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.

What is an ill-posed inverse problem? The answer to this question is the main objective of the present paper and the pre-requisite to follow the material requires only elementary calculus. The first mathematical formulation of an inverse problem, due to N. H. Abel, together with the fundamental work by Jacques Hadamard, are explored at the beginning of the paper. A prototype system is used to consider the regularization concept. Three numerical methods, the Tikhonov regularization, the decomposition into singular values and the Hopfield neural networks, applied to remove the singularity are examined. General aspects of the ill-posed inverse problems in chemistry with emphasis in thermodynamics and a set of general rules for other areas of science are also analyzed.

In this work, a simple derivation of the variable amplitude method using the variation of parameters to solve a differential equation is presented. The variable amplitude method was originally devised by Tikochinsky in 1977, using the quantum theory of scattering. The method is applied to two model potentials, the rectangular potential barrier and the Eckart potential, both with analytical solutions for the reflection coefficient. Numerical results will be compared with the exact values for several energies. The problem of calculating the reflection coefficient, usually involving extensive algebra as described in several textbooks, is reduced to solving a first order differential equation with initial condition. The method is very simple to apply, representing an attractive tool for teaching introductory quantum mechanics. A simple computer code is available from which reflection coefficients for the Eckart potential can be calculated.

Este artigo apresenta a dedução e aplicações da equação da fase variável para o caso de um canal no espalhamento quântico. Esta abordagem foi apresentada pela primeira vez em 1933 por Morse e Allis e baseia-se numa modificação da equação Schrödinger para uma equação diferencial de primeira ordem, adequada para o problema de espalhamento. A dependência do deslocamento de fase com o momento angular e a energia, juntamente com o teorema de Levinson, é discutida. A equação resultante do método da fase variável é de fácil programação e pode ser explorado em cursos introdutórios de mecânica quântica.

This paper presents the derivation and applications of the variable phase equation for single channel quantum scattering. The approach was first presented in 1933 by Morse and Allis and is based on a modification of the Schrödinger equation to a first order differential equation, appropriate to the scattering problem. The dependence of phase shift on angular momentum and energy, together with Levinson's theorem, is discussed. Because the variable phase equation method is easy to program it can be further explored in an introductory quantum mechanics course.

Este trabalho concentra-se no cálculo do segundo coeficiente virial quântico, a partir de um potencial desenvolvido recentemente. Este coeficiente foi determinado com 4-5 algarismos significativos na faixa de temperatura de 3 a 100 K. Nossos resultados estão dentro do erro experimental. Três contribuições para o valor total deste coeficiente são o espalhamento quântico (contribuição de estados no contínuo), o estado ligado (contribuição de estados discretos) e o gás ideal quântico; discutimos estas contribuições separadamente. A contribuição mais importante é do espalhamento quântico, enquanto que as contribuições menores são dos estados discretos. Uma análise da sensibilidade foi realizada em função da temperatura para um parâmetro na região de curto alcance do potencial e para três parâmetros na região de longo alcance do potencial. Para ambas as temperaturas consideradas, 10 e 100 K, o coeficiente de dispersão C6 foi o mais significativo, e o termo dispersão C10 foi o menos significativo para o resultado total. Em geral, a precisão exigida para descrever os potenciais diminui com o aumento da temperatura. A precisão total e a relação dos parâmetros com os erros experimentais são discutidas.

This paper focuses on the calculation of the quantum second virial coefficient, under a recently developed potential. This coefficient was determined to within 4-5 significant figures in the temperature range from 3 to 100 K. Our results are within experimental error. The three contributions to the overall value of the coefficient are the quantum scattering (continuum state contribution), the bound state (discrete state contribution) and the quantum ideal gas; we discuss these contributions separately. The most significant contribution is from the scattering states, whereas the smaller contributions are from the discrete states. A sensitivity analysis was performed as a function of temperature for one parameter in the short-range region of the potential and for three parameters in the long-range regions of the potential. For both temperatures considered, 10 and 100 K, the C6 dispersion coefficient was the most significant, and the C10 dispersion term was the least significant to the overall result. In general, the precision required to describe the potential decays as the temperature increases. The overall accuracy and the relationship of the parameters to the experimental errors are discussed.

A hundred years ago, a twenty-eight year old Danish scientist published a series of three papers in which electron motion was quantized. The Bohr atomic model is surely known by every chemistry student. Nevertheless in this same 1913 trilogy, Bohr studied atoms with several electrons as well as molecules. Chemistry students, in general, are not aware of the Bohr molecule. The present paper aims at rescuing this important classical model. A review of the Bohr atomic model for both one and several electrons is discussed, together with a theoretical presentation of the Bohr molecule.

Methane combustion was studied by the Westbrook and Dryer model. This well-established simplified mechanism is very useful in combustion science, for computational effort can be notably reduced. In the inversion procedure to be studied, rate constants are obtained from [CO] concentration data. However, when inherent experimental errors in chemical concentrations are considered, an ill-conditioned inverse problem must be solved for which appropriate mathematical algorithms are needed. A recurrent neural network was chosen due to its numerical stability and robustness. The proposed methodology was compared against Simplex and Levenberg-Marquardt, the most used methods for optimization problems.

Potential parameters sensitivity analysis for helium unlike molecules, HeNe, HeAr, HeKr and HeXe is the subject of this work. Number of bound states these rare gas dimers can support, for different angular momentum, will be presented and discussed. The variable phase method, together with the Levinson's theorem, is used to explore the quantum scattering process at very low collision energy using the Tang and Toennies potential. These diatomic dimers can support a bound state even for relative angular momentum equal to five, as in HeXe. Vibrational excited states, with zero angular momentum, are also possible for HeKr and HeXe. Results from sensitive analysis will give acceptable order of magnitude on potentials parameters.

A methodology for ion-polymer interaction estimation is discussed in the present work. This method is based on the inversion of experimental spin echo NMR data using Hopfield neural network to retrieve transverse relaxation time distributions. The adopted model systems consist of aqueous solutions of poly (ethylene oxide), molar mass 1500, 4000 and 35000 γ mol-1 and sodium nitroprusside (NP) at different concentrations. Dipolar interaction is investigated in this work through the reduction in the transverse relaxation time and increase in the area under the distribution curves of the PEO protons which presented a linear correlation with the NP concentration. Neural network results were compared with the Simplex optimization procedure and experimental NMR values. The proposed methodology is robust, stable, non restrictive in relation to the system and more efficient to handle experimental data.

Analytical solutions of a cubic equation with real coefficients are established using the Cardano method. The method is first applied to simple third order equation. Calculation of volume in the van der Waals equation of state is afterwards established. These results are exemplified to calculate the volumes below and above critical temperatures. Analytical and numerical values for the compressibility factor are presented as a function of the pressure. As a final example, coexistence volumes in the liquid-vapor equilibrium are calculated. The Cardano approach is very simple to apply, requiring only elementary operations, indicating an attractive method to be used in teaching elementary thermodynamics.

The classical interpretations of Nicolas Léonard Sadi Carnot on some physical principles involved in the operation of heat engines were fundamental to the development and formulation of the Second Law of Thermodynamics. Moreover, an accurate historical survey clearly reveals that Carnot was, by that time, also well aware about some new concepts, which were further worked out by other scientists to lead to what was, some time later, known as the mechanical equivalent of heat and the conservation of energy. Benoit Paul Émile Clapeyron recognized these original concepts in the first of Carnot´s monographs, published in 1824, but no explicit citation is found in any post-Carnot classical texts dealing with the First Law of Thermodynamics, including those by Julius Robert Mayer, James Prescott Joule and Hermann Ludwig Ferdinand von Helmholtz. The main objective of the present work is to point out some historical evidences of the pioneering contribution of Carnot to the modern concept of the First Law of Thermodynamics.