Computational Chemistry: A DFT crash course

DFT

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
Electron
s

6. Schroedinger’s Equation
 
iiii
rRrRV
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

iiii
rRrR .,

8. H.K. Theorem
The ground state is unequivocally
defined by the electronic density
rrrdvFE
v 
 
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)
 rrV
m
iiieff
 .
2
2










rr
rr
r
rr 

XCeff
dVV 



 
i
i
2
rr
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
N
O

11. DFT energy functional





XCNI
EdddvTE 


 
rr
rr
r
rrr
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:
 ,
XCE
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 E
xc
Simplistic approximations of E
xc 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
a
i
ijji
ba .2
Reciprocal Space
b
iBrillouin
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
Molecul
e
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: E
cut

19. Solid State: Summary
Quantities can be calculated in the
direct or reciprocal space
k-point Mesh
Plane wave basis set, E
cut

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

 rrrF dv
iii
