TheoryFrom Quantum Mechanics to Density Functional Theory

[based on Chapter 1, Sholl & Steckel (but at a more advanced level)]

• Quantum mechanics (~1920s)

• Hartree & Hartree-Fock-Slater methods (~1930s-1950s)

• Density functional theory (~1960s)

Quantum theory

• Wave-particle duality of light (“wave”) and electrons (“particle”)

• Many quantities are “quantized” (e.g., energy, momentum, conductivity, magnetic moment, etc.)

• For “matter waves”: Using only three pieces of information (electronic charge, electronic mass, Planck’s constant), the properties of atoms, molecules and solids can be accurately determined (in principle)!

Quantum theory – Light as particles• Max Planck (~1900): energy of electromagnetic (EM) waves can

take on only discrete values: E = nħ– Why? To fix the “ultraviolet catastrophe”

– Classically, EM energy density, ~ 2avg = 2(kT)

– But experimental results could be recovered only if energy of a mode is an integer multiple of ħ as

Classical (~2kT)

experimental

avg (n)e n / kT

n

e n / kT

n

e / kT 1

from density of states

from equipartition theorem

The ultraviolet catastrophe

Quantum theory – Light as particles

• Einstein (1905): photoelectric effect– No matter how intense light is, if < c no photoelectrons

– No matter how low the intensity is, if > c, photoelectrons result

– Light must come in packets (E = nħ)

• Compton scattering (1923): establishes that photons have momentum!

– Scattering of x-rays of a single frequency by electrons in a graphite target resulted in scattered x-rays

– This made sense only if the energy and the momentum were conserved, with the momentum given by p = h/ = ħk (k = 2/, with being the wavelength)

• By now, it is accepted that waves may display particle features …

Quantum theory – Electrons as waves

• Rutherford (~1911): Experiments indicate that atoms are composed of positively charged nuclei surrounded by a cloud of “orbiting” electrons. But,– Orbiting (or accelerating charge radiates energy

electrons should spiral into nucleus all of matter should be unstable!)

– Spectroscopy results of H (Rydberg states) indicated that energy of an electron in H could only be -13.6/n2 eV (n = 1,2,3,…)

Quantum theory – Electrons as waves

• Bohr (~1913):– Postulates “stationary states” or “orbits”, allowed only if electron’s

angular momentum L is quantized by ħ, i.e., L = nħ implies that E = -13.6/n2 eV

– Proof: • centripetal force on electron with mass m and charge e, orbiting with velocity v at

radius r is balanced by electrostatic attraction between electron and nucleus mv2/r = e2/(40r2) v = sqrt(e2/(40mr))

• Total energy at any radius, E = 0.5mv2 - e2/(40r) = -e2/(80r)

• L = nħ mvr = nħ sqrt(e2mr/(40)) = nħ allowed orbit radius, r = 40n2ħ2/(e2m) = a0n2 (this defines the Bohr radius a0 = 0.529 Å)

• Finally, E = -e2/(80r) = -(e4m/(802h2)).(1/n2) = -13.6/n2 eV

– The only non-classical concept introduced (without justification): L = nħ

Quantum theory – Electrons as waves

• de Broglie (~1923): Justification: L = nħ is equivalent to n = 2r (i.e., circumference is integer multiple of wavelength) if = h/p (i.e., if we can “assign” a wavelength to a particle as per the Compton analysis for waves)!– Proof: n = 2r n(h/(mv)) = 2r n(h/2) = mvr nħ = L

• It all fits, if we assume that electrons are waves!

Quantum theory – Electrons as wavesThe Schrodinger equation: the jewel of the crown

• Schrodinger (~1925-1926): writes down “wave equation” for any single particle that obeys the new quantum rules (not just an electron)

• A “proof”, while remembering: E = ħ & p = h/ = ħk– For a free electron “wave” with a wave function Ψ(x,t) = e i(kx-t), energy is purely kinetic

– Thus, E = p2/(2m) ħ = ħ2k2/(2m)

– A wave equation that will give this result for the choice of e i(kx-t) as the wave function is

• Schrodinger then “generalizes” his equation for a bound particle

it(x, t)

2

2m

2

x 2 (x,t)

it(x, t)

2

2m

2

x 2 V (x)

(x, t)

K.E. P.E.Hamiltonian operator

The Schrodinger equation• In 3-d, the time-dependent Schrodinger equation is

• Writing Ψ(x,y,z,t) = ψ(x,y,z)w(t), we get the time-independent Schrodinger equation

• Note that E is the total energy that we seek, and

Ψ(x,y,z,t) = ψ(x,y,z)e-iEt/ħ

it(x,y,z, t)

2

2m

2

x 2 2

y 2 2

z2

V (x, y,z)

(x, y,z, t)

2

2m

2

x 2 2

y 2 2

z2

V (x, y,z)

(x,y,z) E(x, y,z)

Hamiltonian, H

The Schrodinger equation

• An eigenvalue problem– Has infinite number of solutions, with the solutions being Ei

and i

– The solution corresponding to the lowest Ei is the ground state

– Ei is a scalar while i is a vector

– The is are orthonormal, i.e., Int{i(r)j(r)d3r} = ij

– If H is hermitian, Ei are all real (although i are complex)

– Can be cast as a differential equation (Schrodinger) or a matrix equation (Heisenberg)

– ||2 is interpreted as a probability density, or charge density

EH

Applications of 1-particle Schrodinger equation

• Initial applications– Hydrogen atom, Harmonic oscillator, Particle in a box

• The hydrogen atom problem

2

2m

2

x 2 2

y 2 2

z2

V (x, y,z)

nlm (x,y,z) Enlmnlm (x, y,z)

2 1

r2

r

r2 dr

1

r2 sin

sin

d

1

r2 sin22

2

e2

4 0

r

Solutions: Enlm = -13.6/n2 eV; ψnlm(r,θ,ϕ) = Rn(r)Ylm(θ,ϕ)

http://www.falstad.com/qmatom/http://panda.unm.edu/Courses/Finley/P262/Hydrogen/WaveFcns.html

The many-particle Schrodinger equation

• The N-electron, M-nuclei Schrodinger (eigenvalue) equation:

),...,,,,...,,(),...,,,,...,,( 21212121 MNMN RRRrrrERRRrrr

M

I

N

i iI

IN

i

N

Ij ji

M

I

M

IJ JI

JIN

ii

M

II

I rR

eZ

rr

e

RR

eZZ

mM 1 1

2

1

2

1

2

1

22

1

22

2

1

2

1

22

The total energy that we seekThe N-electron, M-nuclei wave function

The N-electron, M-nuclei Hamiltonian

Nuclear kinetic energy

Electronic kinetic energy

Nuclear-nuclear repulsion

Electron-electron repulsion

Electron-nuclear attraction

• The problem is completely parameter-free, but formidable!– Cannot be solved analytically when N > 1– Too many variables – for a 100 atom Pt cluster, the wave function is a

function of 23,000 variables!!!

The Born-Oppenheimer approximation• Electronic mass (m) is ~1/1800 times that of a nucleon mass (MI)• Hence, nuclear degrees of freedom may be factored out• For a fixed configuration of nuclei, nuclear kinetic energy is zero

and nuclear-nuclear repulsion is a constant; thus

),...,,(),...,,( 2121 NelecNelec rrrErrrH

M

I

N

i iI

IN

i

N

Ij ji

N

iielec rR

eZ

rr

e

mH

1 1

2

1

2

1

22

2

1

2

M

I

M

IJ JI

JIelec RR

eZZEE

1

2

2

1

Electronic eigenvalue problem is still difficult to solve! Can this be done numerically though? That is, what if we chose a known functional form for in terms of a set of adjustable parameters, and figure out a way of determining these parameters? In comes the variational theorem

The variational theorem• Casts the electronic eigenvalue problem into a minimization problem• Lets introduce the Dirac notation

• Note that the above eigenvalue equation has infinite solutions: E0, E1, E2, … & correspondingly 0, 1, 2, …

• Our goal is to find the ground state (i.e., the lowest energy state)• Variational theorem

– choose any normalized function containing adjustable parameters, and determine the parameters that minimize <|Helec|>

– The absolute minimum of <|Helec|> will occur when = 0

– Note that E0 = <0|Helec|0> thus, strategy available to solve our problem!

NNN rdrdrdrrrrrr 32

31

32121

* ...),...,,(),...,,(...

HrdrdrdrrrHrrr NNN 32

31

32121

* ...),...,,(),...,,(...

),...,,( 21 Nrrr elecelec EH

The Hartree method• The first attempt to solve the Schrodinger equation for atoms other than hydrogen

(i.e., containing more than one electron) was by Hartree & Hartree (father & son)• Hartree suggested that• Thus )()...()(),...,,( 221121 NNNH rrrrrr

),...,,(),...,,( 2121 NHHNHelec rrrErrrH

N

i

N

Ij

N

ielec

N

i

N

Ij ji

N

i

M

I

N

i iI

Iielec

jivihH

rr

e

rR

eZ

mH

11

1

2

1 1 1

22

2

),(2

1)(

2

1

2

• Apply variational theorem: minimize <H|Helec|H> subject to the constraint <H|H> = 1 (normalization)

• This converts the many-electron Schrodinger equation to a set of 1-electron equations [proved by Slater] which are much easier to solve

Involves only one electron

Involves two electrons

The Hartree method (contd.)

N

i

N

Ijij

N

iiiHelecHH

ijji

ji

jjiijijjiijjii

NNNNNHH

iiiiiii

NNNNNHH

N

i

N

IjHH

N

iHHHelecHH

JEHE

Thus

Jrdrdrr

rnrnerdrdrrjivrr

rdrdrdrrrjivrrrjiv

Erdrihr

rdrdrdrrrihrrrih

jivihHE

11

33233**

32

31

32211

*2

*21

*1

3*

32

31

32211

*2

*21

*1

11

2

1

,

)()()()(),()()(

...)()...()(),()()...()(...),(

)()()(

...)()...()()()()...()(...)(

),(2

1)(

We are not done yet (!), as we still need to know all the s to determine EH

(We need to use the variational theorem)

Electronic energy when electrons do not interact with each other

Classical electrostatic interaction between electrons i and j

ni(ri) = i*(ri)i (ri)

The Hartree method (contd.)• Minimize EH with respect to each of the s, say *i(ri), subject to the

constraint <i|i> = 1

€

∂∂ i

*(ri)E H − ε j ϕ j

* (rj )ϕ j (rj )d3rj∫j

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥= 0

h(i)ϕ i (ri) +j ≠ i

∑ ϕ j* (rj )v(i, j)ϕ j (rj )∫ ϕ i (ri)d

3rj −ε iϕ i (ri) ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥= 0

h(i) +j ≠ i

∑ ϕ j* (rj )v(i, j)ϕ j (rj )∫ d3rj

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ϕ i (ri) = ε iϕ i (ri)

• Resubstituting h(i) and v(i,j), we get the 1-electron Hartree equation

)()()(

232

22

2

iiiiiij

j

ji

jj

I iI

Ii rrrd

rr

rne

rR

eZ

m

),...,,(),...,,( 2121 NHHNHelec rrrErrrH

Compare!

The “Hartree” potential:The electrostatic potential seen by an electron i due to all other electrons (note the summation over j)

Lagrange multipliers

Hartree Hamiltonian, hH

nj(rj) = j*(rj)j (rj)

The Hatree method (contd.)• The Hartree 1-electron equation needs to be solved “self-consistently” to

obtain the solutions (i.e., i and i) for all the electrons! Why?• Because the Hartree potential is written in terms of the solutions• Thus, Hartree “guessed” the solutions, used these guesses to compute

the Hartree potential, after which they solved the equation to get new solutions, used these to calculate the new Hartree potential, and so on, till the input and output solutions were close to each other self-consistency

• Finally, the total energy is given by

€

E = E H +1

2

ZI ZJe2

RI − RJJ ≠I

∑I

∑ = E ii +i

∑ 1

2Jij

j ≠ i

∑i

∑ +1

2

ZI ZJe2

RI − RJJ ≠I

∑I

∑

But,

ε i = ϕ i hH ϕ i = E ii + Jijj ≠ i

∑

Thus,

E = ε i −1

2i

∑ Jijj ≠ i

∑i

∑ +1

2

ZI ZJe2

RI − RJJ ≠I

∑I

∑

Equation A

Equation B

The Hartree prescription for the total energy

)()()(

232

22

2

iiiiiij

j

ji

jj

I iI

Ii rrrd

rr

rne

rR

eZ

m

Guess i(ri) for all the electronsRemember that i(ri) is a 1-electron wave function

Is new i(ri) close to old i(ri) ?

Calculate total energy

I IJ JI

JI

jiij

ii RR

eZZJE

2

, 2

1

2

1

Yes

No

ni(ri) = i*(ri)i (ri)

Solve!

The Hartree-Fock-Slater method• The Hartree method has deficiencies: the wave function does not obey

the Pauli exclusion principle

• The Pauli principle can be stated in many ways– No 2 electrons can be in the same state– 2 electrons with the same spin cannot be in the same spatial orbital– Exchange of 2 electrons will result in a sign change of the total wave function– (we need to explicitly consider spin, but we are going to get by without it!)

(r1,r2,...,rN ) (r2,r1,...,rN )

But,

H (r1,r2,...,rN ) H (r2,r1,...,rN )

as

1(r1)2(r2)...N (rN ) 1(r2)2(r1)...N (rN )

The Slater determinant

• Exchanging 2 rows (or columns) of a determinant results in a sign change of the value of the determinant!

• Using this wave function instead of the Hartree product wave function, and going through the same exercise results in the Hartree-Fock-Slater equation

)()(),()()(ˆ

)()(ˆ)(),()()(

3*

3*

ijjjijjij

iij

iiiiijjjjjjij

rrdrjivrrK

rrKrdrjivrih

The exchange operator(quantum mechanical electron-electron interaction)

Hartree electrostatic interaction(classical electron-electron interaction)

Position & spin variable

Spin orbital

€

(x1,x2,..., xN ) =1

N!

χ1(x1) χ2(x1) . . . χN (x1)

χ1(x2) χ2(x2) . . . χN (x2)

. . . . . .

. . . . . .

. . . . . .

χ1(xN ) χ2(xN ) . . . χN (xN )

€

χ1(x1) = ϕ1(r1)α 1(σ 1)

€

x1 ≡ r1,σ 1{ }

Beyond Hartree-Fock-Slater• The Hartree-Fock-Slater method still has deficiencies as it allows 2

electrons of unlike spin to be at the same spatial location (i.e., it includes the “exchange” interaction exactly, but completely ignores “correlation”)

• This may be overcome be using a linear combination of Slater determinants (“Configuration Interaction”), or by perturbation treatments, which have been shown to provide extremely accurate results (and hence are considered the “gold standard” of electronic structure computations even today)

• A note about scaling: If N is the number of electrons, the Hartree, Hartree-Fock-Slater, and more advanced methods scale roughly as N3, N4, and N6, respectively, in terms of computational time

• Density functional theory (DFT) is an alternative approach, that includes both “exchange” and “correlation” in an approximate way (in practice)

• DFT scales as N3 or better, and comprises the best trade-off between accuracy and practicality at the present time

Density functional theory (DFT)[Hohenberg-Kohn (1965)]

• DFT is a reformulation of Schrodinger’s quantum mechanics

• In Schrodinger’s quantum mechanics, observables are functionals of ψ(r1,r2,…,rN). For e.g., E = <ψ(r1,r2,…,rN)|H|ψ(r1,r2,…,rN)>

• Note: A functional is a function of a function; e.g., E[f(r)] is a functional of f(r), but f(r) is a function of r

– i.e., for different choices of the functional form of f(r), E will take on different values!

• In DFT, the total energy of a system (or any property, including ψ(r1,r2,…,rN)) is a unique functional of the total electronic charge density, n(r) [Theorem 1]

• The correct n(r) minimizes the total energy equivalent to the variational theorem [Theorem 2]

Density functional theory[Hohenberg-Kohn (1965)]

€

Eelec = T[n(r)] +e2

2

n(r)n(r')

r − r'd3rd 3r'+ v(r)n(r)d3r + Exc[n(r)]∫∫∫

The total electronic kinetic energy

External potential (e.g., due to nuclei = ΣIe2ZI/|RI-r|)

The exchange-correlation energy (the only non-classical term, or the sum total of our ignorance!)

Theorem 1: n(r) v(r) H all properties; Thus:

Theorem 2: The correct n(r) minimizes Eelec; Thus:

δEelec/δn(r) = 0 correct ground state Eelec

total electron density

So what? The wave function is a function of 3N variables, but the charge density is a function of only 3 variable! However, the functional form of Exc[n(r)] is not specified

Density functional theory[Kohn-Sham (1965)]

• A unique one-to-one mapping is established between a system containing N interacting electrons with charge density n(r) moving in an external potential, and a fictitious system of N non-interacting electrons also with the same change density n(r)

• What is so great about this? – The problem of N non-interacting electrons is solvable!– Each of the N non-interacting electrons exists in an “external

effective potential” that contain the information about the interactions that have been removed!

Density functional theory[Kohn-Sham (1965)]

N interacting electrons,

with charge density n(r)

v

N noninteracting electrons, with charge density

n(r)

veff = v + vH + vxc

Unique mapping

Theorem 1 v[n(r)], vxc[n(r)]Potential due to nuclei, external fields, etc.

Difficult problem Solvable problem

Classical electron-electron interaction (“Hartree”)

δExc/δn(r)

The Kohn-Sham 1-electron equation

)(

)](['

'

)'()()(

)()()(2

32

22

rn

rnErd

rr

rnervrv

rrrvm

xceff

iiieff

Nuclear potential, electromagnetic potential, etc.

Non-interacting electron wave function

Non-interacting electron energy

Since our electrons do not interact with each other, we may write a Schrodinger equation for each one!

vH(r) vxc(r)

The above 1-electron equation is EXACT, if we know vxc(r)Since this is not the case, vxc is approximated (herein lies the division between DFT and quantum chemistry methods …)

Kohn-Sham Hamiltonian, hKS

The total energy

• Similar to the Hartree treatment, the Kohn-Sham equations are solved “self-consistently”– Why? Because veff(r) depends on n(r) which depends on the ψis (which

are the solutions)!

• The resulting self-consistent is and ψis may be used to compute the total energy as follows

rdrnvrnrnrrddrr

rnrneE

RR

eZZEE

xcxc

N

iielec

I IJ JI

JIelec

3332

1

2

)]([)]([)(''

)'()(

2

1

2

1

<ψi|hKS|ψi> rdrnrnrnENote xcxc3)]([)()]([: Equation C

The Hohenberg-Kohn-Sham prescription for the total energy

)()()(2

22

rrrvm iiieff

Guess ψi(r) for all the electronsRemember that ψi(r) is a 1-electron wave function

Is new ψi(r) close to old ψi(r) ?

Calculate total energy

Yes

No

Solve!

energyrepulsionnuclearEE

rdrnvrnrnrrddrr

rnrneE

elec

xcxc

N

iielec

3332

1

)]([)]([)(''

)'()(

2

1

occ

ii rrn

2)(2)(

Density functional theory

• Still parameter-free, but has a few acceptable approximations (next lecture)• DFT is versatile: in principle, it can be used to study any atom, molecule,

liquid, or solid (metals, semiconductors, insulators, polymers, etc.), at any level of dimensionality (0-d, 1-d, 2-d and 3-d)

)()()(2

22

rrrvm iiieff

),...,,,,...,,(),...,,,,...,,( 21212121 MNMN RRRrrrERRRrrr

Density Functional Theory (DFT)[W. Kohn, Chemistry Nobel Prize, 1999]

Energy can be obtained from n(r), or from i and i (i labels electrons)

1-electron wave function(function of 3 variables!)

1-electron energy(band structure energy)

The “average” potential seen by electron i

occ

ii rrn

2)(2)(