04262010 - Many-Electron Atomic States Terms and Levels...

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Many-Electron Atomic States, Terms, and Levels 27th April 2010 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 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 N-electron 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
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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 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 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 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). Let’s consider the meaning of the Coulomb and Exchange terms we have discussed just now. The coulomb integral can be rearranged as: Z Z φ 2 i ( r 1 ) 1 r 1 2 φ 2 j ( r 2 ) d r 1 d r 2 The square of the wavefunction is the probability of finding an electron at a given point in space. So this term is the energy of the Coulombic interac- tion between an electron in orbital i with an electron in orbital j . For this 2
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reason, this integral is called the Coulomb Integral. Because the Coulomb potential is always positive for like charges, and the square of the wavefunc- tion is always positive, this term contributes a positive energy to the toatl energy. This is a destabilizing energy contribution (arising from unfavorable
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