Fundamentals of DFT R. Wentzcovitch U of Minnesota VLab Tutorial

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Fundamentals of DFTR. WentzcovitchU of
MinnesotaVLab Tutorial

• Hohemberg-Kohn and Kohn-Sham theorems
• Self-consistency cycle
• Extensions of DFT

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BO approximation
- Basic equations for interacting electrons and
nuclei Ions
(RI ) electrons (ri )
This is the quantity calculated by total energy
codes.
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Pseudopotentials
1.0
3s orbital of Si
rRl (r)
0.5
Pseudoatom
Real atom
0.0
-0.5
0
1
2
3
4
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Radial distance (a.u.)
V(r)
r
1/2 Bond length
Nucleus Core electrons Valence electrons
Pseudopotential
Ion potential
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BO approximation
Born-Oppenheimer approximation (1927)
Ions (RI ) electrons (ri )
phonons
forces
stresses
Molecular dynamics
Lattice dynamics
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Electronic Density Functional Theory (DFT)
(T 0 K)
Hohemberg and Kohn (1964). Exact theory of
many-body systems.
Theorem I For any system of interacting
particles in an external
potential Vext(r), the potential Vext(r) is
determined uniquely, except
for a constant, by the ground state
electronic density n0(r). Theorem II A
universal functional for the energy En in terms
of the density n(r) can be
defined, valid for any external
potential Vext(r). For any particular Vext(r),
the exact ground state energy
is the global minimum value of this
functional, and the density n(r), that
minimizes the functional is
the ground state density n0(r).
DFT1
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• Proof of theorem I
• Assume Vext(1)(r) and Vext(2)(r) differ by
  more than a constant and produce the same n(r).
  Vext(1)(r) and Vext(2)(r) produce H(1) and H(2) ,
  which have different ground state wavefunctions,
  ?(1) and ?(2)
• which are hypothesized to have the same
  charge density n(r).
• 
  It follows that
• Then
• and
• Adding both
  which is an absurd!

Hohemberg and Kohn, Phys. Rev. 136, B864 (1964)
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• Proof of theorem II
• Each Vext(r) has its ?(R) and n(r). Therefore
  the energy Eel(r) can be viewed as a functional
  of the density.
• Consider
• and a different n(2)(r) corresponding to a
  different
• It follows that
  

Hohemberg and Kohn, Phys. Rev. 136, B864 (1964)
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The Kohn-Sham Ansatz
Kohn and Sham, Phys. Rev. 140, A1133 (1965)
Hohemberg-Kohn functional
How to find n?
Replacing one problem with another(auxiliary and
tractable non-interacting system) Kohn and
Sham(1965)
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dft2
Kohn-Sham equations (one electron
equation)
Minimizing En expressed in terms of the
non-interacting system w.r.t. ?s, while
constraining ?s to be orthogonal
With eis as Lagrange multipliers associated with
the orthonormalization constraint and
and
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Exchange correlation energy and potential
By separating out the independent particle
kinetic energy and the long range Hartree
term, the remaining exchange correlation
functional Excn can reasonably be approximated
as a local or nearly local functional of the
density.
and
with
Local density approximation (LDA) uses excn
calculagted exactly for the homogeneous
electron system
Quantum Monte Carlo by Ceperley and Alder, 1980
Generalized gradient approximation (GGA)
includes density gradients in excn,n
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• Meaning of the eigenvalues and eigenfunctions
• Eigenvalues and eigenfunctions have only
  mathematical meaning in the KS approach. However,
  they are useful quantities and often have good
  correspondence to experimental excitation
  energies and real charge densities. There is,
  however, one important formal identity
• These eigenvalues and eigenfunctions are used for
  more accurate
• calculations of total energies and
  excitation energy.
• The Hohemberg-Kohn-Sham functional concerns only
  ground state
• properties.
• The Kohn-Sham equations must be solved
  self-consistently

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Self consistency cycle
until
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Extensions of the HKS functional

• Spin density functional theory
• The HK theorem can be generalized to several
  types of particles. The most important example is
  given by spin polarized systems.

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• Finite T and ensemble density functional theory
• The HK theorem has been generalized to finite
  temperatures.
• This is the Mermin functional. This is an even
  stronger generalization of density functional.

D. Mermim, Phys. Rev. 137, A1441 (1965)
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Use of the Mermin functional is recommended in
the study of metals. Even at 300 K, states above
the Fermi level are partially occupied. It helps
tremendously one to achieve self-consistency.
(It stops electrons from jumping from occupied
to empty states in one step of the cycle to the
next.) This was a simulation of liquid
metallic Li at P0 GPa. The quantity that is
conserved when the energy levels are occupied
according to the Fermi-Dirac distribution is
the Mermin free energy, Fn,T.
Wentzcovitch, Martins, Allen, PRB 1991
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Dissociation phase boundary
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Umemoto, Wentzcovitch, Allen Science, 2006
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• Few references
• -Theory of the Inhomogeneous electron gas, ed. by
• S. Lundquist and N. March, Plenum (1983).
• Density-Functional Theory of Atoms and Molecules,
  
• R. Parr and W. Yang, International Series of
  Monographs
• on Chemistry, Oxford Press (1989).
• - A Chemists Guide to Density Fucntional Theory,
  
• W. Koch, M. C. Holthause, Wiley-VCH (2002).
• Much more ahead