  We revisit the accuracy of the variational Monte Carlo (VMC) method by taking
an example of ground state properties for the one-dimensional Hubbard model. We
start from the variational wave functions with the Gutzwiller and long-range
Jastrow factor introduced by Capello et al. [Phys. Rev. B 72, 085121 (2005)]
and further improve it by considering several quantum-number projections and a
generalized one-body wave function. We find that the quantum spin projection
and total momentum projection greatly improve the accuracy of the ground state
energy within 0.5% error, for both small and large systems at half filling.
Besides, the momentum distribution function n(k) at quarter filling calculated
up to 196 sites allows us direct estimate of the critical exponents of the
charge correlations from the power-law behavior of n(k) near the Fermi wave
vector. Estimated critical exponents well reproduce those predicted by the
Tomonaga-Luttinger theory.
