Quantum mechanics of particles constrained to spiral curves with application to polyene chains.

CONTEXT: Due to advances in synthesizing lower-dimensional materials, there is the challenge of finding the wave equation that effectively describes quantum particles moving on 1D and 2D domains. Jensen and Koppe and Da Costa independently introduced a confining potential formalism showing that the effective constrained dynamics is subjected to a scalar geometry-induced potential; for the confinement to a curve, the potential depends on the curve's curvature function. METHOD: To characterize the π electrons in polyenes, we follow two approaches. First, we utilize a weakened Coulomb potential associated with a spiral curve. The solution to the Schrödinger equation with Dirichlet boundary conditions yields Bessel functions, and the spectrum is obtained analytically. We employ the particle-in-a-box model in the second approach, incorporating effective mass corrections. The π - π ∗ transitions of polyenes were calculated in good experimental agreement with both approaches, although with different wave functions.

Rotovibrational states of the water molecule on the sun.

The infrared spectrum of water observed in sunspots is complex and dense, with bands separated by approximately 0.01 cm-1. For top asymmetrical molecules, there is no theoretical approach that allows for the calculation of rotovibrational energy with such precision. Experimentally derived rotovibracional energy levels of water at high temperatures combined with variational calculations have been used for the band assignments. These energy levels are employed to refine the analysis of a small portion of the infrared absorption spectrum. Such procedure has allowed for the identification of additional 55 bands to the 70 already identified as rotovibrational transitions of the water molecule. Our new assignments, which include pure and cross transitions, offer additional evidence of the existence of water on the sun, but above all they illustrate the complexity of the solar spectrum that involves states with higher levels of rotational excitation. Given the conditions on the sun, more molecules of water would occur in excited electronic states, which include apolar and paramagnetic states, generating intense bands in the spectrum. Since there is an analytical solution for the rotovibrational transitions of linear molecules, we were able to identify 16 bands relative to the excited electronic states 1B2 and 3A1 in the sunspot spectrum. Density functional B3LYP/AUG-cc-pVTZ calculations of the electric and magnetic dipole are employed to discuss some consequences of the presence of excited states of water in the dynamics of sunspots and solar magnetic field.