A sequential strategy for searching out the eigenvalues and eigenvectors of a real symmetricHamiltonianmatrix byGeneticAlgorithm (GA) is proposed and its workability demonstrated. The fitness landscape is generated from the gradient of the Rayleigh quotient (RQ) for the state concerned in such a way that the achievement of fitness maximum condition isolates the corresponding eigenvalue and eigenvector . An avenue for computing the excited eigenvalues is suggested and an inverse iteration scheme is invoked to accelerate the convergence. The non-linear problem of diagonalization of the Hamiltonian with simultaneous optimization of the basis set is explored. © 2009 Springer-Verlag Berlin Heidelberg.