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ps1_solution - Solution to 10.675 Assignment#1 1 Derive an...

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0 Solution to 10.675 Assignment #1 1. Derive an expression for the total ground state energy of a two-electron system using Hartree-Fock theory. This should be done in terms of spin orbitals. Identify the coulomb and exchange integrals, and explain their physical significance. Solution: Two-electron system is fairly easy to deal with. Since we are doing the general derivation for N-electron system in the next problem, we will omit the repeated work here. Please refer to chapter 2.3.1 in Szabo and Ostlund’s Modern Quantum Chemistry (pages 64-66) for the detailed derivation of two-electron system. 2. Make yourself comfortable with the general derivation of this expression for N electrons. Solution: For N-electron system, the Hamiltonian is: N N N 1 ( H = i h ) + r ij (1) i = 1 i = 1 j > i Where, i h ) = 1 i 2 Z A ( 2 A r iA In Hartree-Fock theory, we choose the form of the wave function to be that of single determinant, formed from spin orbitals: N ! | ψ >= | χ (1) χ (2) ⋅⋅⋅ χ ( N ) >= 1 ( 1) P n P { χ (1) χ (2) ⋅⋅⋅ χ ( N )} (2) m n k N !
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