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 /ri  Eii
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