Chemistry Reference
In-Depth Information
4. If the spin orbitals found in step 3 are consistent with orbitals used in
step 2, then these are the solutions to the HF problem we set out to
calculate. If not, then a new estimate for the spin orbitals must be
made and we then return to step 2.
This procedure is extremely similar to the iterative method we outlined in
Section 1.4 for solving the Kohn-Sham equations within a DFT calculation.
Just as in our discussion in Section 1.4, we have glossed over many details that
are of great importance for actually doing an HF calculation. To identify just a
few of these details: How do we decide if two sets of spin orbitals are similar
enough to be called consistent? How can we update the spin orbitals in step 4
so that the overall calculation will actually converge to a solution? How large
should a basis set be? How can we form a useful initial estimate of the spin
orbitals? How do we efficiently find the expansion coefficients that define
the solutions to the single-electron equations? Delving into the details of
these issues would take us well beyond our aim in this section of giving an
overview of quantum chemistry methods, but we hope that you can appreciate
that reasonable answers to each of these questions can be found that allow HF
calculations to be performed for physically interesting materials.
1.6.4 Beyond Hartree-Fock
The Hartree-Fock method provides an exact description of electron exchange.
This means that wave functions from HF calculations have exactly the same
properties when the coordinates of two or more electrons are exchanged as
the true solutions of the full Schr¨dinger equation. If HF calculations were
possible using an infinitely large basis set, the energy of
N
electrons that
would be calculated is known as the
Hartree-Fock limit
. This energy is not
the same as the energy for the true electron wave function because the
HF method does not correctly describe how electrons influence other
electrons. More succinctly,
the HF method does not deal with electron
correlations.
As we hinted at in the previous sections, writing down the physical laws that
govern electron correlation is straightforward, but finding an exact description
of electron correlation is intractable for any but the simplest systems. For the
purposes of quantum chemistry, the energy due to electron correlation is
defined in a specific way: the electron correlation energy is the difference
between the Hartree-Fock limit and the true (non-relativistic) ground-state
energy. Quantum chemistry approaches that are more sophisticated than the
HF method for approximately solving the Schr¨dinger equation capture
some part of the electron correlation energy by improving in some way
upon one of the assumptions that were adopted in the Hartree-Fock approach.
