Analytic evaluation of Coulomb integrals for one, two and three-electron distance operators

<p> The state of the art for integral evaluation is that analytical solutions to integrals are far more useful than numerical solutions. We evaluate certain integrals analytically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the (n,m)=(0,0) case is trivial, the (n,m)=(1,0) and (0,1) cases are well known, fundamental milestone in integration and widely used in computation chemistry, as well as based on Laplace transformation with integrand exp(-a²t²). The rest of the cases are new and need the other Laplace transformation with integrand exp(-a²t) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations. </p>

Analytic evaluation of Coulomb integrals for one, two and three-electron distance operators

<p> The state of the art for integral evaluation is that analytical solutions to integrals are far more useful than numerical solutions. We evaluate certain integrals analytically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the (n,m)=(0,0) case is trivial, the (n,m)=(1,0) and (0,1) cases are well known, fundamental milestone in integration and widely used in computation chemistry, as well as based on Laplace transformation with integrand exp(-a²t²). The rest of the cases are new and need the other Laplace transformation with integrand exp(-a²t) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations. </p>

Developmental Changes in Laryngeal and Respiratory Function With Variations in Sound Pressure Level

The development of the speech production system was investigated using a crosssectional design that included children aged 4–14 years and adults. Given that the size and internal structure of the laryngeal and respiratory systems differ between children and adults, it was predicted that children would show functional distinctions from adults during speech. Aerodynamic, acoustic, and respiratory kinematic techniques were used to assess laryngeal and respiratory function while participants varied their sound pressure level. In general, the aerodynamic and acoustic results show that men and 14-year-old boys function differently than women and all other groups of children. For the respiratory function data, children's values are similar to adults' by the time they are 12–14 years of age. These changes correspond closely to developmental laryngeal and respiratory anatomic data. All participants used a combination of laryngeal and respiratory mechanisms to increase sound pressure level, but the combination of mechanisms differed across age groups. These data emphasize that the laryngeal and respiratory behavior of children is not easily predicted from an adult model.