigenbasis of A , exploiting completeness the variational Theorem
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