Transcription of VASP: Basics (DFT, PW, PAW, … )
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VASP: Basics (DFT, PW, PAW, … )
VASP: Basics (DFT, PW, PAW, .. )University of Vienna,Faculty of Physics and Center for Computational Materials Science,Vienna, AustriaOutline Density functional theory Translational invariance and periodic boundary conditions Plane wave basis set The Projector- augmented -Wave method Electronic minimizationThe Many-Body Schr dinger equation H (r1,..,rN)=E (r1,..,rN)0@ 12Xi i+XiV(ri)+Xi6=j1|ri rj|1A (r1,..,rN)=E (r1,..,rN) (r1,..,rN)!{ 1(r), 2(r),.., N(r)}For instance, many-body WF storage demands are prohibitive:A solution: map onto one-electron theory:5 electrons on a 10 10 10 grid ~ 10 PetaBytes! (r1,..,rN)(#grid points)NHohenberg-Kohn-Sham DFTE[ ]=Ts[{ i[ ]}]+EH[ ]+Exc[ ]+EZ[ ]+U[Z] (r1,..,rN)=NYi i(ri) (r)=NXi| i(r)|2 12 +VZ(r)+VH[ ](r)+Vxc[ ](r) i(r)= i i(r) (r1,..,rN)!{ 1(r), 2(r),.., N(r)}Map onto one-electron theory:Total energy is a functional of the density:The density is computed using the one-electron orbitals:The one-electron orbitals are the solutions of the Kohn-Sham equation:BUT:Exc[ ]=?
The Projector-Augmented-Wave method The number of plane waves needed to describe • tightly bound (spatially strongly localized) states, • and rapid oscillations (nodal features) of the orbitals near the nucleus exceeds any practical limit, except maybe for Li and H. The common solution: • Introduce the frozen core approximation:
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