The Rayleigh-Ritz (RR) procedure, including orthogonalization, constitutes a major bottleneck in computing relatively high-dimensional eigenspaces of large sparse matrices. Although operations involved in RR steps can be parallelized to an extent, their parallel scalability, limited by some inherent sequentiality, is lower than dense matrix-matrix multiplications. The primary motivation of this paper is to develop a methodology that reduces the use of the RR procedure in exchange for matrix-matrix multiplications. We propose an unconstrained penalty model and establish its equivalence to the eigenvalue problem. This model enables us to deploy gradient-type algorithms heavily dominated by dense matrixmatrix multiplications. Although the proposed algorithm does not necessarily reduce the total number of arithmetic operations, it leverages highly optimized operations on modern high performance computers to achieve parallel scalability. Numerical results based on a preliminary implementation, parallelized using OpenMP, show that our approach is promising.