CHEM6670Notes2 - Hartree ‐ Fock equation 1 CHEM 6670...

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Unformatted text preview: Hartree ‐ Fock equation 1 CHEM 6670 Computational Chemistry & Biophysics January 17, 2008 Course instructor: Dr. Jana Khandogin [email protected] Course web site: http://ccb.ou.edu/teaching.aspx Topics • Review ydrogen atom • Hydrogen atom • Bra ‐ ket notation • Variational theorem • Derivation of Hartree ‐ Fock equation 2 Quantum mechanics Postulates (wavefunction, operator, expectation value) Separation of space and time t i f l d l t i ti t t r i t r H ∂ Ψ ∂ = Ψ ) , ( ) , ( ˆ r h r ) ( ) ( ˆ r E r H r r Ψ = Ψ 3 Separation of nuclear and electronic motion ) ; ( ) ( ) ; ( ˆ R r R E R r H e e e e r r r r r ψ ψ = R r : nuclear coordinates Solving the electronic Schrödinger equation forms the core of computational quantum chemistry. Dirac (bra ‐ ket) notation ≡ = ψ ψ ψ ψ : ket : bra * r d H H r d r r ∫ ∫ ≡ ≡ ψ ψ ψ ψ ψ ψ ψ ψ ˆ ˆ : element matrix : element overlap * * * ra Bra ‐ ket notation using index only: 4 r d H n H m r d n m n m m n m n m r r ∫ ∫ ≡ ≡ ≡ = n * * ˆ ˆ : element matrix : element overlap : ket : bra ψ ψ ψ ψ ψ ψ Hartree ‐ Fock equation 2 Postulates in quantum mechanics 1. Wave function tells us everything about a particle system. mathematical operator is associated with every physical 2. A mathematical operator is associated with every physical observable. Associated with the operator is a set of eigenfunctions and eigenvalues. Eigenvalues are the definitive values a measurement returns. n n n a A ψ ψ = ˆ 3. Any operator associated with a physical observable is hermitian 5 n m m n n m A A A ψ ψ ψ ψ ψ ψ ˆ ˆ ˆ * = = Postulates in quantum mechanics 4. The set of eigenfunctions of operator forms a complete set f linearly independent functions (basis set). A ˆ of linearly independent functions (basis set). • This also means that any wave function can be expanded in this basis set ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≠ = = = n m n m mn n m for for 1 δ ψ ψ c ~ 5. The average measured value of a physical observable is the expectation value of the associated operator. 6 ψ ψ A A ˆ = ∑ m m m ψ ψ Quantum vs. classical mechanics 7 *Note: atomic units are not used here. Hydrogen ‐ like atom • System: 1 electron and 1 nucleus with charge Z he electronic Schrödinger equation is • The electronic Schrödinger equation is • By switching from the Cartesian to spherical polar coordinate system we can solve the above partial differential equation....
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This note was uploaded on 04/29/2008 for the course CHEM 6670 taught by Professor Janakhandogin during the Spring '08 term at The University of Oklahoma.

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CHEM6670Notes2 - Hartree ‐ Fock equation 1 CHEM 6670...

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