BSE & TDDFT

• BSE - $G({\bf r},t,{\bf r}',t')$

  The theoretical description of electronic excitations in the framework of many-body perturbation theory has undergone a rapid development since new approaches and the increase of computer power have made numerical calculations feasible for real systems. Within many-body perturbation theory one can calculate with a good precision charged excitations (i.e. electron addition and removal energies), using e.g. Hedin's GW approximation¹ for the electron self-energy. In the same framework, neutral excitations (e.g. optical and energy-loss spectra) are also well described today through the solution of the Bethe-Salpeter equation (BSE). An optical absorption experiment creates an interacting electron-hole pair, the exciton.
  Good agreement between theory and experiment can only be achieved taking into account the electron-hole interaction, especially if the system is a semiconductor or an insulator (small-gap semiconductors and metals, instead, screen this electron-hole interaction, and the resulting contribution can therefore be negligible). The Bethe-Salpeter couples indeed the electron and the hole, and it has been very successful for the calculation of absorption spectra of a large variety of systems: insulators, semiconductors, atoms, clusters, surfaces or polymers.
  However, the intrinsic two-particle nature of the Bethe-Salpeter equation makes the calculations very cumbersome, since a four-point equation (due to the propagation of two particles) has to be solved.Therefore, in spite of the excellent results obtained with the BSE for moderately simple systems, the efficient description of electron-hole excited states in realistic materials is still considered to be an unsolved problem in condensed matter theory.

• TDDFT $n ({\bf r},t)$

  A quite different approach, the time-dependent density-functional theory (TDDFT)² might be an advantageous alternative to the BSE formalism because, as in the case of the very successful static density-functional theory (DFT), this theory relies on the (now time-dependent) electron density $n ({\bf r},t)$ instead of the one-particle Green function, the propagator $G({\bf r},t,{\bf r}',t')$. Two-point response functions are involved in the formalism instead of the four-point ones required in the BSE approach. However, the time-dependent exchange-correlation (xc) potential $V_{xc}[n]({\bf r},t)$ is unknown, as well as its density variations. In the framework of linear response, the quantity to be described is the xc kernel $f_{xc}[n]({\bf r},t,{\bf r}',t')=\delta v_{xc}[n]({\bf r},t)/\delta n({\bf r}',t')$.
  First applications of TDDFT concerned finite systems³, and were performed in the adiabatic local density approximation (ALDA), with encouraging results. Subsequently, a large number of ALDA calculations have been performed for atoms, molecules, and clusters (see Review⁴ and references therein). Although the ALDA shows shortcomings (e.g., it does not capture Rydberg series), it is able to describe a large class of finite systems, and systematically improves results with respect to the random phase approximation (RPA), where exchange and correlation (xc) effects in the response are neglected. However, this statement is not true for extended systems. In fact, it has been found for simple semiconductors and insulators that absorption spectra calculated in ALDA are generally very close to RPA ones, and result hence in significant disagreement with experiment. Note that the same is not true for electron energy loss spectra, which are often in reasonable agreement with experiment even on the RPA or (better) ALDA level.

• A new way: TDDFT from BSE

  Recently, a class of exchange-correlation kernels has been proposed that turned out to be very efficient in the description of solids. They are directly derived from the Bethe-Salpeter equation of many-body perturbation theory. A parameter-free ab initio expression has been obtained in several different ways, always leading to the same formula⁵. The results using this kernel in conjunction with a quasi-particle bandstructure are in excellent agreement with those of the Bethe-Salpeter equation, with a potentially reduced computational effort; still, the calculations are significantly more cumbersome than those in RPA or ALDA. Other examples can be found in literature. This ref.⁶ contains a many-body diagrammatic expansion for the echange-correlation kernel and discusses some fundamental questions, like the non-locality of $f_{xc}$

• Future: TDDFT or BSE? which one more promising?

  One important question now is actually which way one should go-is the Bethe-Salpeter equation or TDDFT more promising? Since both approaches are exact, the answer depends of course on whether-and when-the remaining open questions in the two approaches can be solved. Some questions are common to
  both. An important point is whether pseudopotentials are always adequate for describing spectra with the precision one asks for today.
  Other questions are specific to the Bethe-Salpeter or the TDDFT approach. The Bethe-Salpeter approach offers a clear physical picture and straightforward possibilities for the analysis of results. It seems to work over a wide range of systems. The TDDFT approach, on the other hand, is appealing since it calculates things in a more direct way (without passing through electron addition and removal energies) and is, in principle, easier to use.
  Open problems in the Bethe-Salpeter approach include, of course, the need for technical developments to overcome the bottlenecks linked to the four-point equations. Moreover, just as in GW, not all the “ingredients” of the method are uniquely defined: how, for example, should vertex corrections and dynamical screening be included consistently? It is clear that the level of approximation which should be used for each ingredient in the various Green's-function-based approaches is a delicate and nontrivial question.
  In TDDFT, calculations are less cumbersome (if the two-point representation is chosen) than in the Bethe-Salpeter method at the moment, i.e., using the simple kernels, like ALDA. Better
  exchange-correlation potentials and better kernels have to be found, especially if TDDFT is meant to become a method for the calculation of absorption spectra in solids. These kernels might well turn out to be very complicated (see references cited above), and as difficult to treat as the
  Bethe-Salpeter equation. There is some hope that, at least in certain cases, relatively simple solutions can be found. The Bethe-Salpeter equation seems to provide a good starting point for the derivation of such effective kernels, since it contains the essential physics in a structure that is actually close to that of TDDFT.
  In conclusion, it seems reasonable to suppose that progress will come from a common effort of people working in the fields of the Bethe-Salpeter equation and of TDDFT..

Footnotes

¹

  L. Hedin, Phys. Rev. 139, A796 (1969).

²

  E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984).

³

  A. Zangwill and P. Soven, Phys. Rev. A 21, 1561 (1980).

⁴

  G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys 74, 601 (2002).

⁵

  L. Reining, V. Olevano, A. Rubio, and G. Onida, Phys. Rev. Lett. 88, 066404 (2002).
     F. Sottile, V. Olevano, and L. Reining, Phys. Rev. Lett. 91, 056402 (2003).
     G. Adragna, R. Del Sole, and A. Marini, Phys. Rev. B 68, 165108 (2003).
     A. Marini, R. Del Sole, and A. Rubio, Phys. Rev. Lett. 91, 256402 (2003).

⁶

  I. V. Tokatly, and O. Pankratov, Phys. Rev. Lett. 91, 056402 (2003).