Ab_init stress perturbation theory - Department of Physics and ...

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Strain tensor h ab as a perturbation, continued • Existing formulation (1) – Introduce fictitious strained self-consistent Hamiltonian obtained by a scale transformation of the unstrained Hamiltonian, H! ! ∇= ⎡ ⎣ + ⋅ + ⋅∇⎤ ⎦ η −1 SCF (, r ) HSCF (1 η) r,(1 η) H η SCF H! η – obeys the same boundary conditions as the strained Hamiltonian – has the same spectrum unstrained Hamiltonian, and a simply related SCF electron density n! η () r – Energy difference between unstrained crystal and fictitious strained crystal described by ! is easily obtained H η SCF • Problem: Hartree and XC potentials of H! η are not those generated by SCF ! so ! is not a genuine Kohn-Sham Hamiltonian n η () r H η SCF – Must introduce modified external potential ext to compensate η – Then use DFPT treating ! η as the perturbation V ext −V ext • Complex two-step process apparently changes the structure of DPFT from that for other perturbations (1) S. Baroni, P. Giannozzzi, and A. Testa, Phys. Rev. Lett. 59, 2662 (1987). ! V η
Abinit reduced coordinate (~) formulation • Every lattice, unstrained or strained, is a unit cube in reduced coordinates. – Primitive real and reciprocal lattice vectors define the transformations: ∑ ∑ ∑ X = R i X! , K ≡ ( k + G ) = G K! , R G = 2πδ P P P P α α i α α α αi i αi α j ij i i α α, β ," = 1,3 i, j ," = 1,3 – Cartesian indices and reduced indices • Every term in the DFT functional can be expressed in terms of dot products and the unit cell volume W. – Dot products and W in reduced coordinates are computed with metric tensors, ∑ X′ ⋅ X= X! ′ Ξ X! , K′ ⋅ K = K! ′ ϒ K! , Ω= (det[ Ξ ]) ij i ij j i ij j ij ij • Strain is now a “simple” parameter of a density functional whose wave functions have invariant boundary conditions. ∑ 1/2
Page 1 and 2: Implementation of the Strain Pertur
Page 3: Strain tensor h ab as a perturbatio
Page 7 and 8: Stress and strain notation Cartesia
Page 9 and 10: DFPT for elastic and piezoelectric
Page 11 and 12: Design of development process • F
Page 13 and 14: Design of development process (cont
Page 15 and 16: Subroutines for strain perturbation
Page 17 and 18: Nonlocal pseudopotential strain der
Page 19 and 20: XC non-linear core correction • O
Page 21 and 22: Strain perturbation for metals •
Page 23 and 24: 2DTE terms in output file • Mix o
Page 25 and 26: Input file for anaddb run dieflag 3
Page 27 and 28: Global comparison with numerical de
Page 29 and 30: Future development, continued • P
Page 31 and 32: Mathematica for nonlocal psp, , con

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