573lec30

573lec30 - 5.73 Lecture #30 30 - 1 Matrix Elements of...

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30 - 1 5.73 Lecture #30 updated September 19, 4. , H in terms of , parameters eff Slater-Condon 1/r ij εζ n kk n FG ll 123 Matrix Elements of Many-Electron Wavefunctions Last time: need both f and g to satisfy boundary condition for E < 0 as r ν ν =− µ πµ () n fr l is phase shift of , infinite set of integer-spaced ν -values that satisfy r boundary condition Wave emerges from core with ν -independent phase. Core transforms wave with correct r 0 limiting behavior into one that exits imaginary sphere of radius r 0 , which contains the core region, with πµ l phase shift. Core sampled by set of different l ’s. Today: Wavefunctions and Energy States of many-electron atoms orbital energy spin- orbit next few lectures ν E gr r n , / , , l l l 12 0 noninteger principal quantum number solutions to Schröd. Eq. outside sphere of radius A. Normalization B. Matrix Elements of one- e Operators: e.g. C. Matrix Elements of two- e Operators: e.g. Hs Hr SO i ii i e ij ij ar e = () ⋅ = > l 2 1. , orbitals configurations L - S states 2. electrons are Fermions must be “ ”: KEY PROBLEM 3. Slater determinants are antisymmetric wrt all permutations →→ −− ψ antisymmetrized ee (a very bad “perturbation”) Page 31-9 is an example of what we will be able to do. * Interpretable trends: Periodic Table * Atomic energy levels: mysterious code — no atom-to-atom relationships evident without magic decoder ring.
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30 - 2 5.73 Lecture #30 updated September 19, Many-electron H How do we set up matrix representation of this H ? H (0) defines basis set (complete, orthonormal, …) spatial part spin part the φ i (r i )’s could be hydrogenic or shielded-core Rydberg-like orbitals. H as sum, E as sum, ψ as product Electronic Configuration : list of orbital occupancies e.g. C 1s 2 2s 2 2p 2 six e not sufficient to specify state of system Hh s H H = i=1 N (0) (1) ∑∑ () ++ >= i e r ar ij N ij i ii i 12 43 4 4444 3 1 2 ll sum of hydrogenic 1- e terms: unshielded orbital energies =− Z n n 2 2 ε (0) = ↔= [] = = i N i N s i r sm i 1 1 ψφ “spin-orbital” hh h h 1 2 1 1 2 2 2 1 + ()() = + φφ φ ()()()
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573lec30 - 5.73 Lecture #30 30 - 1 Matrix Elements of...

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