In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the motion of atomic nuclei and electrons in a molecule can be separated. The approach is named after Max Born, and J. Robert Oppenheimer. In mathematical terms, it allows the wavefunction of a molecule to be broken into its electronic and nuclear (vibrational, rotational) components.
they're two approximations not directly related in the way you seem to be asking (I could be misunderstanding what you're asking).
start with the full non-relativistic hamiltonian, which includes the terms thrown-out/neglected when the born-oppenheimer/born-huang approximation is invoked. getting rid of those terms will give you an approximate hamiltonian. now you can use something like the hartree-fock approximation to solve for the many-electron wave function for that hamiltonian. put in another hamiltonian that involves fermions (if it's bosons you just need the hartree approximation or hartree product) but with different approximations to the hamiltonian and you get out a hartree-fock wave function within that paradigm (the details of how the calculation is carried out will likely be different, so probably safer to call it hartree-fock-like?).
The Born-Oppenheimer approximation is not only part of the Hartree-Fock method, it's a prerequisite. Without the assumption of stationary atomic nuclei, you wouldn't be able to get a solution for the electronic wavefunctions.
Hartree-Fock is a method able to compute approximate wave functions and energies of multi-electron systems, assuming the nuclei are fixed. Thus, it implies taking the Born-Oppenheimer approximation (separation of nuclear and electron movements) as previous step.
The Hartree-Fock approximation searches the energy minimum of the electronic part of a molecule in the space of single Slater Determinants whereas the Born-Oppenheimer approximation approximates the wave function of a molecule as a product of the electronic part times the wave function of the nuclei. In most numerical calculations the nucleons are treated as being at fixed position and the total energy minimum gives the equilibrium position of the nuclei
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