2Particle density: Correlated motion of electrons
The basic definitions of the 1 and 2particle densities for given
Nparticle state are
The 2particle density represents the probability to find one particle
with arbitrary spin at position r and simultaneously a second
particle with arbitrary spin at position r'.
In the case of noninteracting spin1/2 particles the
Nparticle state reduces to a Slater determinant
The first term in the 2particle density is the direct product of the
probability for finding one particle at point r with the
probability to find a particle at r'.
If the motion of the particles was completely uncorrelated this
product of singleparticle probabilities would be the exact result.
However, in contrast to classical particles the spin1/2 fermions
are correlated by the Pauli principle, even if a particleparticle
interaction is not present.
This socalled Pauli correlation is provided by the second
term.
For Helium the exact ground state wavefunction has the form
Due to the spherical symmetry only 3 spatial coordinates are relevant.
One usually chooses the distances
r_{1} and
r_{2} between the electrons
and the nucleus (located at the origin) as well as the distance
between the two electrons.
The 1 and 2particle densities are then simply given by
Within an effective singleparticle description, as the HartreeFock
or the KohnSham approach, the Helium ground state reduces to a product
of singleparticle orbitals
While the correlated motion of the two electrons essentially does not
show up in the 1particle density  at least as long as a high quality
effective singleparticle method is used  it is immediately visible in
the 2particle density, as demonstrated subsequently.
In all plots the position of the nucleus defines the origin.
The first electron is kept fixed at different positions on the
xaxis (leftright axis), while the position of the
second electron is varied.
Shown is the probability to find the second electron anywhere in the
plane defined by the positions of the three particles.
The precise quantity plotted is the radial 2particle density
which is chosen so that integration over
r_{2}
yields the standard radial density
For the exact ground state a very accurate variational wavefunction [1]
is utilized (see Appendix).
The effective singleparticle wavefunction has been obtained via the
Optimized Potential Method, using the exact
exchange functional of density functional theory.
Atomic units are used for all length scales.
I) Exact 2particle density of Helium:
r_{1}=(0.559 Bohr, 0, 0) 
complete view
x [Bohr]
II) Exact 2particle density of Helium:
r_{1}=(0.559 Bohr, 0, 0) 
half plane only
x [Bohr]
III) KohnSham 2particle density of Helium:
r_{1}=(0.559 Bohr, 0, 0) 
half plane only
x [Bohr]
IV) Exact 2particle density of Helium:
r_{1}=(1.0 Bohr, 0, 0)
x [Bohr]
V) Exact 2particle density of Helium:
r_{1}=(2.0 Bohr, 0, 0)
x [Bohr]
Observations:

The two electrons preferably move on opposite sides of the nucleus:
The likelihood to come close to each other is only half as large as
that of remaining on opposite sides
(Coulomb correlation).

The 2particle density clearly shows the electronelectron cusp at
r_{1}=
r_{2},
reflecting the shortrange Coulomb repulsion.

While the KS ground state reproduces the exact ground state density
exactly, the KS determinant can not reproduce the exact 2particle
density even qualitatively.
The motion of the two electrons (KohnSham particles) is uncorrelated.
The correlated wavefunction used for this comparison is the most
elaborate variational wavefunction (38 parameters) given in the
original work of Kinoshita [1].
It has the form
with
and
,
= variational parameters

T. Kinoshita,
Phys. Rev. 105, 1490 (1957).
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Last modified: Mar 31, 2004
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