  Quantum mechanics for many-body systems may be reduced to the evaluation of
integrals in 3N dimensions using Monte-Carlo, providing the Quantum Monte Carlo
ab initio methods. Here we limit ourselves to expectation values for trial
wavefunctions, that is to Variational quantum Monte Carlo. Almost all previous
implementations employ samples distributed as the physical probability density
of the trial wavefunction, and assume the Central Limit Theorem to be valid. In
this paper we provide an analysis of random error in estimation and
optimisation that leads naturally to new sampling strategies with improved
computational and statistical properties. A rigorous lower limit to the random
error is derived, and an efficient sampling strategy presented that
significantly increases computational efficiency. In addition the infinite
variance heavy tailed random errors of optimum parameters in conventional
methods are replaced with a Normal random error, strengthening the theoretical
basis of optimisation. The method is applied to a number of first-row systems,
and compared with previously published results.
