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Comparison between VMC and DMC methods

The Jastrow trial-wavefunction obtained in the previous section givers a very good approximation for the ground state wavefunction of dilute Bose gases. One can appreciate this by comparing the VMC and DMC energies (Fig. 3.1). On this plot the relative error $\triangle E/E$ of the energy estimate is presented as a function of the density. The coincidence of DMC and VMC results is very good in the density region $n<10^{-4}$ and remains quite good for higher densities.

Figure 3.1: Energy comparison between VMC and DMC calculations $\triangle {E} = E^{VMC}-E^{DMC}$ at different densities $na^3$
\includegraphics[width=\textwidth]{Ecompare.eps}

The energy obtained in VMC is always larger than the ground state energy (see Fig. 3.1). In other words, variational simulations always give an upper bound to the ground state energy.
$\displaystyle E^{VMC} \ge E$     (3.43)