Density functional theory: fundamentals and applications

Manoj K. Harbola

Department of Physics

Indian Institute of Technology, Kanpur

1HRI, 31 March 2017

The many-electron problem

2

∑ ∑≠

+

+∇−=

iji

i,j ijiexti r)r(vH 1

21

21 2

The Hamiltonian

ΨΨ EH =

The Schrodinger equation

).......,,( 21 Nrrr Ψ=Ψ

The wavefunction

( )14unitsatomic 0 ==== πεeme

The simplest many-electron systems: Helium-like atoms

),(),(121

21

21212121

22

21 rrErr

rrrZ

rZ

iii

Ψ=Ψ

−+−−∇−∇−

No exact solution exists for this equation.

Note: Although the equation can be solved numerically using Mote-Carlo techniques but these calculations cannot be done routinely.

Approximate solution for the ground-states using the variational principle for the energy

groundapproxapprox

approxapprox EH

≥ΨΨ

ΨΨ

approxΨ is constructed using physics of the system

It is then optimized to minimize the expectation value of the Hamiltonian

Two approximate wavefunctions for the Helium atom

21),( 21rr

approx eerr ξξ −−=Ψ

au8477.2−=approxE

21122211),( 21rrrr

approx eeeerr ξξξξ −−−− +=Ψ

au8757.2−=approxE

au9037.2exp −=E

• Wavefunction is not a product wavefunction. It has a built-in correlation between the two electrons.

• Parameters ξ1 and ξ2 represent effective nuclearcharge taking into account the correlation betweenmotion of the electrons.

• The wavefunction can be generalized to morenumber of electrons but the approach becomescomplicated.

• Need a systematic way of constructing manyelectron wavefunctions.

21122211),( 21rrrr

approx eeeerr ξξξξ −−−− +=Ψ

Hartree theoryApproximate wavefunction

)(..........)()().......,,( 221121 NNN rrrrrr φφφ=Ψ

Electrons are treated like moving independently in an average field

)()(||

|)(|)()(v21

11

2112 rrrd

rrrrr iii

iext

φεφφρ

=

−

−++∇− ∫

∑=

φ=ρN

ii |)r(|)r(

1

2Density

Question: Can the orbital-dependent potential be replaced by a local (multiplicative) potential?

Energy of the system

rdrdrr

rrrdrrvE

jiji

jiext

iiiHartree ′

′−

′++∇−= ∫ ∑∫∫∑

≠

,

22

2)()(

21)()(

21 φφ

ρφφ

Variational derivation

{ } 01].....,[ 21 =

−−∑

jjjjN

i

E φφεφφφδφδ

Question: Can we work within the domain of productwavefunctions and still get good wavefunctions forelectrons?

• The answer is YES because electrons satisfy Pauli exclusion principle.

• Because of this principle, electrons of the same spin stay away from each other (recall orbital filling – no more than two electrons per orbital)

• The principle requires that the wavefunction be ANTISYMMETRIC with respect to exchange of two electrons.

Hartree-Fock theoryApproximate Wavefunction

)()()(

)()()()()()(

!1

21

22212

12111

NNNN

N

N

HF

xxx

xxxxxx

N

χχχ

χχχχχχ

=Ψ

)()(),( σαφσχ α rr ii

= )()(),( σβφσχ β rr ii

=or

∑ ∫∫

∑ ∑ ∫∫

−−

−+

+∇−=

ji

jiji

ji

i ji

ji

ii

ji

jiji

i ji

jiiextiHF

rdrdrr

rrrr

rdrdrr

rrrH

σσ

σσσσσσ

σ σσ

σσσσ

φφφφδ

φφφφ

,;,1

1

11**

,

, ,;,1

1

21

22

||

)()()()(

21

||

|)(||)(|

21)(v

21

Energy

Hartree-Fock equations

)(||

)()()()(

||)()(v

21

11

11*

11

12 rrdrr

rrrrrd

rrrr ii

j

jijiext

φε

φφφφρ

σ

σσσ =−

−

−

++∇− ∑∑∫∫

Hartree-Fock theory is the best mean-field prescription of a many-electron system

Question: Can the non-local exchange potential be replaced by a local (multiplicative) potential?

New features : Exchange effects

Exchange-energy

∑∑∫∫ −−=

σ

σσσσ φφφφ

ji

jijiX rdrd

rrrrrr

E,

11

11**

||)()()()(

21

Exchange-potential

∑∑∫ −−

σ

σσσ φφφ

j

jij rdrr

rrr1

1

11*

||)()()(

Energy of some atoms in Hartree-Fock theory

Atom Hartree-Fockenergy(a.u.)

Expt. Energy(a.u.)

%corr. Energy#

H -0.5 -0.5 0He -2.863 -2.903 1.4Be -14.572 -14.667 0.65Ne -128.548 -128.927 0.29Mg -199.614 -200.044 0.22Ar -526.817 -527.548 0.14

FockHartreencorrelatio EEE −−= .# experient

Koopmans’ theorem

13

Orbital energies are close to removal energies

Atom -εmax (Ryd.) Ion. Pot. (Ryd.)

Li 0.393 0.396

Be 0.619 0.685

Ne 1.701 1.585

Cl 0.867 0.828

Zn 0.585 0.690

Relative contributions of different components of energy

14

Hartree-Fock level calculation for Ne atom

Total Energy = -128.542 a.u.

Kinetic energy = 128.542 a.u.

Nucleus-electron energy = -311.384 a.u.

Hartree energy = 66.422 a.u.

Exchange energy = -12.122 a.u.

What Hartree-Fock theory does not do?

15

Hartree-Fock theory neglects instantaneous correlationsbetween electrons since it treats interactions in anaverage way (mean-field theory).

Two particles in a box

Hartree-Fock theory Expected distribution

Why is correlation important?

H− (anion) has energy higher than H atom in Hartree-Fock theory. This implies that H anion is unstable. Onthe other hand, H anion is known to exist.

Dissociation of H2 molecule is not described properly in Hartree-Fock theory

The density of states of free-electron gas (alkali metals) vanishes at the Fermi level in Hartree-Fock theory. This means that the electronic specific heat vanishes as T→0 whereas it is known to depend on T linearly.

Band gaps of solids obtained via Hartree-Fock theory are too high.

Beyond Hartree-Fock theoryExpand the wavefunction in terms of many Slater determinants

ORWrite a correlated wavefunction incorporating effect of electron repulsion

1and00 1212

122121

→∞→→→φφ=Ψ

)r(f)r(f)r(f)r()r()r,r(

This lowers the energy compared to Hartree-Fock

FockHartreeicrelativistnon

totalncorrelatio EEE −− −=

Example of constructing a correlated wavefunction for two-electron atoms

R.S. Chauhan and M.K. Harbola, Chem. Phys. Lett. 639, 248 (2015)

[ ][ ]12

12

212121

501

)(cosh)(cosh)()()(brer.

ararrφrφr,rΨ−+×

×+=

Parameters a and b and the orbitals are determined variationally

ϕ

Energies of Helium-like ions (a.u)

R.S. Chauhan and M.K. Harbola, Chem. Phys. Lett. 639, 248 (2015) 19

Atom/Ion

a b -Energy(present work)

-Energy(Literature)

H- 0.62 0.06 0.5271 0.5277

He 0.93 0.20 2.9028 2.9037

Li+ 1.19 0.36 7.2788 7.2799

Be2+ 1.48 0.54 13.6544 13.6555

B3+ 1.72 0.70 22.0297 22.0309

Ne8+ 2.78 1.54 93.9054 93.9068

Inclusion of exchange and correlation energies viawavefunctional approach is very difficult, particularlyif the number of electrons becomes large.

A different approach is needed.

Density-Functional Theory provides that alternatemethod by reformulating the many-electron problemin terms of its density.

Whereas in wavefunctional approach a large numberof orbitals in terms of 3N coordinates of N electronsare involved, in density-functional theory only onevariable – the density of electrons – in terms of only3 coordinates in involved.

Historical backgroundThomas-Fermi theory (approximate treatment of kinetic energy)

rdrdrr

rrrdrrvrdrCE extkTF ′′−′

++= ∫ ∫ ∫∫

)()(

21)()()(][ 3

5 ρρρρρ

Density-variational principle

( ){ } 0)(][)(

=−− ∫ NrdrEr TF

ρµρ

δρδ

Equation of motion

( ) 312

2

)(3)(

)()(2

)(

rrk

rdrr

rrvrk

F

extF

ρπ

µρ

=

=′′−′

++ ∫

Time-dependent theory

∫∫ ∫ ∫∫ +′′−′

++= rdurdrdrr

rrrdrrvrdrCE extk

23

5

21)()(

21)()()(][ ρρρρρρ

ρ and S are canonical variables

∫ ∇+′′−′

++==∂∂

−22

21)()(

2)( Srd

rrrrvrkE

tS

extF

ρ

δρδ

Equation applied to obtain universal photo absorption curve for atoms

),( trSu ∇= ( )

),( trj

SSE

t

⋅∇−=

∇⋅∇−==∂∂ ρ

δδρ

If then

Modern density-functional theory

Hohenberg-Kohn theorem: For a given particle-particleinteraction, the ground-state density ρ(r) of a system givesthe external potential vext(r) or the ground-statewavefunction ψ uniquely.

Proof: Reductio ad absurdum

Assume two different potentials (Hamiltonians) v1(r) and v2(r) give the same ground-state density ρ(r)

2/12/1 ˆˆˆˆ vVTH ee ++=

Let the corresponding wavefunctions be ψ1 and ψ2

ψ1 and ψ2 are different but give the same density ρ(r)

( )∫ −+=

Ψ+−Ψ=

ΨΨ<ΨΨ=

rdrrvrvE

vvH

HHE

)()()(

ˆˆˆ

ˆˆ

212

21222

2121111

ρ

( )∫ −−=

Ψ+−Ψ=

ΨΨ<ΨΨ=

rdrrvrvE

vvH

HHE

)()()(

ˆˆˆ

ˆˆ

211

12111

1212222

ρ

and

Add the two equations to get

2121 EEEE +<+

This is a contradiction

Conclusion: two different potentials cannot give the same ground-state density.

Ψ→→

→ρ

H)r(v

N)r(

ψ(r;[ρ]) is a functional of the ground-state density ρ(r)

Properties of a system can be expressed as a functional of the ground-state density ρ(r)

Variational principle in terms of density

∫∫ ′+′Ψ′+′Ψ′<+Ψ+Ψ rdrrvVTrdrrvVT eeee )()(][ˆˆ][)()(][ˆˆ][ ρρρρρρ

∫ ∫ ′+′<+= rdrrvFrdrrvFEv )()(][)()(][][ ρρρρρ

0)(][

=− µδρ

ρδr

Ev

µδρ

ρδ=+ )(

)(][ rv

rF

The equation is reminiscent of the Thomas-Fermi equation

The Euler equation for the ground-state density

Interpretation of µ

N N+1N-1

E-A

E

E+I ∫ == NrdrE µδµδρδ )(

NE∂∂

=µ = chemical potential

Since electrons come in full

INN

NENE−=

−−−−

=1

)()1(µ for ionization

ANN

NENE−=

−+−+

=1

)()1(µfor adding an electron

2AI

average+

−=µ

(Mulliken electronegativity)

Mapping ground-state density to the wavefunction and external potential

1. M. Levy, Proc. Natl. Acad. Sci. 76, 6062 (1979); 2.E.H. Lieb, Int. J Quant. Chem. 29, 93 (1983) 28

The wavefunction1

[ ] Ψ+Ψρ→Ψ

=ρ eeVTinf

F

The external potential2

[ ] [ ] ( ) ( ){ }∫ ρ−=ρ rrvrdvEsupF

Components of the energy

∫

∫∫=ΨΨ

+−

=ΨΨ

=ΨΨ

rdrrvV

Erdrdrr

rrV

TT

ext

XCee

)()(][][

][''

)'()(21][ˆ][

][][ˆ][

ρρρ

ρρρρρ

ρρρ

Euler equation for the density

µδρ

ρδρδρ

ρδ=+′

′−′

++ ∫ )(][)()(

)(][

rErd

rrrrv

rT XC

Approximations needed to solve the Euler equation

Kinetic energy has to be approximated(as in Thomas-Fermi theory).

Coulomb energy is calculated exactly.

Exchange and correlation energies are approximated.

Since kinetic energy is a big component of the total energy, its approximation affects results significantly: Shell structure of atoms is not obtained in Thomas-Fermi theory.

Need to treat kinetic energy better.

Kohn-Sham formulation

Look for a noninteracting system of Fermions that gives the same density as the interacting system

∫∫ ∫++′′−′

+= rdrrvErdrdrr

rrTE XC

)()(][)()(21][][ ρρρρρρ

∫∫ ∫++′′−′

+= rdrrvErdrdrr

rrTE DFTxcS

)()(][)()(21][][ ρρρρρρ

TS[ρ] = kinetic energy of noninteracting electrons of density ρ

Euler equation for the density

µδρ

ρδρδρ

ρδ=+′

′−′

++ ∫ )(][)()(

)(][

rE

rdrr

rrvr

T DFTxcS

The chemical potential is the same by construction

noninteracting kinetic energy

ii

iST φφρ 2

21][ ∇−= ∑

The Euler equation is equivalent to solving (because of the noninteracting kinetic energy in it)

)()()(21 2 rrrv iiieff

φεφ =

+∇−

)r(][Erd

rr)r()r(v)r(v

DFTxc

eff

δρρδ

+′′−′ρ

+= ∫

∑=i

i rr 2)()( φρ

The equation is to be solved self-consistently

Energy in Kohn-Sham theory

∫∫ ∫∑ ++′′−′

+∇−= rdrrvErdrdrr

rrE DFTxc

iii

)()(][)()(21

21][ 2 ρρρρφφρ

Chemical potential in Kohn-Sham theory

εmax

-εmaxNN EE +−=− max1 ε

max1 εµ =−= −NN EE

but chemical potential is the same as –II−=maxε

So the exact Kohn-Sham theory also gives the ionization potential

Examples: Construction of exact Kohn-Sam system for Be and Ne

(systems with known exact densities)

Be 1s22s2 Ne 1s22s22p6

εmax −0.687 Ry −1.545 Ry

Iexpt 0.685 Ry 1.585 Ry

Texact 14.67 au 128.93 au

TS 14.59 au 128.60 au

Exact density and exchange-correlation potential for Be and Ne

Be Ne

Wavefunctional construction ofthe Kohn-Sham system

M.K. Harbola and V. Sahni, J. Chem. Ed. 70, 920 (1993) 36

Probability of finding an electron at r and another at r’

( ) ( ) ( ) ( ) Ψ−δ−δ−

Ψ= ∑∑=

−

=

N

i

N

jji rrrr

NN'r,rP

1

1

12 1

11

The exchange-correlation hole and the exchange-correlation energy:

37

( ) ( )( ) ( ) ( )[ ]'r,r'rNN

r'r,rP xc

ρ+ρ−

ρ=

11

2

( ) 1−=ρ∫ 'r,r'rd xc

Normalization

( ) ( )∫∫ −

ρρ=

'rr'r,rr'rdrdE xc

xc

21Exchange-correlation

energy

Exchange-correlation hole and exchange-correlation energy in density-functional theory:

38

( ) ( )SCC

xcDFTxc TTT;T

'rr'r,rr'rdrdE −=+

−ρρ

= ∫∫

21

( ) ( )∫∫ −

ρρ=

'rr'r,rr'rdrd

DFTxc

21

Exchange-correlation potential:

M.K. Harbola and V. Sahni, Phys. Rev. Lett. 62, 489 (1989); A. Holas and N.H. March, Phys. Rev. A 51, 2040(1995)

39

( ) ( )∫ −

ρ=

'rr'r,r'rdrV

DFTxc

xc

Is ?

( ) ( ) ( )'rr'rr

'r,r'rdr xcxc

−−

ρ= ∫ 3E

( ) ( ) ( )rT'r'ldrV C

r

xcxc

δρδ

+⋅−= ∫∞

E

Generalization of DFT to fractional number of electrons

J.P. Perdew, R.G. Parr, M. Levy and J.L. Balduz, Phys. Rev. Lett. 49, 1691 (1982) 40

Consider densities such that:

( ) ωNrrd +=ρ∫

Such densities are produced by a statisticalmixing of ground-state wavefunctions withdifferent number of particles

Minimization of energy with respect tocoefficient of mixing leads to:

41

( ) 11 ++ ρ+ρ−=ρ NNωN ωω

( ) 11 ++ +−= NNωN ωEEωE

N N+1N-1

E-A

E

E+I

Discontinuity in the exchange-correlation potential across integer number of electrons

J.P. Perdew, R.G. Parr, M. Levy and J.L. Balduz, Phys. Rev. Lett. 49, 1691 (1982) 42

The Kohn-Sham equation

( ) ( ) ( ) )r()r(rv'rr

'r'rdrv iiixcext

φε=φ

+

−ρ

++∇− ∫2

21

εmax changes discontinuously across integer number of electrons from -I to -A

This can come about if the exchange correlationpotential vxc changes by a constant across integernumber of electrons

Discontinuity in the exchange-correlation potential for He

M.K. Harbola, Phys. Rev. A 57, 4253 (1998) 43

Implementaion of the Kohn-Sham method

44

∫∫ ∫∑ ++′′−′

+∇−= rdrrvErdrdrr

rrE DFTxc

iii

)()(][)()(21

21][ 2 ρρρρφφρ

)()()(21 2 rrrv iiieff

φεφ =

+∇− ∑=

ii rr 2)()( φρ

)(][)()()(

rErd

rrrrvrv

DFTxc

eff

δρρδρ

+′′−′

+= ∫

The equation is to be solved self-consistently

Exchange-correlation energy is to be approximated

The simplest functional: the local-density approximation (LDA)

45

rdrrrE xcLDAxc

)()](;[][ ρρερ ∫=

εxc[r;ρ(r)] = energy per electron for homogeneous electron gas of density ρ(r)

Analytic result for the exchange energy of homogeneous electrons gas is known in terms of its density

Highly accurate numerical results for correlation energy are known for homogeneous electron has and can be parameterized in terms of density

31σLSD

σx,

−=

πr6ρrv )()(

Exchange-potential

Exchange-energy functional

∫

−= rrr 34

31LDAx dρ

π3

43ρE )()]([

][][ βLDAxα

LDAx

LSDx 2ρE

212ρE

21E +=

Results of some atoms (eV) with the LDA

47

Atom -Energy(LDA) -Energy (Expt.) -εmax (LDA) Ion. Pot.

H 12.5 13.6 6.6 13.6

He 77.8 79.0 15.9 25.4

Be 394.5 399.1 5.8 9.3

Mg 3493.9 3508.1 13.9 21.6

Ne 5396.8 5443.2 5.0 7.7

Ar 14319.7 14354.6 10.7 15.8

Calculations for equilibrium lattice parameters (Å) of solids

P. Singh, M.K. Harbola, B. Sanyal and A. Mookerjee, Phys. Rev. B 87, 235110 (2013) 48

Element LDA Experiment

C 3.60 3.57

Si 5.51 5.43

AlN 4.38 4.36

BP 4.60 4.54

Li 3.27 3.45

Na 4.03 4.21

Al 3.99 4.02

Calculations of band gaps(eV) of solids

P. Singh, M.K. Harbola, B. Sanyal and A. Mookerjee, Phys. Rev. B 87, 235110 (2013) 49

Element LDA ExperimentC 2.70 5.48Si 0.49 1.17

AlN 2.44 5.11AlP 1.16 2.51BP 1.38 2.00

Going beyond the LDA

• Include gradient corrections in terms of gradient of the density

• By satisfying certain exact properties, make corrections to the LDA funtionals – these give the generalized gradient approximations or the GGA

Calculations of band gaps(eV) of solids

P. Singh, M.K. Harbola, B. Sanyal and A. Mookerjee, Phys. Rev. B 87, 235110 (2013) 51

Element LDA HS LB Expt.

C 2.70 5.47 5.18 5.48

Si 0.49 1.24 1.21 1.17

AlN 2.44 5.05 5.13 5.11

AlP 1.16 2.53 2.75 2.51

BP 1.38 2.22 2.09 2.00

Concluding Remarks

52

•An overview of density functional theory has been given;

•Focus has been on stationary-state ground-state theory;

•Time-dependent density-functional theory also exist and has been used.

Some of the challenges are:

•to develop excited-state theory;

•to explore fundamental aspects of theory to make it more accurate;

•to develop functionals for strongly-correlated systems.

53

Thank you