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The Plane-Wave Pseudopotential Method The Plane-Wave Pseudopotential Method Eckhard Pehlke, Institut für Laser- und Plasmaphysik, Universität Essen, 45117 Essen, Germany. Starting Point: Electronic structure problem from physics, chemistry, materials science, geology, biology, ... which can be solved by total-energy calculations. Topics of this (i) how to get rid of the "core electrons": talk: the pseudopotential concept (ii) the plane-wave basis-set and its advantages (iii) supercells, Bloch theorem and Brillouin zone integrals choices we have... Treatment of electron-electron interaction and decisions to make... Hartree-Fock (HF) ¿ ÒÌ Ò· Ú ´ÖµÒ´Öµ Ö Ú ¼ ¼ ½ Ò´ÖµÒ´Ö µ ¿ ¿ ¼ · Ö Ö · Ò Configuration-Interaction (CI) ¼ ¾ Ö Ö Quantum Monte-Carlo (QMC) ÑÒ Ò ¼ Ú Ê ¿ Ò´Ö µ ÖÆ Density-Functional Theory (DFT) ¼ ½ Ò´Ö µ ¾ ¿ ¼ ´ Ö · Ú ´Öµ· Ö · Ú ´Öµµ ´Öµ ´Öµ ¼ ¾ Ö Ö Æ Ò ¾ Ò´Öµ ´Öµ Ú ´Öµ Æ´Öµ Problem: Approximation to XC functional. decisions to make... Simulation of Atomic Geometries Example: chemisorption site & energy of a particular atom on a surface = ? How to simulate adsorption geometry? single molecule or cluster Use Bloch theorem. periodically repeated supercell, slab-geometry Efficient Brillouin zone integration schemes. true half-space geometry, Green-function methods decisions to make... Basis Set to Expand Wave-Functions ½ ¡Ê ´Øµ ´ Öµ Ù ´Ö Ê µ Ô Æ linear combination of Ê atomic orbitals (LCAO) ´ Öµ ´µ ´ Öµ simple, unbiased, ½ ´·µ¡Ö ´ Öµ ´ · µ plane waves (PW) Ô independent of ª atomic positions augmented plane waves . (APW) etc. etc. I. The Pseudopotential Concept Core-States and Chemical Bonding? Validity of the Frozen-Core Approximation U. von Barth, C.D. Gelatt, Phys. Rev. B 21, 2222 (1980). bcc <-> fcc Mo, transformation energy 0.5 eV/atom core kinetic energy change of 2.7 eV but: error of total energy due to frozen-core approximation is small, less than 2% of structural energy change reason: frozen-core error of the total energy is of second order ½ ¼ £ ¼ ¿ Æ ´ µ´Ú Ú µ Ö « « ¾ Remove Core-States from the Spectrum: Construct a Pseudo-Hamiltonian occupation, eigenvalue 3p 1 -2.7 eV 2p 1 -2.7 eV 3s 2 -7.8 eV valence 1s 2 -7.8 eV Al 2p 6 -69.8 eV pseudo- Z = 13 2s 2 -108 eV core- Al states Z = 3 1s 2 -1512 eV ½ ½ ¾ ´Ô×µ ´Ô×µ ´Ô×µ ¾ ´ Ö · Ú µ ´ Ö · Ú µ « « ¾ ¾ V. Heine, "The Pseudopotential Concept", in: Solid State Physics 24, pages 1-36, Ed. Ehrenreich, Seitz, Turnbull (Academic Press, New York, 1970). Orthogonalized Plane Waves (OPW) 3p 1 -2.7 eV 2p 1 -2.7 eV How to construct 3s 2 -7.8 eV valence 1s 2 -7.8 eV (ps) Al v 2p 6 -69.8 eV pseudo- 2s 2 -108 eV core- Al in principle ? states 1s 2 -1512 eV (actual proc. -> Martin Fuchs) ½ ½ ¾ ´Ô×µ ´Ô×µ ´Ô×µ ¾ ´ Ö · Ú µ ´ Ö · Ú µ « « ¾ ¾ ÇÈ Ï · ÈÏ · · ´ · µ Definition of OPWs: ´ · µ ÈÏ · with ´ · µ ÇÈ Ï · Expansion of eigenstate in terms of OPWs: ¼ Secular equation: Ø ÇÈ Ï · À ÇÈ Ï · ¼ ´Ô×µ ¼ Re-interpretation: Ø ÈÏ · À ÈÏ · ¼ ´Ô×µ ´ · µ ÈÏ · Pseudo-wavefunction: ´Ô×µÇÈÏ Ú Ú · ´ µ OPW-Pseupopotential: Pseudopotentials and Pseudo-wavefunctions 0.7 Pseudopotentials are softer 1s rc=1.242 3s than all-electron potentials. 0.5 Al (Pseudopotentials do not created with have core-eigenstates.) 0.3 pseudo fhi98PP wavefunction u(r) 0.1 Cancellation Theorem: −0.1 0.15bohr If the pseudizing radius is taken all−electron wavefunction −0.3 as about the core radius, then 0246 v (ps) is small in the core region. r (bohr) ´Ô×µÇÈÏ Pseudo-wavefunction is node-less. Ú ¨ Ú ¨ Ú ¨ Plane-wave basis-set feasible. V. Heine, Solid State Physics 24, 1 (1970). Justification of NFE model. Computation of Total-Energy Differences Ge atom slab, ~ 50 Ge atoms 5 all-electron atom: Etotal = -2096 H ~ 10 H (Z = 32) 2 pseudo-atom: E total = -3.8 H ~ 10 H (Z’ = 4) typical structural few 100 meV total-energy difference: -2 ~ 10 H (dimer buckling,...) II. The Plane-Wave Expansion of the Total Energy J. Ihm, A. Zunger, M.L. Cohen, J. Phys. C 12, 4409 (1979). M. Bockstedte, A. Kley, J. Neugebauer, M. Scheffler, CPC 107, 187 (1997). Plane-Wave Expansion of Kohn-Sham-Wavefunctions Translationally invariant system (supercell) --> Bloch theorem (k: Blochvector) ¡Ê ´ Ö · ʵ ´ Öµ Plane-wave expansion of Kohn-Sham states (G: reciprocal lattice vectors) ´·µ¡Ö ´ Öµ ´ · µ Ô ª Electron density follows from sum over all occupied states: Ó ¿ ¾ Ò´Öµ ´ Öµ (for semiconductors) ª ª Kohn-Sham equation in reciprocal space: ½ ¾ ¼ ¼ · Æ · Ú ´ µ ´ · µ ´µ ´ · µ ¼ « ¾ ¼ Hohenberg-Kohn Functional in Momentum Space Total-energy functional: ´ · Ê µ Ì · · · · · ØÓØÐ Ë Ô×ÐÓ Ô×ÒÓÒ ÐÓ À ÁÓÒ ÁÓÒ Obtain individually convergent energy terms by adding or subtracting superposition of Gaussian charges at the atomic positions: ¾ ¾ Ö Ê Ö Ù×× Ù×× Ò ´Öµ ¿ ¿¾ Ö Ù×× Ê Define valence charge difference wrt. above Gaussian charge density: Ù×× Ò ´Ö µ Ò´Öµ Ò ´Öµ The total energy can thus be written as the sum of individually well defined energies: Ì · · · Ò · · ØÓØÐ Ë Ô×ÐÓ Ô×ÒÓÒ ÐÓ À ÁÓÒ ÁÓÒ ×Ð Hohenberg-Kohn Functional in Momentum Space (continued) Ó ¿ ¾ · ¾ Ì ´ · µ kinetic energy: Ë ª ¾ ª ª Ë ´µ Ú Ò local pseudopotential energy: Ô×ÐÓ (only one kind of atoms) Ù××Ø ¼ Ò ´Ö µ Ô×ÓÒ ¿ ¼ ¡ Ú ´Ö µ Ú ´Ö µ Ö Ë ´µ with¼ and structure factor Ö Ö non-local pseudopotential energy ª ¾ Ò Ò À Hartree energy: ¾ ¾ ¼ exchange-correlation Ä ÓÖ ¿ ÓÖ ÓÑ ÓÖ Ò · Ò Ö ´Ò´Öµ · Ò ´Öµµ ¯ ´Ò´Öµ · Ò ´Öµµ energy: ª ¼ ¼ ½½ ¼ ½ Ê ¼ ½ Ö Õ ÁÓÒ ÁÓÒ Ion-Ion Coulomb interaction: ¼ ¾ ¾ ¾ Ê Ö · Ö ¼ ¼ ¼ Ê ·Ê Ù×× Ù×× ¾ ½ Ô Gaussian self energy: ×Ð Ö ¾ Ù×× Kinetic Energy Cut-Off and Basis-Set Convergence Size of plane-wave basis-set limited Basis-set convergence of total energy: by the kinetic-energy cut-off energy: −2.047 kin E 5 Ry Ô cut · ¾ ÙØ Al (Note: Conventionally, cut-off energy −2.057 is given in Ry, then factor "2" is obsolete.) 7 Ry Efficient calculation of convolutions: total energy [H] −2.067 ´µ ´Ú µ´µ ÐÓ 9 Ry FFT FFT-1 11 Ry mult. 13 Ry Ú ´Ö µ ´Ö µ ÐÓ Ò Ò ´Ö µ Ò 15 Ry −2.077 7.0 7.2 7.4 7.6 7.8 8.0 lattice constant [bohr] Real space mesh fixed by sampling theorem. Advantages of Plane-Wave Basis-Set (1) basis set is independent of atom positions and species, unbiased (2) forces acting on atoms are equal to Hellmann-Feynman forces, no basis-set corrections to the forces (no Pulay forces) (3) efficient calculation of convolutions, use FFT to switch between real space mesh and reciprocal space (4) systematic improvement (decrease) of total energy with increasing size of the basis set (increasing cut-off energy): can control basis-set convergence Remark: When the volume of the supercell is varied, the number of plane- wave component varies discontinuously. Basis-set corrections are available (G.P. Francis, M.C. Payne, J. Phys. Cond. Matt. 2, 4395 (1990).) III. Brillouin Zone Integration and Special k-Point Sets (i) General Considerations (ii) Semiconductors & Insulators (iii) Metals D.J. Chadi, M.L.Cohen, Phys. Rev. B 8, 5747 (1973). H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13, 5188 (1976). R.A. Evarestov, V.P. Smirnov, phys. stat. sol. (b) 119, 9 (1983). The Brillouin Zone Integration Make use of supercells and exploit translational invariance; apply Bloch theorem. Charge density and other quantities are represented by Brillouin-zone integrals: Ó ¿ ¾ Ò´Öµ ´ Öµ (for semiconductors / insulators) ª ª Smooth integrand => approximate integral by a weighted sum over special points: Æ ÔØ Ó ¾ Ò´Öµ Û ´ Öµ Ò Ò Ò½ This is the "trick" by which we get rid of the many degrees of freedom from the crystal electrons! Special Points for Efficient Brillouin-Zone Integration ¿ Calculate integrals of the type ´µ with ª ª f(k) periodic in reciprocal space, ´ · µ ´µ , f(k) symmetric with respect to all point group symmetries. ½ ´µ · ´µ Expand f(k) into a FOURIER series: Ñ Ñ Ñ½ ½ ¡´«Ê µ Ñ ´µ Ñ ´µ ½ with and ¼ ¼ «¾ ¼ Choose special points k (i=1, ..., N) and weights w such that i i Æ Û ´ µÆ ÓÖ Ñ ¼ ½ Å ½ Ñ Ñ¼ ½ for M as large as possible. The real-space vectors R m are assumed to be ordered according to their length. Æ ½ Æ Û ´ µ · Û ´ µ Then: Ñ Ñ ½ ½ ÑÅ error of BZ integration scheme Special k-Points for 2D Square Lattice Let point group be C4 (not C4v). Lattice vector a12 = aex , a = aey ki wi NM _1 _1 11 3 ()4 4 1 1 1 1 1 _ 0 _ _ _ _ 2 6 ()4 ()2 4 22 12 00 _ _ __14 2 4 ()()55 55 1 1 1 331 3 3 1 1 1 1 _ _ _ _ _ _ _ _ _ _ _ _ 4 12 ()8888 ()()88 ()88 4 4 4 4 1 1 1 1 1 1 1 1 2221 2 _ 0 _ _ _ _ _ 0 _ _ _____ 515 ()6() 63362()()()23 99999 (__13 __) (__ 39 __) (__ 55 __) 20 20 20 20 20 20 1 1 1 1 1 _ _ _ _ _ 5 16 __7 __1 __97 __ 5 5 5 5 5 (20 20) (20 20) R.A. Evarestov, V.P. Smirnov, phys.
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