This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Many-Electron Atomic States, Terms, and Levels 5th May 2009 I. Hartree-Fock with Antisymmetrized Wavefunctions Recall the earlier discussion of the Hartree-Fock self-consistent method. For the purposes of that introduction, we used trial wavefunctions that were simple products of single-electron orbitals. We did not account for antisym- metry and Pauli exclusion. Here, we will briefly formulate the Hartree-Fock method with anti-symmetric wavefunctions. The results of this analysis will give rise to the already determined orbital energies and electron-electron repulsion terms (Coulomb integral), as well as a new term arising from the antisymmetric nature of the wavefunctionexchange 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 N-electron Slater determinantal wavefunction form, recall, is: ( r 1 , 1 , r 2 , 2 , ..., r N , N ) = 1 N ! u 1 ( r 1 , 1 ) u 2 ( r 1 , 1 ) . . . u N ( r 1 , 1 ) u 1 ( r 2 , 2 ) u 2 ( r 2 , 2 ) . . . u N ( r 2 , 2 ) . . . . . . . . . u 1 ( r N , N ) u 2 ( r N , N ) . . . u N ( r N , N ) For closed shell systems each spatial orbital is occupied by 2 electrons (of opposite spin). Thus we require a single Slater determinant: ( r 1 , 1 , r 2 , 2 , ..., r 2 N , 2 N ) = 1 p (2 N )! 1 ( r 1 ) ( 1 ) 2 ( r 1 ) ( 1 ) . . . N ( r 1 ) ( 1 ) N ( r 1 ) ( 1 ) 1 ( r 2 ) ( 2 ) 2 ( r 2 ) ( 2 ) . . . N ( r 2 ) ( 2 ) N ( r 2 ) ( 2 ) . . . . . . . . . . . . 1 ( r 2 N ) ( 2 N ) 2 ( r 2 N ) ( 2 N ) . . . N ( r 2 N ) ( 2 N ) N ( r 2 N ) ( 2 N ) 1 The Hamiltonian operating only on spatial coordinates in atomic units is: 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 Hartree-Fock total energies, for this closed-shell 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 * j ( r j ) "- 2 j 2- Z r j # j ( r j ) d r j J ij = Z Z * i ( r 1 ) * j ( r 2 ) 1 r 12 i ( r 1 ) j ( r 2 ) d r 1 d r 2 Coulomb Integral K ij = Z Z * i ( r 1 ) * j ( r 2 ) 1 r 12 i ( r 2 ) j ( r 1 ) d r 1 d r 2 Exchange Integral i 6 = j NOTE: In the definitions of the various one- and two-electron 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)....
View Full Document