the pseudopotential Hamiltonian is just a projection operator in the relative angular momentum space, the metric in $V_1 \mathcal{L}_1(\mathbf{q}^2)$ is the same as that originating from the cyclotron orbits.

![Dependence of the intrinsic metric $\alpha$ on the mass metric $\alpha_m$ (Coulomb metric is set to identity).[]{data-label="fig_intrinsic"}](fig4.eps){width="0.8\linewidth"}

In Fig. \[fig\_optimize\] we pick the ground state of the Coulomb interaction with fixed mass anisotropy $\alpha_0=2, \phi_0=0$ (the metric of the dielectric tensor is implicitly assumed to be $\alpha=1, \phi=0$), and we evaluate the overlap with a family of Laughlin states generated by varying $\alpha, \phi$. The overlap $|\langle \Psi_L^{\alpha,\phi}| \Psi_C^{\alpha_0=2,\phi_0=0}\rangle|$ is plotted as a function of $\alpha$ and $\phi$. We observe that the principal axis of the Laughlin state is aligned with that of the Coulomb state (maximum overlap occurs for $\phi=\phi_0=0$). Interestingly, the maximum overlap does *not* occur for $\alpha=\alpha_0$, but for some value of the anisotropy that is a “compromise" between the dielectric $\alpha=1$ and a cyclotron one $\alpha=2$. The value of the anisotropy that defines the intrinsic metric depends linearly on the band mass anisotropy (Fig. \[fig\_intrinsic\]). This result illustrates the ability of the Laughlin state to optimize the shape of its fundamental droplets and maximize the overlap with 