Exact Exchange in Density Functional Calculations

28 4. Extended SystemsThe first is that two states having almost the same k index are very similar, justifying that thestates are only calculated for a representative number of k-points within the first BZ. Especially forinfinitely periodic systems there are an infinity of occupied states, which in the Bloch representationtranslates to a continuum of k-values, but still only a finite number of occupied bands for eachk point. Calculating an observable, e.g. the total energy, for an infinite or finite periodic systemthus amounts to picking a representative set of k vectors, solving the Kohn-Sham equation for the(few) occupied bands, and making an appropriate average over the k vectors.Many different schemes for picking a representative set of k points exists, see e.g. ref. [23, 24].An additional feature of the Bloch representation is that symmetry properties of the unit cellgreatly reduce the number of distinct k vectors. The only k vectors needed are those of theirreducible BZ, which is usually only a fraction of the BZ in size; existence of a mirror symmetryfor example reduces the BZ by a factor of two.The Bloch representation also implies a reduction of the basis set necessary for representingthe wave functions, as will be demonstrated in the following section.4.2 Basis Sets and Boundary ConditionsSince the Kohn-Sham equations are second order differential equations, two sets of linearly independentboundary conditions (BC’s) must be specified.The appropriate BC’s depend on the the nature of the considered system.Plain WavesThe advantage of Bloch’s theorem, when using plain wave basis sets, is that the function u n,k (r)of (4.1b) is cell-periodic, and as such can be represented by the discrete expansionu n,k (r) = ∑ Gc n,k+G · e iG·r ⇒ ψ n,k (r) = ∑ Gc n,k+G · e i(k+G)·r (4.2)where the G’s are reciprocal lattice vectors (i.e. G · R = 2πp, p ∈ Z). In principle the expansion(4.2) requires an infinite sum, but the low energy (small G values) terms will typically be dominant,so in practice the expansion is truncated beyond some large G. Had Bloch’s theorem not beenapplied, a plane wave expansion would have to be continuous and would therefore require aninfinite basis set despite the truncation of the expansion.Using plane waves as an expansion results in the particularly simple secular version of theKohn-Sham equation∑ [ 12 |k + G′ | 2 δ GG ′ + VG−G eff ] ′ cn,k+G ′ = ɛ n · c n,k+G (4.3)G ′where VGeff are the expansion coefficients of V eff(r). The term in square brackets represents theHamiltonian matrix H k+G,k+G ′. A nice feature is that the kinetic terms are diagonal. For largeG ′ vectors, the kinetic energy E kin = 1 2 |k + G′ | 2 will dominate, and the truncation of the planewave expansion, is usually done by choosing a cutoff energy E c = 1 2 |k + G c| 2 , beyond which allterms of (4.3) are truncated. The size of the Hamiltonian matrix is thus (slightly) k dependent.Real Space GridsOne can also choose to represent the wave functions on real space grids. When using this representation,one works directly with the (non-periodic) Bloch states ψ n,k , utilizing eq. (4.1a) to obtainthe appropriate BC’s:e ik·R ψ n,k (r) = ψ n,k (r + R)e ik·Rˆn(r) · ∇ψ n,k (r) = −ˆn(r + R) · ∇ψ n,k (r + R)(4.4a)(4.4b)