Overview of Hartree-Fock (HF) Approximation


Hartree-Fock (HF) Approximation

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  • The Hartree Fock Approximation

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    Key Points


    The original Hartree method expresses the total wavefunction of the system as a product of one-electron orbitals. In the Hartree-Fock method, the wavefunction is an antisymmetrized determinantal product of one-electron orbitals (the "Slater" determinant). Schrodinger's equation is transformed into a set of Hartree-Fock equations. The Hartree-Fock approximation is also known at the self-consistent field (SCF) method
    1. begin with a set of approximate orbitals for all the electrons in the system
    2. one electron is selected, and the potential in which it moves is calculated by freezing the distribution of all the other electrons and treating their averaged distribution as the centrosymmetric source of potential
    3. the Schrodinger equation is solved for this potential, which gives a new orbital for it
    4. the procedure is repeated for all the other electrons in the system, using the electrons in the frozen orbitals as the source of the potential
    5. at the end of one cycle, there are new orbitals from the original set
    6. the process is repeated until there is little or no change in the orbitals
  • defined as the wavefunction of an electron in an atom
  • square of the AO gives the probability density
  • any electron described by an orbital is said to occupy that orbital
  • atomic orbitals have a well-defined significance only for hydrogenic atoms
  • hydrogenic atoms are used in many of the computational approximations
  • the orbital is a mathematical expression, describing the probability of finding an electron at some point near the nucleus
    1. the amplitude of an atomic orbital varies with distance from the nucleus in a not quite exponential fashion
    2. generally, MOs are expressed as linear combinations of atomic orbitals (LCAO), the sum of atomic orbitals centered on each nucleus
  • the accuracy of the representation of the true molecular orbital as an LCAO increases with the size of the basis set
    1. basis set is the number of atomic orbitals employed in the approximation
  • basic features of LCAO MOs:
    1. MOs are formed from all the valence orbitals, but do not refer to the number of electrons that are to be accommodated
    2. N atomic orbitals overlap to form N molecular orbitals
    3. for a given set of contributing atomic orbitals, the greater the number of interatomic nodes, the greater the energy
    4. the appropriate number of electrons is accommodated in accord with the rules of the building up principle.

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