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Approximations to Excited States

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Computing the Optical Properties of Large Systems

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Abstract

In this chapter, the focus lies on the approximate evaluation of excited states in the many-electron system. The discussion is limited to neutral excitations in which the many-electron system couples to light by absorbing photons of specific wavelengths.

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Notes

  1. 1.

    Here, we limit the discussion to vertical excitations of closed-shell systems. DFT can be used to yield excitation energies in cases where different spin configurations yield different ground state energies. The energy difference between two different configurations can then be interpreted as the excitation energy of the system going from one symmetry to the other.

  2. 2.

    Although, if spin-orbit effects are small, which is the requirement for the Hamiltonian in Eq. (3.1) to be appropriate for the system at hand, the fine splitting can be obtained perturbatively (see for example [1]).

  3. 3.

    In principle, requiring the electron density to vanish at infinity means that the proof of the Runge–Gross theorem presented here is not valid for infinite systems. This limitation is avoided in TDCDFT, as the mapping between the current density and external potential can be proven without relying on any quantity vanishing at infinity (see Eq.  (3.9)).

  4. 4.

    The quantity in brackets linking \(\rho ^{\{1\}}_{ij}(\omega )\) and \(\delta V^\mathrm{{{pert}}}_{ij}(\omega )\) actually corresponds to the inverse of \(\chi (\omega )\) in matrix form.

  5. 5.

    Here, the factor of 2 originates from an implied summation over spin indices. The spin structure of the equation is discussed in some more detail in Sect. 3.2.5.

  6. 6.

    Note that such a measure of basis set quality requires the calculation of all possible excitations and thus a full diagonalisation of the entire 2-particle Hamiltonian and is thus not practical for larger systems.

  7. 7.

    In order to perform the inverse Fourier transform, one has to make use of Cauchy’s theorem and consider \(G(\omega )\) as a function of the complex variable \(\omega \) in the lower half of the complex plane. The contour chosen for the integration is a semicircle in the lower half plane, where the factor \(\mathrm {i}\delta \) ensures that the pole is inside the contour region.

  8. 8.

    It should be noted that there is no full formal justification why the perturbed density matrix should show the same sparsity properties as the ground state density matrix. A more detailed discussion on the question of sparsity of response density matrices is delivered in Sect. 5.2.

References

  1. de Carvalho, F.F., Curchod, B.F.E., Penfold, T.J., Tavernelli, I.: Derivation of spin-orbit couplings in collinear linear-response TDDFT: a rigorous formulation. J. Chem. Phys. 140, 144103 (2014)

    Google Scholar 

  2. Runge, E., Gross, E.K.U.: Density functional theory for time-dependent systems. Phys. Rev. Lett. 52, 997 (1984)

    Article  ADS  Google Scholar 

  3. Ghosh, S.K., Dhara, A.K.: Density-functional theory of many-electron systems subjected to time-dependent electric and magnetic fields. Phys. Rev. A 38, 1149 (1988)

    Article  ADS  Google Scholar 

  4. Vignale, G.: Mapping from current densities to vector potentials in time-dependent current density functional theory. Phys. Rev. B 70, 201102(R) (2004)

    Article  ADS  Google Scholar 

  5. Kohl, H., Dreizler, R.M.: Time-dependent density-functional theory: conceptual and practical aspects. Phys. Rev. Lett. 56, 1993 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  6. van Leeuwen, R.: Causality and symmetry in time-dependent density-functional theory. Phys. Rev. Lett. 80, 1280 (1998)

    Article  ADS  Google Scholar 

  7. Marques, M.A.L., Gross, E.K.U.: Time-dependent density-functional theory. Annu. Rev. Phys. Chem. 55, 427 (2004)

    Article  ADS  Google Scholar 

  8. Yabana, K., Bertsch, G.F.: Time-dependent local-density approximation in real time. Phys. Rev. B 54, 4484 (1996)

    Article  ADS  Google Scholar 

  9. Castro, A., Marques, M.A.L., Rubio, A.: Propagators for the time-dependent Kohn-Sham equations. J. Chem. Phys. 121, 3425 (2004)

    Article  ADS  Google Scholar 

  10. Onida, G., Reining, L., Rubio, A.: Electronic excitations: density-functional versus many-body Green’s-function approaches. Rev. Mod. Phys. 74, 601 (2002)

    Article  ADS  Google Scholar 

  11. Petersilka, M., Grossmann, U.J., Gross, E.K.U.: Excitation energies from time-dependent density-functional theory. Phys. Rev. Lett. 76, 1212 (1996)

    Article  ADS  Google Scholar 

  12. Casida, M.E.: Recent Advances in Density Functional Methods. In: Chong, D.P. (ed.) vol. 1. World Scientific, Singapore (1995)

    Google Scholar 

  13. Hübener, H., Giustino, F.: Linear optical response of finite systems using multishift linear system solvers. J. Chem. Phys. 141, 044117 (2014)

    Article  ADS  Google Scholar 

  14. Berkowitz, J.: Photoabsorption, Photoionization, and Photoelectron Spectroscopy. Academic Press, New York (1979)

    Google Scholar 

  15. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. McGraw-Hill, New York (1971)

    Google Scholar 

  16. Hirata, S., Head-Gordon, M.: Time-dependent density functional theory within the Tamm-Dancoff approximation. Chem. Phys. Lett. 314, 291–299 (1999)

    Article  ADS  Google Scholar 

  17. Rohlfing, M., Louie, S.G.: Electron-hole excitations and optical spectra from first principles. Phys. Rev. B 62, 4927 (2000)

    Article  ADS  Google Scholar 

  18. Casida, M.E.: Time-dependent density-functional theory for molecules and molecular solids. J. Mol. Struc.-THEOCHEM 914, 3–18 (2009)

    Google Scholar 

  19. Peach, M.J.G., Benfield, P., Helgaker, T., Tozer, D.J.: Excitation energies in density functional theory: an evaluation and a diagnostic test. J. Chem. Phys. 128, 044118 (2008)

    Article  ADS  Google Scholar 

  20. Sadovskii, M.V.: Diagrammatics: Lectures on Selected Problems in Condensed Matter Theory. World Scientific Publishing Co Pte Ltd, Singapore (2006)

    Book  Google Scholar 

  21. Lifshitz, E.M., Pitaevskii, L.P.: Statistical Physics: Theory of the Condensed State. Course of Theoretical Physics, vol. 9. Pergamon Press, Oxford (1980)

    Google Scholar 

  22. Hedin, L.: New method for calculating the one-particle green’s function with application to the electron-gas problem. Phys. Rev. 139, A796 (1965)

    Article  ADS  Google Scholar 

  23. Hybertsen, M.S., Louie, S.G.: Electron correlation in semiconductors and insulators: band gaps and quasiparticle energies. Phys. Rev. B 34, 5390 (1986)

    Article  ADS  Google Scholar 

  24. Brar, V.W., Wickenburg, S., Panlasigui, M., Park, C.-H., Wehling, T.O., Zhang, Y., Decker, R., Girit, Ç., Balatsky, A.V., Louie, S.G., Zettl, A., Crommie, M.F.: Observation of carrier-density-dependent many-body effects in graphene via tunneling spectroscopy. Phys. Rev. Lett. 104, 036805 (2010)

    Article  ADS  Google Scholar 

  25. Reining, L., Olevano, V., Rubio, A., Onida, G.: Excitonic effects in solids described by time-dependent density-functional theory. Phys. Rev. Lett. 88, 066404 (2002)

    Article  ADS  Google Scholar 

  26. Foerster, D., Koval, P., Sánches-Portal, D.: An \(O(N^3)\) implementation of Hedin’s GW approximation for molecules. J. Chem. Phys. 135, 074105 (2011)

    Article  ADS  Google Scholar 

  27. Umari, P., Stenuit, G., Baroni, S.: GW quasiparticle spectra from occupied states only. Phys. Rev. B 81, 115104 (2010)

    Article  ADS  Google Scholar 

  28. Giustino, F., Cohen, M.L., Louie, S.G.: GW method with the self-consistent Sternheimer equation. Phys. Rev. B 81, 115105 (2010)

    Article  ADS  Google Scholar 

  29. Lambert, H., Giustino, F.: Ab initio Sternheimer-GW method for quasiparticle calculations using plane waves. Phys. Rev. B 88, 075117 (2013)

    Article  ADS  Google Scholar 

  30. Yam, C., Yokojima, S., Chen, G.: Linear-scaling time-dependent density-functional theory Phys. Rev. B 68, 153105 (2003)

    Article  Google Scholar 

  31. Tretiak, S., Isborn, C.M., Niklasson, A.M.N., Challacombe, M.: Representation independent algorithms for molecular response calculations in time-dependent self-consistent field theories. J. Chem. Phys. 130, 054111 (2009)

    Article  ADS  Google Scholar 

  32. Zuehlsdorff, T.J., Hine, N.D.M., Spencer, J.S., Harrison, N.M., Riley, D.J., Haynes, P.D.: Linear-scaling time-dependent density-functional theory in the linear response formalism. J. Chem. Phys. 139, 064104 (2013)

    Article  ADS  Google Scholar 

  33. Challacombe, M.: Linear scaling solution of the time-dependent self-consistent-field equations. Computation 2, 1–11 (2014)

    Article  Google Scholar 

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  1. Cavendish Laboratory, University of Cambridge, Cambridge, UK

    Tim Joachim Zuehlsdorff

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Zuehlsdorff, T.J. (2015). Approximations to Excited States. In: Computing the Optical Properties of Large Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-19770-8_3

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