Density Functional Theory and the Family of (L)APW-methods: a step-by-step introduction
Stefaan Cottenier
1 Density Functional Theory as a way to solve the quantum many body problem 1
1.1 Level 1: The Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . 1
1.2 Level 2: Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 The theorems of Hohenberg and Kohn . . . . . . . . . . . . . . . . . . . 2
1.2.2 The Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 The exchange-correlation functional . . . . . . . . . . . . . . . . . . . . . 7
1.3 Level 3: Solving the equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 The pseudopotential method (in brief) 11
3 The APW method 15
4 The LAPW method 21
4.1 The regular LAPW method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 LAPW with Local Orbitals (LAPW+LO) . . . . . . . . . . . . . . . . . . . . . 22
5 The APW+lo method 25
5.1 The ‘pure’ APW+lo basis set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Mixed LAPW/APW+lo basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3 APW+lo with Local Orbitals (APW+lo+LO) . . . . . . . . . . . . . . . . . . . 26
6 The PAW method (in brief) 27
7 Examples for WIEN2k 31
7.1 Linearization energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.1.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.1.2 Interpreting input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.1.3 Interpreting output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.1.4 The -in1orig and -in1new options . . . . . . . . . . . . . . . . . . . . . 40
7.1.5 high-lying local orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.2 Finding the best RmtKmax and k-mesh . . . . . . . . . . . . . . . . . . . . . . . 42
7.2.1 Why do we need this? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.2.2 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.2.3 RmtKmax-values per element: overview . . . . . . . . . . . . . . . . . . . 48
7.2.4 Translation of tests to other cells . . . . . . . . . . . . . . . . . . . . . . 48
A Fourier transforms, plane waves, the reciprocal lattice and Bloch’s theorem 51
A.1 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
A.2 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
A.3 The reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.4 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
B Quantum numbers and the Density Of States 57
B.1 Familiar examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B.2 Crystalline solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
C The eigenvalue problem 63
C.1 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
C.2 Basis transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
C.3 A practical procedure to find eigenvalues and eigenvectors . . . . . . . . . . . . 66
D Solutions of the radial part of the Schr¨odinger equation 69
E The homogeneous electron gas 71
F Functionals 73
F.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
F.2 Functional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
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