Monte Carlo methods for using the ground state variational principle to find bounds on the ground state energy by minimizing Et = (ψt|H|ψt)/(ψt|ψt) are plagued by the intrinsic randomness of the Monte Carlo estimate of Et. These do not give Et, but rather Ēt, ± σt, with Ēt the Monte Carlo estimate of Et and σt the estimate for the standard deviation. When ψt is an cigenfunction of H, it is possible to find σt = 0. It is also possible to calculate differences such as (σt − σt′) over the same points so that their difference is much more accurate than would seem possible from the knowledge of σt and σt′, alone. This allows us to vary ψt so as to minimize σt, or Ēt + nσt, rather than Ēt alone. The method has been tested on the ground state of a single lithium atom. A trial wave function, ψt. involving a very general form and 33 constants was found for which Et = −14.898 ± 0.003 Ry which is lower than the Hartree-Fock energy bound of −14.865 Ry (C. Clementi, IBM J. Res. Develop. (Suppl.) 9, 2 (1965)) and above the true energy of −14.956 Ry (S. Larsson, Phys. Rev. 169, 49 (1968)). The small Monte Carlo error relative to the cohesive energy of solid lithium, −0.512 Ry (V. Heine, in The Physics of Metals, J. Ziman, Ed. (Cambridge Univ. Press, England, 1969)), along with the inherent flexibility of the method implies that it should be possible to extend the work to treat collections of M nuclei and 3M electrons. The nuclear- electronic probability distribution for atomic lithium was also evaluated. A comparison with that for the Hartree-Fock wave function shows small differences in accord with those expected from the inclusion of electron- electron parts in the trial wave functions considered here which are theoretically present but are omitted in the Hartrce Fock approximation.