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##

Three Body Jastrow term

In order to describe well the correlation between electrons the simple two-body
Jastrow factor is not sufficient. Indeed it takes into account only
the electron-electron separation and not the individual electronic position
and
. It is expected that close to atoms the
correlation effects
deviate significantly from the translational invariant Jastrow.
For this reason we introduce a factor, often called three body
(electron-electron-nucleus) Jastrow,
that explicitly depends on both electronic positions
and
.
The three body Jastrow is chosen to satisfy the
following requirements:
- The cusp conditions set up by the two-body Jastrow term and by the one-body term are preserved.
- Similarly to the two-body we do not include any spin dependency in the three-body Jastrow. In the way the wave-function remains a spin singlet.
- Whenever the atomic distances are large it factorizes
into a product of independent contributions located near each atom,
an important requirement to satisfy the
size consistency of the variational wave function.

Analogously to the pairing trial function in Eq. 1.19
we define a three body factor as:

where indices
and
indicate different orbitals located around the atoms
and
respectively.
Each Jastrow orbital
is centred on
the corresponding atomic position
.
We have used Gaussian and exponential orbitals multiplied by appropriate
polynomials of the electronic coordinates, related to different
spherical harmonics with given angular momentum,
as in the usual Slater basis.
The chosen form for the 3-body Jastrow (1.28)
has very appealing features: it easily allows to include the
symmetries of the system by imposing them on the matrix
exactly as it is possible for the
pairing part (e.g. by replacing
with
in Eq. 1.20).
It is size consistent, namely the atomic limit
can be smoothly recovered with the same trial function
when the matrix terms
for
approach zero in this limit (see Ref. (15)). Notice that a small non zero value
of
for
acting on p-wave orbitals
can correctly describe
a weak interaction between electrons such as
the Van der Waals forces.

** Next:** Optimization Methods
** Up:** Functional form of the
** Previous:** Two body Jastrow term
** Contents**
Claudio Attaccalite
2005-11-07