The one body density matrix (OBDM) of a homogeneous system described by the
many body wavefunction
is defined as follows
(3.82)
Since in DMC calculation one can not sample the ground-state
probability distribution , instead one samples the mixed
probability and one can calculate the mixed
one-body density matrix
(3.83)
This formula can be further developed
(3.84)
where we have used the asymptotic formula (3.46). If the
trial wavefunction is chosen as a product of pair functions (see
eq. (3.31) then using the notation
) one has
(3.85)
Then, the ratio of trial wavefunction appearing in (3.84)
becomes
(3.86)
In order to gain better statistics one can average over all particles
The asymptotic limit of the OBDM gives the condensate density
(3.87)
and the condensate fraction is obtained by the calculating the
asymptotic ratio