A direct-operation Lanczos approach for calculating energy levels

A direct-operation Lanczos method is suggested for the variational calculation of polyatomic vibrational levels. It exploits the simple structure of the polynomial basis set representations of the operators from which many Hamiltonian operators are composed. A key feature of the method is that no matrix elements and no matrix-vector products are computed but rather the action of the Hamiltonian on a vector is evaluated directly. The cost of the calculation therefore scales linearly with the basis size. The relative efficiency of the approach depends on the number of terms in the Hamiltonian and on the number of factors in each term. The method is tested both in one (Morse oscillator) and in three (SO2) dimensions and is compared to the discrete variable representation-Lanczos approach. The direct method is the most efficient for the one-dimensional case regardless of the basis size. It is also the most efficient in three dimensions when the basis size exceeds a certain minimal size.