Computational Physics

6.6 The linearised APW (LAPW) method 143 with γl=R′l(R)R ̇l(R)[jl′(qR)jl(q′R)+jl(qR)jl′(q′R)] −[R′l(R)R ̇′l(R)jl(qR)jl(q′R)+Rl(R ...

144 Solving the Schrödinger equation in periodic solids V V 1 2 2 1 r V ψ ψ Figure 6.11. The principle of the pseudopotential. T ...

6.7 The pseudopotential method 145 Figure 6.11which shows two different potentials and their solutions (for the same energy). Th ...

146 Solving the Schrödinger equation in periodic solids d x y z Figure 6.12. The diamond structure. There are numerous review ar ...

6.7 The pseudopotential method 147 d1,2=± 1 / 8 (a 1 +a 2 +a 3 )relative to the lattice points: Vps(tot)(K)=Vps(at)(K)(eiK·d^1 + ...

148 Solving the Schrödinger equation in periodic solids Γ XW L Γ K k E 0.0 0.5 Figure 6.13. Band structure of silicon. 6.7.2 Acc ...

6.7 The pseudopotential method 149 If we now consider an energy-dependent pseudopotentialVpswith eigenstateφ, we obtain, aside f ...

150 Solving the Schrödinger equation in periodic solids ourselves to cubic unit cells. Although all quantities are expanded in a ...

6.7 The pseudopotential method 151 wherefj is theoccupancy, which is usually the Fermi–Dirac function with an additional factor ...

152 Solving the Schrödinger equation in periodic solids Even if both|K|and|K′|are smaller thanKmax, their difference can attain ...

6.7 The pseudopotential method 153 (^0) C 2 Gridsize 1 C 1 UC Figure 6.14. Two-dimensional representation of the reciprocal grid ...

154 Solving the Schrödinger equation in periodic solids Note thatKandK′are indices of the Hamiltonian – therefore they lie insid ...

6.7 The pseudopotential method 155 Table 6.1.Parameters for the GTH pseudopotential Hydrogen Silicon ξ 0.2 0.44 C 1 −4.0663326 − ...

156 Solving the Schrödinger equation in periodic solids The projector functions have the formpli: p^01 = 1 √ 4 rs √ 2 rsπ^5 /^4 ...

6.7 The pseudopotential method 157 Table 6.2. Parameters for the GTH parametrisation of the exchange-correlation energy. a 1 0.4 ...

158 Solving the Schrödinger equation in periodic solids 6.7.7 Evaluating the energy If you have obtained the correct spectrum, t ...

6.7 The pseudopotential method 159 where the first term is the Hartree electrostatic energy due to the electrons, the second ter ...

160 Solving the Schrödinger equation in periodic solids Table 6.3.Contributions to electronic energy of hydrogen atom. Contribut ...

6.8 Extracting information from band structures 161 betweenEandE+dE, divided by dE. Another quantity of interest is the charge d ...

162 Solving the Schrödinger equation in periodic solids 6.9 Some additional remarks In this chapter we have described how the no ...