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Hartree-Fock Theory - Chemistry

Hartree-Fock Theory - Chemistry

Hartree-Fock Theory -

Hartree-Fock Theory I. He atom revisited H = h(1) + h(2) + e 2 r 12 (1) Consider a more general form for ψ (1, 2), i.e., ⎛ αβ − βα ψ (1, 2) = φ 1s (1)φ 1s (2) ⎜ ⎝ 2 ⎞ ⎟ (2) ⎠ Seek best form for φ 15 . Hartree theory can be shown that optimum φ 15 satisfies ⎡ ⎤ ⎢ h(1) + φ (2) e2 φ(2) − ε r 1 ⎥ ⎣ ⎢ 12 ⎦ ⎥ φ 1s (1) = 0 ⎡ h(2) + φ(1) e 2 ⎤ ⎢ φ(1) − ε r 2 ⎥ ⎣ ⎢ 12 ⎦ ⎥ φ 1s (2) = 0 These are coupled integro-differential equations. Difficult to solve directly. Instead, use an iterative method. Describe this later. (3a) (3b) First, rewrite (3) as [ h eff (1) − ε 1 ]φ 1s ( r 1 ) = 0 (4a) [ h eff (2) − ε 2 ]φ 1s (r 2 ) = 0 (4b) where h eff (i) = −h2 2m ∇ 2 i − Ze2 r i + φ 1s (j) e2 r 12 φ 1s ( j) . (5) Hartree-Fock Theory 1 of 1

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