A high coordination lattice model for simulating coarse-grained rotational isomeric state (RIS) chains has been under development recently. Initially, the model was developed for chains with symmetric torsional potential energy functions, E(phi)=-E(phi). A single-bead move Monte Carlo algorithm was used and found to be effective in simulating polyethylene chains. A modification was subsequently developed to allow for the simulation of chains with an asymmetric torsional potential, E(phi)not equal-E(phi). The single-bead move Monte Carlo (MC) algorithm employed previously was found to be ineffective following this modification. Similar kinetic effects have been seen previously with single-bead MC moves on the cubic lattice, which lead to the Hilhorst-Deutch modification (two-bead crankshaft MC move) of the Verdier-Stockmayer (single-bead move MC) algorithm. A reptation MC move applied to this model appears problematic. A multiple-bead MC move is developed using the pivot algorithm formalism in order to avoid the lattice model specific kinetic effect seen with only single-bead MC moves. This allows for the effective simulation of vinyl polymers with asymmetric torsional potentials such as polypropylene and polyvinyl chloride. Polypropylene (PP) and poly(vinyl chloride) (PVC) chains of varying stereochemical structure are simulated. The chains are found to relax with reasonable efficiency. Polypropylene and polyvinyl chloride chains are reverse mapped back to the fully atomistic description. The solubility parameters of the reverse-mapped atomistic structures are found to agree reasonably well with experimental values.