Pseudopotentials (Part I): Georg KRESSE Institut f ur - PowerPoint PPT Presentation

Pseudopotentials (Part I): Georg KRESSE Institut f¨ ur Materialphysik and Center for Computational Materials Science Universit¨ at Wien, Sensengasse 8, A-1090 Wien, Austria b-initio ackage imulation ienna G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 1

� � � Overview the very basics – periodic boundary conditions – the Bloch theorem – plane waves – pseudopotentials determining the electronic groundstate effective forces on the ions general road-map to the things you will hear in more detail later G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 2

� � � Periodic boundary conditions as almost all plane wave codes VASP uses al- ways periodic boundary conditions the interaction between repeated images must be handled by a sufficiently large vacuum re- gion sounds disastrous for the treatment of molecules but large molecules can be handled with a comparable or even better performance than by e.g. Gaussian G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 3

✂ ✁ ✄ ☎ ✂ � ✞ ✁ � � ✄ ✂ ✁ The Bloch theorem the translational invariance implies that a good quantum number exists, which is usually termed k k corresponds to a vector in the Brillouin zone all electronic states can be indexed by this quantum number Ψ k in a one-electron theory, one can introduce a second index, corresponding to the one electron band n ψ n r k the Bloch theorem implies that the (single electron) wavefunctions observe the equations e i k τ ψ n τ ψ n r r k k ✄✝✆ where τ is any translational vector leaving the Hamiltonian invariant G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 4

✂ � ✂ ✂ ✂ ✌ ✁ ✞ ✡ ✂ ✄ ✄ ✄ ☛ ✄ ✂ ☎ ✁ ☞ ✡ ✁ ✂ ✁ ☎ ✏ ✂ ✄ ✞ ✌ ✎ � ✍ ✄ ✍ ✂ ✄ ✄ ✂ ✂ ✂ ☎ ✍ ✑ ✟ ✍ ✁ ✡ ✁ ✑ ✄ ✂ ✄ ✍ ✠ ✂ ✁ ☎ ✂ ✄ ✄ ✍ ✂ ✁ ✎ ✆ ✄ The DFT Hamiltonian the charge density is determined by integrating over the entire Brillouin zone and summing over the filled bands ∑ ∞ ρ e k ψ n ψ d 3 k f n r r r k n n k ✄✝✆ where the charge density is cell periodic (can be seen by inserting the Bloch theorem) 1 are the Fermi-weights β ε n ε Fermi and f n 1 exp k k the KS-DFT equations (Schr¨ odinger like) are given by h 2 2 m e ∆ ρ e ψ n ε n k ψ n ¯ V eff r r r r k k ✄✝✆ ρ e ρ ion r r ρ e ρ e V eff e 2 d 3 r r r V xc r ✄✝✆ r r ρ ion is the ionic charge distribution G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 5

✔ ✞ ✁ ✑ ✂ ☎ ✑ ✑ ✒ ☎ ✑ � ✞ ✓ ✁ ✂ ✂ ✞ ✄ ✆ � ✂ ✒ ✕ ✖ � ✂ ✄ ✄ ✁ ✂ ✁ ✁ ✂ ✁ ✞ Plane waves introduce the cell periodic part u n k of the wavefunctions ψ n e i kr r u n r k k ✄✝✆ r is cell periodic (insert into Bloch theorem) u n k all cell periodic functions are now written as a sum of plane waves 1 1 2 ∑ 2 ∑ ψ n C G n k e i Gr C G n k e i G k r u n r r k k Ω 1 Ω 1 ✄✝✆ G G ∑ ρ ρ G e i Gr r ✄✝✆ G ∑ V G e i Gr V r ✄✝✆ G in practice only those plane waves are included which satisfy G k h 2 ¯ 2 G k E cutoff 2 m e G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 6

✘ ✗ ✙ ✚ ✢ ✗ ✛ ✗ ✜ Fast Fourier transformation FFT real space reciprocal space τ 2 G cut b2 0 1 2 3 0 1 2 3 4 5 0 N−1 0 −4 −3 −2 −1 τ 1 b1 N/2 −N/2+1 τ 1 π / τ x = n / N g = n 2 1 1 1 1 1 1 1 ∑ N FFT ∑ ψ n C G n k e i Gr i Gr 2 C r n k e i kr C r n k C G n k C r n k e r k Ω 1 r G G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 7

� � � Why are plane waves so convenient historical reason: many elements exhibit a band-structure that can be interpreted in a free electron picture (metallic s and p elements) the pseudopotential theory was initially developed to cope with these elements (pseudopotential perturbation theory) practicle reason: the total energy expressions and the Hamiltonian H are dead simple to implement a working pseudopotential program can be written in a few weeks using a modern rapid prototyping language computational reason: because of it’s simplicity the evaluations of H ψ is exceedingly efficient using FFT’s G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 8

✣ ✑ ✣ ✑ ✑ ✁ ☎ ✑ ✡ ☎ ✄ ✥ � ✓ ✁ ✔ ✑ ✕ ☎ ☎ ✑ ✣ ✁ ✁ ✧ ✂ ✧ ✄ ✡ � ☛ ✑ ✂ ☎ ✑ ✂ ☎ ✄ ✣ ✁ � ✦ Computational reason evaluation of H ψ n r k h 2 ¯ ∆ ψ n V r r k 2 m e ψ n 1 2 e i G k r and using the convention r G k G k C G n k k Ω 1 ✤✝✆ ✤✝✆ kinetic energy: h 2 h 2 2 ¯ ¯ G k ∆ ψ n G k C G n k N planewaves k ✤✝✆ 2 m e 2 m e local potential: 1 N FFT ∑ ψ n i Gr G k V V r C r n k e N FFT log N FFT k ✤✝✆ r if H would be stored as a matrix with N planewaves N planewaves components N planewaves operations would be required N planewaves G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 9

Local part of Hamiltonian Gcut FFT ψ G+k ψ r V r FFT ψ r ψ V <k+G V > r G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 10

� ★ � � Pseudopotential approximation the number of plane waves would exceed any practicle limits except for H and Li pseudopotentials instead of exact potentials must be applied three different types of potentials are supported by VASP – norm-conserving pseudopotentials – ultra-soft pseudopotentials – PAW potentials they will be discussed in more details in later sessions all three methods have in common that they are presently frozen core methods i.e. the core electrons are pre-calculated in an atomic environment and kept frozen in the course of the remaining calculations G. K RESSE , P SEUDOPOTENTIALS (P ART I) Page 11