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lecture28_umn - Lecture 28 The Hartree-Fock Self-Consistent...

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Lecture 28 November 18, 2009 The Hartree-Fock Self-Consistent Field Procedure Gaussian Orbitals as Basis Functions
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Hartree Fock Molecular Orbitals Solve the HF secular determinant and find its various roots. (28-1) We know how to compute overlap integrals. Fock matrix elements are defined by (28-2) F 11 ES 11 F 12 ES 12 L F 1 N ES 1 N F 21 ES 21 F 22 ES 22 L F 2 N ES 2 N M M O M F N 1 ES N 1 F N 2 ES N 2 L F NN ES NN = 0 F μ ! = μ 1 2 " 2 ! Z k k nuclei # μ 1 r k ! + P $% $% # μ ! $% ( ) 1 2 μ $ !% ( ) ( )
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Self-Consistent Field Procedure Density Matrix Elements (28-3) a : coefficients of the basis functions in the occupied molecular orbitals. Point of solving the secular equation: find those coefficients . If we do not know them already, how can we compute the matrix elements, and if we do know them already, why are we doing this in the first place? Same paradox as in the Hartree formalism. HF method follows a self-consistent field (SCF) procedure : first guess the orbital coefficients and then iterate to convergence P !" = 2 a ! i i occupied # a " i
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Choose a basis set Choose a molecular geometry Compute Store integrals Guess initial density matrix P (0) Construct and solve HF secular equation Construct P From occupied MOs Is P (n) sufficiently similar to P (n-1)? Optimize molecular geometry? Optimization criteria OK? Output data single point Output for optimized structure Choose new geometry Replace P (n-1) with P (n) no yes no yes no yes
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HF Energy After convergence of the MOs: compute the HF energy by evaluating the Hamiltonian operator for the HF determinant . Alternative way : Fock operator for a particular electron (28-4) It only operates on an electron in the i th orbital Expectation value of the Fock operator over that orbital: ε i It represents the total energy of an electron in that orbital. Add up all the ε values for all of the occupied orbitals and double that (two electrons in every orbital). f i = ! 1 2 " i 2 Z k r ik k nuclei # + V i HF j { }
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Computing Total Energy O vercounting of electron-electron repulsion For an electron in orbital 1, evaluate the repulsion that electron feels from an electron in orbital 2 (and 3, and 4, etc.) For electron in orbital 2 (and 3, and 4, etc.): account for that interaction again in dictating the energy of electron 2 (etc.) Double count all of the electron repulsions.
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lecture28_umn - Lecture 28 The Hartree-Fock Self-Consistent...

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