FUNDAMENTALS FUNDAMENTALS The quantum-mechanical The quantum-mechanical many-electron problem many-electron problem
and and Density Functional TheoryDensity Functional Theory
Emilio ArtachoDepartment of Earth Sciences
University of Cambridge
Summer school 2002Linear-scaling ab initio molecular
modelling of environmental processes
First-principles calculations
• Fundamental laws of physics
• Set of “accepted” approximations to solve the corresponding equations on a computer
• No empirical input
PREDICTIVE POWER
Artillery
F = m a
Approximations
•Flat Earth
•Constant g
(air friction: phenomenological)
Fundamental laws for the properties of matter at low
energiesAtomic scale (chemical bonds etc.)
Yes BUT
Electrons and nuclei(simple Coulomb interactions)
=> Quantum Mechanics })({})({ˆii rErHrr
Ψ=Ψ
Many-particle problemSchroedinger’s equation is exactly solvable for - Two particles (analytically) - Very few particles (numerically)The number of electrons and nuclei in a pebble is ~10
=> APPROXIMATIONS
23
),...,,(),...,,(ˆ2121 NN rrrErrrH
rrrrrrΨ=Ψ
Born-Oppenheimer
(1) (2)
1>>e
n
mm
Nuclei are much slower than electrons
electronic/nuclear decoupling
∑∑∑∑∑>>
+−+∇−∇−=μνμ μν
νμ
μ μ
μμ
μ μ ,,,
22 1
2
1
2
1ˆR
ZZ
r
Z
rMH
i iiji iji
i
})({})({})({ˆ}{,}{,}{ i
elRn
elni
elRn
elR
rRErHrrr
rrrμμμ μ Ψ=Ψ
})({1
2
1ˆ}{
,
2
}{ iextR
iji iji
i
elR
rVr
Hr
rrμμ
++∇−= ∑∑>
electronselectronselectronselectrons
})({2
1ˆ 2μμ
μ μ
REM
H eln
r+∇−= ∑
nucleinucleinucleinuclei
})({0 μν
ν RER
F elr
rr
∂
∂=Classical =>Classical =>Classical =>Classical =>
Many-electron problemOld and extremely hard
problem!Different approaches
• Quantum Chemistry (Hartree-Fock, CI…)
• Quantum Monte Carlo
• Perturbation theory (propagators)
• Density Functional Theory (DFT)
Very efficient and general
BUT implementations are approximate and hard to improve (no systematic improvement) (… actually running out of ideas …)
Density-Functional Theory
2. As if non-interacting electrons in an effective (self-consistent) potential
)(})({ rrirr ρ→Ψ1. particle particle
densitydensityparticle particle densitydensity
Hohenberg - Kohn )(})({ rri
rr ρ→Ψ
∑=
++=N
iiextee rVVTH
1
)(ˆ rFor our many-electron problemFor our many-electron problemFor our many-electron problemFor our many-electron problem
GSGS ErE =)]([r
ρ2.2.2.2.
)]([)()(3 rFrrVrd ext
rrrr ρρ +≡∫ GSE≥1.1.1.1. )]([ rErρ
(universal functional)(depends on nuclear positions)
PROBLEM:PROBLEM:Functional Functional unknown!unknown!
PROBLEM:PROBLEM:Functional Functional unknown!unknown!
Kohn - ShamIndependent particles in an effective Independent particles in an effective
potentialpotentialIndependent particles in an effective Independent particles in an effective
potentialpotential
][)]()([)(][][ 213
0 ρρρρ xcext ErrVrrdTE +Φ++= ∫rrrr
They rewrote the functional They rewrote the functional as:as:They rewrote the functional They rewrote the functional as:as:
Kinetic energy for Kinetic energy for system with no e-e system with no e-e interactionsinteractions
Kinetic energy for Kinetic energy for system with no e-e system with no e-e interactionsinteractions
Hartree Hartree potentialpotentialHartree Hartree potentialpotential
The rest: The rest: exchange exchange correlationcorrelation
The rest: The rest: exchange exchange correlationcorrelation
Equivalent to independent Equivalent to independent particles under the particles under the potentialpotential
Equivalent to independent Equivalent to independent particles under the particles under the potentialpotential
)(
][)()()(
r
ErrVrV xc
ext rrrr
δρρδ
+Φ+=
Exc & Vxc )(
][
r
EV xcxc rδρ
ρδ≡
Local Density Approximation Local Density Approximation (LDA)(LDA)
Local Density Approximation Local Density Approximation (LDA)(LDA)))((][ rVV xcxc
rρρ ≈ (function parameterised for the (function parameterised for the homogeneous electron liquid as obtained homogeneous electron liquid as obtained
from QMC)from QMC)
(function parameterised for the (function parameterised for the homogeneous electron liquid as obtained homogeneous electron liquid as obtained
from QMC)from QMC)
Generalised Gradient Approximation Generalised Gradient Approximation (GGA)(GGA)Generalised Gradient Approximation Generalised Gradient Approximation (GGA)(GGA)
))(),((][ rrVV xcxc
rr ρρρ ∇≈(new terms parameterised for heterogeneous (new terms parameterised for heterogeneous electron systems (atoms) as obtained from QC)electron systems (atoms) as obtained from QC)(new terms parameterised for heterogeneous (new terms parameterised for heterogeneous electron systems (atoms) as obtained from QC)electron systems (atoms) as obtained from QC)
Independent particles
)(2
1ˆ 2 rVhr
+∇−=
)()(ˆ rrh nnn
rr ψεψ =
ε2|)(|)( ∑=
occ
nn rr
rrψρ
Self-consistency
PROBLEM: The potential (input) depends PROBLEM: The potential (input) depends on the density (output)on the density (output)
PROBLEM: The potential (input) depends PROBLEM: The potential (input) depends on the density (output)on the density (output)
inρ V ρ outρ
ερρ >− − || 1nn
Solving: 1. Basis set
∑≈μ
μμ φψ )()( rcr nn
rrExpand in terms of a finite set Expand in terms of a finite set
of known wave-functionsof known wave-functionsExpand in terms of a finite set Expand in terms of a finite set
of known wave-functionsof known wave-functions )(rr
μφ unknownunknown unknownunknown
~ - ~-~ - ~-~ - ~-~ - ~-HCHCnn==ɛɛnnSCSCnnHCHCnn==ɛɛnnSCSCnn
)()(ˆ rrh nnn
rr ψεψ = ∑∑ =μ
μμμ
μμ φεφ )()(ˆ rcrhc nnn
rr
rdrhrhrrr 3* )(ˆ)( μννμ φφ∫≡ rdrrS
rrr 3* )()( μννμ φφ∫≡DefDefDefDef andandandand
∑∑ =μ
μνμμ
μνμ ε nnn cSch
Basis set: Atomic orbitals
s
p
d
f Strictly localised
(zero beyond cut-off radius)
Solving: 2. Boundary conditions
•Isolated object (atom, molecule, Isolated object (atom, molecule, cluster): cluster): open boundary conditionsopen boundary conditions (defined at infinity)(defined at infinity)
•3D Periodic object (crystal):3D Periodic object (crystal): Periodic Boundary ConditionsPeriodic Boundary Conditions
• Mixed: 1D periodic (chains)Mixed: 1D periodic (chains) 2D periodic (slabs)2D periodic (slabs)
•Isolated object (atom, molecule, Isolated object (atom, molecule, cluster): cluster): open boundary conditionsopen boundary conditions (defined at infinity)(defined at infinity)
•3D Periodic object (crystal):3D Periodic object (crystal): Periodic Boundary ConditionsPeriodic Boundary Conditions
• Mixed: 1D periodic (chains)Mixed: 1D periodic (chains) 2D periodic (slabs)2D periodic (slabs)
k-point sampling
Electronic quantum states in a periodic solid labelled Electronic quantum states in a periodic solid labelled by:by:
• Band indexBand index
• k-vector: k-vector: vector in reciprocal space within the first Brillouinvector in reciprocal space within the first Brillouin zone (Wigner-Seitz cell in reciprocal space)zone (Wigner-Seitz cell in reciprocal space)
• Other symmetriesOther symmetries (spin, point-group representation…) (spin, point-group representation…)
Electronic quantum states in a periodic solid labelled Electronic quantum states in a periodic solid labelled by:by:
• Band indexBand index
• k-vector: k-vector: vector in reciprocal space within the first Brillouinvector in reciprocal space within the first Brillouin zone (Wigner-Seitz cell in reciprocal space)zone (Wigner-Seitz cell in reciprocal space)
• Other symmetriesOther symmetries (spin, point-group representation…) (spin, point-group representation…)
∫∑∈
⇒=ZBk
occ
nn kdrr
.
32|)(|)(r
rrrψρ
Approximated by Approximated by sums over sums over
selected k pointsselected k points
Approximated by Approximated by sums over sums over
selected k pointsselected k points
Some materials’ properties
Exp. LAPW Other PW
PW DZP
a (Å) 3.57 3.54 3.54 3.53 3.54
C B (GPa) 442 470 436 459 453
Ec (eV) 7.37 10.13 8.96 8.89 8.81
a (Å) 5.43 5.41 5.38 5.38 5.40
Si B (GPa) 99 96 94 96 97
Ec (eV) 4.63 5.28 5.34 5.40 5.31
a (Å) 4.23 4.05 3.98 3.95 3.98
Na B (GPa) 6.9 9.2 8.7 8.7 9.2
Ec (eV) 1.11 1.44 1.28 1.22 1.22
a (Å) 3.60 3.52 3.56 - 3.57
Cu B (GPa) 138 192 172 - 165
Ec (eV) 3.50 4.29 4.24 - 4.37
a (Å) 4.08 4.05 4.07 4.05 4.07
Au B (GPa) 173 198 190 195 188
Ec (eV) 3.81 - - 4.36 4.13
Absence of DC conductivity in -DNA
λP. J. de Pablo et al. Phys. Rev. Lett. 85, 4992 (2000)
Effect of sequence disorder and vibrations on the electronic structure
=> Band-like conduction is extremely unlikely: DNA is not a wire
Pressing nanotubes for a switch
M. Fuhrer et al. Science 288, 494 (2000)
Y.-G. Yoon et al. Phys. Rev. Lett. 86, 688 (2001)
Pushed them together, relaxed &calculated conduction at the contact: SWITCH
Pyrophyllite, illite & smectite
Structural effects of octahedral cation substitutions C. I. Sainz-Diaz et al. (American Mineralogist, 2002)
Organic molecules intercalated between layers M. Craig et al. (Phys. Chem. Miner. 2002)
WET SURFACES
Recap• Born-Oppenheimer: electron-nuclear
decoupling
• Many-electron -> DFT (LDA, GGA)
• One-particle problem in effective self-consistent potential (iterate)
• Basis set => Solving in two steps: 1. Calculation of matrix elements of H and S
2. Diagonalisation
• Extended crystals: PBC + k sampling