CHEM6670Notes3 - HartreeFock method Topics CHEM 6670...

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Hartree Fock method 1 CHEM 6670 Computational Chemistry & Biophysics January 22, 2008 Course instructor: Dr. Jana Khandogin Jana.Khandogin@ou.edu Course web site: http://ccb.ou.edu/teaching.aspx Topics Review: Hartree Fock equation troduce basis functions Introduce basis functions Roothaan equation SCF procedure Basis sets Exchange and correlation effects Limitation of HF method tat o o et od 2 Hartree Fock (HF) theory e e e e E H ψ = ˆ ˆ ˆ N e Schrödinger equation H H E ~ ~ = ( ) = N ij ij j i ij E L δ φ λ ; ~ SD Φ = ij j i = N ) ( ) ( ) 2 ( ) 2 ( ) 1 ( ) 1 ( ! 1 1 1 1 N N N N N N 3 ( ) 0 = = ij j i j i ij E L δφ N i F i i i i ,..., 1 ˆ = = ε 1 e HF equation For convenience we have dropped the prime from the spin orbitals. Hartree Fock equation A set of HF equations i i i i F = ˆ Fock operator ne lectron operator () 1 ˆ ˆ ˆ ˆ ˆ 2 M a N j j j i i Z h K J h F nuc = + = Fock operator Canonical MOs MO energies Oeeecto ope ato Coulomb operator Exchange operator 4 ) 1 ( ) 2 ( 1 ) 2 ( ) 1 ( ˆ ) 2 ( 1 ) 2 ( ˆ 2 j i ij j i j j ij j j a ia i i r K r J r = =
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Hartree Fock method 2 Introduce basis functions To solve the HF equation, we expand the MO’s in a set of nown functions (basis functions). known functions (basis functions). Linear combination of atomic orbitals (LCAO): HF equation in the basis of atomic orbitals, basis M i i M i c basis ,..., 1 = = α χ φ M M basis basis ˆ Multiply both sides by β and integrate, 5 basis i i i i M i c c F ,... 1 = = ε = basis basis M i i M i i c c F * * ˆ HF equation in a basis set The last equation can also be written as, Define Fock matrix Define overlap matrix = basis basis M i i M i i c F c * * ˆ βα = = * * ˆ S F F i F S In the new notation, the above equation becomes, 6 ∑∑ = basis basis MM i i i c S c F αα row col row col Refresh a little linear algebra Suppose A and B are N x N matrices. Then the product of A nd B is and B is Example: mn N i in mi C B A C B A = = N N N b a b a b a 7 = N i i i N i i i N i i i N i i i N i i i N i i i i i i i i i i i i b a b a b a b a b a b a b b b b b b b b b a a a a a a a a a 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 31 31 31 21 21 21 13 12 11 31 31 31 21 21 21 13 12 11 Refresh a little linear algebra Suppose A: N x N matrix and B: N x 1 matrix (vector). The roduct of A and B is product of A and B is Example: m N i i mi C B A C B A = = N 8 = N i i i N i i i i i i b a b a b a b b b a a a a a a a a a 3 2 1 3 2 1 31 31 31 21 21 21 13 12 11
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Hartree Fock method 3 HF equation in a basis set The HF equation in a basis set ∑∑ = basis basis MM i i i c S c F α βα ε is the same as the matrix equation, where C i is a vector of expansion coefficients for orbital i , If basis functions form an orthonormal set (overlap matrix S is unity), the above equation is a matrix eigenvalue equation i i i C S C F = } { i c The unknown C
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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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CHEM6670Notes3 - HartreeFock method Topics CHEM 6670...

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