Zero Monte Carlo error or quantum mechanics is easier

Monte Carlo methods for using the ground state variational principle to find bounds on the ground state energy by minimizing E t = (ψ t | H |ψ t )/(ψ t |ψ t ) are plagued by the intrinsic randomness of the Monte Carlo estimate of E t . These do not give E t , but rather Ē t , ± σ t , with Ē t the Monte Carlo estimate of E t and σ t the estimate for the standard deviation. When ψ t is an cigenfunction of H , it is possible to find σ t = 0. It is also possible to calculate differences such as (σ t − σ t′ ) over the same points so that their difference is much more accurate than would seem possible from the knowledge of σ t and σ t′ , alone. This allows us to vary ψ t so as to minimize σ t , or Ē t + n σ t , rather than Ē t alone. The method has been tested on the ground state of a single lithium atom. A trial wave function, ψ t . involving a very general form and 33 constants was found for which E t = −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 3 M 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.