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
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