  Within the Tamm-Dancoff approximation ab initio approaches describe excitons
as packets of electron-hole pairs propagating only forward in time. However, we
show that in nanoscale materials excitons and plasmons hybridize, creating
exciton--plasmon states where the electron-hole pairs oscillate back and forth
in time. Then, as exemplified by the trans-azobenzene molecule and carbon
nanotubes, the Tamm-Dancoff approximation yields errors as large as the
accuracy claimed in ab initio calculations. Instead, we propose a general and
efficient approach that avoids the Tamm--Dancoff approximation, and correctly
describes excitons, plasmons and exciton-plasmon states.
