V(V + 1)], in analogy with the regular-spin Casimir operator, S 2 = S 2 x + S 2 y + S 2 z [with eigenvalues S(S + 1)]. An electron having V z = ±1/2 means that it lies in the lowor high-energy valley, respectively. An expanded presentation of the mathematics of the VFCI approach is given in Sec. III.