Density Functional Theory:
a first look
Patrick Briddon
Theory of Condensed Matter
Department of Physics,
University of Newcastle,
UK.
Contents
Density Functional Theory
– Hohenberg Kohn Theorems
– Thomas Fermi Theory
– Kohn-Sham Equations
– Self Consistency
– Approximations for Exc.
Density Functional Theory
Work with n(r) instead of 
Standard approach of QM :
Vext r   S.E.()  E, nr 
DFT : work in terms of density :
E  E nr 
N.B. : few IFs and BUTs here
3 Important Questions
Three important questions:
• Can we really write E[n]?
• If so, how can we find n(r)?
• What is the functional E[n] ?
1st Hohenberg Kohn Theorem
The external potential V(r) is determined to within
a constant by the ground state charge density of
a system.
i.e. one-to-one relationship
nr   Vext r 
This is an astonishing statement!
Why?
1st Hohenberg Kohn Theorem
Proof:
Suppose we have two systems
Hamiltonians H1, H2
External potentials V1 , V2
GS wavefunctions: 1, 2
But the same GS density n(r)
E1  1 H1 1  2 H1 2
We clearly have
But:
H1  Tˆ  V1
 Tˆ  V2  V1  V2 
 H 2  V1  V2 
So:
E1  2 H 2 2  2 V1  V2 2

E1  E2  nr V1  V2  dr

Swap 1 and 2: E1  E2  nr V1  V2  dr
Contradiction!
Our starting point was wrong!
We cannot have two different systems with
the same GS density.
Importance of this: we can write E[n].
Now move on the second question – how
can we find the density?
Hohenberg – Kohn’s second theorem.
2nd Hohenberg Kohn Theorem
“The true ground state charge density is
that which minimises the total energy.”
An equivalent to the usual variational
principle of quantum mechanics.
Proof
We have (variational principle)
EGS  min

H


Define En  min  H 
n
Then
min  H   min En
EGS  min
n n
n
Some problems
V-representability
(only minimise over densities which can
arise from GS wavefunctions of real systems
Levy showed that the densities need only
satisfy a weaker condition that they can be
obtained from an antisymmetric
wavefunction (N-representable).
n must be +ve, continuous, normalised
Some extensions
Spin dependent potentials
e.g. magnetic fields
E[n]  E[n, n - n] or E[n, n]
Main advantage: better description of
open shell systems.
3 Important Questions
Three important questions:
• Can we really write E[n]?
• If so, how can we find n(r)?
• What is the functional E[n] ?
Now for the last question.
What is the formula!
What is the functional?
Difficult : still not answered exactly!
E  T V
  nr V
ext
ext
1
r   

i  j ri  r j
r  dr  difficult terms
Problem is that other two terms are very
large - any attempt at approximation must
be good.
Thomas-Fermi Method
Classical expression for electronelectron term:
1
1 n r  n r

  
dr dr
2
r  r
i  j ri  r j
Thomas-Fermi Method
Statistical idea for KE based on uniform
electron gas result:
 Tˆ  

k kF
KE per electron =
2 2
 k
53
 An
2m
An
23
Thomas-Fermi Method
What about a non uniform system?
A. Assume that things vary slowly:
DV
DT  nr 
23
r
 nr  DV
Total is thus T   An r  dr
53
Thomas-Fermi Method
Final energy is thus:
E n   A n r  dr   n r V
53
1 n r  n r
 
dr dr
2
r  r
How useful is this?
ext
r  dr
Thomas-Fermi Method
What is the conclusion?
• Energies quite good (error < 1%).
• Difference of energies not good enough to
describe bonding.
• How can we improve this?
Thomas-Fermi Method
Add exchange/correlation (missing do far).
• Try to take account of non-uniform system.
• Write T[n] as T[n, grad |n|]
• All failures!
Kohn-Sham method
Phys Rev 140, 1133A (1966)
Realised that approximation must be
made to terms that are small: KE is big!
Improving T[n] did not work.
Need a completely different approach.
Second half of HK paper therefore
discarded.
Kohn-Sham method
Phys Rev 140, 1133A (1966)
Introduce a system which:
1. Is non-interacting
2. Has same n(r) as the real system.
[non-interacting N-representability
- an assumption! ]
Kohn-Sham contd.
 Tˆ   Ts n  DT
where Ts[n] is the KE of the non-interacting
system and the final term, DT, is small.
E  Ts n   nr V ext r  dr 
1 nr  nr
 
dr dr  E xc n
2 r  r
Kohn-Sham contd.
Exc[n] includes both DT and contributions
to el-el energy beyond the Hartree term.
The key hope is that this is
• small
• less sensitive to external potential
These mean differences are accurate.
We now have two questions:
(a) how to find Ts[n] ?
(b) what is Exc[n] ?
For a non-interacting system it is exactly
true that the many electron wavefunction
is a single Slater determinant.
1
r1 ,, rN   det  ri 
N!
For this:
nr  
  r 
2
and:
Ts n      

1
2
2
The (r) must be found from
a self consistent solution of:

1
2

  Vs r     
2
Vs r   V
ext
nr
Exc n
r   
dr 
r  r
 nr 
nr     r 
2

These are called the Kohn-Sham
equations. Solve iteratively:
Guess: nr   n r 
in
Construct Vs r   V
Solve

1
2
ext
nr
Exc n
r  
dr 
r  r
nr 


  Vs r     
2
Find new density: n r  
out
Look at n
out
r   n r 
  r 
in
Form a better input and continue.
2
Self Consistent Cycle
• This process is called the self-consistent
cycle.
• Starting guess is a superposition of atomic
charge densities (or a restart dump).
• AIMPRO produces output showing how
the energy converged and how the input
and output densities come together.
AIMPRO SCF
etot,echerr
etot,echerr
etot,echerr
etot,echerr
etot,echerr
1
2
3
4
5
-1.1289007706
-1.1319461182
-1.1361047998
-1.1360826939
-1.1361076063
0.0547166884
0.0303263020
0.0000338275
0.0001723143
0.0000002649
0.884981 2.95 106.1 120.7
0.488911 3.05 106.1 120.7
0.000689 3.00 106.1 120.7
0.002700 2.99 106.1 120.7
0.000004 3.00 106.1 120.7
The numbers are:
• Total energy (reduces to converged value)
• 2 measures of
• Time taken per iteration
• Current memory being used (MB)
• Max memory used so far (MB)
Kohn-Sham Levels
We got the Kohn Sham eqn:

1
2
  Vs r      
2
Q: what exactly are the  and ?
A: the eigenvalues and eigenfunctions of a
ficticious non-interacting system which has
the same density as the real system.
KS Levels Contd.
They are not the energies of quasiparticles.
Typical semiconductor results:
Lattice constant to 1%
Bulk modulus to 1%
Phonon frequencies to 5%
LDA “gap” for Si is 0.6eV; 0.1 eV for Ge!
KS Levels Contd.
Bandstructures are qualitatively correct.
(Scissors operator).
Physical nature of the KS eigenfunctions
sensible.
P in Si - get state just below conduction band
Dangling bonds - localised states in mid gap.
AIMPRO and KS levels
spin, kpoint :
1 -10.0938
2.0000
1
1.2658
0.0000
1
3.7422
0.0000
3.7422
0.0000
5.9056
0.0000
8.7912
0.0000
The KS levels in eV.
Used in “bandstructure” plots.
Occupancies also given (this is a spin averaged
calculation)
3 Important Questions
Three important questions:
• Can we really write E[n]?
• If so, how can we find n(r)?
• What is the functional E[n] ?
• All remaining questions are in Exc[n].
Now finally we look at what this is.
Exchange correlation energy
Our DFT total energy is:
E  Ts n   Vext r  nr  dr
1  nr  nr
 
drdr
2  r  r 
 E xc n
What about the last term?
The Local Density
Approximation (LDA)
Write

Exc  nr  xc nr  dr
where xc(n) is the exchange-correlation
energy per electron for a uniform electron
gas.
This seems a bit rough and very similar to
Thomas Fermi, but this term is now very
much smaller.
Exc for Homogeneous electron
gas
• By simple analytical treatment for the
exchange energy.
• Using many body perturbation theory
(for various limits of correlation energy)
• By looking at quantum Monte-Carlo
calculations and parametrising them
• Intelligent interpolation between these
Exc for Homogeneous electron
gas contd.
Simple analysis for exchange part gives
13
3 3 
Ex    
2  4 

43
n
43
 n

Correlation is harder, see::
• Perdew Zunger (1981)
• Vosko, Wilk, Nusair (1980)
• Perdew, Wang (1992)
Simple Tests : Molecules
Example : water H20
Property
Calc.
Expt.
R(O–H) (Å)
0.967
0.957
(H–O–H) (deg)
105.7
104.5
1 (as str : cm-1)
3874
3757
2 (sym cm-1)
3773
3652
3 (bend cm-1)
1586
1596
dip. mom. (a.u.)
0.735
0.730
Simple tests : solids
• Standard “bulk” calculations :
– lattice constant (Si : 1%)
– bulk modulus (Si : 2%)
– phonon spectra (2 %)
– formation energies (LDA : 20 %)
– excitation energies (50 %)
Phonon Spectrum
Material : GaAs. All frequencies in cm-1
Mode Calc (5) Calc (6) Expt
278
276 267,285

TA(X)
88
90
79
TA(L)
66
66
62
LA(X)
224
225
227
LA(L)
217
216
209
TO(X)
256
255
252
TO(L)
267
265
261
LO(X)
246
244
241
LO(L)
238
239
238
How to Improve?
Next step is to move beyond the LDA:
Exc n, n   E
n
  C nr  4 3 dr
n
2
LDA
xc
The Gradient expansion Approximation
(GEA).
Early history of these was bad. Calculations
made worse, not better.
Generalised Gradient
Approximations (GGA)
Idea is to ensure that
• Sum rules are obeyed correctly
• scaling behaviour of exchange
correlation energy correct
• Various limiting forms
• Bounds (Lieb-Oxford)
Popular GGAs
•
•
•
•
B88 (empirical, chemistry)
BLYP (chemistry)
PW91 (physics, poor form)
PBE96 (physics, easier to use)
Atomisation energies (kcal/mol)
(1 eV = 23 kcal/mol)
HF
LSD GGA expt
84
113
105
109
CH4 328
462
420
419
H20 155
267
234
232
Cl2 17
81
63
58
H2
Generalised Gradient
Approximations (GGA)
In general, GGA weakens bonds slightly.
It improves results for:
• binding energies of molecules
• description of surfaces
• H-bonding
THE CONCLUSION
• An absolutely huge success
• 1988 two groups in UK doing DFT
– Cambridge (TCM)
– Exeter
• Today: every department?
• Chemistry/engineering too!
• Applications in huge variety of areas.
Work to do!
•
•
•
•
•
•
Kittel Ch 6: “Free Electron Fermi Gas”
Hohenberg and Kohn, PR (1964)
Kohn-Sham, PR (1965)
Perdew Zunger, PRB (1981)
Perdew, Wang, PRB (1992)
Perdew, Burke, Enzerhoff PRL (1996)