computational chemistry computational materials physics computational solid state physics

1. Computational Chemistry:
A DFT crash course

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. …

4. Outline: Part2
Using DFT
Practical Issues
 Input File(s)
 Output files
 Configuration
 K-points mesh
 Pseudopotentials
 Control Parameters
 LDA/GGA
 ‘Diagonalisation’
Applications
 Isolated molecule
 Bulk
 Surface

5. The Basic Problem
Dangerously
classical
representation
Cores
Electrons

6. Schroedinger’s Equation
   
i
i
i
i r
R
r
R
V
m
,
.
,
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
     
i
i
i
i r
R
r
R 
 .
, 


8. H.K. Theorem
The ground state is unequivocally
defined by the electronic density
        r
r
r d
v
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)
   
r
r
V
m
i
i
i
eff 

 .
2
2











        
r
r
r
r
r
r
r 


XC
eff d
V
V 





 
   


i
i
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
r
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
ai
ij
j
i b
a 
.
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

16. Band structure
Molecule
E
Crystal
Energy levels
(eigenvalues of SE)

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

19. Solid State: Summary
Quantities can be calculated in the
direct or reciprocal space
k-point Mesh
Plane wave basis set, 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
   
 



 r
r
r
F d
v i
i
i 