Density Functionals: Basic DFT Theory

Sergio AragonSan Francisco State University

CalTech PASI January 4-16, 2004

DFT Publications

Outline• Wavefunction methods: Hartree-Fock• To know ground state E, what else do we

need to know? Electron Density.• Basic DFT Theorems• Kohn-Sham Equations• Exchange Correlation Functionals• Excited States

Molecular Schroedinger Equation

H (x1,x2,…,xN; R1,…,RNn) = E (x1,x2,…,xN; R1,…,RNn)

H = Te +Ven +Vee +Vnn

Te =

N

em 1

22

2

e nN N

k k

k

RreZ

1 1 0

2

||4 Ven =

e eN N

rre

1 1 0

2

||4 Vee =

n nN

j

N

k kj

jk

RR

eZZ

1 1 0

2

||4

Vnn =

Born-Oppenheimer Aproximation

Hartree-Fock Theory

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

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

!1

1

221

11211

NNN

N

N

xx

xxxxx

N

HF =

Vnnrr

h

N N

N

]|1||1|[21

||

121 1 12

1

EHF=<HF|H|HF

nN

k k

k

rZh

1

2

21

)}({ xVh HF

Fictitious system of non-interacting electrons.

Correlation Energy• Exchange Correlation: Fermions of the same spin cannot move

independently of one another. is zero for two Fermions of the same spin located at the same point, regardless of their electric charge.

• HF Theory achieves exact Exchange correlation in a non-local manner (the Exchange operator acting on a given orbital requires information everywhere in space of that orbital.)

• Coulomb Correlation: Electrons repel each other and therefore do not move independently even if they have the different spin state.

• HF Theory misses 100% of the Coulomb Correlation. Chemical accuracy requires post-Hartree Fock methods such as MP2 or CI.

Information: Basic Problem

N = 42 electrons

Number of terms in :

N! = 1.4 1051

Number of Cartesian dimensions in which lives:

3N = 126

is an extremely complex object and contains more information that we need!

Benzene

Can we make do with less??

The one-electron density!

NNnN xdxddRRxxx

...|......| 212

121 (r1) = N

This object is merely 3-dimensional, regardless of the number of electrons or the size of the molecule.

Electron Density of Water

Hohenberg-Kohn Theorems

Hohenberg, P. & Kohn, W. “The inhomogeneous electron gas”, Phys. Rev. 136, B864 (1964).

Theorem 1: The external potential, Ven is determined, up to an additive constant, by the electron density. Since (r1) determines the number of electrons N, then the density also determines the ground state wavefunction and all other electronic properties of the system.

Theorem 2: There exists a Variational Principle for the density.The correct density is the one that produces the minimum energy.

HK Proof IProve by contradiction. Assume that there exist two different external potentials V=Ven +Vnn and V’=Ven’ +Vnn’ which both give the same electron density (r1). Then we have two Hamiltonians, and H’ with the same ground state density and different and ’. Now we use the variational principle, taking ’ as a trial function for the H Hamiltonian, to obtain:

E0 < < ’|H|’> = < ’|H'|’> + < ’|(H-H')|’>= E0' + < ’|(V-V')|’>

In addition, we can take the as a trial function for the H’ Hamiltonian, to obtain:E0’ < < |H'|> = < |H|> + < |(H’-H)|>

= E0 + < |(V’-V)|>Now we recognize that the expectation value of the difference in the external potentials differ only in sign because we assumed that the electron density is the same. When we add the two equations, we then obtain the contradiction:

E0 + E0’ < E0’ + E0

Thus we conclude there there exists a unique map between the external potential V and the ground state density. This implies that all the energies, including the total energy, is a functional of the density. We write this as E = E[]. The density determines the Hamiltonian, and thereby, the wavefunction.

HK Proof IISuppose we have a trial density ’. Then this density defines its own wavefunction ’, and the expectation value of the true Hamiltonian

satisfies the variational principle:

<’’Vee[E0[0] = < 0|H|0>.

Thus, the correct density is the one that produces the minimum energy.

rdrVr

)()( ’]

What is a Functional?

We give an intuitive definition by comparing the situation to the more familiar notion of function. A function f: R R is a map (or assignment) between one number of the set R (say the real numbers) to another number of the set R. We usually denote this assignment by y = f(x), and we agree that to each value of x there corresponds only one value of y.

On the other hand, a functional is a map between a set of functions and a set of num-bers; F: {f(x)} R, and we denote this by F[f(x)] = y. A simple example is the definite integration operator:

b

a

dxxggF )(][

The specification of the integrable function g(x) produces a number defined in terms of the constants a and b.

Thomas-Fermi-Dirac ModelThe uniform electron gas consists of N=Z electrons in a volume which isuniformily positive so as to keep the entire system electrically neutral. For this system, Thomas and Fermi (1927) computed the kinetic energy, and Dirac (1930) computed the Exchange correlation energy:

ETFD[] = T[] + Ven[] + Vee[] + Vx[] =

rdrCrdrdr

rrrd

rrZrdrC XF

)(

)()(21)()( 3/4

2112

213/5

Coulomb correlation for the electron gas was computed by Monte Carlo methods by Ceperly & Alder, Phys. Rev. Lett. 45, 566 (1980).

TFD Atomic Density

TFD

HF

1.0 Sqrt[r] a.u. 2.0

r2r

Kinetic Energy poorly represented!

Kohn-Sham Reference System

N

jjjjSST

1

2 ||21][

Introduce a fictitious reference system of non-interacting particles with density S. Then, a single Slater determinant gives the exact solution to the fictitious system, and,

2

1

|),(|)(

N

jjS rr

E[] = TS[] + J[] + Ven+Vnn + Exc[]

Adjust the potential that these particles move under so that they have a density = S and write:

Exc[] = T[] - TS[] + Eee[] - J[].

rdrdr

rr

112

21 )()(21

All unknowns here!

Kohn-Sham Equations

N

j

N

iijij

N

jjjj Exc

rE

1 1 121

2 ][|1|21||

21][

Nn

a

Nn

b ab

baj

aj

Nn

a

N

ja rr 1 111 1

|1|

jjjKS

Nn

a aj

a rVrZ

)}(21{ 1

1

21

)()(

)( 1212

21 rVxcrd

rr

rVKS

Variation principle

EXACT EQUATION!

HF and DFT Compared

• HF is never exact. DFT could be exact.• HF formally scales as N4 while DFT scales as N3.• HF misses Coulomb correlation. DFT includes both

Coulomb & Exchange.• Both DFT & HF have kinetic energy error.• HF can be systematically improved.• DFT can be improved mostly by guessing!• Scaling in DFT is independent of Vxc improvements.• HF scaling increases dramatically as improved.• DFT performs as HF//MP2 or better, with less effort.• DFT can do transition metals, HF can’t.

HF/KS Orbitals Compared

• Koopman’s theorem applies to all HF orbital energies.

• Orbitals yield approx. wavefunction.

• Orbital energies lack correlation, probably not physical.

• HF orbitals do not yield correct density.

• The DFT maximum filled orbital energy is minus the Ionization energy.

• Orbitals do not yield wavefunction.

• Orbital energies are probably physical.

• KS orbitals yield the correct density.

DFTHF

Exchange-Correlation Functionals

• Uniform electron gas basis: LDA• Separate the Exchange and Coulomb

correlation parts.• Add terms depending explicitly on the

gradient: GGA• Make hybrid functionals containing a

percentage of HF exchange: B3LYP

The Local Density Approximation

The LDA Approximation (Koch & Holthausen)

LDA: Quantitatively

rdrrE xcLDAxc

))(()(][

))(())(())(( rrr cxxc

3/1)(343))((

rrx

Coulomb correlation interpolated from Ceperly & Alder.

Generalized Gradient Approximation

rdrsFEE LDAx

CGAx

)()( 3/4

3/4

||

s

)(61 1

sSinhss

F B

Coulomb Correlation: Ec

LYP : Lee, Parr &Yang (1988)PW91: Perdew & Wang (1991)P86: Perdew (1986)

Exchange Correlation: Ex

15/164286 2.014296.11 xxxF

3/12 )24( sx

Perdew (1986) : Becke (1988):

Becke & Perdew/Wang: BPW91Becke & Lee/Parr/Yang: BLYP

Hybrid Functionals

LSDc

LYPc

Bcxc

LSDx

LYPBxc EccEbEaEEaE )1()1( 8803

Add a percentage of HF Exchange since the systematic errors of HF and DFT are in opposite directions.

The famous B3LYP functional:

The constants a =.2, b=.8, c=.7 are obtained from fits to a reference set of atoms & compounds. Semi-empirical?

Excited States

• The variational principle will always yield the lowest energy of a manifold of states of the same symmetry. Any excited states which have a different symmetry from the ground state can also be found.

• In general, the application of Time Dependent DFT is the most promising for excited states. This technique uses the idea of linear response and evaluates the excited state in terms of the ground state density. TDDFT is implemented in G98.

H2 Potential Curve