This investigation combines the concept of the seniority number (defined as the total number of singly occupied orbitals in a determinant) with the energy renormalization group (ERG) approach to obtain the lowest-energy electronic states on molecular potential energy surfaces. The proposed -ERG method uses Slater determinants that are ordered according to seniority number in ascending order. In the -ERG procedure, the active system consists of M (N-electron) states and K additional complement (N-electron) states (complement-system). Among the M states in the active system the lowest-energy m states represent target states of interest (target-states), thus m M. The environment consists of Full Configuration Interaction (FCI) determinants that represent a reservoir from which the complement-states K are being selected. The goal of the -ERG procedure is to obtain lowest-energy target states m of FCI quality in an iterative way at a reduced computational cost. In general, the convergence rate of -ERG energies towards FCI values depends on m and M, thus, the notation -ERG(m,M) is used. It is found that the -ERG(m,M) method can be very effective for calculating lowestenergy m (ground and excited) target states when a sufficiently large number of sweeps is used. We find that the fastest convergence is observed when M > m. The performance of the -ERG(m,M) procedure in describing strongly correlated molecular systems has been illustrated by examining bond-breaking processes in N2, H8, H2O and C2. The present, proof-of-principle study yields encouraging results for calculating multiple electronic states on potential energy surfaces with near Full CI quality.