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Unformatted text preview: ManyElectron Atomic States, Terms, and Levels 27th April 2010 I. HartreeFock with Antisymmetrized Wavefunctions Recall the earlier discussion of the HartreeFock selfconsistent method. For the purposes of that introduction, we used trial wavefunctions that were simple products of singleelectron orbitals. We did not account for antisym metry and Pauli exclusion. Here, we will briefly formulate the HartreeFock method with antisymmetric wavefunctions. The results of this analysis will give rise to the already determined orbital energies and electronelectron repulsion terms (Coulomb integral), as well as a new term arising from the antisymmetric nature of the wavefunction—exchange integral. Keep in mind that the following does still not consider explicitly effects of electron corre lation (though, depending on the source, the exchange term is thought to contribute some amount to correlation). The Nelectron Slater determinantal wavefunction form, recall, is: Ψ( r 1 , σ 1 , r 2 , σ 2 , ..., r N , σ N ) = 1 √ N ! u 1 ( r 1 , σ 1 ) u 2 ( r 1 , σ 1 ) . . . u N ( r 1 , σ 1 ) u 1 ( r 2 , σ 2 ) u 2 ( r 2 , σ 2 ) . . . u N ( r 2 , σ 2 ) . . . . . . . . . u 1 ( r N , σ N ) u 2 ( r N , σ N ) . . . u N ( r N , σ N ) For closed shell systems each spatial orbital is occupied by 2 electrons (of opposite spin). Thus we require a single Slater determinant: Ψ( r 1 , σ 1 , r 2 , σ 2 , ..., r 2 N , σ 2 N ) = 1 p (2 N )! φ 1 ( r 1 ) α ( σ 1 ) φ 2 ( r 1 ) β ( σ 1 ) . . . φ N ( r 1 ) α ( σ 1 ) φ N ( r 1 ) β ( σ 1 ) φ 1 ( r 2 ) α ( σ 2 ) φ 2 ( r 2 ) β ( σ 2 ) . . . φ N ( r 2 ) α ( σ 2 ) φ N ( r 2 ) β ( σ 2 ) . . . . . . . . . . . . φ 1 ( r 2 N ) α ( σ 2 N ) φ 2 ( r 2 N ) β ( σ 2 N ) . . . φ N ( r 2 N ) α ( σ 2 N ) φ N ( r 2 N ) β ( σ 2 N ) 1 The Hamiltonian operating only on spatial coordinates in atomic units is: ˆ H = 1 2 2 N X j =1 ∇ 2 j 2 N X j =1 Z r j + 2 N X i =1 2 N X j>i 1 r ij The HartreeFock total energies, for this closedshell configuration much like we have seen earlier are: E = 2 N X j =1 I j + N X i =1 N X j =1 (2 J ij K ij ) The various terms in the energy expression are: I j = Z φ * j ( r j ) " ∇ 2 j 2 Z r j # φ j ( r j ) d r j J ij = Z Z φ * i ( r 1 ) φ * j ( r 2 ) 1 r 12 φ i ( r 1 ) φ j ( r 2 ) d r 1 d r 2 Coulomb Integral K ij = Z Z φ * i ( r 1 ) φ * j ( r 2 ) 1 r 12 φ i ( r 2 ) φ j ( r 1 ) d r 1 d r 2 Exchange Integral i 6 = j NOTE: In the definitions of the various one and twoelectron integrals listed im mediately above, the summations are effectively over orbitals . These can be written in terms of summations over electrons with minor modifications in the leading multiplicative factors (see Szabo and Ostlund for further details)....
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This note was uploaded on 02/02/2012 for the course CHEM 444 taught by Professor Dybowski,c during the Fall '08 term at University of Delaware.
 Fall '08
 Dybowski,C
 Physical chemistry, Electron, pH

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