Establishment of the radial distribution function by solving the Ornstein-Zernike equation is still an important problem, even more than a hundred years after the original paper publication. New strategies and approximations are common in the literature. A crucial step in this process consists in defining a closure relation which retrieves correlation functions in agreement with experiments or molecular simulations. In this paper, the functional Taylor expansion, as proposed by J. K. Percus, is applied to introduce two new closure relations: one that modifies the Percus Yevick closure relation and another one modifying the Hypernetted-Chain approximation. These new approximations will be applied to a hard sphere system. An improvement for the radial distribution function is observed in both cases. For some densities a greater accuracy, by a factor of five times compared to the original approximations, was obtained.

A general kinetic equation to simulate differential scanning calorimetry (DSC) data was employed along this work. Random noises are used to generate a thousand data, which are considered to evaluate the performance of Levenberg-Marquardt (LM) and a Hopfield neural network (HNN) based algorithm in the fitting process. The HNN-based algorithm showed better results for two different initial conditions: exact and approximated values. After this statistical analysis, DSC experimental data at three heating rates for losartan potassium, an antihypertensive drug, was adjusted by the HNN method using different initial conditions to obtain the activation energy and frequency factor. Additionally, it was possible to recover the parameters for the kinetic model with accuracy, showing that the conversion is described by a complex process, once these values do not correspond to any ideal models described in the literature.

The quantum mechanics of two electrons atoms, as suggested by E. A. Hylleraas in the period between 1928-1930, is discussed in the present work. Hylleraas introduced explicit correlations in the wavefunction by treating the coordinates between the electrons as an independent variable of the problem, an approach that gives very precise energy values. Solution of the problem is given for one, two, three and six basis functions, as in the original work. The solution is also carried out with 140 basis functions, showing that the energy for two electron atoms can be calculated with seven significant figures. Numerical methods presented here are accessible to undergraduated level students.