Basics of Density Functional Theory & the Kohn-Sham … · Schrödinger’s wave function theory...

Basics of Density Functional Theory & the Kohn-Sham scheme

Νικήτας Γιδόπουλος Durham University

Schrödinger’s wave function theory (non-relativistic)

N-electron system described by wf Ψ(r1,σ1;r2,σ2…rN,σN)

Eigenvalue equation for Ψ:

(T + Vee + Ven) Ψ(r1,σ1;…rN,σN) = E Ψ(r1,σ1;…rN,σN)

T = − !2

2m∇i2

∑ , Vee = 12ʹΣ

N 1rij, Ven =Σ

−Σα

Zα| ri −Rα |

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

V (ri )" #$$ %$$

Shortly after Schrödinger’s equation had been proposed and validated for small systems like He and H2, Dirac declared that ... chemistry had come to an end - its content was entirely contained in that powerful equation. Too bad, he is said to have added, that in almost all cases, this equation was far too complex to allow solution.

Solution? How about recording the wf Ψ ?

Suppose we managed to solve Schrödinger’s equation for the ground state Ψ of an atom (e.g. Fe, 26 electrons), and feeling ambitious, we want to record Ψ(r1,σ1;…r26,σ26) :

N = 26 electrons → 26×3 = 78 degrees of freedom (ignoring spin, otherwise 156).

10 grid points/degree of freedom → 100026=1078 grid points

Real number ~ 4 bytes, 1DVD (~10g) ~ 4Gbytes, we need about 1078/109 = 1069 DVDs, weighing 1067kg ~ 5×1036 solar masses.

Some sort of approximation is desirable, which will not involve the many-body-wf Ψ.

Introduction to Density Functional Theory

•  Aim: to obtain the properties (not just density & total energy!) of the ground state of an interacting electronic system without solving Schrödinger’s equation.

Definition of single-particle density:

22212 ),,,;,()( NNNddN σσσρ

σrrrrrr …! Ψ= ∫∫Σ

Properties: Any function which satisfies these two properties is representable by an N-particle fermionic wf.

0)(,)( ≥=∫ rrr ρρ Nd

ΨΨ=→Ψ

||0 HMinEN

Η : Hamiltonian of N interacting electrons Ε0: N-electron ground state energy of Η Ψ : any N-electron state

H = T +Vee+ V (ri )i∑

Ψ V Ψ = driV (ri )∫i=1

∑ dr1!dri−1 dri+1!drN∫∫∫ Ψ r1,r2,…,rN( )2

ρ(ri )N

! "######## $########= drV (r)∫ ρ(r)

We split the minimization in two steps:

E0 =Minρ→NMinΨ→ρ

Ψ |H |Ψ⎡⎣⎢

⎤⎦⎥

Ψ |H |Ψ⎡⎣⎢

⎤⎦⎥

MinΨ→ρ

Ψ |H |Ψ =MinΨ→ρ

Ψ |T +Vee |Ψ

F[ρ]! "### $###

+ dr ρ(r)V (r)∫

Ψ |H |Ψ⎡⎣⎢

⎤⎦⎥

MinΨ→ρ

Ψ |H |Ψ =MinΨ→ρ

Ψ |T +Vee |Ψ

F[ρ]! "### $###

+ dr ρ(r)V (r)∫

So, we have: E0 =Minρ→NF[ρ]+ dr ρ(r)V (r)∫⎡⎣

⎤⎦

⇒ 0 = δF[ρ]δρ(r) ρ0

+V (r)−µ(DFT’s basic eqn.)

0 = δF[ρ]δρ(r) ρ0

+V (r)−µ

Hohenberg-Kohn theorem:

• If we know the ground state density ρ0(r) we may determine the external potential V(r). • If we know the external potential V(r) we may determine the ground state density ρ0(r).

0 = δF[ρ]δρ(r) ρ0

+V (r)−µ

Hohenberg-Kohn theorem:

• If we know the ground state density ρ0(r) we may determine the external potential V(r). • If we know the external potential V(r) we may determine the ground state density ρ0(r).

Provided F[ρ]=<Ψρ|T+Vee|Ψρ> is known! So, a calculation has already been performed.

Formalism has physical meaning if we do not have to calculate F[ρ] separately for each system!

µδρ

−+= )()(][0

DFT’s basic equation. (It is scaling linearly with N) • In DFT this is always the equation we solve in one way or another.

• If we do not solve this equation, then the method is not within DFT.

Corollary from DFT’s basic equation: Thomas Fermi theory (predated DFT)

F[ρ]=MinΨ→ρ

Ψ |T +Vee |Ψ = Ψρ |T |Ψρ

310(3π 2 )2/3 dr∫ ρ(r)5/3! "## $##

+ Ψρ |Vee |Ψρ

ρ(r)ρ(r ')| r− r ' |∫∫

! "## $##

Very crude approximation for the kinetic energy! The Coulomb (Hartree) energy is not sufficiently accurate either.

Introduction to the theory of Kohn-Sham (KS)

F[ρ]=MinΨ→ρ

Ψ |T +Vee |Ψ

= Ψρ |T |Ψρ + Ψρ |Vee |Ψρ

We approximate kinetic energy of N interacting electrons with the kinetic energy of a fictitious system of

N non-interacting electrons with the same density ρ0(r)

F[ρ]= Ts[ρ]+ 12

ρ(r)ρ(r ')| r − r ' |

+Exc[ρ]∫

F[ρ]=MinΨ→ρ

Ψ |T +Vee |Ψ

F[ρ]= Ts[ρ]+ 12

ρ(r)ρ(r ')| r − r ' |

+Exc[ρ]∫

So, δF[ρ]δρ(r)µ−V (r)!"#

=δTs[ρ]δρ(r)

µs −Vs(r)!"#

+dr 'ρ(r ')| r− r ' |∫ +

δExc[ρ]δρ(r)Vxc(r)!"$ #$

(DFT’s basic equation, still linear scaling.)

F[ρ]=MinΨ→ρ

Ψ |T +Vee |Ψ

The fictitious system of N non-interacting electrons bound by the local potential Vs(r)

so that its density is equal to the exact ρ0(r) is …

the Kohn-Sham system.

Basic equation of DFT (to be solved self-consistently):

Vs(r) =Ven(r)+ dr 'ρ(r ')| r− r ' |∫ +

δExc[ρ]δρ(r)

+µ ''

So, we start with trial ρ(r), and calculate r.h.s. and Vs(r). Then? How do we obtain new density to iterate?

−!2∇2

2m+Vs(r)

⎣⎢

⎦⎥φn (r) = εn φn (r) , ρ(r) = φn (r)

Vs(r) =Ven(r)+ dr 'ρ(r ')| r− r ' |∫ +

δExc[ρ]δρ(r)

+µ ''

KS single-particle equations for the non-interacting KS system.

(But they do not scale linearly any more.)

So, we start with trial ρ(r), and calculate r.h.s. and Vs(r). Then? How do we obtain new density to iterate?

Questions •  Physically, is it a good choice for the non-interacting

system to have the same density as the interacting one?

•  Could the condition of exact density result by minimizing a total energy rather than in an ad hoc manner?

•  Do the KS Slater-determinant, orbitals and energy levels have physical meaning?

Remember the Hartree-Fock equations: The Slater determinant (non-interacting w.f.)

Antisymmetric many-electron product w.f.

Φ = Det[ϕ1,...,ϕN ]=1N!

ϕ1(r1,σ1) ϕ1(r2,σ 2 ) … ϕ1(rN ,σ N )ϕ2 (r1,σ1) ϕ2 (r2,σ 2 ) … ϕ2 (rN ,σ N )! ! " !

ϕN (r1,σ1) ϕN (r2,σ 2 ) … ϕN (rN ,σ N )

Example: expectation value of H in terms of Slater determinant

Φ H Φ =i,σ∑ dr∫ ϕi

σ*(r) − !2∇2

2m+V (r)

⎣⎢

⎦⎥ϕi

σ (r)+ 12

ρ(r)ρ(r ')r− r '∫∫ −

ρσ (r,r ')2

r− r '∫∫σ

ρσ (r,r ') = ϕiσ (r)ϕi

σ *(r ')i=1

∑ , ρ(r) = ρσ (r,r)σ

Varying the orbitals to minimize the energy we obtain single-particle equations:

−!2∇2

2m+Ven(r)+ ρ(r ')

r− r '∫"#$ %$

⎢⎢⎢⎢

⎥⎥⎥⎥

ϕiσ (r)− ρσ (r,r ')ϕi

σ (r ')r− r '∫

" #$$$ %$$$= λi

σ ϕiσ (r)

Hartree-Fock Equations

Equations to be solved self-consistently. Start with trial orbitals φi

σ construct density matrix & density. Solve equations and obtain new orbitals and iterate.

ρσ (r,r ') = ϕiσ (r)ϕi

σ *(r ')i=1

∑ , ρ(r) = ϕiσ (r)

Exchange term Hartree term

Compare with KS scheme

Single-particle equations for the non-interacting KS system

−!2∇2

2m+Vs(r)

⎣⎢

⎦⎥ϕi

σ (r) = εiσϕi

σ (r) , ρ(r) = ϕnσ (r)

Vs(r) =Ven(r)+ dr 'ρ(r ')| r− r ' |∫

! "# $#+δExc[ρ]δρ(r)Vxc(r)!"# $#

+µ ''

Hartree term

Comparison between HF & KS: • Exchange potential is local in KS, non-local in HF

• HF derived by minimizing total energy (“best energy criterion”),

• KS derived by imposing condition of exact density (“best density criterion”). Therefore, the g.s. Slater determinant in HF seems to have a more clear physical meaning than the g.s. Slater determinant in KS.

• KS accounts for correlation with Exc, which must be approximated. • HF does not include correlation, “post-HF” methods for correlation, MP, CI.

• HF equations treat occupied-unoccupied orbitals differently.

• KS equations treat occupied-unoccupied orbitals in a symmetric way. The unoccupied single-particle energy levels in HF are wrong!

• HF: Koopmans’ theorem. • In KS theory, single-particle levels have no physical meaning (except HOMO: The ionization energy is I = –εN). In particular, the unoccupied energy levels also have no meaning. Nevertheless, they are routinely used.

Approximations for Exc

Local density approximation (LDA)

Treat the (inhomogeneous) density of an electronic system as locally homogenous, and employ Exc energy as calculated exactly from the uniform

electron gas model.

Exc[ρ]= F[ρ]−Ts[ρ]− 12

ρ(r)ρ(r ')| r− r ' |∫

F[ρ]= Ψρ T +Vee Ψρ

Ψρ is the ground state of uniform electron gas model with density ρ.

Approximations for Exc

Generalised Gradient Approximations, GGAs (e.g. PBE)

Taylor expansions of Exc[ρ] for inhomogeneous densities around LDA. (The expansion is not unique).

It favours homogeneous systems

Over-binds molecules and solids

Chemical trends are usually correct.

In favourable systems (covalent, ionic, metallic bonds) geometries are good, bond lengths, angles and phonon frequencies are within a few percent, dielectric properties are overestimated by about 10%.

In weakly bound systems, bond lengths are short (overbinding).

In finite systems, the XC potential does not decay as –e2/r in the vacuum region, thus affecting the dissociation limit and ionization energies. Both LDA and LSDA fail to cancel the self-interaction included in the Hartree term of the energy. Probably the most severe limitation.

When does LDA fail?

In atomic systems (large variations in density and self-interaction important)

In weak molecular bonds, for example hydrogen bonds, because binding is dominated by inhomogeneities.

In van der Waals (closed shell) systems.

In metallic surfaces, because the XC potential decays exponentially, while it should follow a power law (image potential)

In negatively charged ions, because the LDA fails to cancel exactly the self-interaction.

The energy band gap in semiconductors turns out to be very small.

Taylor series expansions of the Kohn-Sham

potential: Exchange potential.

Work in collaboration with:

Tom Hollins, DurhamStewart Clark, DurhamKeith Refson, Royal HollowayAlexander Blair, DurhamAristeidis Kroukis, Durham

• Taylor series expansions of the KS potential

• Ways to calculate exchange accurately in KS equations.

The KS system is traditionally defined as the virtual non-interactingsystem having the ground state density of the interacting systemunder study.

The KS potential is the potential that binds the non-interactingsystem giving it the right density.

The KS system was introduced in order to obtain a good ap-proximation for the kinetic energy in DFT.

The KS system is traditionally defined as the virtual non-interactingsystem having the ground state density of the interacting systemunder study.

The KS potential is the potential that binds the non-interactingsystem giving it the right density.

The KS system was introduced in order to obtain a good ap-proximation for the kinetic energy in DFT.

Remember from Introduction:

• Physically, is it a good choice for the non-interacting systemto have the same density as the interacting one? (e.g. vsHF)

• Could the condition of exact density result by minimizing thetotal energy rather than in an ad hoc manner?

• Do the KS Slater-determinant, orbitals and energy levels havephysical meaning?

The KS potential can also be defined without fixing the density:

It is the local potential in that e↵ective Hamiltonian, whichadopts optimally the interacting ground state as its own ap-proximate ground state.

NIG, Phys Rev A 83, 040502(R) (2011)

is not the gs of e↵ective Hamiltonian Hv = T + Ven +P

i v(ri)

) h |Hv| i � Ev > 0

How to determine v(r), so that Hv will best adopt as itsapproximate ground state?

Minimize the energy di↵erence on the l.h.s.:

�v(r)

hh |Hv| i � Ev

i= ⇢ (r)� ⇢v(r) = 0

i v(ri)

) h |Hv| i � Ev > 0

�v(r)

hh |Hv| i � Ev

i= ⇢ (r)� ⇢v(r) = 0

i v(ri)

) h |Hv| i � Ev > 0

�v(r)

hh |Hv| i � Ev

i= ⇢ (r)� ⇢v(r) = 0

HF, OEP and KS theories from WFT viewpoint:

�HF minimises : h�|H|�i � E > 0

vxOEP minimises : h�v|H|�vi � E > 0

vKS minimises : h |Hv| i � Ev > 0

Minimisation of h�|H|�i leads to HF equations.

Minimisation of h�v|H|�vi gives the functional derivative:

�h�v|H|�vi�v(r)

ha|J �K� v|ii✏a � ✏i

�i(r)�a(r)

Minimisation of h |Hv| i � Ev gives functional derivative:

�v(r)

hh |Hv| i � Ev

i= ⇢ (r)� ⇢v(r)

In the last two cases, the orbitals �i satisfy:"

2+ ven(r) + v(r)

�i(r) = ✏i �i(r)

Minimisation of h�|H|�i leads to HF equations.

Minimisation of h�v|H|�vi gives the functional derivative:

�h�v|H|�vi�v(r)

ha|J �K� v|ii✏a � ✏i

�i(r)�a(r)

Minimisation of h |Hv| i � Ev gives functional derivative:

�v(r)

hh |Hv| i � Ev

i= ⇢ (r)� ⇢v(r)

In the last two cases, the orbitals �i satisfy:"

2+ ven(r) + v(r)

�i(r) = ✏i �i(r)

T [v].= h |Hv| i � Ev > 0

Exact is not known! Use PT to approximate :

= (0) + � (1) + �2 (2) + . . .

If we sub this expansion in T [v] and optimise for v, we

obtain a corresponding expansion for the KS potential vs:

vs(r) = v(0)s (r) + � v(1)s (r) + �2 v(2)s (r) + . . .

Two interesting examples for the expansion of :

Møller-Plesset (MP) expansion of the KS potential:

Consider Hartree Fock as zero order.

T�HF[v]

.= h�HF |Hv|�HF i � Ev > 0

� T�HF[v]/�v(r) = ⇢HF (r)� ⇢v(r)

The zero-order potential has the HF density.

(Hartree + Local Fock exchange potential (LFX) )

Because single excitations do not couple directly with �HF , thedensity of the MP ground state does not change to first order.

The first order correction potential vanishes v(1)s (r) = 0.

Another choice for the zero-order Hamiltonian (with a local

potential) is xOEP:

The xOEP potential (+Hartree) is the zero-order KS potentialv(0)s (r), for which the first order correction vanishes v(1)s (r) = 0.

Algorithm to determine the LFX potential:

• Start with a trial potential v with ground state density ⇢v.

• If ⇢v 6= ⇢HF , correct the potential in the direction,

v(r) ! v(r)� ✏Zdr0

⇢HF (r0)� ⇢v(r0)

|r� r0|(1)

where ✏ > 0 is a small, positive real number.

• Recalculate ⇢v for corrected v(r).

• Iterate to convergence.

For small enough ✏, the energy di↵erence T�HF[v] always de-

creases.

The algorithm stops only when the two densities become equal.

The ground state density of the resulting potential is equal tothe HF density.

(More about the algorithm if there is interest.)

The LFX potential and xOEP are equivalent physically.

Both potentials are zero-order terms, v(0)s (r), in Taylor seriesexpansions of the KS potential, where the first-order correctionsvanish, v(1)s (r) = 0.

The physical content of LFX and xOEP is the same.

(From other points of view as well).

Some results.

We implemented in the plane wave pseudopotential code CASTEPthe EXX (xOEP) and the LFX potentials.

We performed calculations for a broad range of systems for

which either HF, or LDA/GGA, or both, fail.

For LFX: we obtained the HF density and then found the poten-tial with the same ground state density.

We then performed KS “exact exchange” (EXX or xOEP) cal-culations for the same systems.

CdTe CaO

l CZnS

andgap

0 5 10

Experimental Bandgap (eV)

In the implementation ofEXX in CASTEP, the Hylleraas

variational principle allowedto bypass the computationally

expensive infinite sum overvirtual orbitals in the response

function by solving insteada Sternheimer equation.

Bandgaps from EXXimplementation in CASTEP.

T.W. Hollins, S.J. Clark,K. Refson & NIG

Phys Rev B 85, 235126 (2012).

L Γ X W K-15

SiRed lines: EXX, Blue Lines: LFX

A H K Γ M L ΓΓ-20

ZnORed lines: EXX, Blue Lines: LFX

W L Γ X W K-15

CdTeEXX: Blue, LFX: Red

A H K Γ M L HΓ-15

GaNEXX: Blue, LFX: Red

�E (eV) Bandgap (eV)xOEP LFX CEDA xOEP LFX CEDA Exp.

Ge 0.432 0.441 0.724 0.91 0.91 0.39 0.79InN 0.467 0.480 0.700 1.36 1.32 0.73 0.93Si 0.213 0.213 0.299 1.18 1.18 0.71 1.16

GaAs 0.428 0.444 0.718 1.89 1.85 1.00 1.52CdTe 0.390 0.397 0.572 2.22 2.16 1.57 1.61ZnSe 0.472 0.485 0.702 2.89 2.85 2.14 2.80GaN 0.401 0.416 0.652 3.29 3.27 2.65 3.39ZnO 0.381 0.391 0.539 3.49 3.41 2.88 3.43C 0.159 0.160 0.224 4.77 4.78 4.25 5.47

CaO 0.258 0.264 0.546 6.08 5.93 4.73 8.97NaCl 0.050 0.050 0.112 6.28 6.23 5.47 7.09MnO 0.823 0.951 1.833 3.85 3.30 3.36 3.9FeO 1.438 1.595 (2.994) 1.21 0.72 (0.36) 2.4CoO 1.595 1.698 (3.193) 2.26 1.95 (1.11) 2.5NiO 1.647 1.717 (3.403) 3.93 3.74 (2.72) 4.0

Jellium 0.000 0.000Na 0.010 0.010Al 0.030 0.031

(KS bandgaps = CBmin - VBmax)

Y Γ R Y S Γ-10

FeOEXX: Blue, LFX: Red

L Γ Z L F Γ-20

NiO, EXX FunctionalBandgap = 3.93eV

L Γ Z L F Γ-20

NiO, LFX FunctionalBandgap = 3.74eV

Simple metals

The HF approximation famously gives a qualitatively flawed de-scription of metals.

If we solve the HF equations for the jellium model, we obtainplane wave solutions for the HF orbitals, with dispersion:

HF predicts jellium to be almost an insulator! (well-known failureassociated with lack of screening, i.e., lack of correlation.)

Supposedly, the lack of correlation is behind the poor descriptionof metals for HF.

What do xOEP and LFX give for simple metals?

So, ... we calculated exchange accurately, still ignoring correla-tion but, unlike HF, we treated occupied and unoccupied orbitalsin a symmetric way.

N Γ H N P-10

NaHF: Red, LFX: Blue, LDA: Green

W L Γ X W K-20

AlHF: Red, LFX: Blue, LDA:, Green

Conclusions

We presented a method to construct power series expansions ofthe KS potential by minimizing a non-negative energy di↵erence.

The minimum of the second order energy di↵erence serves asthe correlation energy.

Two expansions of the KS potential stand out: The MP expan-sion and the expansion with Hartree + xOEP as the zero-orderpotential.

In both expansions, the first order correction vanishes.

From several di↵erent viewpoints follows that the LFX potentialand the xOEP potential are physically equivalent.

We applied the two exchange methods to a large number ofsystems, ranging from metals to semiconductors, to wide-bandinsulators and to transition metal oxides.

With the plane-wave basis, we did not encounter any issues withfinite basis errors.

For the systems we tried, the results from the two calculationsare very similar and almost indistinguishable.

The small di↵erences between LFX and xOEP in some systems,especially the transition metal monoxides, suggest that correla-tion plays a more significant role in these systems. (But thesesystem do not quite exhibit strong correlations.)

Self-interactions are important: treat exchange accurately,

with a local potential or with a nonlocal potential, but use

a method that deals in a symmetric way with the occupied

and unoccupied electrons.

In many calculations in the literature using LDA/GGA,

what appears as strong correlation (i.e. beyond the current

accuracy of DFT approximations) is often just inaccurate

description of exchange.

Thank you!

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