To properly apply the DQMC method, one must allow for the nodes produced by the antisymmetry requirement.
(a) Consider a system of three electrons in a one-dimensional box, where we shall pretend that the interelectronic repulsions are small enough to be neglected. By analogy to the Li zeroth-order wave function (10.48), write down the ground-state wave function for this system.
(b) Use orthogonality of the three different three-electron spin factors that multiply a, b, and c in Prob. 10.15 to show that each spatial factor that corresponds to a, b, or c is an eigenfunction of H. Hence, in doing the DQMC computer simulation of the imaginary-time Schrödinger equation, we need deal with only one of the spatial factors, say, the one corresponding to a. For our system of electrons in a box, there are three spatial variables, the coordinates x1, x2, and x3 of the three electrons, and the DQMC simulation is done in a three-dimensional space bounded by the sides of a cube, on which the wave function vanishes. Show that the spatial factor corresponding to a has a nodal surface defined by x2 = x3 and this is the only nodal surface within the cube.
(c) Show that the nodal surface x2 = x3 is a plane that divides the cube into two regions of equal volume; that in one of these regions (the one with x3 > x2) the wave function is positive, and in the other region, the wave function is negative. Also show that for each point P in one region, there is a corresponding point (the one with the values of x2 and x3 interchanged) where the wave function has minus its value at P. In doing the DQMC simulation, one works entirely within one region and eliminates any walker that crosses the nodal surface.