Fundamental Problems in Density-Functional Theory

Density-functional theory (DFT) offers a powerful and elegant method for calculating the ground-state total energy and electron density of a system of interacting electrons. The system may range in complexity from a single atom to a complex system such as the gas molecules shown in the Figure, together with the atoms of the solid surface on which they are about to be adsorbed and where they will react with one another, guided by the total energy. The whole theory is based on functionals of the electron density, which therefore plays the central role. However, the key functional, which describes the total energy of the electrons as a functional of their density, is not known exactly: the part of it which describes electronic exchange and correlation has to be approximated in practical calculations.

For some years I have been investigating aspects of the exact functionals that are not reflected in the commonly used approximations. In 1986, Michael Schlüter, Lu Sham and I [1] calculated an accurate exchange-correlation potential for silicon using many-body perturbation theory, and we showed that the "band-gap problem" (the observation that the electronic band gap of semiconductors in density-functional-theory calculations was only about 50% of the experimental band gap) was present even with our more accurate exchange-correlation potential, and therefore corresponded to a non-analyticity in the functional, rather than an inadequacy of the local-density approximation normally used in density-functional theory.

Related work includes an investigation of the DFT band-gap problem as a semiconductor is compressed to the metallic state [2], a study of the behaviour of the exact exchange-correlation potential at semiconductor interfaces [3,4], an investigation of exact DFT for a model semiconducting wire using Monte Carlo methods [5,6], a study of exact DFT in the presence of a macroscopic electric field [7,9,10] (which for an infinite solid requires DFT to be augmented to become a density-polarisation functional theory), and an investigation of DFT for a Hubbard model [8].  We have also recently developed a new type of density-functional theory [11], based on a simplified self-energy approach, which initial tests indicate to outperform conventional Kohn-Sham DFT at a very similar cost.

Most recently, we have turned our attention to the description of electronic quantum transport in nanosystems, which takes us into the domain of time-dependent DFT.  Accurate description of the effects of electron-electron interaction requires sophisticated aspects to the exchange-correlation functional, which now depends on the time-dependent density (and/or current).  Our studies [12,13,14] are helping to illuminate the (often severe) deficiencies of the commonly used functionals, and contribute to the development of improved TDDFT functionals.

Illustrative publications

1. "Accurate exchange-correlation potential for silicon and its discontinuity on addition of an electron", R.W. Godby, M. Schlüter and L.J. Sham, Phys. Rev. Lett. 56 2415 (1986). Abstract

2. "The metal-insulator transition in quasiparticle theory and Kohn-Sham theory", R.W. Godby and R.J. Needs, Phys. Rev. Lett. 62 1169 (1989).  Abstract

3. "Exchange and correlation in Schottky barriers and heterojunctions", R.W. Godby, L.J. Sham and M. Schlüter, Phys. Rev. Lett. 65 2083 (1990).  Abstract

4. "Exchange-correlation potentials at semiconductor interfaces", R.W. Godby and L.J. Sham, Phys. Rev. B 49 1849 (1994). Abstract

5. "Investigating exact density-functional theory of a model semiconductor", W. Knorr and R.W. Godby, Phys. Rev. Lett. 68 639 (1992). Abstract

6. "A quantum Monte Carlo study of density-functional theory for a semiconducting wire", W. Knorr and R.W. Godby, Phys. Rev. B. 50 1779 (1994). Abstract

7. "Density-polarisation functional theory of the response of a periodic insulating solid to an electric field", X. Gonze, P. Ghosez and R.W. Godby, Phys. Rev. Lett. 74 4035 (1995). Abstract

8. "Density-functional theory and the v-representability problem for model strongly correlated electron-systems", A. Schindlmayr and R.W. Godby, Phys. Rev. B 51 10427 (1995). Abstract

9. "Density-functional theory of polar insulators", X. Gonze, Ph. Ghosez and R.W. Godby, Phys. Rev. Lett. 78, 294 (1997).  Abstract

10. "Polarization-Dependence of the Exchange Energy", X. Gonze, Ph. Ghosez and R.W. Godby, Phys. Rev. Lett. 78 2029 (1997).  Abstract

11. "Efficient total energy calculations from self-energy models", Paula Sánchez-Friera and R.W. Godby, Phys. Rev. Lett. 85 5611-5614 (2000).   Abstract

12. "Conductance and polarization in quantum junctions", P. Bokes and R.W. Godby, Phys. Rev. B 69 245420 (2004) (8 pages).   Abstract 

13. "Stroboscopic wavepacket description of non-equilibrium many-electron problems", P. Bokes, F. Corsetti and R. W. Godby, Phys. Rev. Lett. 101 046402 (2008) (4 pages).  Abstract  

14. "First-principles conductance of nanoscale junctions from the polarizability of finite systems", Matthieu J. Verstraete, P. Bokes and R.W. Godby, Journal of Chemical Physics 130 124715 (2009) (8 pages).  Abstract

Full list of publications