6.3 The Hamiltonian within periodic boundary conditions

The many-body Hamiltonian for a system of electrons at positions ${\bf r}_i$ and static ions at positions ${\bf x}_\alpha$ is

$$\hat{H} = \sum_{i}-\frac{1}{2}\nabla^2_i + \sum_i\sum_{\alpha}v_{\alpha}({\bf r}_i,{\bf x}_\alpha)$$

$$+ \frac{1}{2}\sum_i\sum_{j\neq i}v({\bf r}_i,{\bf r}_j) + \frac{1}{... ...a}\sum_{\beta\neq \alpha}v_{\alpha\beta}({\bf x}_{\alpha},{\bf x}_{\beta})\;\;.$$ (6.1)

An infinite system is normally modeled by a finite simulation cell subject to periodic boundary conditions. The model interaction terms, $v_{\alpha}$, $v$, and $v_{\alpha\beta}$, are chosen such that the potential energy of the model system, which involves only the positions of the particles in the finite simulation cell, mimics the potential energy of the infinite system as closely as possible. Since the potential energy of the infinite system depends on the positions of all the charges in the solid, only a few of which are included in the simulation, the model interaction energy is approximate even in crystalline solids. To enforce the periodic boundary conditions the functions $v_{\alpha}$, $v$, and $v_{\alpha\beta}$ must be invariant under the translation of either argument by a member of the set of translation vectors of the simulation cell lattice, $\{{\bf R}\}$. The standard approach is to choose the model Hamiltonian such that the full potential energy of equation 6.1, evaluated by summing the model interactions between all pairs of particles in the simulation cell, equals the potential energy per cell of an infinite array of identical copies of the simulation cell. However, even when each simulation cell in the array of copies is restricted to be overall charge neutral, the sum of inter-particle Coulomb $1/r$ interactions is only conditionally convergent, [76] and to define this model interaction uniquely the boundary conditions at infinity must be specified. The standard procedure is to define the potential by solving Poisson's equation subject to periodic boundary conditions, in which case the model interaction is the Ewald interaction. [51] Further details of the Ewald interaction and the supercell approach were given in chapter 4.