Density Functional Theory Richard M. Martin University of Illinois Cu d orbitals Electron density in La2CuO4 - difference from sum of atom densities - J. M. Zuo (UIUC) Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 Outline 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 Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 Questions for you Why were “orbitals” mentioned on the introductory slide and not simply “density” Can you tell whether La2CuO4 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? Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Assumes non-degenerate ground state H-K Functional Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 Wavefunctions with density n( r ) Density Functional Theory IPAM 2002

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 Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 Kohn-Sham Ansatz If you don’t 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 Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 Auxiliary System Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 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 Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Negative energy: electron – positive hole Kinetic energy: positive Density Functional Theory IPAM 2002

Exchange-correlation hole in homogeneous electron gas Exchange dominates at high density (small rs) Correlation dominates at low density (large rs) Gori-Giorgi, Sacchetti and Bachelet, PRB 61, 7353 (2000). Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Exchange hole in Ne atom Gunnarsson, et al, PRB 20, 3136 (79). Spherical average close to LDA! Density Functional Theory IPAM 2002

Exchange hole in Si Crystal Variational Monte Carlo Hood, et al, PRB 57, 8972(98). Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 Examples of Results Hydrogen molecules - using the LSDA (from O. Gunnarsson) Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 Examples of Results Phase transformations of Si, Ge from Yin and Cohen (1982) Needs and Mujica (1995) Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 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 Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Less compressible than Diamond Bulk Modulus B (Gpa) Exp Th (LDA) C 444 467 Os 462 444 Cynn, et al, PRL March 14 (2002) Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 Slater average exchange Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 Phonons - LDA and GGA Baroni, et al, RMP 73, 515 (2000). Calculated by response function method LDA GGA Exp Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 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 Naïve use of the eigenvalues as exciation energies is the famous “band gap problem” To understand the effcets we first examine the potential Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Exchange potential in atoms 2-electron systems LDA Vxc is too shallow Almbladh and Pedroza, PR A 29, 2322 (84). Density Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 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” Ge is a metal in LDA! M. Staedele et al, PRL 79, 2089 (1997) Density Functional Theory IPAM 2002

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? Density Functional Theory IPAM 2002

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 Density Functional Theory IPAM 2002

Linear Scaling ‘Order-N’ Methods Computational complexity ~ N = number of atoms (Current methods scale as N2 or N3) “Divide and Conquer” Green’s Function Fermi Operator Expansion Density matrix “purification” Generalized Wannier Functions Spectral “Telescoping” (Review by S. Goedecker in Rev Mod Phys) Density Functional Theory IPAM 2002

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). Density Functional Theory IPAM 2002

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 Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 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 Density Functional Theory IPAM 2002

Density Functional Theory IPAM 2002 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 Density Functional Theory IPAM 2002