Chem120A+Notes+10-15-08

Chem120A+Notes+10-15-08 - Chem120A Notes 10/15/08...

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Unformatted text preview: Chem120A Notes 10/15/08 Variational Principle → For ground state, approximation of the wave function gives energy higher than the actual energy ⇒ Ψ = Ψ ˆ E H ground state, is the eigenfunction for the ground state Ψ E is the exact ground state energy Variational Principle * * ~ ~ ~ ˆ ~ ~ E d d H E ≥ Ψ Ψ Ψ Ψ = ∫ ∫ τ τ ( 1 ) → Ψ is approximation to the ground state energy, with corresponding energy ~ E ~ L + Ψ + Ψ + Ψ = Ψ 3 3 1 1 ~ c c c ( 2 ) → Assume are a set of real, orthonormal set functions (such as eigenfunctions) K , , , 2 1 Ψ Ψ Ψ ∫ = Ψ Ψ ∴ mn n m d δ τ ( 3 ) → Assume are real coefficients K , , , 2 1 c c c Substituting (2) into (1) gives: [ ] [ ] [ ] [ ] ∫ ∫ + Ψ + Ψ + Ψ + Ψ + Ψ + Ψ + Ψ + Ψ + Ψ + Ψ + Ψ + Ψ = τ τ d c c c c c c d c c c H c c c E L L L L 3 3 1 1 * 3 3 1 1 3 3 1 1 * 3 3 1 1 ˆ ~ L L + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ τ τ τ τ τ τ τ τ d c c d c c d c c d c c d H c c d H c c d H c c d H c c E 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ˆ ˆ ˆ ˆ ~ L L + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ τ τ τ τ τ τ τ τ d c d c c d c c d c d E c d E c c d E c c d E c E 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 2 ~ L L + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ τ τ τ τ...
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This note was uploaded on 09/28/2009 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at Berkeley.

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Chem120A+Notes+10-15-08 - Chem120A Notes 10/15/08...

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