Density Functional Theory - I partDFT-1.pdf · Density Functional Theory - 1 Basics Electron...

Density Functional Theory - 1

Density Functional Theory - I part

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Overview

BasicsMotivationWave function methodsElectron densityAncient history

FoundationsH-K theorem IFunctionalsH-K theorem II


Notation

~r position vector~x position vector + spin labelΦ(~x1, ~x2, . . . , ~xN) wave function for N electronχr (~x) r -th spin orbitalφr (~r) r -th molecular orbitalϕr (~r) r -th basis (atomic) orbitalH electronic Hamiltonian


Basics

Motivation

I Density-functional theory is nowadays the most popular quantummechanical approach to study molecules and materials .

I applications relevant for fields traditionally considered moredistant from quantum mechanics, such as biology andmineralogy , are beginning to appear

I phenomena likeI superconductivityI atoms in the focus of strong laser pulsesI relativistic effects in heavy elements and in atomic nucleiI classical liquidsI magnetic properties of materials

can be addressed with formalisms based on DFT.


Basics

Motivation

A. Becke, J. Chem. Phys. 140, 18A301 (2014)


Basics

Motivation


Basics

Wave function methods

Where we areNon relativistically, the electronic wave function of an (isolated) system of Nelectrons interacting with M nuclei is calculated from Schrödinger’s equation(from the complete Schrödinger’s equation with the Born-Oppenheimerapproximation), in atomic units

HΦ(~x1, ~x2, . . . , ~xN) = EΦ(~x1, ~x2, . . . , ~xN)

H =12

N∑i=1

∇2i +

12

N,N∑i 6=j

1rij−

N∑i=1

M∑a=1

Za

ria= T + U + V

where we define

the kinetic energy operator T = 12

∑Ni=1∇

2i

the electron-electron potential energy operator U = 12

∑N,Ni 6=j

1rij

the electron-nuclei potential energy operator

V = −N∑

i=1

M∑a=1

Za

ria=

N∑i=1

u(~ri )


Basics


Remember

I ~xi = (~ri , si ), where ~ri is the position of the i-th electron, si is the spinstate (si = −1/2, 1/2)

I 〈f |g〉 =∫

f ∗(~x1~x2 . . . ~xN)g(~x1~x2 . . . ~xN)d~x1d~x2 . . . d~xN ;I∫

f (~x)d~x =∫

f (~r ,−1/2)d~r +∫

f (~r , 1/2)d~rI Φ can be (and usually is) written in the general form of a sum of Slater

determinants

Φ(~x1, ~x2, . . . , ~xN) =∞∑

k=1

ck Φk (~x1, ~x2, . . . , ~xN)

Φk (~x1, ~x2, . . . , ~xN) =

∣∣∣∣∣∣∣∣∣χ1(~x1) χ2(~x1) . . . χN(~x1)χ1(~x2) χ2(~x2) . . . χN(~x2)

......

. . ....

χ1(~xN) χ2(~xN) . . . χN(~xN)

∣∣∣∣∣∣∣∣∣I the spin-orbitals are one-electron functions χj (~xi ) = ϕ(~ri )s(si ) with

s(1/2) = α or s(−1/2) = β


Basics


Wave-function methods"Since its formal inception in 1964–1965, Kohn-Sham density-functionaltheory (KS-DFT) has become the most popular electronic structure method incomputational physics and chemistry." (A. Becke)

1. In the HF method an exact equation needs to be solved ...

HΦ = EΦ

2. ... to find the wave function Φ(~x1, ~x2, . . . , ~xN) (WF ) ...3. ... using an approximate approach based on single electron

functions, i.e. molecular orbitals φ(~ri ) ...4. ... which can be upgraded till a numerically exact (in principle)

solution is found (post-HF methods)5. the role of the external potential is fundamental: for a given N, to

change V means to change univocally the wave function andexpectation values of properties

V solve SE−→ Φevaluate properties−→ 〈Φ|O|Φ〉


Basics


What’s bothering us?

ProblemsI Φ(~x1, ~x2, . . . , ~xN) is complicatedI Φ(~x1, ~x2, . . . , ~xN) is difficult to interpret physicallyI Φ(~x1, ~x2, . . . , ~xN) has ’strange’ dimensions [Φ] = [L]−3N/2

I the HF method scales as N4

I the HF is difficult to employ in periodic infinite systems (crystals)I therefore post HF methods, e.g. CI, often are required (excited

states, computational spectroscopy)

Change of perspective : a simpler unknown (the electron density) anda more versatile and computationally efficient approach to find it

Price(s) paid : i) the wave-function has more information and the ii)combination of HF + post-HF is, in principle, exact


Basics


A test

I Suppose we can ignore spin and let’s consider a Φ for N electrona real 3N-coordinates function;

I for each coordinates let’s define Φ on a grid of 20 points: wewould need 203N points ...

I ... and only 203 for ρI for N = 10 (NH3 molecule): 2024 ≈ 1.6× 1032 vs. 203 = 8000


Basics


M.G. Medvedev et al., Science 355, 49–52 (2017)


Basics

Electron density

Defining the electron density

The (total) electron density ρ(~x) is a a real function depending on ~x

ρ(~x) = N∫ ∣∣Φ(~x , ~x2, . . . , ~xN)

∣∣2 d~x2 . . . d~xN (1)

N.B. The integral of ρ is the number of electrons∫ρ(~x)d~x = N

∫ ∣∣Φ(~x , ~x2, . . . , ~xN)∣∣2 d~xd~x2 . . . d~xN = N

since the overall integrated probability is 1 (normalization condition forthe WF).


Basics

Electron density

Information content of the electron densityThe electron density displays a cusp at the position of each nucleus,represented in the Hamiltonian by an electron-nucleus Coulomb term

H =12

N∑i=1

∇2i +

12

N,N∑i 6=j

1rij−

N∑i=1

M∑a=1

Za

ria

and the shape of the cusp is related to the atomic number (charge) of thenucleus, according to Kato’s theorem : if ρ(~r) is the average of the densityalong all angles around point ~r then

limr→ra

(∂

∂r+ 2Za

)ρ(~r) = 0


Basics

Ancient history

Early approximations

Let us consider a generic molecular Hamiltonian H = T + VNe + Vee

I this corresponds to a classical distribution of negative charges(electrons) and positive charges (nuclei)

I can we use ρ to determine the energy of the system?

The Thomas-Fermi model is an early attempt to address the question[Thomas (1927), Fermi (1927), Block (1929), Dirac (1929)]

I energy is separable into kinetic and potential components.I use only the electron density to determine the molecular energy,

considering the system as classicalI introduce corrections as needed, shake (not stir)


Basics

Ancient history

Thomas-Fermi model

The potential energy VNe is easy (and exact even for quantumsystems)

VNe[ρ(~x)] =M∑

a=1

∫Za

|~r −~ra|ρ(~x)d~x

The potential energy Vee is easy, too (but valid only for classicalsystems)

V classicalee [ρ(~x)] =

12

∫1

r12ρ(~x1)ρ(~x2)d~x1d~x2

The kinetic energy is difficult: not enough information! How are theparticles moving?


Basics

Ancient history

Thomas-Fermi modelJellium : infinite number of electrons moving in an infinite volume of spacewith uniformly distributed positive charge (uniform electron gas model)

Thomas and Fermi (1927), using statistical mechanics (non interactivefermions in a box) derive the kinetic energy for jellium

T ueg[ρ(~x)] =310

(3π2)2/3∫ρ5/3(~x)d~x (2)

The repulsion potential energy can be corrected to try to take into accountexchange and correlation, using a hole function

VQMee [ρ(~x)] = 〈Φ|Vee|Φ〉 ≈

12

∫1

r12ρ(~x1)ρ(~x2)d~x1d~x2+

12

∫1

r12ρ(~x1)h(~r1,~r2)d~x1d~x2


Basics

Ancient history

Thomas-Fermi model

Notes on Eq. (2)

I The second term on the r.h.s. of (2) corrects for the errors in thefirst term (the classical expression).

I The hole function h is centred on the position of electron 1 and isevaluated from there as a function of the remaining spatialcoordinates defining ~r2

I The value and form of h varies as a function of ~r2 for a givenvalue of ~r1

I For a one electron system the l.h.s. of(2) must be zero.I In the one-electron case h is −ρ; in the many-electron case the

exact form of h i not knownI h both corrects for self-interaction error (SIE) and accounts for

exchange and correlation energy in a many-electron system.


Basics

Ancient history

Thomas-Fermi model

I Slater (1951): one of the consequences of the Pauli principle isthat the Fermi exchange hole is larger than the correlation hole ...

I ... this is a consequence of the Pauli exclusion principle , whichcauses a reduced probability of finding two electrons of the samespin close to one another

I Exchange corrections to classical interelectronic repulsion arelarger than correlation corrections (one or two order ofmagnitude).

I Slater decided to ignore correlation corrections , and simplify theexchange corrections.

I Slater exchange energy

Ex =9α8

(3π

)1/3 ∫ρ4/3(~x)d~x


Basics

Ancient history

Xα

I Slater assumed α = 1I Many early workers saw fit to treat alpha as an empirical value.I HF calculations can be made evaluating Ex using Slater’s

formula instead of the standard (exact) approach Xα orHartree-Fock-Slater method).

I Empirical analysis in a variety of systems suggested thatα = 3/4 provides more accurate results

I at this stage DFT (and HFS) models gave generally large errorsin molecular calculations, but was useful in solid state physics .


Foundations

A rational basis for DFT

To make DFT a "... legitimate quantum mechanical methodology ..."(C.J. Cramer) one needs

1. to demonstrate that ρ is uniquely determined by the externalpotential (i.e. the ensemble of nuclei making the molecularstructure under study) V I Hohenberg-Kohn Theorem

2. to show that a variational criterion based on ρ can be established,as was the case of Φ V II Hohenberg-Kohn Theorem

3. to implement the variational criterion in an efficient computationalstrategy V Kohn-Sham Self-consistent Field Method


Foundations

H-K theorem I

Hohenberg-Kohn theorem

I In 1964 Hohenberg and Kohn show that the total ground-state electrondensity ρ0 determines the ground-state properties of an N-electronsystem .

I ρ0 is the fundamental “variable” in electronic structure theory and theground-state wave function Φ0 can be considered, in principle,superfluous.

I the proof of this statement is obtained by showing that ρ0 is uniquelyassociated to the external potential v of a N-electron system ...

I ... i.e. the sum of potential interactions with all the nuclei for eachelectron)

I hence ρ0 is uniquely associated to a Hamiltonian H, hence to aground-state wave function Φ0, which determines all the properties ofthe system (in the ground-state)

I the fact that the ground-state is not degenerate is crucial in thederivation

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).


Foundations

H-K theorem I


1. Suppose that two different external potentials va and vb (i.e. twodifferent arrangements of nuclei) have the same ground-state density ρ0

for the N-electron system2. to each external potential we can associate a ground-state wave

function and its associated eigenvalue

va ↔ Ha ↔ E0,a,Φ0,a : HaΦ0,a = E0,aΦ0,a

vb ↔ Hb ↔ E0,b,Φ0,b : HbΦ0,b = E0,bΦ0,b

N.B. Φ0,a 6= Φ0,b by Hp.3. from the variational theorem

E0,a < 〈Φ0,b|Ha|Φ0,b〉

and we can write

E0,a < 〈Φ0,b|(Ha − Hb + Hb)|Φ0,b〉< 〈Φ0,b|(Ha − Hb)|Φ0,b〉+ 〈Φ0,b|Hb|Φ0,b〉= 〈Φ0,b|(va − vb)|Φ0,b〉+ E0,b


Foundations

H-K theorem I


1. The first term in the last equation can be written in terms of ρ0, since va

and vb are sums of one-electron functions

va =N∑

i=1

ua(~ri )

2. thus we obtain (making the same considerations starting from E0,b)

E0,a <

∫[ua(~r)− ub(~r)]ρ0(~x)d~x + E0,b

E0,b <

∫[ub(~r)− ua(~r)]ρ0(~x)d~x + E0,a

3. adding together the two inequalities we get

E0,a+E0,b <

∫[ua(~r)−ub(~r)]ρ0(~x)d~x+E0,b+

∫[ub(~r)−ua(~r)]ρ0(~x)d~x+E0,a

i.e.E0,a + E0,b < E0,b + E0,a

this is absurd: the original Hp. (va 6= vb, i.e. Ha 6= Hb) must be wrong .


Foundations

H-K theorem I


I Since the ground-state density ρ0 determines uniquely H, it alsoimplicitly determines all the excited-states properties as well

I what about the densities of excited states?I the fact that in the ground state Φ0 is non-degenerate, and so

therefore is ρ0, is indispensable to establish the variationaltheorem

I it is possible to extend the same conclusions to the lowestenergy non degenerate excited state within each irreduciblerepresentation of the molecular point group

O. Gunnarsson and B.L. Lundqvist, Phys. Rev. B, 13, 4274


Foundations

Functionals

To say that the total ground-state electron density ρ0 determines theground-state properties of an N-electron system can be stated, moreprecisely, as

Φ0 is a functional of ρ0, i.e. Φ0 = Φ0[ρ0](~x1, ~x2, . . . , ~xN)

So we need to talk about functionals and functionals derivatives ...


Foundations

Functionals

In mathematical analysis a functional associates to a function a realnumber (R) or sometimes a complex number (C).

For instance the total number of electrons is associated to ρ0 via thefunctional relation

N =

∫ρ0(~x)d~x

To say that Φ0 = Φ0[ρ0](~x1, ~x2, . . . , ~xN), means that the form of Φ0 (inevery point ~x1, ~x2, . . . , ~xN ) depends from the function ρ0 as a whole.

A functional variation is the variation of a functional due to a variationof its argument.


Foundations

Functionals

For instance, consider a function f (x); the variation of the function is (to 1storder)

f (x + dx) = f (x) +dfdx

dx + O(dx2)

Consider now a functional F = F [f ] of f (x). The variation of F is (to 1st order)

F [f + δf ] = F [f ] +

∫δFδfδf (x)dx + O(δf 2)

I The first-order coefficient (or ‘functional slope’) δFδf is the functional

derivative .I Specific rules exist to evaluate functional derivatives (calculus of

variations ). For instance if F [f ] =∫

g(f , f ′, f ′′, . . . , x)dx with f = f (x)

δFδf

=∂g∂f− d

dx∂g∂f ′

+d2

dx2∂g∂f ′′

+ . . .


Foundations

H-K theorem II


Recall that the total electron density ground state ρ0(~x) is a functionalof Φ0

ρ0(~x) = N∫ ∣∣Φ0(~x , ~x2, . . . , ~xN)

∣∣2 d~x2 . . . d~xN

HK theorem in essence, states that the last equation can be inverted,i.e. that Φ0 is a functional of ρ0

Φ0(~x1, ~x2, . . . , ~xN) = Φ0[ρ0(~x)] (3)

Eq. (3) is the basis of DFT and is usually called, expecially in thequantum chemistry community the first Hohenberg-Kohn theorem .


Foundations

H-K theorem II


Consider now a generic observable, represented by an operator O: theexpectation value in the ground state is a functional of Φ0; therefore it is afunctional of ρ0

O0 = 〈Φ0[ρ0]|O|Φ0[ρ0]〉 = O0[ρ0] (4)

The most relevant observable (to a chemist) is the ground state energy

E0 = 〈Φ0[ρ0]|H|Φ0[ρ0]〉 = E0[ρ0]

Consider a generic Φ 6= Φ0, where Φ = Φ[ρ] and ρ 6= ρ0; the expectationvalue is a functional of ρ

E = 〈Φ|H|Φ〉 = E [ρ]

But the variational principle states that E0 is always smaller that E , i.e.

E0[ρ0] ≤ E [ρ] (5)

Therefore the functional E [ρ] has an absolute minimum for ρ = ρ0. Eq. (5) isusually called the second Hohenberg-Kohn theorem .


Foundations

H-K theorem II

I The energy functional to be minimized is

E0[ρ] = T [ρ] + Vee[ρ] + VNe[ρ]

VNe[ρ] =

∫u(~r)ρ(~x)d~x

T [ρ] + Vee[ρ] is universal while VNe[ρ] is specific and known sinceit is the expectation value of the external potential

VNe =N∑

i=1

M∑a=1

Za

ria=

N∑i=1

u(~ri )↔ u(~ri ) = −M∑

a=1

Za

ria

I when minimizing E0[ρ] with respect to ρ this constraint must hold

N =

∫ρ(~x)d~x


Foundations

H-K theorem II

RepresentabilityGiven a generic function ρ(~x) one says that ρ is

N-representable if it can be obtained from an antisymmetric wave functionΦ(~x1, ~x2, . . . , ~xN) according to Eq. (1)

v -representable if, being N-representable, is the ground state density of aN-electron system with external potential defined by u(~r)

I Why it is relevant? If a numerical procedure to find a minimum of E0

would give a certain solution ρ which is not N-representable, it would notbe a physically acceptable solution.

I Today, we know that any non-negative function can be obtained from anantisymmetric wave function Φ(~x1, ~x2, . . . , ~xN) according to Eq. (1), i.e. itis N-representable

I v -representability is more difficult: the HK theorem only guarantees thatthere cannot be more than one potential for each density, but does notexclude the possibility that there is less than one, i.e., zero, potentialscapable of producing that density.

J.E. Harriman, Phys. Rev. A 24, 680 (1964).

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