CHEM6670Notes6 - Density functional theory Topics CHEM 6670...

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Density functional theory 1 CHEM 6670 omputational Chemistry & Biophysics Computational Chemistry & Biophysics Density functional theory January 31, 2008 Course instructor: Dr. Jana Khandogin [email protected] Course web site: http://ccb.ou.edu/teaching.aspx Topics Review: DFT erivation of KS equation Derivation of KS equation Comparison of KS and HF equations Local density approximation Generalized gradient approximation Hybrid functionals Performance of DFT methods eo ac e o et ods Limitations of current DFT methods 2 Density functional methods Hohenberg Kohn theorems: 1) Ground state energy is a unique functional of electron density. Kohn Sham formulation: A set of Kohn Sham equations Effective one electron KS operator Slater determinant with molecular orbitals 2) The exact density minimizes the ground state energy. 3 basis functions Matrix equation Kohn Sham orbitals, energies SCF procedure Derivation of Kohn Sham equation Similar to the derivation of HF equation, we write down the grange function, assuming a single determinant wave Lagrange function, assuming a single determinant wave function and orthonormal orbitals. ( ) ∑∑ = = N i s i i N j i ij j i ij s r r r E L 2 , ) , ( ) ( )] ( [ r r r φ ρ δ λ The variation of the Lagrange with respect to the orbitals, leads to the equations, 4 0 = L = + = N j j ij i eff i KS v h 2 2 1 ˆ
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Density functional theory 2 Kohn Sham equation After a uniform transformation of the Lagrange multiplier atrix we obtain the Kohn am orbital equation in the matrix λ ij , we obtain the Kohn Sham orbital equation in the canonical form, ) ( ' ' ) ' ( ' ' 2 1 2 r v r d r r r R r Z v v xc M a eff i i i eff r r r r r r r + + = = + ρ φ ε where i ‘ are called canonical orbitals. i are the orbital energies. We will assume canonical orbitals and omit prime from now on. 5 a a i KS and HF equations Kohn Sham equation Hartree Fock equation ) ( ' ' ) ' ( 2 1 2 r v r d r r r R r Z v v xc M a a i a eff i i i eff i r r r r r r r + + = = + N N M a a i a i i i i i N j j j i J R r Z h K J h = = = + 1 ˆ 2 1 ˆ ) ) ˆ ˆ ( ˆ ( 2 r r r r 6 j i ij ij j j i j N j j j j j i j j j P P r r K r r = = ˆ ˆ 1 ˆ r r Exchange correlation potential is unknown!
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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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CHEM6670Notes6 - Density functional theory Topics CHEM 6670...

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