With the initial guess for the potential calculated, HADOKEN computes initial wavefunctions and energies from the Schrödinger equation (Eq. ( )) using the Arnoldi algorithm within the MATLAB PDE Toolbox pdeeig function . For both charge neutral and Fermi-pinned hexagonal geometries, the output of the previously discussed V_conduction_band.m is used as input for the "a" coefficient in pdeeig. For both the Ga-and N-face triangular geometries, the "a" coefficient uses the output of the V_total_piezo.m m-file, which calculates the sum of V_poisson (containing the spontaneous and piezoelectric polarization contributions) and the conduction band edge profile, V tri CB (x, y), given by Eq. ( ). Since the effective mass has a spatial dependence in the Schrödinger equation, the heaviside_core_schrod.m m-file (which has a similar functionality as the heaviside_core_poiss.m m-file discussed above) is used to construct an appropriately scaled m 0 /m * ( x, ỹ) term, which is used as the "c" coefficient in the pdeeig PDE solver. A two-element vector, r, containing the range of eigenvalues to compute is also used by the pdeeig command: for hexagonal cross-sections obeying a charge-neutrality constraint, r is initially set to [0, C]; however, for NWs with a Fermi-pinning constraint, the lower limit of r is initially set to the minimum of the electrostatic potential energy, and the upper limit is set to C or 10C above the previous lower limit for Ferm-pinned hexagonal or triangular geometries, respectively. A boundary condition matrix, b, which sets the wavefunctions to zero at the outer shell edge, is used as input to pdeeig. Finally, at least two wavefunctions and their associated energy levels are computed by incrementally changing the lower and upper limit of the r vector in subsequent function calls to pdeeig. The normalize_and_sqrt_m_triangular.m m-file calculates the normalization constant and d x d ỹ |ψ n ( x, ỹ)| 2 m * ( x, ỹ)/m 0 for each of the n wavefunctions (the latter expression is used evaluate n e ( x, ỹ) in Eq. ( )).