1. Comp. Mat. Science School 20011 Electrons in Materials Density Functional Theory Richard M. Martin Electron density in La 2 CuO 4 - difference from sum of atom densities - J. M. Zuo (UIUC) d orbitals

2. Comp. Mat. Science School 20012 Many Body Problem! Density Functional Theory Kohn-Sham Equations allow in principle exact solution for ground state of many-body system using independent particle methods Approximate LDA, GGA functionals Examples of Results from practical calculations Pseudopotentials - needed for plane wave calculations Next Time - Bloch Theorem, Bands in crystals, Plane wave calculations, Iterative methods Outline

3. Comp. Mat. Science School 20013 Ab Initio Simulations of Matter Why is this a hard problem? Many-Body Problem - Electrons/ Nuclei Must be Accurate --- Computation Emphasize here: Density Functional Theory –Numerical Algorithms –Some recent results

4. Comp. Mat. Science School 20014 Eigenstates of electrons For optical absortion, etc., one needs the spectrum of excited states For thermodynamics and chemistry the lowest states are most important In many problems the temperature is low compared to characteristic electronic energies and we need only the ground state –Phase transitions –Phonons, etc.

5. Comp. Mat. Science School 20015 The Ground State General idea: Can use minmization methods to get the lowest energy state Why is this difficult ? It is a Many-Body Problem  i ( r 1, r 2, r 3, r 4, r 5,... ) How to minimize in such a large space

6. Comp. Mat. Science School 20016 The Ground State How to minimize in such a large space –Methods of Quantum Chemistry- expand in extremely large bases - Billions - grows exponentially with size of system Limited to small molecules –Quantum Monte Carlo - statistical sampling of high-dimensional spaces Exact for Bosons (Helium 4) Fermion sign problem for Electrons

7. Comp. Mat. Science School 20017 Quantum Monte Carlo Variational - Guess form for  ( r 1, r 2, …) Minimize total energy with respect to all parameters in  Carry out the integrals by Monte Carlo Diffusion Monte Carlo - Start with VMC and apply operator e -H   to project out an improved ground state  0 Exact for Bosons (Helium 4) Fermion sign problem for Electrons E 0 =  dr 1 dr 2 dr 3 …  H 

8. Comp. Mat. Science School 20018 Density Functional Theory 1998 Nobel Prize in Chemistry to Walter Kohn and John Pople Key point - the ground state energy for the hard many-body problem can in principle be found by solving non-interacting electron equations in an effective potential determined only by the density Recently accurate approximations for the functionals of the density have been found H  i (x,y,z) = E i  i (x,y,z), H = - + V(x,y,z) h2h2 2 m  2

9. Comp. Mat. Science School 20019 Density Functional Theory Must solve N equations, I = 1, N with a self-consistent potential V(x,y,z) that depends upon the density of the electrons Text-Book - Find the eigenstates More efficient Modern Algorithms –Minimize total energy for N states subject to the condition that they must be orthonormal Conjugate Gradient with constraints –Recent “Order N” Linear scaling methods H  i (x,y,z) = E i  i (x,y,z), H = - + V(x,y,z) h2h2 2 m  2

10. Comp. Mat. Science School 200110 Examples of Results Hydrogen molecules - using the LSDA (from O. Gunnarsson)

11. Comp. Mat. Science School 200111 Examples of Results Phase transformations of Si, Ge from Yin and Cohen (1982) Needs and Mujica (1995)

12. Comp. Mat. Science School 200112 Enthalpy vs pressure H = E + PV - equilibrium structure at a fixed pressure P is the one with minimum H Transition pressures slightly below experiment 80 kbar vs ~100kbar Simple Hexagonal Cubic Diamond Needs and Mujica (1995)

13. Comp. Mat. Science School 200113 Graphite vs Diamond A very severe test Fahy, Louie, Cohen calculated energy along a path connecting the phases Most important - energy of grahite and diamond essentially the same! ~ 0. 3 eV/atom barrier

14. Comp. Mat. Science School 200114 A new phase of Nitrogen Published in Nature this week. Reported in the NY Times - Dense, metastable semiconductor Predicted by theory ~10 years ago! “Cubic Gauche” Polymeric form with 3 coordination Mailhiot, et al 1992 Molecular form

15. Comp. Mat. Science School 200115 The Great Failures 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!

16. Comp. Mat. Science School 200116 Conclusions The ground state properties are predicted with remarkable success by the simple LDA and GGAs. Structures, phonons (~5%), …. Excitations are NOT well-predicted by the usual LDA, GGA forms of DFT The “Band Gap Problem” Orbital dependant functionals increase the gaps - agree well with experiment - now a research topic