Ice Ih is comprised of orientationally disordered water molecules giving rise to positional disorder of the hydrogen atoms in the hydrogen bonded network of the lattice. Here we arrive at a first principles determination of the surface energy of ice Ih and suggest that the surface of ice is significantly more proton ordered than the bulk. We predict that the proton order-disorder transition, which occurs in the bulk at $72 K, will not occur at the surface at any temperature below surface melting. An order parameter which defines the surface energy of ice Ih surfaces is also identified. DOI: 10.1103/PhysRevLett.101.155703 PACS numbers: 64.60.Cn, 68.47.Àb, 82.65.+r Ice Ih, the normal form of ice, is built from water molecules arranged on a tetrahedral lattice, subject to a set of rules known as the ice rules [1]. These rules allow for orientational disorder of the molecules, which is discussed in terms of the positional disorder of the hydrogen atoms, a phenomenon known as proton disorder. Under ambient pressures, ice Ih is the stable phase down to $72 K [2], below which a proton ordered (ferroelectric) phase known as ice XI becomes the ground state. In practice, the ordering transformation from ice Ih to the lowest total energy ice XI phase rarely happens since, as the temperature is lowered to the transition temperature, proton motion comes to a halt. This leaves the crystal out of equilibrium at below 72 K with residual entropy. Indeed, ice Ih is so resistant to transforming from ice Ih into ice XI that a dopant that accelerates proton rearrangement (KOH) is usually added to make it occur.Given the experimental difficulties in probing the ice Ih-XI transition, molecular simulations are very useful [3][4][5]. In particular, in a tour de force study, density functional theory (DFT) was combined with a framework based on graph invariants to simulate the bulk ice Ih-XI transition. A transition temperature of 98 K was predicted, within 30 K of the experimental transition [5]. This demonstrates that DFT is appropriate for tackling the subtle question of proton order and disorder in ice, something which cannot necessarily be said about the many empirical potentials that otherwise describe many parts of the phase diagram of water well [6].Unlike in bulk ice, the understanding of the energetics of proton order at the surface is in its infancy [7,8]. This is true despite widespread interest in ice surfaces at low temperatures, motivated mainly by their catalytic role in atmospheric chemistries such as ozone depletion (see, e.g., [9]). Indeed, the understanding is so incomplete that the value of the surface energy-one of the most important quantities of any material-is well-established neither experimentally nor theoretically and how it is influenced by proton order is not known. Here, in arriving at an ab initio determination of the surface energy of the (0001) (basal plane) of ice Ih, we demonstrate that the energetics of proton ordering differs significantly at the surface compared to the bulk. Indeed, the range of...