chapter  18
24 Pages

Electronic Excitation Energies of Molecular Systems from the Bethe–Salpeter Equation: Example of the H2 Molecule

ByExample of the H Molecule Elisa Rebolini, Julien Toulouse, and Andreas Savin

Time-dependent density functional theory (TDDFT)[1] within the linear response formalism[2-4] is nowadays the most widely used approach to the calculation of electronic excitation energies of molecules and solids. Applied within the adiabatic approximation and with the usual local or semilocal density functionals, TDDFT

18.1 Introduction .................................................................................................. 367 18.2 Review of Green’s Function Many-Body Theory ......................................... 368

18.2.1 One-Particle Green’s Function ......................................................... 368 18.2.2 Two-Particle Green’s Function ......................................................... 369 18.2.3 Dyson Equation ................................................................................ 370 18.2.4 Bethe-Salpeter Equation .................................................................. 372 18.2.5 Hedin’s Equations ............................................................................. 372 18.2.6 Static GW Approximation ................................................................ 374

18.3 Expressions in Finite Orbital Basis .............................................................. 375 18.3.1 Spin-Orbital Basis ............................................................................. 375 18.3.2 Spin Adaptation ................................................................................ 377

18.4 Example of H2 in a Minimal Basis ............................................................... 379 18.4.1 BSE-GW Method Using the Noninteracting Green’s Function ....... 379 18.4.2 BSE-GW Method Using the Exact Green’s Function....................... 383

18.4.2.1 Independent-Particle Response Function ........................... 383 18.4.2.2 Excitation Energies ............................................................ 385

18.5 Conclusion .................................................................................................... 387 Acknowledgments .................................................................................................. 388 References .............................................................................................................. 388

gives indeed in many cases excitation energies with reasonable accuracy and low computational cost. However, several serious limitations of these approximations are known, for example, for molecules, too low charge-transfer excitation energies,[5] lack of double excitations,[6] and wrong behavior of the excited-state surface along a bond-breaking coordinate (see, e.g., reference [7]). Several remedies to these problems are actively being explored, including long-range corrected TDDFT,[8, 9] which improves charge-transfer excitation energies; dressed TDDFT,[6, 10, 11] which includes double excitations; and time-dependent density-matrix functional theory (TDDMFT),[12-16] which tries to address all these problems.