L_Quantum_Chemistry

L_Quantum_Chemistry - The Hartree-Fock approximation...

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The Hartree-Fock approximation Prof. D.R. Hartree, Cambridge, UK, 1897- 1958 Prof. V.A.Fock, St.Petersburg, Russia 1898-1974 Approximation for the many-body electronic wavefunction which is expressed as an antisymmetrized product of N orthonormal single particle orbitals, each written as a product of a spatial orbital and a spin function:
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The Hartree-Fock approximation If we approximate the many-body wavefunction by a Slater determinant, the energy expectation value E HF is: The H i , J ij and K ij integrals are written in terms of single particle wavefunctions; J ij are called Coulomb integrals and K ij are called exchange integrals
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The Hartree-Fock differential equations Minimization of E HF subject to orthonormalization conditions for the single particle orbitals gives the Hartree-Fock differential equations:
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The Hartree-Fock differential equations Minimization of E HF subject to orthonormalization conditions for the single particle orbitals gives the Hartree-Fock differential equations: Note that depends on the orbitals and thus the Fock operator depends on the orbitals. Therefore solutions of the Hartree-Fock equations must proceed iteratively . The Fock operator is a non linear operator. Consequently the Hartree- Fock method is called a non-linear “self-consistent” ±eld method.
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Orbital energies
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Systems with an even number of electrons: Restricted Hartree-Fock approximation (RHF) The single particles orbitals ψ i are taken to comprise N/2 orbitals of the form φ k α (s) [ spin up orbitals] and N/2 orbitals of the form φ k β (s) [ spin down orbitals]; the radial part φ k is the same for up and down spins. Now the integrals H i , J ij and K ij are written in terms of the radial part of the single particle wavefunctions and the Hartree-Fock equation reads Since a Slater determinant is invariant by unitary transformation within the subspace of occupied single particle orbitals (but for a phase factor) and so are the operators and the Fock operator , one may choose a unitary transformation which diagonalizes the Hermitian matrix .
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Systems with an even number of electrons: Restricted Hartree-Fock approximation (RHF) The single particle orbitals ψ i are taken to comprise N/2 orbitals of the form φ k α (s) [ spin up orbitals] and N/2 orbitals of the form φ k β (s) [ spin down orbitals]; the radial part φ k is the same for up and down spins. Now the integrals H i , J ij and K ij are written in terms of the radial part of the single particle wavefunctions and the Hartree-Fock equation reads:
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Open shell systems: UHF vs. RHF A system is open shell if it contains one or more unpaired electrons. There are two ways to treat open shell systems: -- Spin restricted (or RHF, for Restricted Hartree-Fock) This approach uses combinations of singly and doubly occupied molecular orbitals. Doubly occupied orbitals use the same spatial functions for electrons of both spins (say, α and β ).
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L_Quantum_Chemistry - The Hartree-Fock approximation...

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