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2. Useful Material
Books
A chemist’s guide to density-functional theory
Wolfram Koch and Max C. Holthausen (second edition,
Wiley)
The theory of the cohesive energies of solids
G. P. Srivastava and D. Weaire
Advances in Physics 36 (1987) 463-517
Gulliver among the atoms
Mike Gillan, New Scientist 138 (1993) 34
Web
www.nobel.se/chemistry/laureates/1998/
www.abinit.org
Version 4.2.3 compiled for windows, install and good
tutorial
3. Outline: Part 1,
The Framework of DFT
DFT: the theory
Schroedinger’s equation
Hohenberg-Kohn Theorem
Kohn-Sham Theorem
Simplifying Schroedinger’s
LDA, GGA
Elements of Solid State Physics
Reciprocal space
Band structure
Plane waves
And then ?
Forces (Hellmann-Feynman theorem)
E.O., M.D., M.C. …
6. Schroedinger’s Equation
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r
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m
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2
2
Hamiltonian operator
Kinetic Energy
Potential Energy
Coulombic interaction
External Fields
Very Complex many body Problem !!
(Because everything interacts)
Wave function
Energy levels
7. First approximations
Adiabatic (or Born-Openheimer)
Electrons are much lighter, and faster
Decoupling in the wave function
Nuclei are treated classically
They go in the external potential
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r
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,
8. H.K. Theorem
The ground state is unequivocally
defined by the electronic density
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r
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F
Ev
Universal functional
•Functional ?? Function of a function
•No more wave functions here
•But still too complex
9. K.S. Formulation
Use an auxiliary system
Non interacting electrons
Same Density
=> Back to wave functions, but simpler this time
(a lot more though)
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r
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m
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i
i
eff
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2
2
r
r
r
r
r
r
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XC
eff d
V
V
i
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2
r
r
N K.S. equations
(ONE particle in a box really)
(KS3)
(KS2)
(KS1)
Exchange correlation potential
10. Self consistent loop
Solve the independents K.S.
=>wave functions
From density, work out
Effective potential
New density ‘=‘
input density ??
Deduce new density from w.f.
Initial density
Finita la musica
YES
NO
11. DFT energy functional
XC
NI E
d
d
d
v
T
E
r
r
r
r
r
r
r
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2
1
Exchange correlation funtional
Contains:
Exchange
Correlation
Interacting part of K.E.
Electrons are fermions
(antisymmetric wave function)
12. Exchange correlation functional
At this stage, the only thing we need is:
XC
E
Still a functional (way too many variables)
#1 approximation, Local Density Approximation:
Homogeneous electron gas
Functional becomes function !! (see KS3)
Very good parameterisation for
XC
E
Generalised Gradient Approximation:
,
XC
E
GGA
LDA
13. DFT: Summary
The ground state energy depends only on
the electronic density (H.K.)
One can formally replace the SE for the
system by a set of SE for non-interacting
electrons (K.S.)
Everything hard is dumped into Exc
Simplistic approximations of Exc work !
LDA or GGA
14. And now, for something completely different:
A little bit of Solid State Physics
Crystal structure Periodicity
15. Reciprocal space
Real Space
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ij
j
i b
a
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2
Reciprocal Space
bi
Brillouin Zone
(Inverting effect)
k-vector (or k-point)
sin(k.r)
See X-Ray diffraction for instance
Also, Fourier transform and Bloch theorem
17. The k-point mesh
Brillouin Zone
(6x6) mesh
Corresponds to a supercell
36 time bigger than the
primitive cell
Question:
Which require a finer mesh,
Metals or Insulators ??
18. Plane waves
Project the wave functions on a basis set
Tricky integrals become linear algebra
Plane Wave for Solid State
Could be localised (ex: Gaussians)
+ + =
Sum of plane waves of increasing
frequency (or energy)
One has to stop: Ecut
20. Now what ?
We have access to the energy of a system,
without any empirical input
With little efforts, the forces can be computed,
Hellman-Feynman theorem
Then, the methodologies discussed for atomistic potential
can be used
Energy Optimisation
Monte Carlo
Molecular dynamics
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