CHEM6670Notes2

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 7

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online