Short Course on Density Functional Theory and Applications

I. Basics of Time-independent DFT

Quantum Theory ProjectDept. of Physics and Dept. of Chemistry

trickey@qtp.ufl.edu

Samuel B. Trickey©Sept. 2008

Scientific Acknowledgments� DFT: Valentin Karasiev, Brian Weiner, Frank Harris, Randy Jones, Alberto Vela, Klaus Capelle, Mel Levy, John Perdew� GTO: Jonathan Boettger, Joe Callaway, Notker Rösch, John W. Mintmire, Jack Sabin, Uwe Birkenheuer, Richard Mathar, Ashley Alford, Juan Torras Costa, Wuming Zhu, Jin-Zhong Wu� FLAPW : Karlheinz Schwarz, Peter Blaha, Peter Sorantin, Claudia Ambrosch-Draxl, David Singh, Eric Wimmer� QTP:

Funding Acknowledgments: U.S. National Science Foundation

[Earlier – U.S. Army Research Office, IBM Corp. ]

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49th Sanibel Symposium -

February 26 – March 3, 2009; St. Simons Island Georgia

Graduate and Undergraduate Student research posters are encouraged

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Graduate and Undergraduate Student research posters are encouragedvia unique 2 minute oral abstacts; poster sessions are notin parallel withany other Symposium activity.

http://www.qtp.ufl.edu/sanibel

Quantum Mechanics, Condensed Matter, Materials, and Chemistry

There can be no atomic shell structure in classical mechanics, so the periodic table is itself powerful testimony that QM determines chemistry, hence determines material properties. Adapted from David Singh

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The many-electron problem in QM is difficult. What is “many”?Here’s a table of generally solvable problems:

Classical mechanics

0-body

1-body

Quantum mechanics

0-body

1-body

Quantum field theory

0-body

1-body

Marriage & Family

0-body

1-body

Many-electron Quantum Mechanics is Challenging

1-body

2-body

3-body

1-body

2-body

3-body

1-body

2-body

3-body

1-body

2-body ?

3-body

Credit: So Hirata

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• The time-independent Schrödinger equation & Hamiltonian for the Ne-electron problem with N fixed nuclei (Hartree atomic units):

Many-electron Quantum Mechanics is Challenging (cont’d.)

{ } ( ) { }( ) { } { }( )

{ } ( )

( ) ( )

1 0 1 1 0 1 10;

,21 1

2 211 ,

12

1

ˆ , , , , ; , , , ;

1ˆ ,

: ,

e e e e e

e e e

e

e e

N N N N N

N N N NI

N ii i I i ji I i j

N N

i i ji i j

Z

h g

σ σ σ σ

= ≠

= ≠

Ψ = Ψ

= − ∇ − +− −

= +

∑ ∑ ∑

∑ ∑

R R

R

r r r r R r r R

r rr R r r

r r r

… … …

…

H E

H

Sum of 1- & 2-body Hamiltonians

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� Quantum chemistry (Hartree atomic units and traditional units)vs. materials physics units (Rydberg, Hartree, & traditional units)� Hartree & traditional :

2

1

1E 27.2116 eV 1 au.=0.5292 Å

1One-electron KE: r2

electron electron

Hartree

m q

d φ φ

= = ==

− ∇∫

ℏ

� Rydberg & traditional units:

2

2 / 2 1

1 E 13.6058 eV 1 au.=0.5292 Å

One-electron KE: r

electron electron

Rydberg

m q

d φ φ

= = ==

− ∇∫

ℏ

Nuclear positions (Born-Oppenheimer approximation); suppressed hereafter unless directly relevant.

For a normalized ground state, the energy is therefore

( ) ( )( ) ( )

*0 0 1 0 1 1

*0 1 0 1 1

, , , ,

, , , , 1

e e e

e e e

N N N

N N N

H d d

d d

= Ψ Ψ

Ψ Ψ =

∫

∫

r r r r r r

r r r r r r

⌢… … …

… … …

E

Since the state-function is anti-symmetric under particle exchange and the Hamiltonian is a symmetric sum of 1- and 2-body interactions, we end up doing integrals over all but one or two of the coordinates just to eliminate them, thus (introducing spin here):

Many-electron Quantum Mechanics is Challenging (cont’d.)

2-particle reduced density matrix “2-RDM”

( ) ( )

( ) ( ) ( ) ( )( )

1 1;0 01 1

2 2

0

0

(2)0 1 1 1 1 1 2 1 2 1 2 1 2

(2) *1 2 1 2 0 1 1 2 2 3 3 0 1 1 2 2 3 3 3

*1 1 0 1 1 2 2 3 3

( ) | ( , ) |

1| : , , , , , , , ,

2

| : , , , ,

N e N e ee e

N

x xx x

x x

e eN N N

e

h x x dx g x x x x dx dx

N Nx x x x dx dx

x x N

γ

σ σ σ σ σ σ σ σ

γ σ σ σ

′=Ψ Ψ′= ′ =

Ψ

Ψ

′ ′ ′= + Γ

− ′ ′′ ′ ′ ′Γ = Ψ Ψ

′′ ′= Ψ

∫ ∫

∫

r r r

r r r r r r r r

r r r r

… … …

…

E

( ) ( )0 1 1 2 2 3 3, 2 3, , ,e e e ee

N N N Ndx dx dxσ σ σ σ σΨ∫ r r r r… …

1-particle reduced density matrix “1-RDM”

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Sum over spins

Brute force doesn’t work! Suppose Ne = 10. Then we have to do a 3 x 8 = 24- dimensional integral without coordinates 1, 2 followed by a 6-D integral involving the operator coordinates 1,2 (or some equivalent strategy).

Many-electron Quantum Mechanics is Challenging (cont’d.)

( ) ( ) ( ) ( )( ) ( ) ( )

( )

0

0

(2) *1 2 1 2 0 1 1 2 2 3 3 0 1 1 2 2 3 3, 3

*1 1 0 1 1 2 2 3 3 0 1 1 2 2 3 3, 2 3

(2)1 2 1 2 2

1| : , , , , , , ,

2

| : , , , , , , ,

2|

1

N e e e ee

N e e e ee

e eN N N N

e N N N N

e

N Nx x x x dx dx

x x N dx dx dx

x x x x dxN

σ σ σ σ σ σ σ σ

γ σ σ σ σ σ σ σ σ

Ψ

Ψ

− ′ ′′ ′ ′ ′Γ = Ψ Ψ

′′ ′= Ψ Ψ

′= Γ−

∫

∫

∫

r r r r r r r r

r r r r r r r r

… … …

… … …

involving the operator coordinates 1,2 (or some equivalent strategy).

Guessing a form for the 2-RDM and using it in the variational principle doesn’t work either. This is the notorious N-representability problem. How can we know for certain that a guessed 2-RDM actually came from a properly anti-symmetric Ne particle wave function? The answer is knownbut using it is essentially as intractable as using the many-electron wave function.

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Simplest physically acceptable approximate wave function is Hartree-Fock. The outline is to assume a trial variational wave-function of determinantal form:

( ) ( ) ( ) ( )1,2, , 1 2e a b z eN Nψ ϕ ϕ ϕ ≈ … ⋯A

Forced separation of variables = “mean field”Fully coupled electrons

Some assumed functional form

Challenges of the QM Many-Electron Problem (continued)

( ) ( ) ( ) ( )1,2, , 1 2e a b z eN A Nϕ ϕ ϕ Ψ ≈ ɶ ɶ ɶ… ⋯

Orbital optimization

Variationally optimized

SCF solution

Credit: So HirataAnother shorthand notation:1 11 ,σ→ r

Antisymmetrizer

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Hartree-Fock: Example of Orbital Optimization by Functional DifferentiationThe variational principle

is equivalent to the Schrödinger equation in that the S.E. arises from requiring the first variation of Etrial to be zero

*1

0 *1

e

e

trial trial N

trial

trial trial N

H d d

d d

Ψ Ψ≤ =

Ψ Ψ∫∫

r r

r r

⌢…

…E E

*

*1

* 0

( ) 0e

trial trial

trial trial N

trial

trial

H dr dr

H

δδδ δ

δ

Ψ= Ψ =

Ψ − Ψ =

⇒ Ψ = Ψ∫

⌢ � �…

⌢

EE

E

E Lagrange Multiplier

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( ){ } ( ) ( ){ }*0 1 1

ˆ, , ,

0 Variational minimum (extremum)

e eHF HF N HF N

HF

l

d d d

l

ψ ϕ σ ψ ϕ σ σ

δδϕ

≤ =

= ∀ ⇒

∏∫ ii

r r r r r r… …E E H

E

Because of the variational principle, the H-F energy is above the true ground state energy. The task is to make the difference as small as possible by picking optimum spin-orbitals. This leads to a similar functional derivative:

0 0 0H⇒ Ψ = Ψ⌢

E Lagrange Multiplier

Functional Derivatives• A functional F [f ] prescribes how to get a number, F , from a function f.For example, the variational principle prescribes how to get the trial energy from the trial wave function.

Analogy from ordinary calculus: differential of a function of several variables

( ) ( )f f dδδ δδ = ∫ r r rFF

• A functional derivative gives the change in the numerical value F that depends linearly on a change in f , thus

1 2( , , )L

L j

FdF x x x dx

x

∂=∂∑…

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( ) ( ) ( )2 2, , , : , , ,n n n d f n n n′ ′ ′ ′ ∇ ∇ = ∇ ∇ ∫ r r r r… …F

1 21

( , , )L jj j

dF x x x dxx=

=∂∑…

• The following expression will be useful in DFT. Given a functional of a function and its derivatives

( )( )

( )( )

( )( )

2

22

, , ,n n n f f f

n n n n

δδ

∇ ∇ ∂ ∂ ∂ = − ∇ + ∇ − + ∂ ∂∇ ∂∇

r r r

r r r

…i …

F

its functional derivative is

Approximate versus exact:

( ) ( ) ( ) ( )1,2, , 1 2e a b z eN Nψ ϕ ϕ ϕ ≈ ɶ ɶ ɶ… ⋯A

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

0

1 1

1,2, , 1 2

2

e a b z e

b z eA

N c N

c N

ϕ ϕ ϕ

ϕ ϕ

ϕ

ϕ

ϕϕ

Ψ =

+

+

ɶ ɶ ɶ… ⋯

ɶ ɶ⋯

ɶ

ɶ

ɶ⋯ɶ

A

A

HF is approximate

All “excited” Slater

Full CI is exact

Challenges of the QM Many-Electron Problem (continued)

( ) ( ) ( )2 1 2a z eBc Nϕ ϕϕ +

+

ɶ ɶ⋯ɶ

…

AAll “excited” Slater determinants →→→→“configuration interaction”

Question 1: is this expansion exact? YESQuestion 2: is this expansion rapidly converging? NO

Credit: So HirataJ.C. Slater (by SBT)J.C. Slater (by SBT)J.C. Slater (by SBT)J.C. Slater (by SBT)

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( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

1,2, , (55%) 1 2

(30%) 2

( %

1

10 2) 1

A

B

e a b z e

b z e

a z e

N N

N

N

ϕ ϕ ϕ

ϕ ϕ

ϕ ϕ

ϕ

ϕ

Ψ =

+

+ +

ɶ ɶ ɶ… ⋯

ɶ ɶ⋯

ɶ ɶ⋯ …

ɶ

ɶ

A

A

A

“Non-dynamical” correlation

[coefficients arbitrary for sake of illustration]

There is another complication! Quantum chemistry has a strange terminology for an important insight. There are two kinds of correlation contribution. • One comes from the failure of a single Slater determinant to describe the ground state well:

Challenges of the QM Many-Electron Problem (continued)

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

1,2, , (99%) 1 2

(0.6%) 2

(0.3%

1

) 21

A

e a b z

z

eB

e

b e

a z

N N

N

N

ϕ ϕ ϕ

ϕϕ

ϕ

ϕ

ϕ ϕ

Ψ =

+

+ +

ɶ ɶ ɶ…

ɶ

ɶ

⋯

ɶ ɶ⋯

ɶ ɶ⋯ …

A

A

A

“Dynamical” correlation

[coefficients arbitrary for sake of illustration]

for sake of illustration]

• The other kind of correlation contribution comes from electron positions not handled properly in the mean field:

Credit: So HirataCredit: So HirataCredit: So HirataCredit: So Hirata

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Reminder about wave-function-based methods: Many-electron ground state is approximated by a trial wave-function:

{ }( ); ,σ σ σΨ R r r r⋯

Limitations of Typical Wave-function Methods

Even today, many quantum chemists have a strong preference for wave-function-based, so-called “ab initio” methods.

Reminder: “ ab initio” is an archaic Latin phrase meaning “expensive” (J.W.D. Connolly, about 1973)

{ }( )1 1 2 2; ,e eN Nσ σ σΨ R r r r⋯

Hartree Fock: is a single determinant of one-particle spin orbitalsdetermined by variational principle. Gives bad physics for extended systems: Fermi-level pathology for metals, excessively large band gaps for insulators.

Ψ

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Limitations of Typical Wave-function Methods (contd.)

Various wave-function methods much-used in quantum chemistry are not feasible in surfaces, solids, immense polymers, and biomolecules because of immense computational cost. They include•Configuration Interaction – variational principle expanded in a linear combination of determinants of spin orbitals. Expensive, not size-consistent in a finite basis.• Many-body Perturbation Theory (“ Møller-PlessetPert. Theory”) –

{ }( ) ˆ1 1 2 2; , cc

e eN NT

Referenceeσ σ σ ψΨ =R r r r⋯

• Many-body Perturbation Theory (“ Møller-PlessetPert. Theory”) –•doesn’t converge well order by order (MBPT-2 about as good as MBPT-3 or -4)• Coupled Cluster Theory- Superb accuracy, limited in practice to closed shells (the “multi-reference problem” ) and about 100 electrons:

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The Density (Properly, the Electron Number Density)

Can we use the density to find the ground state? Seems implausiblein the face of the N-representability problem for the 2-RDM. But thereis a physical argument (due to E.B. Wilson) in support of the idea. Some preliminaries:

( ) ( )

( ) ( ) ( )*1 1 1 1 2 2 3 3 1 1 2 2 3 3, 2 3

| (chemists use symbol )

| : , , , , , , ,N e e e ee

e N N N N

n

x x N dx dx dx

ψ ψσ

ψ

γ σ σ ρ

γ ψ σ σ σ σ ψ σ σ σ σ

σ

=

′′ ′=

=

∑

∫

r r r

r r r r r r r r… … …

The [electron number] density is the “diagonal element” of the 1-RDM:

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: ,i i ix σ= rNote that the probability of finding an electron at position r is ( )1

eN nψ− r

and that ( ) ed n Nψ =∫ r r

The spin density is The total density is

( ) ( )( ) ( ) ( )

|n

n n n

ψ ψ

ψ ψ ψ

σ γ σ σα β

=

= +

r r r

r r r

E.B. Wilson’s Argument1. We know we are dealing with electrons (back to the periodic table argument), so we know the form of the Hamiltonian.

2. If we know the electron number density at all points in space, we can getthe number of electrons:

3. In the neighborhood of nucleus I , the Kato cusp condition gives:

( ) ( )21 2 In Z O− +r r r∼

( ) ed n N=∫ r r

so the density n(r) identifies the location and charge of every nucleus.

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4. Therefore, knowledge of the density n(r) is equivalent to knowing allthe system-dependent constants in the many-electron Hamiltonian

{ } ( ) ( ) ( ),

21 1 12 2 21

1 , 1

1ˆ , : ,e e e e e

e

N N N N N NI

N i i i ji i I i j i i ji I i j

Zh g

= ≠ = ≠

= − ∇ − + = +− −∑ ∑ ∑ ∑ ∑R r r r r r

r R r r…H

5. Since the many-electron Hamiltonian determines the ground state, it follows that the density determines the ground state.

Hohenberg-Kohn Theorems – consider the Ne -electron Hamiltonian, which includes an external potential (for us, the nuclear-electron attraction)

{ } ( ),

21 12 21

1 ,

1ˆ ,

ˆ ˆˆ ˆ ˆ ˆ:

e e e

e

N N N NI

N ii i j i I i Ii j

ee ext ee Nuc electr

Z

= ≠

−

= − ∇ + −−−

= + + ≡ + +

∑ ∑ ∑R r rr Rr r

…H

�T V V T V V

HK-I: “A given ground state density n0 (r) determines the ground state wave function and hence all the ground state properties of an Ne -electron system.”

Rudiments of Density Functional Theory

e

system.”Original Proof: by contradiction [ P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)Modern Proof: Levy-Lieb constrained search (sequential application of the Variational Principle density by density)[M. Levy, Proc. Natl. Acad. Sci. USA 76, 6062 (1979); L. Lieb, Internat. J. Quantum Chem. 24, 243 (1983)]

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HK-I Proof by Contradiction

0ˆ ˆ,ext ext n′ ֏V V

1. Consider a non-degenerate ground state. Suppose that there are two external potentials that yield the same ground-state density:

2. Then by the variational principle

( ) ( ) ( )0 0 0 0 0 0 0

0 0 0 0

ˆ ˆ

ˆ ˆext ext

ext ext ext extd n v v

′ ′ ′ ′ ′ ′= Ψ Ψ < Ψ Ψ = Ψ + − Ψ

′ ′= + Ψ − Ψ = + − ∫ r r r r

⌢ ⌢ ⌢E H H H V V

E V V E

3. But the argument can be done reversing primed and unprimed quantities.

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3. But the argument can be done reversing primed and unprimed quantities.This gives ( ) ( ) ( )0 0 ext extd n v v′ ′= + − ∫ r r r rE E

4. Adding these last two results gives the contradiction0 0 0 0′ ′+ < +E E E E

5. Therefore the ground state density determines the external potential and hence, the ground state wave function.

Remark: this proof does not address the possibility of a density n which isnot associated with any vext (the v-representability problem)

Constrained Search [Parr and Yang; Fig. 3.1(b), p. 59]

Suppose there is an elementary school with six classrooms, 1st through 6th Grade.How could one find the tallest pupil and know his/her identity?– The “Rayleigh-Ritz variational approach”: gather all the pupils (e.g., in the cafeteria) and measure each. Keep a record of the greatest height found so far, along with the name and grade of that pupil. Except for “degeneracies” (students of the same height), that record never would contain more than one name.– The “constrained search approach”: search each grade for the tallest student in that room, then combine the results into a single list and search that for the tallest member. If the list has more than 6 entries, then those sets of students who have the same height belong to equivalence classes.

credit N. Rösch

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HK-I Proof by Constrained Search

2. Minimize this functional over all the Ne electron states that give n0

[ ]( ) ( )

0

0

ˆ ˆmin

|

ee

n

ψ

ψ

ψ ψ ψ

γ

= +

∋ =r r r

T VE

3. Then at most ψψψψ0 is a normalized linear combination of ground states, since

1. Consider the positive operator ˆee+T V and form

[ ] ˆeeψ ψ ψ= +T VE

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3. Then at most ψψψψ0 is a normalized linear combination of ground states, since the external potential contribution to the total energy depends only on the density: ( ) ( )0ext extd n v= ∫ r r rE

4. Therefore the ground state density determines the ground state wave function.Remarks: No v-representability problem, no restriction to non-degenerateground states. We have not shown that, subject to mild conditions, every densityis associated with at least one Ne electron state. In fact, there are infinitely many such states for each density (Harriman, Phys. Rev. A 24, 680 (1981)).

HK-II: “For an Ne –electron system with an external potential vext(r), thereexists a universal (i.e., independent of vext ) functional F[n] with thefollowing properties:

Original Proof: Essentially by announcement; the paper assumes that HK-I holds for non-ground-state densities.

Modern Proof: Levy-Lieb Constrained Search

Rudiments of DFT (continued)

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

0

0 0( )

min physically acceptable ( )

ext

ext ext

v ext

v vn r

E n F n d n v E

E n E E n n

= + ≥

= = ∀∫ r r r

r�

Modern Proof: Levy-Lieb Constrained SearchForm

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[ ] ; ;ˆ ˆˆ ˆ: min ee min n ee min n

nF n

ψψ ψ ψ ψ= + ≡ +

֏

T V T V

[ ] ( ) ( ) [ ]0 0 0 ; ;

ˆ ˆˆ ˆ ˆ ˆ

:ext

ee ext min n ee ext min n

ext v

E

F n d n v E n

ψ ψ= Ψ + + Ψ ≤ + +

= + =∫ r r r

V V V VT T

Then

which is the first piece of the theorem.

Now apply the variational principle again

HK-II proof (continued)

0 0

0 0

0 0 0 ; ;

0 0 ; ;

ˆ ˆˆ ˆ ˆ ˆ

ˆ ˆˆ ˆ

ee ext min n ee ext min n

ee min n ee min n

E ψ ψ

ψ ψ

= Ψ + + Ψ ≤ + +

⇒ Ψ + Ψ ≤ +

T T

T T

V V V V

V V

But the definition of ψψψψmin;n means that0 0; ; 0 0

ˆ ˆˆ ˆmin n ee min n eeψ ψ+ ≤ Ψ + ΨT TV V

Taken together, these give bracketing inequalities

0 0 00 0 ; ; 0 0ˆ ˆ ˆˆ ˆ ˆ

ee min n ee min n eeF nψ ψ Ψ + Ψ ≤ + = ≤ Ψ + Ψ T T TV V V

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0 0 0

0 0 0 0 0ˆ ˆ [ ]

extee vF n E n E

⇒ = Ψ + Ψ ⇒ = T V

Remark: This F[n] is NOT the same mathematical object as the oneoriginally defined by HK. In particular, there is no v-representability issue with regard to the variation over n nor restriction to non-degenerate ground state. This functional does fulfill the role of the one defined by HK.

Bijective mapping

The original proof by contradiction of HK-I establishes a bijective mappingbetween n(r) and vext(r). The Levy-Lieb constrained search seems to losethat but, at least formally, the ground state part can be recovered. Assume that n0 and Ψ0 been found as the minimum of

0 0 0 0 0 0

0 0 0 0 0 0 0 0

[ ]and [ ]

ˆˆ ˆ[ ] [ ] [ ] ( ) [ ]ext ee

n E E n

n E n n n

= Ψ =

⇒ = Ψ − + Ψ

Ψ

ΨV T V

( ) ( )[ ] extF n d n v+ ∫ r r rThen

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0 0 0 0 0 0 0 0

0 0 0 00 0

[ ] [ ] [ ] ( ) [ ]

1 ˆˆ ˆ[ ] ( ) [ ][ ]

ext ee

ext ee

n E n n n

E n nn

⇒ = Ψ − + Ψ

= − +

Ψ

ΨΨ

V T V

V T V