The Plane-wave Pseudopotential Method
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1 The Plane-wave Pseudopotential Method k(r) = X G c k,g e i(g+k) r Chris J Pickard
2 Electrons in a Solid
3 Nearly Free Electrons
4 Nearly Free Electrons
5 Nearly Free Electrons
6 Electronic Structures Methods
7 Empirical Pseudopotentials
8 Ab Initio Pseudopotentials
9 Ab Initio Pseudopotentials
10 Ab Initio Pseudopotentials
11 Ab Initio Pseudopotentials
12 Ultrasoft Pseudopotentials (Vanderbilt, 1990)
13 Projector Augmented Waves
14 Deriving the Pseudo Hamiltonian
15 PAW vs PAW C. G. van de Walle and P. E. Bloechl 1993 (I) PAW to calculate all electron properties from PSP calculations" P. E. Bloechl 1994 (II) The PAW electronic structure method" " The two are linked, but should not be confused
16 References David Vanderbilt, Soft self-consistent pseudopotentials in a generalised eigenvalue formalism, PRB " Kari Laasonen et al, Car-Parinello molecular dynamics with Vanderbilt ultrasoft pseudopotentials, PRB " P.E. Bloechl, Projector augmented-wave method, PRB " G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, PRB
17 On-the-fly Pseudopotentials - All quantities for PAW (I) reconstruction available" - Allows automatic consistency of the potentials with functionals" - Even fewer input files required" - The code in CASTEP is a cut down (fast) pseudopotential code" - There are internal databases" - Other generation codes (Vanderbilt s original, OPIUM, LD1) exist
18 It all works!
19 Kohn-Sham Equations
20 Being (Self) Consistent
21 Eigenproblem in a Basis
22 Just a Few Atoms
23 In a Crystal
24 Plane-waves Represent the orbital in Fourier space" Z (r) = (g)e ig r d 3 r For a periodic system we have Bloch s Theorem" k(r) = X G c k,g e i(g+k) r Where the Gs are reciprocal lattice vectors and k is a symmetry label in the 1 st Brillouin zone"
25 Cutoff Energy Limit the number of plane wave components to those such that (G + k) 2 2 apple E cut This defines a length scale = p Ecut
26 Cutoff Energy "" - The minimum length scale depends on the elements in the system through PSPs" "" - Variational principle -> energy monotonically decreases to ground state energy as E cut increases (not strictly for USP/PAW)" "" - Converge required property with respect to cutoff energy "
27 Cutoff Energy "" - The minimum length scale depends on the elements in the system through PSPs" "" - Variational principle -> energy monotonically decreases to ground state energy as E cut increases (not strictly for USP/PAW)" "" - Converge required property with respect to cutoff energy "
28 Cutoff Energy "" - The minimum length scale depends on the elements in the system through PSPs" "" - Variational principle -> energy monotonically decreases to ground state energy as E cut increases (not strictly for USP/PAW)" "" - Converge required property with respect to cutoff energy "
29 Why Plane-waves? - Systematic convergence with respect to single parameter" - Non-local. Cover all space equally" - Cheap forces (no Pulay term)" - No basis-set superposition error " - Numerically efficient " - Use FFTs to transform between real and reciprocal space" - The obvious choice for periodic and works well for aperiodic systems" - Rapid development of new functionality
30 Progress, but
31 The Total Energy
32 Applying H
33 Evaluating the Energy
34 Iterative Diagonalisation
35 Minimisation
36 Plane-wave Codes CASTEP PEtot VASP PARATEC PWscf Da Capo Abinit CPMD Qbox fhi98md PWPAW SFHIngX DOD-PW NWchem Octopus JDFTx
37 CASTEP Original code due to Mike Payne rewritten by CDG, starting 1999, unveiled at 2001 CASTEP Workshop " Modern modular design, using best features of Fortran90 " Focus on robustness, ease of use, and core functionality " Particular strength in spectroscopy (NMR/IR/Raman/EELS) " Marketed commercially by Biovia (formerly Accelrys) " Low cost European Academic licence available
38 The CDG Matthew Segall Main author of the code specification, and responsible for all the low-level communications and basis set coding. Also the population analysis algorithms. " Matt Probert Responsible for the geometry optimisation and molecular dynamics coding, and keeper of the parameters module. " Stewart Clark Responsible for the band structure and exchange-correlation functional coding and keeper of the cell module. Also co-author of the linear response code. " Chris Pickard Responsible all pseudopotential coding, amongst other things. Also co-author of the NMR code. " Phil Hasnip Responsible for the electronic energy minimisation coding (including density mixing and ensemble DFT), band-parallelism, run-time profiling and parallel efficiency report, and wavefunction subroutines. Also works on optimisation and parallel performance. " Keith Refson Responsible for phonon and E-field calculations and co-author of the linear response code. " Jonathan Yates Co-author of the NMR code, responsible for spectral module (dos, optics, eels). " Mike Payne CASTEP creator and general overseer of the whole project.
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