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Electrons in Materials Density Functional Theory Richard M. Martin

Electrons in Materials Density Functional Theory Richard M. Martin. d orbitals. Electron density in La 2 CuO 4 - difference from sum of atom densities - J. M. Zuo (UIUC). Outline. Many Body Problem!

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Electrons in Materials Density Functional Theory Richard M. Martin

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  1. Electrons in MaterialsDensity Functional TheoryRichard M. Martin d orbitals Electron density in La2CuO4 - difference from sum of atom densities - J. M. Zuo (UIUC) Comp. Mat. Science School 2001

  2. Outline • 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 Comp. Mat. Science School 2001

  3. 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 Comp. Mat. Science School 2001

  4. 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 theground state • Phase transitions • Phonons, etc. Comp. Mat. Science School 2001

  5. 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 • Yi ( r1, r2, r3, r4, r5, . . . ) • How to minimize in such a large space Comp. Mat. Science School 2001

  6. 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 Comp. Mat. Science School 2001

  7. Quantum Monte Carlo • Variational - Guess form for Y ( r1, r2, …) • Minimize total energy with respect to all parameters in Y • Carry out the integrals by Monte Carlo • Diffusion Monte Carlo - Start with VMC and apply operator e-HtY to project out an improved ground state Y0 • Exact for Bosons (Helium 4) • Fermion sign problem for Electrons E0 =  dr1 dr2 dr3 …YH Y Comp. Mat. Science School 2001

  8. h2 2 m 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 canin principlebe found by solvingnon-interacting electron equationsin an effective potential determined only by the density • Recently accurate approximations for the functionals of the density have been found H Yi(x,y,z) = EiYi (x,y,z) , H = - + V(x,y,z) D 2 Comp. Mat. Science School 2001

  9. h2 2 m 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 Yi(x,y,z) = EiYi (x,y,z) , H = - + V(x,y,z) D 2 Comp. Mat. Science School 2001

  10. Examples of Results • Hydrogen molecules - using the LSDA(from O. Gunnarsson) Comp. Mat. Science School 2001

  11. Examples of Results • Phase transformations of Si, Ge • from Yin and Cohen (1982) Needs and Mujica (1995) Comp. Mat. Science School 2001

  12. 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 Needs and Mujica (1995) Simple Hexagonal Cubic Diamond Comp. Mat. Science School 2001

  13. 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 Comp. Mat. Science School 2001

  14. A new phase of Nitrogen • Published in Nature this week. Reported in the NY Times - Dense, metastable semiconductor • Predicted by theory ~10 years ago! Molecular form Mailhiot, et al 1992 “Cubic Gauche” Polymeric form with 3 coordination Comp. Mat. Science School 2001

  15. 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” Ge is a metal in LDA! M. Staedele et al, PRL 79, 2089 (1997) Comp. Mat. Science School 2001

  16. 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 DFTThe “Band Gap Problem”Orbital dependant functionals increase the gaps - agree well with experiment - now a research topic Comp. Mat. Science School 2001

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