Triangle Configuration Research Articles

A Test of Gregory's Theory of Primary Constancy Scaling

Gregory's well-known account of the visual illusions depends on the observer's misconstruing certain lines as cues for depth. Proving the existence of these 'inappropriate' depth cues has been difficult. It is shown here that Gregory's major example of a 'primary' depth cue --converging lines --can be tested independently of the illusions by using a triangle configuration but that the outcome does not support the theory of primary constancy scaling.

Elimination of a coordinate singularity in the three-body problem

One of the coordinate systems commonly used in the three-body problem consists of three center-of-mass coordinates, three interparticle separations, and three Euler angles specifying the orientation of the triangle whose vertices are the three particles. The usual specification of the Euler angles for this system, which aligns the axes of the body-fixed coordinate system with the principal axes of the moment of inertia tensor, results in a coordinate singularity whenever two of the moments of inertia are equal. An alternative specification of the Euler angles for the equal mass case which treats the three particles symmetrically and eliminates the coordinate singularity at the equilateral triangle configuration is presented.

Geometry of the LiO2 radical

Ab initio quantum mechanical calculations have been carried out for the two lowest electronic doublet states of the lithium dioxide molecule. The results are pertinent to possible crossed molecular beam experiments and to matrix isolation spectroscopy. A qualitative discussion of the electronic structure changes accompanying the Li + O2 and LiO + O reactions is given. For the quantitative calculations, a contracted Gaussian basis set was used, designated Li (9s 4p/4s 2p), O (9s 5p/4s 3p). For isosceles triangle configurations, the 2A2 state is the electronic ground state, with equilibrium geometry r(LiO) = 1.82 Å and θ(O–Li–O) = 44.5°. The 2B2 state is predicted to lie 14 kcal/mole higher with r(LiO) = 1.76 Å and θ(O–Li–O) = 46.5°. For C∞v geometry the 2Π state bond distances were predicted, R(Li–O) = 1.62 Å and R(O–O) = 1.35 Å. There appears to be little or no barrier between the C2v and C∞v forms.

Localized orbitals for arbitrary molecular wave functions

Localized orbitals are defined for arbitrary wave functions for atoms or molecules with even or odd numbers of electrons by making a particular non-orthogonal transformation among the natural orbitals which preserves the diagonal nature of the electron density. Different definitions of localized orbitals are obtained by relating them to different approximate forms for the wave function. The equilateral triangle configuration of H 3 is treated as an example.