Calculation of absolute spectral densities via stochastic estimators of tr{δ(E - Ĥ)}

The calculation of absolute vibrational spectral densities, tr{δ(E - Ĥ)}, is investigated utilizing the stochastic trace estimator technique of Hutchinson. The spectral density is evaluated by a Monte Carlo scheme in which random vectors are sequentially sampled, their spectral density profiles computed and averaged. The requisite matrix elements of δ(E - Ĥ) are evaluated using a Lanczos projection algorithm. The issue of distinguishing degenerate and replicated eigenvalues generated by the Lanczos algorithm is addressed and can be overcome using a recently-developed filter diagonalization scheme. The resulting method is simple, efficient and converges the density of states remarkably quickly for dense spectra. Illustrative calculations are presented for one- and two-dimensional test cases and finally for nitrogen dioxide in the energy range 0–12000 cm−1 using the V11 diabatic surface of Hirsch et al.