Exact Exchange in Density Functional Calculations

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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)
4.2 Basis Sets and Boundary Conditions 29where ˆn(r) is the (outward) unit normal of the cell boundary at r.Having specified the two BC’s (4.4), the KS equations only have to be solved in between, i.e. ina single unit cell. Had Bloch’s theorem not been applied, one would have to solve the KS equationin the entire domain of the crystal.While the kinetic energy was diagonal in the plain wave representation, the effective KS potentialis diagonal in real space (if it is local, i.e. the HF potential is not diagonal in real spaceeither).Isolated SystemsChoosing a plane wave basis set for the representation of the wave functions requires the applicationof periodic (Born-Von Karman) boundary conditions, thus in practice making any system infinitelyperiodic. This is the natural boundary conditions for solids. For isolated systems, e.g. molecules,the appropriate choice is Dirichlet boundary conditions. When using plain waves, Dirichlet boundaryconditions can be obtained by embedding the system in a sufficiently large super-cell, suchthat the wave functions are essentially zero, and has zero gradient, at the boundaries. In this casethere will be no difference between the wave functions for different k values, and typically onlythe Γ point (k = (0, 0, 0)) is chosen.Enforcing Dirichlet BC’s in real space is not a problem, which is one of the strengths of realspace basis sets. Actually one has the freedom to choose more exotic BC’s like e.g. chiral boundaryconditions, making it possible to represent for example nanotubes with a minimal unit cell.To represent an isolated system correctly, the unit cell must be big enough that the potential is(practically) zero at the boundaries. As the decay of potentials is generally quite slow, this impliesthe use of very large cells. In real space, one can do a multipole expansion of the density, anduse these to enforce the correct boundary conditions, thus reducing the minimal unit cell to onecompletely containing the density (which decays must faster than the potential). This procedurealso allows one to handle charged systems efficiently, which is non-trivial when using plain waves,as a non-zero charge per unit cell in this case implies an infinite charge in the system (see section8.3.1).
Page 1: Exact Exchange in DensityFunctional
Page 5: AbstractThis thesis discuss the the
Page 8 and 9: viii
Page 11 and 12: Chapter 1IntroductionElectronic Str
Page 13: 3magnetic and electric field streng
Page 16 and 17: 6 2. Basic Wave Function TheoryIt d
Page 18 and 19: 8 2. Basic Wave Function Theory2.3
Page 21 and 22: Chapter 3Density Functional TheoryD
Page 23 and 24: 3.2 The Generalized Kohn-Sham Schem
Page 29 and 30: 3.3 Exchange and Correlation 19Ther
Page 31 and 32: 3.3 Exchange and Correlation 21Figu
Page 33 and 34: 3.3 Exchange and Correlation 23is t
Page 35 and 36: 3.3 Exchange and Correlation 25The
Page 37: Chapter 4Extended SystemsAlthough w
Page 42 and 43: 32 5. Orbital Dependent Functionals
Page 44 and 45: 34 5. Orbital Dependent Functionals
Page 47 and 48: Chapter 6Projector Augmented WaveMe
Page 49 and 50: 6.1 The Transformation Operator 39i
Page 51 and 52: 6.4 Densities 41For local operators
Page 53 and 54: 6.5 Total Energies 43where the nota
Page 55 and 56: 6.5 Total Energies 45i.e. that the
Page 57 and 58: 6.6 The Transformed Kohn-Sham Equat
Page 59 and 60: 6.7 Exact Exchange in PAW 49The Val
Page 61 and 62: 6.7 Exact Exchange in PAW 51where t
Page 63 and 64: 6.8 Summary 53where∫ṽ nn ′(r)
Page 65 and 66: Chapter 7Implementing PAWFor an imp
Page 67 and 68: 7.2 The Atomic Data Set of PAW 57Th
Page 69 and 70: 7.2 The Atomic Data Set of PAW 59Th
Page 71 and 72: Chapter 8Numerical ResultsThe main
Page 73 and 74: 8.2 Molecules 63Figure 8.1: Compari
Page 75 and 76: 8.2 Molecules 65Mol Expt LDA PBE
Page 77 and 78: 8.3 Miscellaneous 67Poisson equatio
Page 79 and 80: 8.3 Miscellaneous 69And the express
Page 81: HG¡"!$#&%')(+*,*-/.$0'#&%1"(+*,*>>
Page 85 and 86: References[1] J. P. Perdew, S. Kurt
Page 87 and 88: REFERENCES 77[26] P. E. Blöchl, C.
Page 89 and 90:
REFERENCES 79[52] A. GörlingOrbita
Page 91 and 92:
Appendix AExact ExchangeThe non-int
Page 93 and 94:
Appendix BWannier Functions and Exa
Page 95 and 96:
Appendix CFourier TransformTo repre
Page 97 and 98:
Appendix DMultipoles and SphericalH
Page 99 and 100:
D.3 Spherical Harmonics 89Generaliz
Page 101 and 102:
D.3 Spherical Harmonics 91The RSH s
Page 103:
D.3 Spherical Harmonics 93L l m ȳl
Page 106 and 107:
96 E. Approximations of the Exchang
Page 109 and 110:
Appendix FForces in PAWIn the groun
Page 111 and 112:
Appendix GThe External Potential in
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