CHEM6670Notes5 - 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 29, 2008 Course instructor: Dr. Jana Khandogin [email protected] Course web site: http://ccb.ou.edu/teaching.aspx Topics Review: Hartree Fock theory and electron correlation ethods methods Reduced density matrix Thomas Fermi Dirac model Hohenberg Kohn theorem Kohn Sham theory 2 Electronic structure methods e e e e E H ψ = ˆ Hartree Fock method I MP CC Trial wave function: Single determinant Variational principle Approximations of integrals Adding more determinants Systematic convergence to exact solution 3 Semi empirical methods CI, MP, CC Density functional methods (energy is a functional of electron density; implicitly models electron exchange and correlation) Function and functional A function produces a number from a set of variables. A functional produces a number from a function. Wave function is a function of 3N coordinates (N electrons). ) , , ( 3 2 1 x x x f )] , , ( [ 3 2 1 x x x f F aeuc to s auc o 3 coo d ates ( eecto s) Electron density is a function of 3 coordinates (one electron). 4 ) ,..., ( 1 N r r r r ) ( 1 r r ρ
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Density functional theory 2 Reduced density matrix We define a density matrix as First order reduced density matrix econd rder reduced density matrix N N N r d r d r r r r r r N r r r r r r r r r r r r = 2 2 1 2 1 1 1 1 ) ,... , ' ( ) ,... , ( * ) ' , ( ψ γ ) ' ' , ' ( ) , ( * ) , ' ' ( 2 1 2 1 1 1 1 N N N N r r r r r r r r r r r r r r r r r r r r = Seco dode educed de s ty at 5 N N N r r d r r r r r r N N r r r r r r r r r r r r r r r r = 3 2 1 2 1 2 1 2 1 1 ) ,... ' , ' ( ) ,... , ( * 2 ) 1 ( ) ' , ' , , ( Electron density function Diagonal elements of the first order density matrix give the lectron density function. It is the probability of finding electron density function. It is the probability of finding electron one at = = = N N N r d r d r r r N r d r d r r r r r r N r r r r r r r r r r r r r r r r r r r 2 2 2 1 2 1 1 1 1 1 ) ,... , ( ) ,... , ( ) ,... , ( * ) , ( ) ( ρ 1 r r The probability of finding electron one anywhere is, 6 ... ) ... ( ) ( 2 1 2 1 1 1 N r d r d r d r r N r d r N N = = r r r r r r r N N 2 2 1 , Total number of electrons Electron pair density function Probability of finding electron one at and electron two at iagonal elements of the second rder density matrix by 1 r r 2 r r Diagonal elements of the second order density matrix by setting and .
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CHEM6670Notes5 - Density functional theory Topics CHEM 6670...

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