Lecture 14. Density Functional Theory (DFT)

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Lecture 14. Density Functional Theory (DFT)

• References
• Ratner Ch.11.5, Engel Ch.15.6.3
• Lewars Ch.7, Cramer Ch.8, Jensen Ch.6
• A birds-eye view of density functional theory
  (Klaus Capelle)
• http//arxiv.org/PS_cache/cond-mat/pdf/0211/02114
  43.pdf
• Nobel lecture Electronic structure of matter
  wave functions and density functionals (Walter
  Kohn 1998)
• http//prola.aps.org/pdf/RMP/v71/i5/p1253_1

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Postulate 1 of quantum mechanics

• The state of a quantum mechanical system is
  completely specified by the wavefunction or state
  function that depends on the coordinates
  of the particle(s) and on time.
• The probability density to find the particle in
  the volume element located at r at
  time t is given by .
  (Born interpretation)
• The wavefunction must be single-valued,
  continuous, finite, and normalized (the
  probability of find it somewhere is 1).
• lt??gt

Probability density
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Wave Function vs. Electron Density

• Probability density of finding any electron
  within a volume element dr1

N electrons are indistinguishable
Probability density of finding electron 1 with
arbitrary spin within the volume element dr1
while the N-1 electrons have arbitrary positions
and spin

• Wavefunction
• Function of 3N variables (r1, r2, , rN)
• Not observable

• Function of three spatial variables r
• Observable (measured by diffraction)
• Possible to extend to spin-dependent electron
  density

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Electron Density as the Basic Variable

• Wavefunction as the center quantity
• Cannot be probed experimentally
• Depends on 4N (3N spatial, N spin) variables for
  N-electron system
• Can we replace the wavefunction by a simpler
  quantity?
• Electron density ?(r) as the center quantity
• Depends on 3 spatial variables independent of the
  system size

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Density suffices.

• Unique definition of the molecular system
  (through Schrödinger equation)
• (N, RA, ZA) ? Hamiltonian operator ?
  wavefunction ? properties
• N number of electrons
• RA nuclear positions
• ZA nuclear charges
• Unique definition of the molecular system
  (through density, too)
• (N, RA, ZA) ? electron density ? properties
• ?(r) has maxima (cusps) at RA

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Electron Density as the Basic Variable 1st
Attempt Thomas-Fermi model (1927)

• Kinetic energy based on the uniform electron gas
  (Coarse approximation)
• Classical expression for nuclear-electron and
  electron-electron interaction
• (Exchange-correlation completely neglected)
• The energy is given completely in terms of the
  electron density ?(r).
• The first example of density functional for
  energy.
• No recourse to the wavefunction.

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Slaters Approximation of HF Exchange X? method
(1951)

• Approximation to the non-local exchange
  contribution of the HF scheme
• Interaction between the charge density and the
  Fermi hole (same spin)
• Simple approximation to the Fermi hole
  (spherically symmetric)
• Exchange energy expressed as a density functional
• Semi-empirical parameter ? (2/31) introduced to
  improve the quality

(from uniform electron gas)
X? or Hartree-Fock-Slater (HFS) method
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Thomas-Fermi-Dirac Model

• Combinations of the above two
• Thomas-Fermi model for kinetic classical
  Coulomb contributions
• Modified X? model for exchange contribution
• Pure density functionals
• NOT very successful in chemical application

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The Hohenberg-Kohn Theorems (1964)

• Reference
• P. Hohenberg and W. Kohn, Phys. Rev. (1964) 136,
  B864
• http//prola.aps.org/pdf/PR/v136/i3B/pB864_1

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Hohenberg-Kohn Theorem 1 (1964) Proof of
Existence
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Hohenberg-Kohn Theorem 1 (1964) Proof of
Existence
The ground state electron density ?(r) in fact
uniquely determines the external potential Vext
and thus the Hamilton operator H and thus all
the properties of the system.
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Proof
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Hohenberg-Kohn Functional

• Since the complete ground state energy is a
  functional of the ground state electron density,
  so must be its individual components.

system-independent, i.e. independent of
(N,RA,ZA)
Hohenberg-Kohn functional
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Hohenberg-Kohn Functional Holy Grail of DFT
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Finding Unknown Functional Major Challenge in DFT

• The explicit form of the functionals lies
  completely in the dark.
• Finding explicit forms for the unknown
  functionals represent the major challenge in
  DFT.

Kinetic energy
Non-classical contribution Self-interaction,
exchange, correlation
Classical coulomb interaction
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Hohenberg-Kohn Theorem 2 (1964) Variational
Principle

• FHK? delivers the lowest energy if and only
  if the input density
• is the true ground state density ?0.
• Limited only to the ground state energy. No
  excited state information!

Proof
from the variational principle of wavefunction
theory
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Variational Principle in DFT Levys Constrained
Search (1979)

• Use the variation principle in wavefunction
  theory (Chapter 1)
• Do it in two separate steps
• Search over the subset of all the antisymmetric
  wavefunctions ?X that yield a particular density
  ?X upon quadrature ? Identify ?Xmin which
  delivers the lowest energy EX for the given
  density ?X
• Search over all densities ?? (?A,B,,X,) ?
  Identify the density ?? for which the
  wavefunction ??min from (Step 1) delivers the
  lowest energy of all.

Search over all allowed, antisymmetric N-electron
wavefunction
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Variational Principle in DFT

• Determined simply by the density
• Independent of the wavefunction
• The same for all the wavefunctions
• integrating to a particular density

Universal functional
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HK Theorem in Real Life? Pragmatic Point of View

• The variational principle applies to the exact
  functional only.
• The true functional is not available.
• We use an approximation for F?.
• The variational principle in DFT does not hold
  any more in real life.
• The energies obtained from an approximate
  density functional theory can be lower than the
  exact ones!
• Offers no solution to practical considerations.
  Only of theoretical value.

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The Kohn-Sham Approach (1965)

• Reference
• W. Kohn and L.J. Sham, Phys. Rev. (1965) 140,
  A1133
• http//prola.aps.org/abstract/PR/v140/i4A/pA1133_
  1

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Implement Hohenberg-Kohn Theorems Thomas-Fermi?

• Hohenberg-Kohn theorems
• Hohenberg-Kohn universal functional
• Thomas-Fermi(-Dirac) model for kinetic energy
  fails miserably
• No molecular system is stable with respect to
  its fragments!

Classical coulomb known
Explicit forms remain a mystery.
(from uniform electron gas)
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Hartree-Fock, a Single-Particle Approach Better
than TF
Section IV.C, Nobel lecture Electronic structure
of matter wave functions and density
functionals (Walter Kohn 1998)
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Better Model for the Kinetic Energy Orbitals
Non-Interacting Reference System (HF for DFT?)
Section IV.C, Nobel lecture Electronic structure
of matter wave functions and density
functionals (Walter Kohn 1998)

• A single Slater determinant constructed from N
  spin orbitals (HF scheme)
• Approximation to the true N-electron wavefunction
• Exact wavefunction of a fictitious system of N
  non-interacting electrons (fermions) under an
  effective potential VHF
• The kinetic energy is exactly expressed as
• Use this expression in order to compute the major
  fraction of the kinetic energy of the interacting
  system at hand

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Non-Interacting Reference System Kohn-Sham
Orbital

• Hamiltonian with an effective local potential Vs
  (no e-e interaction)
• The ground state wavefunction (Slater
  determinant)
• One-electron Kohn-Sham orbitals determined by
• with the one-electron Kohn-Sham operator
• satisfy

Vs chosen to satisfy the density condition
Ground state density of the real target system
of interacting electrons
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The Kohn-Sham One-Electron Equations

• If we are not able to accurately determine the
  kinetic energy through an explicit functional, we
  should be a bit less ambitious and
• concentrate on computing as much as we can of the
  true kinetic energy exactly and then
• deal with the remainder in an approximate manner.
• Non-interacting reference system with the same
  density as the real one
• Exchange-Correlation energy (Junkyard of all
  the unknowns)

Kohn-Sham orbitals
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The Kohn-Sham Equations. Vs and SCF

• Energy expression
• Variational principle (minimize E under the
  constraint )

Density-based
Wavefunction -based
only term unknown
iterative solution SCF
where
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The Kohn-Sham Approach Wave Function is Back!
A birds-eye view of density functional theory
(Klaus Capelle), Section 4 http//arxiv.org/PS_cac
he/cond-mat/pdf/0211/0211443.pdf
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The Kohn-Sham Equation is in principle exact!

• Hartree-Fock
• By using a single Slater determinant which cant
  be the true wavefunction, the approximation is
  introduced right from the start
• Kohn-Sham
• If the exact forms of EXC and VXC were known
  (which is not the case), it would lead to the
  exact energy.
• Approximation only enters when we decide on the
  explicit form of the unknown functional, EXC and
  VXC.
• The central goal is to find better approximations
  to those exchange-correlation functionals.

Section IV.C, Nobel lecture Electronic structure
of matter wave functions and density
functionals (Walter Kohn 1998)
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The Kohn-Sham Procedure I
where
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The Kohn-Sham Procedure II
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The Kohn-Sham Procedure III
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The Exchange-Correlation Energy Hartree-Fock
vs. Kohn-Sham

• Hatree-Fock

• Kohn-Sham

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The Quest for Approximate Exchange-Correlation
(XC) Functionals

• The Kohn-Sham approach allows an exact treatment
  of most of the contributions to the electronic
  energy.
• All remaining unknown parts are collective folded
  into the junkyard exchange-correlation
  functional (EXC).
• The Kohn-Sham approach makes sense only if EXC is
  known exactly, which is unfortunately not the
  case.
• The quest for finding better and better XC
  functionals (EXC) is at the very heart of the
  density functional theory.

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Is There a Systematic Strategy?

• Conventional wavefunction theory
• The results solely depends on the choice of the
  approximate wavefunction.
• The true wavefunction can be constructed by
• Full configuration interaction (infinite number
  of Slater determinants)
• Complete (infinite) basis set expansion
• Never realized just because its too complicated
  to be ever solved,
• but we know how it can be improved step by step
  in a systematic manner
• Density functional theory
• The explicit form of the exact functional is a
  total mystery.
• We dont know how to approach toward the exact
  functional.
• There is no systematic way to improve approximate
  functionals.
• However, there are a few physical constraints for
  a reasonable functional.

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The Local Density Approximation (LDA)

• Model
• Hypothetical homogeneous, uniform electron gas
• Model of an idealized simple metal with a
  perfect crystal
• (the positive cores are smeared out to a
  uniform background charge)
• Far from realistic situation (atom,molecule)
  with rapidly varying density
• The only system for which we know EXC exactly
• (Slater or Dirac exchange functional in
  Thomas-Fermi-Dirac model)

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The Local Density Approximation (LDA)

• Exchange
• Correlation
• From numerical simulations,

(VWN)
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Gradient Expansion Approximation (GEA)

• LDA not sufficient for chemical accuracy,
  solid-state application only
• includes only ?(r), the first term of
  Taylor expansion
• In order to account for the non-homogeneity,
• lets supplement with the second term, ??(r)
  (gradient)
• Works when the density is not uniform but very
  slowly varying
• Does not perform well (Even worse than LDA)
• ? Violates basic requirements of true holes (sum
  rules, non-positiveness)
• (LDA meets those requirements.)

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Generalized Gradient Approximation (GGA)

• Contains the gradients of the charge density and
  the hole constraints
• Enforce the restrictions valid for the true holes
• When its not negative, just set it to zero.
• Truncate the holes to satisfy the correct sum
  rules.

Reduced density gradient of spin ? Local
inhomogeneity parameter
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Exact Exchange Approach?

• Exchange contribution is bigger than correlation
  contribution.
• Exchange energy of a Slater determinant can be
  calculated exactly (HF).
• Exact HF exchange approximate functionals only
  for correlation
• 
  (parts missing in HF)
• Good for atoms, Bad for molecules (32 kcal/mol G2
  error) HF 78 kcal/mol
• Why?
• The resulting total hole has the wrong
  characteristics (not localized).
• Local Slater exchange from uniform electron gas
  seems a better model.

This division is artificial anyway.
delocalized (due to a single Slater determinant)
local model functional (should be delocalized
to compensate Ex)
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Slaters Approximation of HF Exchange X? method
(1951)

• Approximation to the non-local exchange
  contribution of the HF scheme
• Interaction between the charge density and the
  Fermi hole (same spin)
• Simple approximation to the Fermi hole
  (spherically symmetric)
• Exchange energy expressed as a density functional
• Semi-empirical parameter ? (2/31) introduced to
  improve the quality

(from uniform electron gas)
X? or Hartree-Fock-Slater (HFS) method
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Beckes Hybrid Functionals Adiabatic Connection
where
Non-interacting, Exchange only (of a Slater
determinant), Exact
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Fully-interacting, Unknown, Approximated with XC
functionals
Empirical Fit (Becke93) (2-3 kcal/mol G2 error)
Half-and-half (Becke93) (6.5 kcal/mol G2 error)
B3LYP
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Summary XC Functionals

• LDA Good structural properties, Fails in
  energies with overbinding
• GGA (BP86, BLYP, BPW91, PBE) Good energetics (lt
  5 kcal/mol wrt G2)
• Hybrid (B3LYP) The most satisfactory results

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Self-Interaction Problem

• Consider a one-electron system (e.g. H?)
• Theres absolutely no electron-electron
  interaction.
• The general KS equation should still hold.
• Classical electron repulsion
• To remove this wrong self-interaction error, we
  must demand
• None of the current approximate XC functions
  (which are set up independent of J?) is
  self-interaction free.

? 0 even for one-electron system
(naturally taken care of in HF)
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Self-Interaction in HF

• Coulomb term J when i j (Coulomb interaction
  with oneself)
• Beautifully cancelled by exchange term K in HF
  scheme
• HF scheme is free of self-interaction errors.

? 0
0
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Self-Interaction Error J?EXC?
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HK Theorem in Real Life?

• The variational principle applies to the exact
  functional only.
• The true functional is not available.
• We use an approximation for F?.
• The variational principle in DFT does not hold
  any more in real life.
• The energies obtained from an approximate
  density functional theory can be lower than the
  exact ones!

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