Hartree-Fock-Wigner correlation models

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 r12 in position space is u and whose separation p12 in momentum space is v. From this, we can derive the action intracule A(w) which gives the quasiprobability that the separation product
r12 p12 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.

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

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).

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.

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.


The Wigner intracule for the beryllium atom


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