Lecture 17. Density Functional Theory (DFT) References A bird’s-eye view of density functional theory (Klaus Capelle) http://arxiv.org/PS_cache/cond-mat/pdf/0211/0211443.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

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). = <|> Probability density

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

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

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}

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.

Slater’s 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  (2/3~1) introduced to improve the quality (from uniform electron gas) X or Hartree-Fock-Slater (HFS) method

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

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

Hohenberg-Kohn Theorem #1 (1964) Proof of Existence

Hohenberg-Kohn Theorem #1 (1964) Proof of Existence There cannot be two different Vext (thus two different wavefuntion ) that yield the same ground state electron density (r). 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.

Proof

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

Hohenberg-Kohn Functional: Holy Grail of DFT

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

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

Variational Principle in DFT Levy’s 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

Variational Principle in DFT Universal functional Determined simply by the density Independent of the wavefunction The same for all the wavefunctions integrating to a particular density

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.

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

Implement Hohenberg-Kohn Theorems: Thomas-Fermi? 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)

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)

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

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

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

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

The Kohn-Sham Approach: Wave Function is Back! A bird’s-eye view of density functional theory (Klaus Capelle), Section 4 http://arxiv.org/PS_cache/cond-mat/pdf/0211/0211443.pdf

The Kohn-Sham Equation is “in principle” exact! Hartree-Fock: By using a single Slater determinant which can’t 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)

The Kohn-Sham Procedure I where

The Kohn-Sham Procedure II

The Kohn-Sham Procedure III

The Exchange-Correlation Energy: Hartree-Fock vs. Kohn-Sham Hatree-Fock Kohn-Sham

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.

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 it’s 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 don’t 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.

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) (constant everywhere)

The Local Density Approximation (LDA) Exchange Correlation From numerical simulations, (VWN)

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, let’s 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.)

Generalized Gradient Approximation (GGA) Contains the gradients of the charge density and the hole constraints Enforce the restrictions valid for the true holes When it’s not negative, just set it to zero. Truncate the holes to satisfy the correct sum rules. Reduced density gradient of spin  Local inhomogeneity parameter

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)

Slater’s 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  (2/3~1) introduced to improve the quality (from uniform electron gas) X or Hartree-Fock-Slater (HFS) method

Becke’s Hybrid Functionals: Adiabatic Connection where Non-interacting, Exchange only (of a Slater determinant), Exact 1 Fully-interacting, Unknown, Approximated with XC functionals linear Empirical Fit (Becke93) (2-3 kcal/mol G2 error) Half-and-half (Becke93) (6.5 kcal/mol G2 error) “B3LYP”

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

Self-Interaction Problem Consider a one-electron system (e.g. H) There’s 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)

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

Self-Interaction Error J[]+EXC[]

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!