Download Chapter 5a: Lecture Notes

Computational Chemistry:
A DFT crash course
Arnaud Marmier
1South 1.19
[email protected]
http://www.bath.ac.uk/~chsam/dft4dummies.bho
Phys. 555/342: Ch.5(2007)
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
Phys. 555/342: Ch.5(2007)
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. …
Phys. 555/342: Ch.5(2007)
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
Phys. 555/342: Ch.5(2007)
The Basic Problem
Dangerously
classical
representation
Cores
Electrons
Phys. 555/342: Ch.5(2007)
Schroedinger’s Equation
' h 2
$
( + V "! (Ri , ri ) = * .! (Ri , ri )
%)
& 2m
#
Kinetic Energy
Wave function
Potential Energy
Coulombic interaction
External Fields
Energy levels
Hamiltonian operator
Very Complex many body Problem !!
(Because everything interacts)
Phys. 555/342: Ch.5(2007)
First approximations
• Adiabatic (or Born-Openheimer)
– Electrons are much lighter, and faster
– Decoupling in the wave function
# (Ri , ri ) = " (Ri ).! (ri )
• Nuclei are treated classically
– They go in the external potential
Phys. 555/342: Ch.5(2007)
Hohenberg-Kohn Theorem
The ground state is unequivocally
defined by the electronic density
Ev [" ]= F [" ]+ ! v(r )" (r )dr
Universal functional
•Functional ?? Function of a function
•No more wave functions here
•But still too complex
Phys. 555/342: Ch.5(2007)
Kohn-Sham Formulation
Use an auxiliary system
– Non interacting electrons
– Same Density
– => Back to wave functions, but simpler this time
(a lot more though)
N K.S. equations
( h 2
(KS1) & * ) + Veff
' 2m
%
#! i (r ) = " i .! i (r )
$
(KS2)
! (r ")
dr " + µ XC [! ](r )
r # r"
Veff (r ) = V (r )+ $
(KS3) # (r )= ! " i (r )
(ONE particle in a box really)
Exchange correlation potential
2
i
Phys. 555/342: Ch.5(2007)
Self consistent loop
Initial density
From density, work out
Effective potential
Solve the independents
K.S. =>wave functions
Deduce new density from w.f.
NO
New density ‘=‘
input density ??
YES
Finita la musica
Phys. 555/342: Ch.5(2007)
DFT energy functional
1
! (r ")
E [! ]= TNI [! ]+ $ ! (r )v(r )dr + $ $
drdr " + E XC [! ]
2 r # r"
Exchange correlation
funtional
Contains:
Exchange
Correlation
Interacting part of K.E.
Electrons are fermions
(antisymmetric wave function)
Phys. 555/342: Ch.5(2007)
Exchange correlation functional
At this stage, the only thing we need is: E XC [! ]
Still a functional (way too many variables)
#1 approximation, Local Density Approximation:
Homogeneous electron gas
Functional becomes function !! (see KS3)
Very good parameterisation for E XC (! )
Generalised Gradient Approximation:
E XC (! , "! )
LDA
GGA
Phys. 555/342: Ch.5(2007)
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
Phys. 555/342: Ch.5(2007)
And now, for something completely different:
A little bit of Solid State Physics
Crystal structure
Periodicity
Phys. 555/342: Ch.5(2007)
Reciprocal space
ai # b j = 2" .! ij
(Inverting effect)
sin(k.r)
Real Space
ai
Reciprocal Space
Brillouin Zone
bi
k-vector (or k-point)
See X-Ray diffraction for instance
Also, Fourier transform and Bloch theorem
Phys. 555/342: Ch.5(2007)
Band structure
We will do this.
E
Energy levels
(eigenvalues of SE)
Molecule
Crystal
Phys. 555/342: Ch.5(2007)
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
??
Phys. 555/342: Ch.5(2007)
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
Phys. 555/342: Ch.5(2007)
Solid State: Summary
• Quantities can be calculated in the direct or
reciprocal space
• k-point Mesh
• Plane wave basis set, Ecut
Phys. 555/342: Ch.5(2007)
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
Fi = # ! % (r )$vi (r # "i )dr
Then, the methodologies discussed for atomistic
potential can be used
Energy Optimisation
Monte Carlo
Molecular dynamics
Phys. 555/342: Ch.5(2007)
Pauli Principle for
Indistinguishable particles
Consider two electron 1 and 2
"(1,2)
Switch 1 and 2 then since they are identical
2
2
| "(1,2) | =| "(2,1) |
Consider two electron 1 and 2 in two states φ and χ
!
!
"(1,2) = # (1) $ (2) % # (2) $ (1)
Need anti-symmetric wave function
goes to zero when states are the same.
Phys. 555/342: Ch.5(2007)
!
Density of states in 3-D
3/2
V $ 2m ' 1/ 2
g(")d" = 2 & 2 ) " d"
2# % h (
"F = ( n3#
!
2 2/3
)
h2
2m
3
< " >= N"F
5
!
Phys. 555/342: Ch.5(2007)
!
Fermi Temperature and kF
Phys. 555/342: Ch.5(2007)
Fermi-Dirac Dist. f(ε)
What is the probability that a state of energy ε is occupied?
T=0 K
f(ε)=1 if ε<εF
f(ε)=0 if ε>εF
T>0 K
1
f (") =
exp(" # µ) + 1
µ chemical pot.
µ(T=0 K)=εF
Phys. 555/342: Ch.5(2007)
!
Fermi-Dirac Distribution function at various T
Chemical Potential
µ defined f(ε)=.5
Phys. 555/342: Ch.5(2007)
FD and MB distribution Functions
FD
MB
Phys. 555/342: Ch.5(2007)
Density of states at finite temperature
N(") = f (")g(")
!
"F
Phys. 555/342: Ch.5(2007)
Temperature dependence of µ
N(",T) = f (",T)g(")
#
N=
$
$
0
1/ 2
f (",T)g(")d"
$
1/ 2
" d"
x dx
3
N!= A %
= BT % (x#& )
+1
0 exp(" # µ)k B T) + 1
0 e
!
µ
.76"F
kB T
.76"F
2/
,
2&
)
%
k
T
B
µ(T) " #F .1$ (
+1
.- 12 ' #F * 10
Phys. 555/342: Ch.5(2007)
Periodic Boundary Conditions
3/2
3-dimensional
2-dimensional
!
1-dimensional
!
V $ 2m ' 1/ 2
g(")d" = 2 & 2 ) " d"
2# % h (
A $m'
g(")d" = & 2 ) d"
# %h (
L
g(")d" =
#
1/ 2
$ 2m '
& 2)
%h (
d"
1/ 2
"
Phys. 555/342: Ch.5(2007)
!
Properties of Electron Gas: Specific Heat
2/
,
2&
)
%
k
T
B
µ(T) " #F .1$ (
+1
.- 12 ' #F * 10
2
# 2
CV " kB g($F )T
3
#
&2
U(T) = $ "f (",T)g(")d" % (k B T) 2 g("F ) + O(T 4 )
6
0
!
Phys. 555/342: Ch.5(2007)
Properties of Electron Gas: Cohesive Energy
Hartree-Fock Theory
h 2 k 2 e 2 kF %
kF2 # k2 k + k F (
"=
# 2 '2 +
ln
*
2m 8$ "0 &
kkF
k # kF )
2
F
2
3"
3e k F
< " >=
#
5 16$ 2"0
!
U Fg
2.21 0.916
= 2 "
" (0.115 " 0.0313ln rs )
rS
rs
1/ 3
1/ 3
!
# 3V &
# 3 &
rs = %
( =%
(
$ 4"N '
$ 4"n '
Phys. 555/342: Ch.5(2007)
Properties of Electron Gas: Bulk Modulus
$ #U 0 '
p = "&
)
% #V ( N
But
"F #V $2 / 3
!
So
2U 0
p=
3V
!$ #p '
$ # 2U 0 ' 10U 0 2N* F 0.586
B = "V & ) = V &
=
=
= 5
2 )
% #V (
3V
rS
% #V ( 9V
!
Phys. 555/342: Ch.5(2007)
Properties of Electron Gas: Bulk Modulus
Phys. 555/342: Ch.5(2007)
Failures of Free Electron Model
Hall Coefficients
Magnetoresistance
The thermoelectric Field
The Wiedemann-Franz Law
Temperature Dependence of DC Conductivity
Directional Dependence of DC Conductivity
AC Conductivity.
Optical Proberties
Cubic term in specific heat
Why are some elements not metallic?
Phys. 555/342: Ch.5(2007)