Density Functional Theory

1
Density Functional Theory

• DFT is an alternative approach to electronic
  structure theory, where the electron density is
  the key variable
• A fundamental theorem shows that we can represent
  total electronic energy in terms of the density
  alone
• BUT, approximations must be made for a practical
  implementation

2
The first step Thomas-Fermi theory
(1920s)

• 2 elecrons are put into each h3 portion of phase
  space
• Effective potential determines overall
  distribution of charge
• Divide space into small cells
• E levels

3
TF theory (contd)

• Total number of states with energy below a cutoff
• Then
• Fermi-Dirac distribution
• Low T? Step function

Exercise plot the FD distribution for low and
high temperatures
4
TF theory (contd)

• Kinetic energy of electrons
• Number of electrons
• Then

Exercise prove these formulas
5
TF theory (contd)

• Continuum limit of kinetic energy
• Total energy

Atomic units
2nd term is el-nuc interaction, 3rd is el-el
interaction
6
TF theory (contd)

• We know
• Variational calc with constraint
• With
• Then in 1964 DFT was invented

(the electrostatic potential due to nucleus and
electrons)
Exercise derive for yourself the above
variational formula
7
Hohenberg-Kohn theorems

• 1st theorem The external potential is
  determined, within a trivial additive constant,
  by the electron density And
  determines the ground state wave function. We
  can measure
• 2nd theorem any trial density not equal to the
  exact density yields a total energy above the
  exact ground state energy

8
Variational statement

• Euler-Lagrange equation
• Similar equations can be formulated for finite
  Ts also (not T0 as here)

That last term is a universal functional of the
density. If we knew it we would know
everything, but we dont know the exact form,
so we need to make approximations.
9
Kohn-Sham equations

• This approach made DFT practical.
• Make up an eigenvalue problem to obtain most of
  the correct kinetic energy for the electrons.
  Then need to approximate the exchange-correlation
  functional.
• The reference system is called a noninteracting
  reference system.

10
KS equations (contd)

• Total energy
• Variational equation
• Constraint
• Noninteracting system

11
KS theory (contd)

• Universal electron energy functional
• Variational statement

Exercise confirm this partitioning
12
KS eigenvalue equations

• Eigenvalue equations to solve
• Total energy

with
Exercise prove these formulas
13
xc potentials

• How can we model ?
• Simplest approximation is the local density
  approximation
• Model the xc potl as that for a uniform electron
  gas (which atoms are NOT!)
• Get the xc potl from QMC simulations (Ceperley
  and Alder).
• Recent history better v_xc approximations,
  gradient expansions (BLYP) and/or inclusion of
  some Hartree-Fock exchange (B3LYP by Becke)

14
Conclusions

• The derivation of the KS equations is exact in
  principle but we dont know the exact xc potl.
• Better and better xc potls are being derived.
• B3LYP can obtain near-chemical accuracies, but
  inclusion of exchange makes that potl more
  difficult to calculate.
• DFT is the choice in terms of good efficiency for
  large systems
• BUT dispersion interactions are not properly
  treated and that is a real problem for condensed
  phases.