  We investigate polyatomic ultralong-range Rydberg molecules consisting of
three ground state atoms bound to a Rydberg atom via $s$- and $p$-wave
interactions. By employing the finite basis set representation of the
unperturbed Rydberg electron Green's function we reduce the computational
effort to solve the electronic problem substantially. This method is
subsequently applied to determine the potential energy surfaces of triatomic
systems in electronic $s$- and $p$-Rydberg states. Their molecular geometry and
resulting vibrational structure are analyzed within an adiabatic approach that
separates the vibrational bending and stretching dynamics. This procedure
yields information on the radial and angular arrangement of the nuclei and
indicates in particular that kinetic couplings between bending and stretching
modes induce a linear structure in triatomic $l=0$ ultralong-range Rydberg
molecules.
