# Hartree-Fock-Wigner correlation models

** Description
**The Heisenberg Uncertainty Principle
informs us that it is not possible simultaneously to measure
exactly both the position and momentum of a particle. However,
we have introduced the Wigner intracule

**W**(

**u**,

**v**), a function that gives the quasiprobability of finding two electrons whose separation

**r**12 in position space is

**u**and whose separation

**p**12 in momentum space is

**v**. From this, we can derive the action intracule

**A**(

**w**) which gives the quasiprobability that the separation product

**r**12

**p**12 is

**w**. These functions give a more detailed picture of the motion of electron pairs than has been previously been available and we have conjectured that the correlation energy – a component of the total energy that is essential for an accurate understanding of bond formation – is given by

where **G**(**w**) is a universal (but as yet
unknown) function called the correlation kernel.

** Applications
**HFW models can be applied to any
chemical problem where one might currently use DFT or wavefunction
methods.

** Uniqueness/Innovation
**HFW theory can be regarded
as a two-electron density functional theory. In that sense,
it lies between DFT (which is a one-electron theory) and
conventional post-Hartree-Fock methods (which are many-electron
theories).

** Competition
**HFW models are not available in any
other commercial package

** Application limit; technical limits
**The computational
cost of a Hartree-Fock-Wigner correlation calculation is
proportional to the cost of a Hartree-Fock calculation.
The proportionality constant is currently quite large but
we are working to reduce this in future releases of the
Q-Chem package.

** Application scope
**The accuracy of HFW theory
depends on the underlying correlation kernel. We have implemented
a simple preliminary kernel (a spherical Bessel function)
in Q-Chem 3.0 and are currently developing more sophisticated
ones.

** Publications
**P.M.W. Gill, D.P. O’Neill
and N.A. Besley,

**Theor. Chem. Acc.**109 (2003) 241–250

N.A. Besley and P.M.W. Gill, **J. Chem. Phys.** 120
(2004) 7290–7297

D.P. O’Neill and P.M.W. Gill, in **Recent Advances
in Electron Correlation Methodology**, in press.

** Graphs**

The Wigner intracule for the beryllium atom

Exact and HFW correlation energies for a selection of atoms and small molecules

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