Density Functionals: Basic DFT Theory

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Density Functionals Basic DFT Theory

• Sergio Aragon
• San Francisco State University
• CalTech PASI January 4-16, 2004

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DFT Publications
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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

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Molecular Schroedinger Equation
H Y(x1,x2,,xN R1,,RNn) E Y (x1,x2,,xN
R1,,RNn)
H Te Ven Vee Vnn
Ven
Te
Vnn
Vee
Born-Oppenheimer Aproximation
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Hartree-Fock Theory
Fictitious system of non-interacting electrons.
YHF
EHFltYHFHYHFgt
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Correlation Energy

• Exchange Correlation Fermions of the same spin
  cannot move independently of one another. Y 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.

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Information Basic Problem
N 42 electrons Number of terms in Y N! 1.4
1051 Number of Cartesian dimensions in which Y
lives 3N 126 Y is an extremely complex object
and contains more information that we need!
Benzene
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Can we make do with less??

• The one-electron density!

r(r1) N
This object is merely 3-dimensional, regardless
of the number of electrons or the size of the
molecule.
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Electron Density of Water
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Hohenberg-Kohn Theorems
Theorem 1 The external potential, Ven is
determined, up to an additive constant, by the
electron density. Since r(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.
Hohenberg, P. Kohn, W. The inhomogeneous
electron gas, Phys. Rev. 136, B864 (1964).
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HK Proof I
Prove by contradiction. Assume that there exist
two different external potentials VVen Vnn and
VVen Vnn which both give the same electron
density r(r1). Then we have two Hamiltonians,
and H with the same ground state density and
different Y and Y. Now we use the variational
principle, taking Y as a trial function for the
H Hamiltonian, to obtain E0 lt lt YHYgt lt
YH'Ygt lt Y(H-H')Ygt E0' lt
Y(V-V')Ygt In addition, we can take the Y as a
trial function for the H Hamiltonian, to
obtain E0 lt lt YH'Ygt lt YHYgt lt
Y(H-H)Ygt E0 lt Y(V-V)Ygt 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 lt 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 Er. The density determines
the Hamiltonian, and thereby, the wavefunction.
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HK Proof II
Suppose we have a trial density r. Then this
density defines its own wavefunction Y, and the
expectation value of the true Hamiltonian
satisfies the variational principle lt
YHYgt Tr Veer E0r0 lt
Y0HY0gt. Thus, the correct density is the
one that produces the minimum energy.
Er

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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 Ff(x) y. A simple
example is the definite integration operator
The specification of the integrable function g(x)
produces a number defined in terms of the
constants a and b.
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Thomas-Fermi-Dirac Model
The uniform electron gas consists of NZ
electrons in a volume which is uniformily
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 ETFDr Tr Venr
Veer Vxr
Coulomb correlation for the electron gas was
computed by Monte Carlo methods by Ceperly
Alder, Phys. Rev. Lett. 45, 566 (1980).
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TFD Atomic Density
HF
TFD
r2r(r)
Kinetic Energy poorly represented!
1.0 Sqrtr a.u. 2.0
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Kohn-Sham Reference System
Introduce a fictitious reference system of
non-interacting particles with density rS. Then,
a single Slater determinant gives the exact
solution to the fictitious system, and,
Adjust the potential that these particles move
under so that they have a density r rS and
write
Er TSr Jr VenVnn Excr
Excr Tr - TSr Eeer - Jr.
All unknowns here!
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Kohn-Sham Equations
Variation principle
EXACT EQUATION!
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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 cant.

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HF/KS Orbitals Compared
DFT
HF

• Koopmans 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.

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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 funtionals containing a percentage of
  HF exchange B3LYP

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The Local Density Approximation
The LDA Approximation (Koch Holthausen)
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LDA Quantitatively
Coulomb correlation interpolated from Ceperly
Alder.
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Generalized Gradient Approximation
Exchange Correlation Ex
Coulomb Correlation Ec LYP Lee, Parr Yang
(1988) PW91 Perdew Wang (1991) P86 Perdew
(1986)
Becke Perdew/Wang BPW91 Becke
Lee/Parr/Yang BLYP
Perdew (1986)
Becke (1988)
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Hybrid Functionals
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?
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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.

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H2 Potential Curve