1. Density Functional Theory Richard M. Martin University of Illinois
Cu d orbitals
Electron density in La2CuO4 - difference from
sum of atom densities - J. M. Zuo (UIUC)
Density Functional Theory IPAM 2002

2. Density Functional Theory IPAM 2002
Outline
DFT is an approach to Interacting Many-Body Problems
Hohenberg-Kohn Theorems & Levy-Lieb Construction
Kohn-Sham Ansatz allows in principle exact solution for ground state of many-body system using independent particle methods
Classes of functionals: LDA, GGA, OEP, ….
Examples of Results
Locality Principles and linear scaling
Electric polarization in crystals - deep issues that bring out stimulating questions about DFT, and the differences between the Hohenberg-Kohn and Kohn-Sham approaches
Density Functional Theory IPAM 2002

3. Density Functional Theory IPAM 2002
Questions for you
Why were “orbitals” mentioned on the introductory slide and not simply “density”
Can you tell whether La2CuO4 is an insulator or a metal just by looking at the density? If so, what aspects of the density?
Is Kohn-Sham theory the same as Density Functional Theory?
If not, what is the difference? What did Kohn-Sham add? What did they subtract?
Do locality principles in independent particle methods carry over to the real many-body world?
Is the electric polarization of a ferroelectric an intrinsic ground state property? Is it determined by the density?
Density Functional Theory IPAM 2002

4. Density Functional Theory IPAM 2002

5. Density Functional Theory IPAM 2002

6. Density Functional Theory IPAM 2002

7. Assumes non-degenerate ground state
H-K Functional
Density Functional Theory IPAM 2002

8. Density Functional Theory IPAM 2002

9. Density Functional Theory IPAM 2002
Wavefunctions with density n( r )
Density Functional Theory IPAM 2002

10. What have we gained so far?
Apparently Nothing!
The only result is that the density determines the potential
We are still left with the original many-body problem
But the proofs suggest(ed) the next step
Density Functional Theory IPAM 2002

11. Density Functional Theory IPAM 2002
Kohn-Sham Ansatz
If you don’t like the answer, change the question
Replace the original interacting-particle problem with a different problem more easily solved
Kohn-Sham auxiliary system: non-interacting ”electrons” assumed to have the same density as the interacting system
Density Functional Theory IPAM 2002

12. Density Functional Theory IPAM 2002
Auxiliary System
Density Functional Theory IPAM 2002

13. Density Functional Theory IPAM 2002
Replace interacting problem with auxiliary non-interacting problem
Each term in figure is uniquely related to each other term!
The ansatz has been shown to be fulfilled in several simple cases – but not in general
We will proceed assuming the ansatz is justiified
Density Functional Theory IPAM 2002

14. Density Functional Theory IPAM 2002

15. Density Functional Theory IPAM 2002

16. Density Functional Theory IPAM 2002

17. Negative energy: electron – positive hole
Kinetic energy: positive
Density Functional Theory IPAM 2002

18. Exchange-correlation hole in homogeneous electron gas
Exchange dominates at high density (small rs)
Correlation dominates at low density (large rs)
Gori-Giorgi, Sacchetti and Bachelet, PRB 61, 7353 (2000).
Density Functional Theory IPAM 2002

19. Density Functional Theory IPAM 2002

20. Exchange hole in Ne atom
Gunnarsson, et al, PRB 20, 3136 (79).
Spherical average close to LDA!
Density Functional Theory IPAM 2002

21. Exchange hole in Si Crystal
Variational Monte Carlo
Hood, et al, PRB 57, 8972(98).
Density Functional Theory IPAM 2002

22. Density Functional Theory IPAM 2002
Examples of Results
Hydrogen molecules - using the LSDA (from O. Gunnarsson)
Density Functional Theory IPAM 2002

23. Density Functional Theory IPAM 2002
Examples of Results
Phase transformations of Si, Ge
from Yin and Cohen (1982) Needs and Mujica (1995)
Density Functional Theory IPAM 2002

24. Density Functional Theory IPAM 2002
Graphite vs Diamond
A very severe test
Fahy, Louie, Cohen calculated energy along a path connecting the phases
Most important - energy of graphite and diamond essentially the same!
~ 0. 3 eV/atom barrier
Density Functional Theory IPAM 2002

25. Density Functional Theory IPAM 2002

26. Less compressible than Diamond
Bulk Modulus B (Gpa) Exp Th (LDA) C Os
Cynn, et al, PRL March 14 (2002)
Density Functional Theory IPAM 2002

27. Density Functional Theory IPAM 2002

28. Density Functional Theory IPAM 2002

29. Density Functional Theory IPAM 2002
Slater average exchange
Density Functional Theory IPAM 2002

30. Density Functional Theory IPAM 2002
Phonons - LDA and GGA
Baroni, et al, RMP 73, 515 (2000).
Calculated by response function method
LDA
GGA
Exp
Density Functional Theory IPAM 2002

31. Density Functional Theory IPAM 2002
The “Band Gap Problem”
Often said that the eigenvalues have no meaning – just Lagrange multipliers
Energy to add or subtract an electron in the non-interacting system - not an excitation energy of the interacting system
Naïve use of the eigenvalues as exciation energies is the famous “band gap problem”
To understand the effcets we first examine the potential
Density Functional Theory IPAM 2002

32. Density Functional Theory IPAM 2002

33. Density Functional Theory IPAM 2002

34. Exchange potential in atoms
2-electron systems
LDA Vxc is too shallow
Almbladh and Pedroza, PR A 29, 2322 (84).
Density
Density Functional Theory IPAM 2002

35. Density Functional Theory IPAM 2002

36. Density Functional Theory IPAM 2002
The “Band Gap Problem”
Excitations are NOT well-predicted by the “standard” LDA, GGA forms of DFT
The “Band Gap Problem”
Orbital dependent DFT is more complicated but
gives improvements -
treat exchange better, e.g,
“Exact Exchange”
Ge is a metal
in LDA!
M. Staedele et al, PRL 79, 2089 (1997)
Density Functional Theory IPAM 2002

37. Status of “Band Gap Problem”
It should be possible to calculate all excitation energies from the Kohn-Sham approach
But not clear how close Kohn-Sham eigenvalues should be to true excitation energies
Not clear how much of the “band gap problem” is due to approximate functionals
Size of derivative discontinuity?
Density Functional Theory IPAM 2002

38. Locality and Linear Scaling
DFT provides a fundamental basis for “nearsightedness” (W. Kohn) -- if properties in a region are determined only by densities in a neighborhood -- so that an “Order N” method must be possible
Used, e.g., by W. Yang in his divide and conquer method
Orbital picture in Kohn-Sham method provides the concrete methods
Density Functional Theory IPAM 2002

39. Linear Scaling ‘Order-N’ Methods
Computational complexity ~ N = number of atoms (Current methods scale as N2 or N3)
“Divide and Conquer”
Green’s Function
Fermi Operator Expansion
Density matrix “purification”
Generalized Wannier Functions
Spectral “Telescoping” (Review by S. Goedecker in Rev Mod Phys)
Density Functional Theory IPAM 2002

40. Example of Our work Prediction of Shapes of Giant Fullerenes
S. Itoh, P. Ordejon, D. A. Drabold and R. M. Martin, Phys Rev B 53, 2132 (1996). See also C. Xu and G. Scuceria, Chem. Phys. Lett. 262, 219 (1996).
Density Functional Theory IPAM 2002

41. Simulations of DNA with the SIESTA code
Machado, Ordejon, Artacho, Sanchez-Portal, Soler
Self-Consistent Local Orbital O(N) Code
Relaxation - ~15-60 min/step (~ 1 day with diagonalization)
Iso-density surfaces
Density Functional Theory IPAM 2002

42. Density Functional Theory IPAM 2002
Conclusions - I
DFT is a general approach to interacting many-body problems Kohn-Sham approach makes it feasible
Ground state properties are predicted with remarkable success by LDA and GGAs. Structures, phonons (~5%), ….
Excitations are NOT well-predicted by the LDA, GGA approximations The “Band Gap Problem” Orbital dependant functionals increase the gaps - agree better with experiment “Derivative discontinuity” natural in orbital functionals
Density Functional Theory IPAM 2002

43. Density Functional Theory IPAM 2002
Conclusions - II
Locality inherent for properties of a region that depend only on the density in a neighborhood Forces, stress, .. “Order N” linear scaling method should be possible
Density matrix shows the locality in the quantum system Several feasible methods for insulators
Carries over to interacting many-body system
Some propreties are not local in real space Fermi surface of a metal, etc. But states near Fermi energy have universal behavior that should make linear scaling possible
When is the functional an extremely non-local functional of the density? A polarized insulator, where the Kohn-Sham theory must be fundamentally revised
Density Functional Theory IPAM 2002