Appearance potential symmetry analysis

Here g-j)presents the /ill -component in the /th well and A, is the temperature dependent Boltzmann population of the /th well. Analysis of the spectra shows that the tetragonal deformation appears abruptly at Tc= 191 K corresponding to the separation energy 8 li2(3) 130 cm-1. 8 1 2(3) gradually increases to 150 cm-1 as the temperature decreases to T 60 K. The corresponding adiabatic potential calculated for the vibronic parameters of cubic symmetry to which the Ae cos cp component (Ae = 8 1 2(3)) is added, is shown in Ref. [20]. The energy difference between the deepest and the two higher wells is equal to 140 cm-1 and the potential barrier between the wells is about 400 cm-1. 

Recently Pack, Wang and Rein51) published a convergence analysis of analytical expansions of the electrostatic potential on parallel lines to the present one. These authors compare with the exact expansion the one-center one and a segmental atomic expansion (centered at the nuclei). Convergence is tested on pyridine (semiempirical iterative extended Hiickel wave function) along the symmetry axis with expansion truncated after the octopole term. Their results are comparable to those reported here in particualr, the segmental expansion appears quite reasonable)). 

The term used in the stochastic literature for these models is intrinsic random fields (IRFs) they appear under that name in textbooks of geostatistics (e.g. Cressie, 1991 Chiles and Delfiner, 1999). This chapter also introduces the intrinsic warps (IWs) intended to replace my principal and partial warps of 1989 for data analysis of landmark-like configurations under symmetries of scale. These IWs could well implement the movable potentially homologous characters that may someday underlie computer-assisted algorithms for stable ordinations or ecophenotypy surveillance much better than existing techniques could likely do. 

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