The electronic structure of materials 2 - DFT - Quantum ...fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_9_The...The ground-state density n(r) of a bound

Contents Density functional theory (DFT) Literature

The electronic structure of materials 2 - DFTQuantum mechanics 2 - Lecture 9

Igor Lukacevic

UJJS, Dept. of Physics, Osijek

December 19, 2012

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 2 - DFT


1 Density functional theory (DFT)

2 Literature




Contents


2 Literature




Historical background

The beginnings:

L. de Broglie (1923)

E. Schrodinger (1925)

W. Pauli (1925)

L. Thomas & E. Fermi(1927)

P. A. M. Dirac (1928)





Quantum mechanics 99K technology transition:

F. Bloch (1928)

R. Wilson (1931) - implications of band theory - insulators/metals

J. Slater (1934-1937) - bands of Na

E. Wigner & F. Seitz (1935) - quantitative calcs on Na

J. Bardeen (1935) - Fermi surface

first understanding of semiconductors (1930’s)

transistor (1940’s)





Electronic structure methods:

E. Hylleraas (1929) - numerically exact solution for H2

J. Slater (1937) - augmented plane waves (APW)

C. Herring (1940) - orthogonalized plane waves (OPW)

S. F. Boys (1950’s) - Gaussian orbitals

J. C. Phillips & L. Kleinman (1950’s) - pseudopotentials

O. K. Andersen (1975) - linear muffin tin orbitals (LMTO)





Density functional theory:

P. Hohenberg & W. Kohn (1964)

W. Kohn & L. J. Sham (1965)

R. Car & M. Parrinello (1985) - CPMD

improved approximations for functionals

evolution of computer power

W, Kohn (1998) - Nobel Prize for Chemistry

widely used codes - abinit, quantum espresso, vaps, castep, wien2k, cpmd,fhi98md, siesta, crystal, fplo, elk,...





From Physics Today, June (2005).




Hohenberg-Kohn ansatz

Starting ideas of Hohenberg & Kohn

create an exact many-body theory around the electronic ground-statedensity n(r)

start from the system of N interacting particles (B-O approx.):

Helec = −N∑

i=1

1

2∆i −

N∑i=1

M∑A=1

ZA

riA+

N∑i=1

N∑j>i

1

rij

A question

What do you think, why did they want n(r) as the main variable?





Reduction of the number of variables

Example: system of 100 particles

positions (r1, . . . , r100) , rk = (xk , yk , zk )

Schrodinger theory: ψ = ψ (r1, . . . , r100)

Questions

1 ψ (r1, . . . , r100) depends on how many variables?

2 if you want to calculate the energy from the variational principle, andguess the initial w.f. with p = 3 parameters, on how many variables willthe minimization of energy depend?









Questions

1 ψ (r1, . . . , r100) depends on how many variables? 300

2 if you want to calculate the energy from the variational principle, andguess the initial w.f. with p = 3 parameters, on how many variables willthe minimization of energy depend? 3300 ≈ 10150!

Exponential wall!









DFT: ψ = ψ [n (r)] , n = n (nx , ny , nz )

Questions

1 ψ [n (r)] depends on how many variables?

2 on how many variables now does the minimization of energyE0 = minn(r) E [n (r)] depend?









1st HK theorem

The ground-state density n(r) of a bound system of interacting electrons insome external potential Vext(r) determines this potential uniquely.





2nd HK theorem

A universal functional E [n] can be defined for any external potential Vext(r).The exact ground state energy of the system is the global minimum of thisfunctional, and the density n(r) that minimizes the functional is the exactground state density n0(r).

E0 = E [n0(r)] = minn(r)

E [n (r)]

E [n (r)] =

∫Vext(r)n (r)d3r + F [n]

F [n] = T [n] + Eint [n]

A question

What about F [n]? Do we know it?





HK DFT pros

an exact theory for ineractingparticle systems

provides a fundamentalunderstanding of physicalquantities like electron density andresponse functions

adds a practical contribution vialowering the number of variables

HK DFT cons

we don’t know F [n]




Kohn-Sham ansatz

Motivation Hartree’s single particle self consistent equations




Kohn-Sham ansatz

Motivation Hartree’s single particle self consistent equations

=⇒ ignore interaction: Vint = 0

=⇒ E [n (r)] =∫Vext(r)n (r)d3r + Ts [n]

Ts [n] = kinetic energy of the ground state of noninteracting electronswith density n (r)




Kohn-Sham ansatz

From this assumption, they got:(−1

2∆ + Vext(r)− εj

)ϕj (r) = 0 ,

E =N∑

j=1

εj ,

n (r) =N∑

j=1

|ϕj (r)|2




Kohn-Sham ansatz

To obtain the connection between HK and KS ansatz, rewrite:

EHK =

∫Vext(r)n (r) d3r + T [n] + Eint [n]︸︷︷︸

EKS =

∫Vext(r)n (r) d3r + Ts [n] + EHartree [n] + Exc [n]︸︷︷︸

Definition of exchange-correlation energy

Exc [n] = T [n] + Eint [n]− (Ts [n] + EHartree [n])




Kohn-Sham ansatz

Rewriting EKS as

EKS =

∫Veff (r)d3r + Ts [n(r)]

gives the equations(−1

2∆ + Veff (r)− εj

)ϕj (r) = 0 ,

n (r) =N∑

j=1

|ϕj (r)|2 ,

Veff = Vext(r) + VHartree(r) + Vxc (r)




Kohn-Sham ansatz

Noninteracting system Interacting system(−1

2∆ + Vext(r)− εj

)ϕj (r) = 0 ⇔

(−1

2∆ + Veff (r)− εj

)ϕj (r) = 0

↓single particle equations for thesystem of interacting particles= noninteracting particles inan effective potential

A question

What about Veff ? (hint: it depends on Vxc )




Kohn-Sham ansatz

KS DFT pros

one N-particle problem None-particle problems

if Exc was known =⇒ exact E0

and n0

KS DFT cons

an exact form of Exc is unknown

A question

What do you think is the meaning of:

a) ϕj (use an analogy with Schrodinger theory)?

b) εj ?




Kohn-Sham ansatz

KS DFT pros

one N-particle problem None-particle problems

if Exc was known =⇒ exact E0

and n0

KS DFT cons

an exact form of Exc is unknown

A question

What do you think is the meaning of:

a) ϕj ? n (r) =∑N

j=1 |ϕj (r)|2

b) εj ? εhighestj = −Eionization. Starting approx. for more precise calculations.




Kohn-Sham ansatz




Approximation to Exc - LDA

LDA = Local Density Approximation

E LDAxc =

∫n(r) εhom

xc [n(r)]︸︷︷︸ d3r

↓xc density at each point same asthat in a homogenous electron gaswith that density

A question

Where do you think LDA performs well and where not?




Approximation to Exc - LDA

LDA precision

Quantity Deviation from exp.

atomic & molecular ground state energies < 0.5%molecular equilibrium distances < 5%band structures of metals few %band gap < 100%lattice constants < 2%Ex O(10%)Ec ×2E atom

ionization & Edissoc & Ecoh 10− 20%




Approximation to Exc - GGA

GGA = General Gradient Approximation

E (0)xc =

∫εhom

xc [n(r)]n(r)d3r (LDA)

E (1)xc =

∫f (1)[n(r), |∇n(r)|]n(r)d3r (GGA)

E (2)xc =

∫f (2)[n(r), |∇n(r)|]∇2n(r)d3r

A question

What do you think why GGA opened DFT to chemists?




Approximation to Exc - GGA

GGA precision

Quantity Deviation wrt LDA

B underestimatevalence bandwidth narrowslattice constants corrects or overcorrectsTO(Γ) νphon underestimateEbinding corrects the overestimationE atom

ionization & Etot corrects




Plane waves

core states

strongly bound to nuclei

atomic-like

valence states

changes in the materials

bonding, electric & opticproperties, magnetism,...




Plane waves

A question

What would be the most appropriate basis for representing the wave functionsin solids?




Plane waves

A question

What would be the most appropriate basis for representing the wave functionsin solids?

Possibilities:

1 plane waves e ikr

2 localized orbitals (gaussians,...)

3 augmented functions (LAPW, PAW,...)




Plane waves

Period solid simulation

Bloch’s theorem:

ψm,k(r + ai ) = e ik·ai um,k(r)

Born-von Karman boundarycondition:

ψm,k(r + Ni ai ) = ψm,k(r)




Plane waves

How to make a plane wave basis?

Plane wave kin. energy

E pwkin = −1

2∆

|k + G|2

2

Plane wave sphere|k + G|2

2< Ecut




Plane waves




Plane waves

But, there are problems!

Si core and valence electron w.f.




Pseudopotentials

Basic assumption

ψcore are the same in atomic or bounding conditions

=⇒ n(r) = ncore(r) + nval (r)

Atomic Si electron energy levels and core w.f.




Pseudopotentials

Basic assumption



A question

What do you think, is this core/valence partitioning obvious for all elements?




Pseudopotentials

Basic assumption



Examples

F atom: (1s)2 + (2s)2(2p)5

IP 1 keV 10 − 100 eV E expion ≈ 18 eV

Ti atom: (1s)2(2s)2(2p)6 + (3s)2(3p)6(4s)2(3d)2 small core

IP 99.2 eV E expion ≈ 7 eV

(1s)2(2s)2(2p)6(3s)2(3p)6 + (4s)2(3d)2 large core

IP 43.3 eV




Pseudopotentials

Basic idea

“Freeze” core electrons =⇒ effective potential

Si 3s and 3s with frozen 3s electron w.f.




Pseudopotentials




Pseudopotentials




Pseudopotentials

Sometimes the core correction is needed. (a) Na as semi-metal (no core correction). (b) Na as insulator (with core correction).




Brillouin zone integration

In order to calculate something in a period case, you have to:

1 sum over bands

2 integrate over the BZ

Expressions for T and n

T =∑

n

1

Ω0k

∫Ω0k

(εF − εnk) 〈ψnk| −1

2∆|ψnk〉dk

n(r) =∑

n

1

Ω0k

∫Ω0k

(εF − εnk)ψ∗nk(r)ψnk(r)dk




Brillouin zone integration

How to calculate an integral over the BZ?

1

Ω0k

∫Ω0k

Xkdk 7→∑k

wkXk

How to choose k andwk?

Monkhorst-Pack grids

tetrahedron methods...

Homogenous sampling of the BZ.




Contents


2 Literature




Literature

1 R. M. Martin, “Electronic Structure - Basic Theory and PracticalMethods”, Cambridge University Press, Cambridge, 2004.

2 W. Kohn “Electronic structure of matter” - Nobel Lecture



http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/kohn-lecture.pdf

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The electronic structure of materials 2 - DFT - Quantum ...fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_9_The...The ground-state density n(r) of a bound