Introduction to Density Functional Theory

Juan Carlos Cuevas

Institut fur Theoretische Festkorperphysik

Universitat Karlsruhe (Germany)

www-tfp.physik.uni-karlsruhe.de/∼cuevas

0.- Motivation I.

Walter Kohn was awarded with the Nobel Prize in Chemistry in 1998 for his develop-ment of the density functional theory.

Walter Kohn receiving his Nobel Prize from HisMajesty the King at the Stockholm Concert

Hall.

The Nobel Prize medal.

0.- Motivation II.

The Density Functional Theory was introduced in two seminal papers in the 60’s:

1. Hohenberg-Kohn (1964): ∼ 4000 citations

2. Kohn-Sham (1965): ∼ 9000 citations

The following figure shows the number of publications where the phrase“density functionaltheory”appears in the title or abstract (taken from the ISI Web of Science).

1980 1984 1988 1992 1996 2000Year

0

500

1000

1500

2000

2500

3000

Num

ber

of P

ublic

atio

ns

0.- Motivation III.

• The density functional theory (DFT) is presently the most successfull (and also the mostpromising) approach to compute the electronic structure of matter.

• Its applicability ranges from atoms, molecules and solids to nuclei and quantum andclassical fluids.

• In its original formulation, the density functional theory provides the ground state prop-erties of a system, and the electron density plays a key role.

• An example: chemistry. DFT predicts a great variety of molecular properties: molecularstructures, vibrational frequencies, atomization energies, ionization energies, electric andmagnetic properties, reaction paths, etc.

• The original density functional theoy has been generalized to deal with many differentsituations: spin polarized systems, multicomponent systems such as nuclei and electronhole droplets, free energy at finite temperatures, superconductors with electronic pairingmechanisms, relativistic electrons, time-dependent phenomena and excited states, bosons,molecular dynamics, etc.

0.- Motivation IV: Molecular Electronics.

The possibility of manipulating singlemolecules has created a new field:

MOLECULAR ELECTRONICS

Goal: understanding of the transportat the molecular scale.

Method: ab initio quantum chemistry (DFT) and Green function techniques.

• First step: Understanding of the relation between conduction channels and molecularorbitals.

G =2e2

h

∑

i

Ti

Landauer formula−3 −2 −1 0 1 2 3

E (eV)

0

2

4

6

Den

sity

of s

tate

s

ab cd(a)

(d)

(e)

(b)

(c)

Heurich, Cuevas, Wenzel and Schon, PRL (2002)

REVIEW: Cuevas, Heurich, Pauly, Wenzel and Schon, Nanotechnology 14, 29 (2003).

0.- Literature.

The goal of this lecture is to give an elementary introduction to density functional theory. Forthose who want to get deeper into the subtleties and performance of this theory, the followingentry points in the literature are strongly adviced:

Original papers

• Inhomogeneous Electron Gas, P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

• Self Consistent Equations Including Exchange and Correlation Effects, W. Kohn and L.J. Sham,Phys. Rev. 140, A1133 (1965).

Review Articles

• Nobel Lecture: Electronic structure of matter–wave functions and density functionals,W. Kohn, Rev. Mod. Phys. 71, 1253 (1998).

• The density functional formalism, its applications and prospects, R.O. Jones and O. Gun-narsson, Rev. Mod. Phys. 61, 689 (1989).

Books

• Density-Functional Theory of Atoms and Molecules, R.G Parr and W. Yang, Oxford Uni-versity Press, New York (1989).

• A Chemist’s Guide to Density Functional Theory, W. Koch and M.C. Holthausen, WILEY-VCH (2001).

Outline of the lecture.

• 0.- Motivation and literature.

• 1.- Elementary quantum mechanics:– 1.1 The Schrodinger equation.– 1.2 The variational principle.– 1.3 The Hartree-Fock approxima-

tion.

• 2.- Early density functional theories:

– 2.1 The electron density.– 2.2 The Thomas-Fermi model.

• 3.- The Hohenberg-Kohn theorems:

– 3.1 The first Hohenberg-Kohn theo-rem.

– 3.2 The second Hohenberg-Kohntheorem.

• 4.- The Kohn-Sham approach:– 4.1 The Kohn-Sham equations.

• 5.- The exchange-correlation func-tionals:– 5.1 LDA approximation.– 5.2 The generalized gradient approx-

imation.

• 6.- The basic machinary of DFT:– 6.1 The LCAO Ansatz in the KS

equations.– 6.2 Basis sets.

• 7.- DFT applications:– 7.1 Applications in quantum chem-

istry.– 7.2 Applications in solid state

physics.– 7.3 DFT in Molecular Electronics.

1.1 The Schrodinger equation.

The ultimate goal of most appproaches in solid state physics and quantum chemistry is thesolution of the time-independent, non-relativistic Schrodinger equation

HΨi(~x1, ~x2, ..., ~xN , ~R1, ~R2, ..., ~RM) = EiΨi(~x1, ~x2, ..., ~xN , ~R1, ~R2, ..., ~RM) (1)

H is the Hamiltonian for a system consisting of M nuclei and N electrons.

H = −1

2

N∑

i=1

∇2i −

1

2

M∑

A=1

1

MA∇2A −

N∑

i=1

M∑

A=1

ZA

riA+

N∑

i=1

N∑

j>i

1

rij+

M∑

A=1

M∑

B>A

ZAZB

RAB(2)

Here, A and B run over the M nuclei while i and j denote the N electrons in the system.

The first two terms describe the kinetic energy of the electrons and nuclei. The other threeterms represent the attractive electrostatic interaction between the nuclei and the electrons andrepulsive potential due to the electron-electron and nucleus-nucleus interactions.

Note: throughout this talk atomic units are used.

1.1 The Schrodinger equation.

Born-Oppenheimer approximation: due to their masses the nuclei move much slower thanthe electrons =⇒ we can consider the electrons as moving in the field of fixed nuclei =⇒ thenuclear kinetic energy is zero and their potential energy is merely a constant. Thus, the electronicHamiltonian reduces to

Helec = −1

2

N∑

i=1

∇2i −

N∑

i=1

M∑

A=1

ZA

riA+

N∑

i=1

N∑

j>i

1

rij= T + VNe + Vee (3)

The solution of the Schrodinger equation with Helec is the electronic wave function Ψelec andthe electronic energy Eelec. The total energy Etot is then the sum of Eelec and the constantnuclear repulsion term Enuc.

HelecΨelec = EelecΨelec (4)

Etot = Eelec + Enuc where Enuc =

M∑

A=1

M∑

B>A

ZAZB

RAB(5)

1.2 The variational principle for the ground state.

When a system is in the state Ψ, the expectation value of the energy is given by

E[Ψ] =〈Ψ|H|Ψ〉〈Ψ|Ψ〉

where 〈Ψ|H|Ψ〉 =

∫

Ψ∗HΨd~x (6)

The variational principle states that the energy computed from a guessed Ψ is an upperbound to the true ground-state energy E0. Full minimization of the functional E[Ψ] withrespect to all allowed N -electrons wave functions will give the true ground state Ψ0 and energyE[Ψ0] = E0; that is

E0 = minΨ→NE[Ψ] = minΨ→N〈Ψ|T + VNe + Vee|Ψ〉 (7)

For a system of N electrons and given nuclear potential Vext, the variational principle defines aprocedure to determine the ground-state wave function Ψ0, the ground-state energyE0[N,Vext],and other properties of interest. In other words, the ground state energy is a functional ofthe number of electrons N and the nuclear potential Vext:

E0 = E[N,Vext] (8)

1.3 The Hartree-Fock approximation.

Suppose that Ψ0 (the ground state wave function) is approximated as an antisymmetrized productofN orthonormal spin orbitals ψi(~x), each a product of a spatial orbital φk(~r) and a spin functionσ(s) = α(s) or β(s), the Slater determinant

Ψ0 ≈ ΨHF =1√N !

∣

∣

∣

∣

∣

∣

∣

∣

ψ1(~x1) ψ2(~x1) ... ψN(~x1)ψ1(~x2) ψ2(~x2) ... ψN(~x2)

... ... ...ψ1(~xN) ψ2(~xN) ... ψN(~xN)

∣

∣

∣

∣

∣

∣

∣

∣

(9)

The Hartree-Fock approximation is the method whereby the orthogonal orbitals ψi are foundthat minimize the energy for this determinantal form of Ψ0:

EHF = min(ΨHF→N)E [ΨHF ] (10)

The expectation value of the Hamiltonian operator with ΨHF is given by

EHF = 〈ΨHF |H|ΨHF 〉 =

N∑

i=1

Hi +1

2

N∑

i,j=1

(Jij −Kij) (11)

Hi ≡∫

ψ∗i (~x)

[

−1

2∇2 − Vext(~x)

]

ψi(~x) d~x (12)

defines the contribution due to the kinetic energy and the electron-nucleus attraction and

1.3 The Hartree-Fock approximation.

Jij =

∫ ∫

ψi(~x1)ψ∗i (~x1)

1

r12ψ∗j(~x2)ψj(~x2)d~x1d~x2 (13)

Kij =

∫ ∫

ψ∗i (~x1)ψj(~x1)

1

r12ψi(~x2)ψ

∗j(~x2)d~x1d~x2 (14)

The integrals are all real, and Jij ≥ Kij ≥ 0. The Jij are called Coulomb integrals, the Kij

are called exchange integrals. We have the property Jii = Kii.

The variational freedom in the expression of the energy [Eq. (11)] is in the choice of the orbitals.The minimization of the energy functional with the normalization conditions

∫

ψ∗i (~x)ψj(~x)d~x =

δij leads to the Hartree-Fock differential equations:

f ψi = εi ψi , i = 1,2, ...,N (15)

TheseN equations have the appearance of eigenvalue equations, where the Lagrangian multipliersεi are the eigenvalues of the operator f . The Fock operator f is an effective one-electron operatordefined as

f = −1

2∇2i −

M∑

A

ZA

riA+ VHF(i) (16)

1.3 The Hartree-Fock approximation.

The first two terms are the kinetic energy and the potential energy due to the electron-nucleusattraction. VHF(i) is the Hartree-Fock potential, the average repulsive potential experience bythe i’th electron due to the remaining N -1 electrons, and it is given by

VHF(~x1) =

N∑

j

(

Jj(~x1) − Kj(~x1))

(17)

Jj(~x1) =

∫

|ψj(~x2)|21

r12d~x2 (18)

The Coulomb operator J represents the potential that an electron at position ~x1 experiences dueto the average charge distribution of another electron in spin orbital ψj.

The second term in Eq. (17) is the exchange contribution to the HF potential. It has no classicalanalog and it is defined through its effect when operating on a spin orbital:

Kj(~x1) ψi(~x1) =

∫

ψ∗j(~x2)

1

r12ψi(~x2) d~x2 ψj(~x1) (19)

• The HF potential is non-local and it depends on the spin orbitals. Thus, the HF equationsmust be solved self-consistently.

• The Koopman’s theorem (1934) provides a physical interpretation of the orbital energies: itstates that the orbital energy εi is an approximation of minus the ionization energy associatedwith the removal of an electron from the orbital ψi, i.e. εi ≈ EN − Ei

N−1 = −IE(i).

2.1 The electron density.

The electron density is the central quantity in DFT. It is defined as the integral over the spincoordinates of all electrons and over all but one of the spatial variables (~x ≡ ~r, s)

ρ(~r) = N

∫

...

∫

|Ψ(~x1, ~x2, ..., ~xN)|2ds1d~x2...d~xN . (20)

ρ(~r) determines the probability of finding any of the N electrons within volumen element d~r.

Some properties of the electron density:

• ρ(~r) is a non-negative function of only the three spatial variables which vanishes at infinityand integrates to the total number of electrons:

ρ(~r → ∞) = 0

∫

ρ(~r)d~r = N (21)

• ρ(~r) is an observable and can be measured experimentally, e.g. by X-ray diffraction.

• At any position of an atom, the gradient of ρ(~r) has a discontinuity and a cusp results:

limriA→0 [∇r + 2ZA] ρ(~r) = 0 (22)

where Z is the nuclear charge and ρ(~r) is the spherical average of ρ(~r).

• The asymptotic exponential decay for large distances from all nuclei:

ρ(~r) ∼ exp[

−2√

2I|~r|]

I is the exact ionization energy (23)

2.2 The Thomas-Fermi model.

The conventional approaches use the wave function Ψ as the central quantity, since Ψ containsthe full information of a system. However, Ψ is a very complicated quantity that cannot beprobed experimentally and that depends on 4N variables, N being the number of electrons.

The Thomas-Fermi model: the first density functional theory (1927).

• Based on the uniform electron gas, they proposed the following functional for the kineticenergy:

TTF [ρ(~r)] =3

10(3π2)2/3

∫

ρ5/3(~r)d~r. (24)

• The energy of an atom is finally obtained using the classical expression for the nuclear-nuclear potential and the electron-electron potential:

ETF [ρ(~r)] =3

10(3π2)2/3

∫

ρ5/3(~r)d~r−Z

∫

ρ(~r)

rd~r+

1

2

∫ ∫

ρ(~r1)ρ(~r2)

r12d~r1d~r2. (25)

The energy is given completely in terms of the electron density!!!

• In order to determine the correct density to be included in Eq. (25), they employed avariational principle. They assumed that the ground state of the system is connected tothe ρ(~r) for which the energy is minimized under the constraint of

∫

ρ(~r)d~r = N .

Does this variational principle make sense?

3.1 The first Hohenberg-Kohn theorem.

The first Hohenberg-Kohn theorem demonstrates that the electron density uniquely determinesthe Hamiltonian operator and thus all the properties of the system.

This first theorem states that the external potential Vext(~r) is (to within a constant) aunique functional of ρ(~r); since, in turn Vext(~r) fixes H we see that the full manyparticle ground state is a unique functional of ρ(~r).

Proof: let us assume that there were two external potential Vext(~r) and V ′ext(~r) differing by more

than a constant, each giving the same ρ(~r) for its ground state, we would have two HamiltoniansH and H ′ whose ground-state densities were the same although the normalized wave functionsΨ and Ψ′ would be different. Taking Ψ′ as a trial wave function for the H problem

E0 < 〈Ψ′|H|Ψ′〉 = 〈Ψ′|H ′|Ψ′〉+〈Ψ′|H−H ′|Ψ′〉 = E′0+

∫

ρ(~r)[

Vext(~r) − V ′ext(~r)

]

d~r,

(26)

where E0 and E′0 are the ground-state energies for H and H ′, respectively. Similarly, taking Ψ

as a trial function for the H ′ problem,

E′0 < 〈Ψ|H ′|Ψ〉 = 〈Ψ|H|Ψ〉 + 〈Ψ|H ′ − H|Ψ〉 = E0 +

∫

ρ(~r)[

Vext(~r) − V ′ext(~r)

]

d~r,

(27)

Adding Eq. (26) and Eq. (27), we would obtain E0 + E′0 < E′

0 + E0, a contradiction, and sothere cannot be two different Vext(~r) that give the same ρ(~r) for their ground state.

3.1 The first Hohenberg-Kohn theorem.

Thus, ρ(~r) determines N and Vext(~r) and hence all the properties of the ground state, forexample the kinetic energy T [ρ], the potential energy V [ρ], and the total energy E[ρ]. Now,we can write the total energy as

E[ρ] = ENe[ρ] + T [ρ] + Eee[ρ] =

∫

ρ(~r)VNe(~r)d~r+ FHK[ρ], (28)

FHK[ρ] = T [ρ] + Eee. (29)

This functional FHK[ρ] is the holy grail of density functional theory. If it were known we wouldhave solved the Schrodinger equation exactly! And, since it is an universal functional completelyindependent of the system at hand, it applies equally well to the hydrogen atom as to giganticmolecules such as, say, DNA! FHK[ρ] contains the functional for the kinetic energy T [ρ] andthat for the electron-electron interaction, Eee[ρ]. The explicit form of both these functional liescompletely in the dark. However, from the latter we can extract at least the classical part J[ρ],

Eee[ρ] =1

2

∫ ∫

ρ(~r1)ρ(~r2)

r12d~r1d~r2 + Encl = J[ρ] +Encl[ρ]. (30)

Encl is the non-classical contribution to the electron-electron interaction: self-interaction correc-tion, exchange and Coulomb correlation.

The explicit form of the functionals T [ρ] and Encl[ρ] is the major challange of DFT.

3.2 The second Hohenberg-Kohn theorem.

How can we be sure that a certain density is the ground-state density that we are looking for?

The second H-K theorem states that FHK[ρ], the functional that delivers the ground stateenergy of the system, delivers the lowest energy if and only if the input density is thetrue ground state density. This is nothing but the variational principle:

E0 ≤ E[ρ] = T [ρ] + ENe[ρ] +Eee[ρ] (31)

In other words this means that for any trial density ρ(~r), which satisfies the necessary boundaryconditions such as ρ(~r) ≥ 0,

∫

ρ(~r)d~r = N , and which is associated with some external

potential Vext, the energy obtained from the functional of Eq. (28) represents an upper boundto the true ground state energy E0. E0 results if and only if the exact ground state density isinserted in Eq. (24).

Proof: the proof of Eq. (31) makes use of the variational principle established for wave functions.

We recall that any trial density ρ defines its own Hamiltonian ˜H and hence its own wave functionΨ. This wave function can now be taken as the trial wave function for the Hamiltonian generatedfrom the true external potential Vext. Thus,

〈Ψ|H|Ψ〉 = T [ρ] + Eee[ρ] +

∫

ρ(~r)Vextd~r = E[ρ] ≥ E0[ρ] = 〈Ψ0|H|Ψ0〉. (32)

What have we learned so far?

Let us summarize what we have shown so far:

• All the properties of a system defined by an external potential Vext are determined by theground state density. In particular, the ground state energy associated with a density ρ isavaible through the functional

∫

ρ(~r)Vextd~r+ FHK[ρ]. (33)

• This functional attains its minimum value with respect to all allowed densities if and only ifthe input density is the true ground state density, i.e. for ρ(~r) ≡ ρ(~r).

• The applicability of the varational principle is limited to the ground state. Hence, we cannoteasily transfer this strategy to the problem of excited states.

• The explicit form of the functional FHK[ρ] is the major challange of DFT.

4.1 The Kohn-Sham equations.

We have seen that the ground state energy of a system can be written as

E0 = minρ→N

(

F [ρ] +

∫

ρ(~r)VNed~r

)

(34)

where the universal functional F [ρ] contains the contributions of the kinetic energy, the classicalCoulomb interaction and the non-classical portion:

F [ρ] = T [ρ] + J[ρ] +Encl[ρ] (35)

Of these, only J[ρ] is known. The main problem is to find the expressions for T [ρ] and Encl[ρ].

The Thomas-Fermi model of section 2.2 provides an example of density functional theory. How-ever, its performance is really bad due to the poor approximation of the kinetic energy. To solvethis problem Kohn and Sham proposed in 1965 the approach described below.

They suggested to calculate the exact kinetic energy of a non-interacting reference system withthe same density as the real, interacting one

TS = −1

2

N∑

i

〈ψi|∇2|ψi〉 ρS(~r) =

N∑

i

∑

s

|ψi(~r, s)|2 = ρ(~r) (36)

4.1 The Kohn-Sham equations.

where the ψi are the orbitals of the non-interacting system. Of course, TS is not equal to the truekinetic energy of the system. Kohn and Sham accounted for that by introducing the followingseparation of the functional F [ρ]

F [ρ] = TS[ρ] + J[ρ] +EXC[ρ], (37)

where EXC , the so-called exchange-correlation energy is defined through Eq. (37) as

EXC[ρ] ≡ (T [ρ] − TS[ρ]) + (Eee[ρ]− J[ρ]) . (38)

The exchange and correlation energy EXC is the functional that contains everything that isunknown.

Now the question is: how can we uniquely determine the orbitals in our non-interacting referencesystem? In other words, how can we define a potential VS such that it provides us with a Slaterdeterminant which is characterized by the same density as our real system? To solve this problem,we write down the expression for the energy of the interacting system in terms of the separationdescribed in Eq. (37)

E[ρ] = TS[ρ] + J[ρ] + EXC[ρ] +ENe[ρ] (39)

4.1 The Kohn-Sham equations.

E[ρ] = TS[ρ] +1

2

∫ ∫

ρ(~r1)ρ(~r2)

r12d~r1d~r2 +EXC[ρ] +

∫

VNeρ(~r)d~r =

−1

2

N∑

i

〈ψi|∇2|ψi〉 +1

2

N∑

i

N∑

j

∫ ∫

|ψi(~r1)|21

r12|ψj(~r2)|2d~r1d~r2 + EXC[ρ]−

−N∑

i

∫ M∑

A

ZA

r1A|ψi(~r1)|2d~r1. (40)

The only term for which no explicit form can be given is EXC . We now apply the variationalprinciple and ask: what condition must the orbitals {ψi} fulfill in order to minimize this energyexpression under the usual constraint 〈ψi|ψj〉 = δij? The resulting equations are the Kohn-Sham equations:

(

−1

2∇2 +

[

∫

ρ(~r2)

r12+ VXC(~r1) −

M∑

A

ZA

r1A

])

ψi =

(

−1

2∇2 + VS(~r1)

)

ψi = εiψi

(41)

VS(~r1) =

∫

ρ(~r2)

r12d~r2 + VXC(~r1) −

M∑

A

ZA

r1A(42)

4.1 The Kohn-Sham equations.

Some comments:

• Once we know the various contributions in Eqs. (41-42) we have a grip on the potential VSwhich we need to insert into the one-particle equations, which in turn determine the orbitalsand hence the ground state density and the ground state energy employing Eq. (40). Noticethat VS depends on the density, and therefore the Kohn-Sham equations have to be solvediteratively.

• The exchange-correlation potential, VXC is defined as the functional derivative of EXCwith respect to ρ, i.e. VXC = δEXC/δρ.

• It is very important to realize that if the exact forms of EXC and VXC were known, theKohn-Sham strategy would lead to the exact energy!!

• Do the Kohn-Sham orbitals mean anything? Strictly speaking, these orbitals have no physi-cal significance, except the highest occupied orbital, εmax, which equals the negative of theexact ionization energy.

5.1 The local density approximation (LDA).

The local density approximation (LDA) is the basis of all approximate exchange-correlation func-tionals. At the center of this model is the idea of an uniform electron gas. This is a system inwhich electrons move on a positive background charge distribution such that the total ensembleis neutral.

The central idea of LDA is the assumption that we can write EXC in the following form

ELDAXC [ρ] =

∫

ρ(~r)εXC (ρ(~r)) d~r (43)

Here, εXC(ρ(~r)) is the exchange-correlation energy per particle of an uniform electron gasof density ρ(~r). This energy per particle is weighted with the probability ρ(~r) that there isan electron at this position. The quantity εXC(ρ(~r)) can be further split into exchange andcorrelation contributions,

εXC(ρ(~r)) = εX(ρ(~r)) + εC(ρ(~r)). (44)

The exchange part, εX , which represents the exchange energy of an electron in a uniform electrongas of a particular density, was originally derived by Bloch and Dirac in the late 1920’s

εX = −3

4

(

3ρ(~r)

π

)1/3

(45)

5.1 The local density approximation (LDA).

No such explicit expression is known for the correlation part, εC . However, highly accurate numer-ical quantum Monte-Carlo simulations of the homogeneous electron gas are available (Ceperly-Alder, 1980).

Some comments:

• The accuracy of the LDA for the exchange energy is typically within 10%, while thenormally much smaller correlation energy is generally overstimated by up to a factor 2. Thetwo errors typically cancle partially.

• Experience has shown that the LDA gives ionization energies of atoms, dissociation energiesof molecules and cohesive energies with a fair accuracy of typically 10-20%. However, theLDA gives bond lengths of molecules and solids typically with an astonishing accuracy of∼ 2%.

• This moderate accuracy that LDA delivers is certainly insufficient for most applications inchemistry.

• LDA can also fail in systems, like heavy fermions, so dominated by electron-electron inter-action effects.

5.2 The generalized gradient approximation (GGA).

The first logical step to go beyond LDA is the use of not only the information about the densityρ(~r) at a particular point ~r, but to supplement the density with information about the gradientof the charge density, ∇ρ(~r) in order to account for the non-homogeneity of the true electrondensity. Thus, we write the exchang-correlation energy in the following form termed generalizedgradient approximation (GGA),

EGEAXC [ρα, ρβ] =

∫

f(ρα, ρβ,∇ρα,∇ρβ) d~r (46)

Thanks to much thoughtful work, important progress has been made in deriving successful GGA’s.Their construction has made use of sum rules, general scaling properites, etc.

In another approach A. Becke introduced a successful hybrid functional:

EhybXC = αEKS

X + (1 − α)EGGAXC , (47)

where EKSX is the exchange calculated with the exact KS wave function, EGGA

XC is an appropiateGGA, and α is a fitting parameter.

5.2 The generalized gradient approximation (GGA).

Some comments:

• GGA’s and hybrid approximations has reduced the LDA errors of atomization energies ofstandard set of small molecules by a factor 3-5. This improved accuracy has made DFT asignificant component of quantum chemistry.

• All the present functionals are inadequate for situations where the density is not a slowlyvarying function. Examples are (a) Wigner crystals; (b) Van der Waals energies betweennonoverlapping subsystems; (c) electronic tails evanescing into the vacuum near the surfacesof bounded electronic systems. However, this does not preclude that DFT with appropiateapproximations can successfully deal with such problems.

6.1 The LCAO Ansatz in the The Kohn-Sham equations.

Recall the central ingredient of the Kohn-Sham approach to density functional theory, i.e. theone-electron KS equations,

−1

2∇2 +

N∑

j

∫ |ψj(~r2)|2r12

d~r2 + VXC(~r1) −M∑

A

ZA

r1A

ψi = εi ψi. (48)

The term in square brackets defines the Kohn-Sham one-electron operator and Eq. (48) can bewritten more compactly as

fKS ψi = εi ψi. (49)

Most of the applications in chemistry of the Kohn-Sham density functional theory make use of theLCAO expansion of the Kohn-Sham orbitals. In this approach we introduce a set of L predefinedbasis functions {ηµ} and linearly expand the K-S orbitals as

ψi =

L∑

µ=1

cµiηµ. (50)

We now insert Eq. (44) into Eq. (43) and obtain in very close analogy to the Hartree-Fock case

fKS(~r1)

L∑

ν=1

cνiην(~r1) = εi

L∑

ν=1

cνiην(~r1). (51)

6.1 The LCAO Ansatz in the The Kohn-Sham equations.

If we now multiply this equation from the left with an arbitrary basis function ηµ and integrateover space we get L equations

L∑

ν=1

cνi

∫

ηµ(~r1)fKS(~r1)ην(~r1)d~r1 = εi

L∑

ν=1

cνi

∫

ηµ(~r1)ην(~r1)d~r1 for 1 ≤ i ≤ L

(52)

The integrals on both sides of this equation define a matrix:

FKSµν =

∫

ηµ(~r1)fKS(~r1)ην(~r1)d~r1 Sµν =

∫

ηµ(~r1)ην(~r1)d~r1, (53)

which are the elements of the Kohn-Sham matrix and the overlap matrix, respectively. Bothmatrices are L× L dimensional. Eqs.(52) can be rewritten compactly as a matrix equation

FKSC = SCε. (54)

Hence, through the LCAO expansion we have translated the non-linear optimization problem intoa linear one, which can be expressed in the language of standard algebra.

By expanding fKS into its components, the individual elements of the KS matrix become

6.1 The LCAO Ansatz in the The Kohn-Sham equations.

FKSµν =

∫

ηµ(~r1)

(

−1

2∇2 −

M∑

A

ZA

r1A+

∫

ρ(~r2)

r12d~r2 + VXC(~r1)

)

ην(~r1)d~r1 (55)

The first two terms describe the kinetic energy and the electron-nuclear interaction, and they areusually combined one-electron integrals

hµν =

∫

ηµ(~r1)

(

−1

2∇2 −

M∑

A

ZA

r1A

)

ην(~r1)d~r1 (56)

For the third term we need the charge density ρ which takes the following form in the LCAOscheme

ρ(~r) =

L∑

i

|ψi(~r)|2 =

N∑

i

L∑

µ

L∑

nu

cµicνiηµ(~r)ην(~r). (57)

The expansion coefficients are usually collected in the so-called density matrix P with elements

Pµν =

N∑

i

cµicνi. (58)

Thus, the Coulomb contribution in Eq. (55) can be expressed as

6.1 The LCAO Ansatz in the The Kohn-Sham equations.

Jµν =

L∑

λ

L∑

σ

Pλσ

∫ ∫

ηµ(~r1)ην(~r1)1

r12ηλ(~r2)ησ(~r2)d~r1d~r2. (59)

Up to this point, exactly the same formulae also apply in the Hartree-Fock case. The differenceis only in the exchange-correlation part. In the Kohn-Sham scheme this is represented by theintegral

V XCµν =

∫

ηµ(~r1)VXC(~r1)ην(~r1)d~r1, (60)

whereas the Hartree-Fock exchange integral is given by

Kµν =

L∑

λ

L∑

σ

Pλσ

∫ ∫

ηµ(~x1)ηλ(~x1)1

r12ην(~x2)ησ(~x2)d~x1d~x2. (61)

The calculation of the L2/2 one-electron integrals contained in hµν can be fairly easily computed.The computational bottle-neck is the calcalution of the ∼ L4 two-electron integrals in theCoulomb term.

6.2 Basis sets.

• Slater-type-orbitals (STO): they seem to be the natural choice for basis functions. Theyare exponential functions that mimic the exact eigenfunctions of the hydrogen atom. Atypical STO is expressed as

ηSTO = Nrn−1 exp [−βr]Ylm(Θ, φ). (62)

Here, n corresponds to the principal quantum number, the orbital exponent is termed β andYlm are the usual spherical harmonics. Unfortunately, many-center integrals are very difficultto compute with STO basis, and they do not play a major role in quantum chemistry.

• Gaussian-type-orbitals (GTO): they are the usual choice in quantum chemistry. Theyhave the following general form

ηGTO = Nxlymzn exp [−αr] . (63)

N is a normalization factor which ensures that 〈ηµ|ηµ〉 = 1, α represents the orbitalexponent. L = l+m+n is used to clasify the GTO as s-functions (L = 0), p-functions(L = 1), etc.

• Sometimes one uses the so-called contracted Gaussian functions (CGF) basis sets, inwhich several primitive Gaussian functions are combined in a fixed linear combination:

ηCGFτ =

A∑

a

daτηGTOa . (64)

7.1 Applications in quantum chemistry.

Molecular structures: DFT gives the bond lengths of a large set of molecules with a precisionof 1-2%. The hybrid functionals have improved the LDA results.

Bond lengths for different bonding situations [A]:

Bond LDA BLYP BP86 ExperimentH-H RH−H 0.765 0.748 0.752 0.741

H3C-CH3 RC−C 1.510 1.542 1.535 1.526RC−H 1.101 1.100 1.102 1.088

HC≡CH RC−C 1.203 1.209 1.210 1.203RC−H 1.073 1.068 1.072 1.061

Vibrational frequencies: DFT predicts the vibrational frequencies of a broad range of molecules

within 5-10% accuracy.

Vibrational frequencies of a set of 122 molecules: method, rms deviations, proportion outside a10% error range and listings of problematic cases (taken from Scott and Radom, 1996).

Method RMS 10% Problematic cases (deviations larger than 100 cm−1)BP86 41 6 142(H2), 115(HF), 106(F2)B3LYP 34 6 132(HF), 125(F2), 121(H2)

7.1 Applications in quantum chemistry.

Atomization energies: the most common way of testing the performance of new functionals isthe comparison with the experimental atomization energies of well-studied sets of small molecules.These comparisons have established the following hierarchy of functionals:

LDA < GGA < hybrid functionals

The hybrid functionals are progressively approaching the desired accuracy in the atomizationenergies, and in many cases they deliver results comparable with highly sophisticated post-HFmethods.

Molecule LDA BLYP Molecule LDA BLYPCH 7 0 F2 47 18CH3 31 -2 O2 57 19CH4 44 -3 N2 32 6C2H2 50 -6 CO 37 1C2H4 86 -6 CO2 82 11

Deviations [kcal/mol] between computed atomization energies and experiment (taken from Johnson et al., 1993).

Ionization and affinity energies: the hybrid functionals can determined these energies with anaverage error of around 0.2 eV for a large variety of molecules.

7.2 Applications in solid state physics.

Example 1: DFT describes very accurately alkali metals. Below, the spin susceptibility of alkali

metals χ/χ0, where χ0 is the Pauli susceptibility of a free electron gas (Vosko et al., 1975):

Metal DFT result (LDA) ExperimentLi 2.66 2.57Na 1.62 1.65K 1.79 1.70

Metal DFT result (LDA) ExperimentRb 1.78 1.72Cs 2.20 2.24

Example 2: DFT fails to describe strongly correlated system such as heavy fermions or hightemperature superconductors.

7.3 DFT in Molecular Electronics I.

[Heurich, Cuevas, Wenzel and Schon, PRL 88, 256803 (2002).]

Three subsystems: central cluster and leads

H = HL + HR + HC + V

��

�

• Central cluster: Density functional calculation (DFT) HC =∑

i εi d†i di

εi −→ molecular levels (Kohn-Sham orbital levels)

d†i −→ molecular orbitals

(a) Basis set: LANL2DZ (relativistic core pseudopotentials)

(b) DFT calculations: B3LYP hybrid functional.

• Leads: The reservoirs are modeled as two perfect semi-infinite crystals using a tight-bindingparameterization (Papaconstantopoulos’ 86).

• Coupling: V =∑

ij vij(d†i cj + h.c.).

vij: hopping elements between the lead orbitals and the MOs of the central cluster (Lowdintransformation).

7.3 DFT in Molecular Electronics II.

Generalization of DFT to nonequilibrium: implementation in TURBOMOLE

• self-consistent determination ofthe electrostatic profile

• inclusion of the leads in the self-consitency

• use the relations:

ρ =

∫

dε[−iG<(ε)/2π]

−iG< = G(f1Γ1 + f2Γ2)G†

Eqm. Stat. Mech.

HF / DFT

NEGF

HF / DFT

2

,µ µ,1(a) (b)

ρ Σ2,Σ1

V

µ 1 µ 2

ρFρ

F, NρF,

F

[ F ]

µ1 − µ2 = eV

7.3 DFT in Molecular Electronics III.

Electrical current:

I =2e

h

∫ ∞

−∞dε Tr

{

tt†}

[f(ε− eV/2) − f(ε+ eV/2)]

Transmission matrix:

t(ε, V ) = 2 Γ1/2L (ε− eV/2) Gr

C(ε, V ) Γ1/2R (ε+ eV/2)

• Scattering rates: ΓL,R(ε) ≡ Im {ΣL,R(ε)} (Σ = self-energy)

ΣL,R(ε) = vCL,R gL,R(ε) vL,RC

• The central cluster Green functions GrC are given by:

GC(ε, V ) =[

ε1 − HC − ΣL(ε− eV/2) − ΣR(ε+ eV/2)]−1

• Dim(tt†) = ML → number of molecular orbitals in the central cluster.

Linear regime: (Ti are the eigenvalues of tt† at the Fermi level)

G =2e2

h

ML∑

i=1

Ti

7.3 DFT in Molecular Electronics IV.

• We can analyze the current in terms of conduction channels, defined as eigenfunctionsof tt†.

• Such analysis allows to quantify the contribution to the transport of every individualmolecular level.

• The channels are a linear combination of the molecular orbitals |φj〉 of the centralcluster, i.e. |c〉 =

∑

j αcj|φj〉.

Cb + ...b

+I CHANNEL > = Ca a• Ultimately, this information concerning the channels can eventually be measured using su-

perconducting electrodes or other means.

Conductance of a hydrogen molecule I.

[R.H.M. Smit et al., Nature 419, 906 (2002).]

counter supports

pushing rodflexible substrate

polyimidesacrificial layer

Aluminum film

Main observations

• The H2 bridge has a nearly perfect conductance of one quantum unit, carried by a singlechannel.

• The presence of the molecule was identified by the analysis of its vibration modes.

• The number of channels was estimated from the analysis of the conductance fluctuations.

Conductance of a hydrogen molecule II.

Simple model:

⇒ Two molecular orbitals

• bonding state at ε0 + t:

• anti-bonding state at ε0 − t:

0

0.2

0.4

0.6

0.8

1

T(ε

)

Γ = 1.0tH

Γ = 0.4tH

Γ = 0.2tH

Γ = 0.05tH

-2 -1 0 1 2(ε − ε0)/|tH|

0

0.5

1

1.5

DO

S(ε)

Approximations:

• flat DOS: ρ(ε) = ρ

• no charge transfer

Result:

• Γ = πt2ρ. Due to the large value oftH ∼ −7 eV, one would expect thecurve Γ = 0.05tH to represent therelevant situation.

Naively, H2 should be insulating!

Conductance of a hydrogen molecule III.

Making the model more realistic:

• account for proper charge transferDFT predicts ∼ 0.1 e−

⇒ EF is shifted down ∼ 6 eV

• replace the flat DOS by the properbulk Pt-Green’s function

-10 -5 0 5 10 15 20ε (eV)

0

0.2

0.4

0.6

0.8

1

Pt B

ulk

DO

S 5d6p6s

0

0.2

0.4

0.6

0.8

1

T(ε

)-10 -5 0 5 10 15 20

ε (eV)

0

0.05

0.1

DO

S(ε)

t = 0.10tH

t = 0.15tH

• estimate the coupling strength ∼ 1 − 2 eV

• the simple model shows a transmission compatible with the experiment

The charge transfer and the strong hybridization with the Pt d-band make H2 a goodconductor!

Conductance of a hydrogen molecule IV.

DFT calculations: [Cuevas, Heurich, Cuevas, Wenzel, Schon, Nanotechnology 14, 29 (2003)]

Top position

0

0.02

0.04

0.06

0.08

LDO

S (

1/eV

)

−6 −5 −4 −3 −2 −1 0 1 2 3ε (eV)

0

0.2

0.4

0.6

0.8

1

1.2

Tra

nsm

issi

on

Ttotal

T1

T2

bonding

Hollow position

−3 −2 −1 0 1 2 3ε (eV)

0

1

2

3

4

Tra

nsm

issi

on

Ttotal

T1

T2

T3

T4

T5

T6

T7

The DFT calculations confirm the conduction mechanism throughhydrogen and provide a good description of the experimental results

Conclusions.

• The density functional theory (DFT) is presently the most successfull (and also the mostpromising) approach to compute the electronic structure of matter.

• Its applicability ranges from atoms, molecules and solids to nuclei and quantum andclassical fluids.

• In its original formulation, the density functional theory provides the ground state prop-erties of a system, and the electron density plays a key role.

• An example: chemistry. DFT predicts a great variety of molecular properties: molecularstructures, vibrational frequencies, atomization energies, ionization energies, electric andmagnetic properties, reaction paths, etc.

• The original density functional theoy has been generalized to deal with many differentsituations: spin polarized systems, multicomponent systems such as nuclei and electronhole droplets, free energy at finite temperatures, superconductors with electronic pairingmechanisms, relativistic electrons, time-dependent phenomena and excited states, bosons,molecular dynamics, etc.