A single Lanczos propagation method for calculating transition amplitudes

A method using only a single Lanczos propagation to determine multiple transition amplitudes without the explicit calculation of the eigenstates is proposed. Comparing with methods requiring multiple propagations, this method is very attractive for large dimensional problems since the propagation is usually the most computationally intensive step. When overlaps between eigenstates and pre-specified quantum states are of interest, it is shown that the “spurious” Lanczos eigenvalues cannot be simply deleted as done in the Cullum–Willoughby procedure. Practical procedures for calculating the overlaps are provided, which take into consideration numerical behaviors of the Lanczos algorithm in finite precision arithmetic, such as the loss of global orthogonality and emergence of “spurious” eigenvalues. Numerical tests in a realistic triatomic system confirm the accuracy of the present method.