  Vibrational properties of ultralong Rydberg Cs$_2$ molecules are investigated
on corresponding potential curves obtained by perturbation theory. The Rydberg
Cs$_2$ molecules are associated by a Rydberg Cs($nS/nP$) atom $(n=30-70)$ and a
ground state Cs($6s$) atom. The starting point for the perturbation treatment
of corresponding Rydberg molecular potential curves is to generate accurate
atomic Rydberg states from realistic {\it ab initio} effective core potential.
The calculated results have similar characteristics with available experimental
and theoretical investigations on Rydberg Rb$_2$ molecules. And this is the
first time that Rydberg molecules are studied at the {\it ab initio} level.

Atomic Physics