6 restrict our attention to ferromagnetic HEGs to investigate the proposed hybrid phase.) The hybrid phase has the same symmetry as the Wigner crystal, but has partially delocalized orbitals. Falakshahi and Waintal generated hybrid orbitals for the ferromagnetic N -electron system by solving the Schr¨ odinger equation for a single, positively charged particle moving in a lattice of negative charges placed at the N Wigner-crystal lattice sites. If the charge of the test particle is zero then the orbitals for the system are plane waves, i.e., fluid orbitals are obtained. If the charge is large then localized (crystal-like) orbitals are obtained. As the charge of the test particle is increased from zero, there must come a point at which a band gap opens up between the N and ( N + 1)th states. The set of orbitals obtained at the point at which the gap opens correspond to the hybrid phase. We have tried to find such a hybrid phase by optimizing orbitals of the form centered on the crystal lattice sites, where P denotes a star of symmetry-equivalent simulation-cell G vectors. The coefficients { c$_{P}$ } and C are optimizable parameters. Equation (1) is a general expansion of Wannier-like orbitals. It can therefore describe the crystal and hybrid phases, but not the fluid, since the latter corresponds to a partially filled band. Our QMC results for ferromagnetic HEGs at r$_{s}$ = 30 a.u. obtained with different orbitals are shown in Table II. Within VMC it is possible to lower the energy significantly by optimizing the plane-wave coefficients in Eq. (1), apparently suggesting that Gaussian crystal orbitals are nonoptimal. This does not carry over to DMC, however. In fact it is more important to use Gaussian exponents optimized within DMC than it is to use either plane-wave expansions in the orbitals or BF functions, suggesting that the fixed-node errors in our crystal DMC energies are very small. We have searched for the hybrid phase by optimizing wave functions with orbitals of the form given in Eq. (1) using different starting points, but have not been able to lower the DMC energy significantly. While this does not prove the nonexistence of the hybrid phase, as searching for minima in a high-dimensional space is a very difficult problem, the fact that our extensive searches have been unable to find it makes its existence unlikely. In summary, we have studied the zero-temperature phase behavior of the 2D HEG using QMC. We find a transition from a paramagnetic fluid to an antiferromagnetic triangular Wigner crystal at r$_{s}$ = 31(1) a.u. We find no region of stability for a ferromagnetic fluid, although we find that the paramagnetic fluid is unstable to the ferromagnetic fluid at about r$_{s}$ = 40 a.u. and the antiferromagnetic crystal is unstable to the ferromagnetic crystal at r$_{s}$ = 38(5) a.u. We find no evidence for the existence of hybrid phases of the type suggested by Falakshahi and Waintal [11], but we φ ( r ) = A exp( − Cr $^{2}$) + ∑ P c$_{P}$ ∑ G ∈ P cos( G · r ) (1)