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Density Functional Theory IPAM 20021 Density Functional Theory Richard M. Martin University of Illinois Electron density in La 2 CuO 4 - difference from sum of atom densities - J. M. Zuo (UIUC) Cu d orbitals Slide 2 Density Functional Theory IPAM 20022 DFT is an approach to Interacting Many-Body Problems Hohenberg-Kohn Theorems & Levy-Lieb Construction Kohn-Sham Ansatz allows in principle exact solution for ground state of many-body system using independent particle methods Classes of functionals: LDA, GGA, OEP, . Examples of Results Locality Principles and linear scaling Electric polarization in crystals - deep issues that bring out stimulating questions about DFT, and the differences between the Hohenberg-Kohn and Kohn-Sham approaches Outline Slide 3 Density Functional Theory IPAM 20023 Why were orbitals mentioned on the introductory slide and not simply density Can you tell whether La 2 CuO 4 is an insulator or a metal just by looking at the density? If so, what aspects of the density? Is Kohn-Sham theory the same as Density Functional Theory? If not, what is the difference? What did Kohn-Sham add? What did they subtract? Do locality principles in independent particle methods carry over to the real many-body world? Is the electric polarization of a ferroelectric an intrinsic ground state property? Is it determined by the density? Questions for you Slide 4 Density Functional Theory IPAM 20024 Slide 5 5 Slide 6 6 Slide 7 7 H-K Functional Assumes non-degenerate ground state Slide 8 Density Functional Theory IPAM 20028 Slide 9 9 Wavefunctions with density n( r ) Slide 10 Density Functional Theory IPAM 200210 What have we gained so far? Apparently Nothing! The only result is that the density determines the potential We are still left with the original many-body problem But the proofs suggest(ed) the next step Slide 11 Density Functional Theory IPAM 200211 Kohn-Sham Ansatz If you dont like the answer, change the question Replace the original interacting-particle problem with a different problem more easily solved Kohn-Sham auxiliary system: non-interacting electrons assumed to have the same density as the interacting system Slide 12 Density Functional Theory IPAM 200212 Auxiliary System Slide 13 Density Functional Theory IPAM 200213 Replace interacting problem with auxiliary non-interacting problem Each term in figure is uniquely related to each other term! The ansatz has been shown to be fulfilled in several simple cases but not in general We will proceed assuming the ansatz is justiified Slide 14 Density Functional Theory IPAM 200214 Slide 15 Density Functional Theory IPAM 200215 Slide 16 Density Functional Theory IPAM 200216 Slide 17 Density Functional Theory IPAM 200217 Negative energy: electron positive hole Kinetic energy: positive Slide 18 Density Functional Theory IPAM 200218 Exchange-correlation hole in homogeneous electron gas Exchange dominates at high density (small r s ) Correlation dominates at low density (large r s ) Gori-Giorgi, Sacchetti and Bachelet, PRB 61, 7353 (2000). Slide 19 Density Functional Theory IPAM 200219 Slide 20 Density Functional Theory IPAM 200220 Exchange hole in Ne atom Spherical average close to LDA! Gunnarsson, et al, PRB 20, 3136 (79). Slide 21 Density Functional Theory IPAM 200221 Exchange hole in Si Crystal Variational Monte Carlo Hood, et al, PRB 57, 8972(98). Slide 22 Density Functional Theory IPAM 200222 Examples of Results Hydrogen molecules - using the LSDA (from O. Gunnarsson) Slide 23 Density Functional Theory IPAM 200223 Examples of Results Phase transformations of Si, Ge from Yin and Cohen (1982) Needs and Mujica (1995) Slide 24 Density Functional Theory IPAM 200224 Graphite vs Diamond A very severe test Fahy, Louie, Cohen calculated energy along a path connecting the phases Most important - energy of graphite and diamond essentially the same! ~ 0. 3 eV/atom barrier Slide 25 Density Functional Theory IPAM 200225 Slide 26 Density Functional Theory IPAM 200226 Less compressible than Diamond Bulk Modulus B (Gpa) ExpTh (LDA) C444467 Os462444 Cynn, et al, PRL March 14 (2002) Slide 27 Density Functional Theory IPAM 200227 Slide 28 Density Functional Theory IPAM 200228 Slide 29 Density Functional Theory IPAM 200229 Slater average exchange Slide 30 Density Functional Theory IPAM 200230 Phonons - LDA and GGA Calculated by response function method Baroni, et al, RMP 73, 515 (2000). LDA GGA Exp Slide 31 Density Functional Theory IPAM 200231 The Band Gap Problem Often said that the eigenvalues have no meaning just Lagrange multipliers Energy to add or subtract an electron in the non-interacting system - not an excitation energy of the interacting system Nave use of the eigenvalues as exciation energies is the famous band gap problem To understand the effcets we first examine the potential Slide 32 Density Functional Theory IPAM 200232 Slide 33 Density Functional Theory IPAM 200233 Slide 34 Density Functional Theory IPAM 200234 Exchange potential in atoms 2-electron systems LDA V xc is too shallow Almbladh and Pedroza, PR A 29, 2322 (84). Density Slide 35 Density Functional Theory IPAM 200235 Slide 36 Density Functional Theory IPAM 200236 The Band Gap Problem Excitations are NOT well-predicted by the standard LDA, GGA forms of DFT The Band Gap Problem Orbital dependent DFT is more complicated but gives improvements - treat exchange better, e.g, Exact Exchange M. Staedele et al, PRL 79, 2089 (1997) Ge is a metal in LDA! Slide 37 Density Functional Theory IPAM 200237 Status of Band Gap Problem It should be possible to calculate all excitation energies from the Kohn-Sham approach But not clear how close Kohn-Sham eigenvalues should be to true excitation energies Not clear how much of the band gap problem is due to approximate functionals Size of derivative discontinuity? Slide 38 Density Functional Theory IPAM 200238 Locality and Linear Scaling DFT provides a fundamental basis for nearsightedness (W. Kohn) -- if properties in a region are determined only by densities in a neighborhood -- so that an Order N method must be possible Used, e.g., by W. Yang in his divide and conquer method Orbital picture in Kohn-Sham method provides the concrete methods Slide 39 Density Functional Theory IPAM 200239 Linear Scaling Order-N Methods Computational complexity ~ N = number of atoms (Current methods scale as N 2 or N 3 ) Divide and Conquer Greens Function Fermi Operator Expansion Density matrix purification Generalized Wannier Functions Spectral Telescoping (Review by S. Goedecker in Rev Mod Phys) Slide 40 Density Functional Theory IPAM 200240 Example of Our work Prediction of Shapes of Giant Fullerenes S. Itoh, P. Ordejon, D. A. Drabold and R. M. Martin, Phys Rev B 53, 2132 (1996). See also C. Xu and G. Scuceria, Chem. Phys. Lett. 262, 219 (1996). Slide 41 Density Functional Theory IPAM 200241 Simulations of DNA with the SIESTA code Machado, Ordejon, Artacho, Sanchez-Portal, Soler Self-Consistent Local Orbital O(N) Code Relaxation - ~15-60 min/step (~ 1 day with diagonalization) Iso-density surfaces Slide 42 Density Functional Theory IPAM 200242 Conclusions - I DFT is a general approach to interacting many-body problems Kohn-Sham approach makes it feasible Ground state properties are predicted with remarkable success by LDA and GGAs. Structures, phonons (~5%), . Excitations are NOT well-predicted by the LDA, GGA approximations The Band Gap Problem Orbital dependant functionals increase the gaps - agree better with experiment Derivative discontinuity natural in orbital functionals Slide 43 Density Functional Theory IPAM 200243 Conclusions - II Locality inherent for properties of a region that depend only on the density in a neighborhood Forces, stress,.. Order N linear scaling method should be possible Density matrix shows the locality in the quantum system Several feasible methods for insulators Carries over to interacting many-body system Some propreties are not local in real space Fermi surface of a metal, etc. But states near Fermi energy have universal behavior that should make linear scaling possible When is the functional an extremely non-local functional of the density? A polarized insulator, where the Kohn- Sham theory must be fundamentally revised