Electronic Structure and First Principles Theory 5/31/2016 CIDER/ITP Short Course Equation of State • Start from fundamental relation • Helmholtz free energy 140 F=af – F=F(V,T,Ni) – F=F(V) • Taylor series expansion • Expansion variable must be V or a function of V – F=af2 + bf3 + … • f = f(V) Eulerian finite strain Pressure (GPa) • Isotherm, fixed composition 120 2 MgSiO3 Perovskite 300 K 100 80 60 40 20 0 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Volume, V/V 0 Microscopic Picture 1 Pair Potential • Assume pairwise interactions • Assume simple functional form – V(r) = exp(-r/ + Z1Z2e2/r – Fast • Fundamental inadequacies – C12=C44 • Empirical inadequacies – N+1th observation – More complex functional forms and/or parameters depend on • Pressure • Temperature • Structure Potential Energy • Advantages Distance • Assume charge density of crystal = that of overlapping, spherical, fully charged, ions • Assume charge density of ions = that in free state • Advantage – Ab initio • Problems – Only ionic bonding – Cauchy violations – O2- not stable in free state • Partial solution – Breathing Charge Density Microscopic Picture 2 Gordon-Kim Distance Ions or electrons? • Pauling/Goldschmidt Model – Hard fully charge spheres – Rationalize/predict low pressure structures • High pressure? – Pbond~eV/Å3=160 GPa~Pmantle – Ions change • Size • Shape • Charge The one electron atom • Exactly soluble i: wave function of state i can have either sign • Charge density, = square of wave function • Ei Energy of state i • States described by three quantum numbers (+ spin) 2 Z /ri Eii Multi-electron Atom in a crystal field Multi-electron atom One-electron atom l=0 n=3 3s l=1 l=2 3p 3d 3d 3p 3s n=2 n=1 2s 1s 2p 3d3z2-r2 3dx2-y2 m=+2 3dyz 3dxz 3dxy m=0 m=-1 m=+1 m=-2 Molecules • Isolated Atoms – One energy level • Molecule – Two energy levels • Bonding • Anti-bonding – Population • Energy difference • Temperature Metallic Solid • Asymptotically continuous band of N states E k cos(kx) • Each state accommodates 2 electrons • Half-filled band • Fermi energy separates occupied from unoccupied states • No gap Energy, E k cos(kx)u(x) Unoccupied Fermi Occupied Energy /a 0 Wavevector, k • Doubled unit cell • Halved Brillouin zone • Folding • Gap Energy, E Covalent Solid Unoccupied Occupied Fermi Energy /a 0 Wavevector, k • Cation and Anion • Lower energy state: valence electrons on anion • ~Flat bands: localized states • Gap Energy, E Ionic Solid Unoccupied Occupied Fermi Energy /a 0 Wavevector, k Density functional theory • No assumption about charge density, type of bonding, … • No experimental input, i.e. no free parameters • Positions and charges of nuclei. • Assumption of nuclear positions is generally relaxed • Not exact Cohen, 1992 Uniform Charge Density • Uniform distribution of atoms – P=RTnA • Uniform distribution of electrons – Kinetic P 2 3 5m 2 2/3 n e5 / 3 – Exchange – Correlation – Ion-electron interaction Nuclei Electrons Uniform Charge Density • EOS depends on Z – Jupiter, Z~1 – Mantle, Z~10 – Core, Z~26 • Calculated density too high • Screening Density Functional Theory • Kohn,Sham,Hohenberg • Ground State Internal Energy a unique functional of the charge density • Approximations – Essential • Exchange-Correlation Functional – Local density approximation – Convenient • Pseudopotential approximation Computational Methods • Pseudopotential • Nuclear potential is hard! • Replace with that of nucleus + core electrons • Represent valence electrons with plane wave basis set Origin of Magnetism electron s=±1/2 Ferromagnet Paramagnet Pauli Paramagnet atomic or local S=2 Bulk f(V) Magnetic Collapse Origin Levels Low Pressure Bands High Pressure Electronic transition in Potassium Potassium shows a fundamental change in its electronic structure at high pressure, from that of an alkali metal to that of a transition metal. 4s electrons are more strongly influenced by compression than the initially unoccupied 3d states, which are increasingly populated at high pressure Large decrease in ionic radius Change in chemical affinity from lithophile to siderophile? Potassium 35 GPa Bukowinski (1976) GRL 3, 491 Phase transition in CaSiO3 perovskite 1 2' Test of fundamental theories Guiding new experiments Interpretation of observations 1000 5' 25' 1 -1 4 12' 1 Frequency (cm ) Prediction of behavior and properties at extreme conditions Origin of behavior and properties at the fundamental level Interplay with experiment 3 2' 5 4' 5 1 15 500 1 5 2 25 1 5 5 5' 5 5' 2' 4' 5 5' 15 5' 4' 0 2 15 200i 25 Shim, Jeanloz, Duffy (2002) GRL 29, 016148 Stixrude, Cohen, Yu, Krakauer (1996) Am. Min. 81, 1293. X Z M R S X R T M