Electronic Structure and First Principles Theory

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Electronic Structure and First Principles Theory
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Equation of State

• Start from fundamental relation
• Helmholtz free energy
• FF(V,T,Ni)
• Isotherm, fixed composition
• FF(V)
• Taylor series expansion
• Expansion variable must be V or a function of V
• Faf2 bf3
• f f(V) Eulerian finite strain

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Microscopic Picture 1Pair Potential

• Assume pairwise interactions
• Assume simple functional form
• V(r) exp(-r/?? Z1Z2e2/r
• Advantages
• Fast
• Fundamental inadequacies
• C12C44
• Empirical inadequacies
• N1th observation
• More complex functional forms and/or parameters
  depend on
• Pressure
• Temperature
• Structure

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Microscopic Picture 2Gordon-Kim

• 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

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Ions or electrons?

• Pauling/Goldschmidt Model
• Hard fully charge spheres
• Rationalize/predict low pressure structures
• High pressure?
• PbondeV/Å3160 GPaPmantle
• Ions change
• Size
• Shape
• Charge

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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)

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Multi-electron Atom in a crystal field
Multi-electron atom
3d3z2-r2
m2
One-electron atom
3dx2-y2
m1
3d
3dyz
m0
l0
l1
l2
3dxz
m-1
3s
3p
3d
3p
3dxy
n3
m-2
3s
2s
2p
n2
1s
n1
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Molecules

• Isolated Atoms
• One energy level
• Molecule
• Two energy levels
• Bonding
• Anti-bonding
• Population
• Energy difference
• Temperature

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Metallic Solid

• Asymptotically continuous band of N states
• Each state accommodates 2 electrons
• Half-filled band
• Fermi energy separates occupied from unoccupied
  states
• No gap

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Covalent Solid

• Doubled unit cell
• Halved Brillouin zone
• Folding
• Gap

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Ionic Solid

• Cation and Anion
• Lower energy state valence electrons on anion
• Flat bands localized states
• Gap

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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
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Uniform Charge Density

• Uniform distribution of atoms
• PRTnA
• Uniform distribution of electrons
• Kinetic
• Exchange
• Correlation
• Ion-electron interaction

Nuclei
Electrons
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Uniform Charge Density

• EOS depends on Z
• Jupiter, Z1
• Mantle, Z10
• Core, Z26
• Calculated density too high
• Screening

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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

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Computational Methods

• Pseudopotential
• Nuclear potential is hard!
• Replace with that of nucleus core electrons
• Represent valence electrons with plane wave basis
  set

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Origin of Magnetism
Bulk f(V)
electron s1/2
atomic or local S2
Ferromagnet
Paramagnet
Pauli Paramagnet
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Magnetic CollapseOrigin
Levels
High Pressure
Low Pressure
Bands
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Electronic transition in Potassium
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?

Bukowinski (1976) GRL 3, 491
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Phase transition in CaSiO3 perovskite

• Prediction of behavior and properties at extreme
  conditions
• Origin of behavior and properties at the
  fundamental level
• Interplay with experiment
• Test of fundamental theories
• Guiding new experiments
• Interpretation of observations