In the context of pulsed electron paramagnetic resonance (EPR), the correlation time (τ_c) and an effective parameter, representing the fluctuation of the director of the malonic-acid molecule due to thermal motion in the directions transverse to the molecular symmetry axis, are calculated. These calculations are made using the experimental electronic and nuclear spin relaxation times of an electron-nuclear spin-coupled system (electron spin S=1/2; nuclear spin I=1/2), as determined from echo electron-electron double-resonance (echo-ELDOR) measurements in a γ-irradiated malonic acid single crystal for a specific orientation of the external magnetic field with respect to the crystal axes. To this end, thermal motion of the molecule is considered to cause fluctuations in the values of the g- and A (hyperfine) matrices of the spin system, which are calculated using the model of Frezzato et al. [J. Phys. Chem. B, 108, 9505 (2004)] as functions of the thermal motion of the malonic-acid molecule in the directions transverse to its symmetry-axis. These fluctuations, i.e. the time-dependent variation from the average values of the spin-Hamiltonian parameters, are then used to calculate the four electronic (T_2e) and two nuclear (T_2n) spin-relaxation times, which constitute six duplicates of diagonal elements of the relaxation matrix in Liouville space for the four-level coupled electron-nuclear spin system, as outlined by Lee et al. [J. Chem. Phys. 98, 3665-3689 (1993); hereafter LPF]. The four electronic- and two nuclear-spin relaxation times, are then found to be functions of τ_c, the correlation time, and a fluctuation-limiting factor, h, which, in turn, are estimated from the two experimentally known average values of the electronic and nuclear spin-relaxation times, (T_2e)_exp and (T_2n)_exp, respectively, determined from the spin-echo correlation spectroscopy (SECSY) and echo-ELDOR signals data [LPF]. A rather narrow region of such τ_c and h values is found, giving a theoretical estimate to the values at its center: τ_c=8.9 × 10^-8s and h=0.11, which are then used to calculate the time-dependent echo-ELDOR signal by exploiting the relevant Liouville-von Neumann equation, whose Fourier transform is found to be in very good agreement with that obtained by the experiment.

We suggest an approach using machine learning random forest algorithms to comparing and calibrating the results of calculations of transition energies in organic molecules by ZINDO/S (Zerner's intermediate neglect of differential overlap) and TDDFT (time-dependent density-functional theory) methods. We show how our machine learning model, trained on a relatively small data set can improve the results of semi-empirical methods and obtain absorption spectra comparable to TDDFT calculations.