  Linear response time dependent density functional theory (TDDFT), which
builds upon configuration interaction singles (CIS) and TD-Hartree-Fock (TDHF),
is the most widely used class of excited state quantum chemistry methods and is
often employed to study photochemical processes. This paper studies the
behavior of the resulting excited state potential energy surfaces beyond the
Coulson-Fisher (CF) point in single bond dissociations, when the optimal
reference determinant is spin-polarized. Many excited states exhibit sharp
kinks at the CF point, and connect to different dissociation limits via a zone
of unphysical concave curvature. In particular, the unrestricted M$_S=0$ lowest
triplet T$_1$ state changes character, and does not dissociate into ground
state fragments. The unrestricted $M_S=\pm 1$ T$_1$ CIS states better
approximate the physical dissociation limit, but their degeneracy is broken
beyond the CF point for most single bond dissociations. On the other hand, the
$M_S=\pm 1$ T$_1$ TDHF states reach the asymptote too soon, by merging with the
ground state from the CF point onwards. Use of local exchange-correlation
functionals causes $M_S=\pm 1$ T$_1$ TDDFT states to resemble their unphysical
$M_S= 0$ counterpart. The 2 orbital, 2-electron model system of minimal basis
H$_2$ is analytically treated to understand the origin of these issues,
revealing that the lack of double excitations is at the root of these
remarkable observations. The behavior of excited state surfaces is also
numerically examined for species like H$_2$, NH$_3$, C$_2$H$_6$ and LiH in
extended basis sets.
