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