Phys560Notes-6 - Electronic States in a Crystal ~1023 interacting electrons and ion cores a big computational problem p2 1 p2 Z Z e2 1 e2 n n n HT

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1 Electronic States in a Crystal ~10 23 interacting electrons and ion cores – a big computational problem.         22 2 2 2 ,, ' , spin-orbit-interaction soi 11 2 2 ... ... in n n n T i i j n nn ni n i n ij el core el core pp Z Z e Z e e H mM HH H H H RR r R rr Frozen core approximation : cores at equilibrium positions; no phonons; no electron- phonon interactions, etc. Ignore spin-orbit-interaction. Te le l c o r e HHH , still cannot be solved exactly even for two electrons because of cross terms. Independent e approximation : assuming   Ti (no cross terms), and i independent of i (all electrons treated equally). A good approximation if i is chosen wisely. One-electron problem (single-particle approximation): 2     2 2 2 HU m r U = crystal potential Hartree approx: U r = potential from ion cores + average potential from all other electrons Hartree-Fock approx     Hartree exchange UU U r (exchange from exclusion principle) Generally, Hartree exchange correlation U U r (correlation = whatever is left) For now, consider U as given (as a parameterized function, for example). Translational symmetry:    + R Bloch theorem: wave function is of the form   i eu kr k , where uu kk R = a periodic function. k = wave vector = quantum number related to translation k crystal momentum
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3 Translating the wave function by R :   i eu krR k rR Equivalent form of Bloch theorem     i e kR r Proof: Define translation operator    T Rr r R . Explicit form of T R from Taylor expansion:        00 11 exp !! n n nn i i pp R r R r R r So,  exp Ti p RR Translation symmetry:      1 HT H T H R r R r ;     ,0 TH R . Also,    TT . H and all T R form a commuting set. Can choose to be a simultaneous eigenfunction. H  ;   Tc . 4 Since      T R R , cc c R R . Let   2 i ix i ce a ; i x = complex in general; i = 1, 2, 3.   3 12 2 2 3 3 3 2 2 22 i ii n in x n x n x x xx i n e e e e Ra Let i x kb    i R      r R r i Te . QED Note k is complex in general. Alternate proof based on group theory: T R form a 3d cyclic group of order. All representations are 1D. Choose the 1D matrix to be 2 x i a . Etc. Another quick proof: Because of translation symmetry, if   r is a solution, is also a solution. Choose the basis set such that c .
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This note was uploaded on 03/21/2012 for the course PHYS 560 taught by Professor Flynn during the Spring '08 term at University of Illinois, Urbana Champaign.

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Phys560Notes-6 - Electronic States in a Crystal ~1023 interacting electrons and ion cores a big computational problem p2 1 p2 Z Z e2 1 e2 n n n HT

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