Linear symmetry

Labels refer to the notation of A-orbitals and (A, S)-terms appropriate for linear symmetry (Doon)- This has the same consequencies for d-orbitals as the trigonal symmetry D31,. 2 is orbitally non-degenerate and 77 and A are doubly degenerate. This is also true for if formed from d-electrons, whereas it could split in the general case of Dsh. 

The notion of a group is a natural mathematical abstraction of physical symmetry. Because quantum mechanical state spaces are linear, symmetries in quantum mechanics have the additional structure of group representations. Formally, a group is a set with a binary operation that satisfies certain criteria, and a representation is a natural function from a group to a set of linear operators. 

Singer Geometry Plane and Fancy. Singer Linearity, Symmetry, and... 

Linearity, symmetry, and prediction in the hydrogen atom / Stephanie Frank Singer. 

The physical properties of the thallium halides have been tabulated.1 Crystalline T1F has a distorted rock-salt lattice, with two independent thallium sites.333-335 The thermodynamic properties of the solid have been reassessed.336 At high temperature, the vapor phase contains TLF and T12F2 molecules,337 and matrix isolation studies have demonstrated the presence of both T1F and T12F2 in the solid the dimers have linear symmetry.338... 

That concludes the description of the input. To do a calculation on the lowest state of 2n symmetry we can use the same input, except that the symmetry of the wave function is now 2 (or 3). We can then include an extra input which forces the n orbitals of symmetry 2 and 3 to be identical. If we do not do that the wave function may not have linear symmetry. However, since we are also going to do calculations for non-linear geometries, we might prefer not to force linear symmetry, since that leads to an energy that does not vary in a continuous way as a function of the bending angle. 

Contrary to the usual situation in hexahalide complexes361 where it is sufficient, to a very good approximation, to consider one-electron energies, it is absolutely necessary to consider the quantum-numbers (written with capital letters, such as S, A, Q in linear symmetries replacing S, L, J in the spherical symmetry of monatomic entities) of the many-electron wave-function32,62) in the case of the uranyl ion. However, it is also necessary to start with the orbitals serving as raw material in L.C. A.O. descriptions, such as the empty shells of the central atom in a picture based on uranium(VI) and oxygen)-II) ... 

Table 1. M.O. energies (all in eV = 8065.48 cm-1) calculated for the uranyl ion, and in the first column, monatomic U+2 and oxygen atoms. The half-numbered quantum number is a> of linear symmetries. The two last columns give M.O. energies, neglecting effects of spin-orbital coupling, and hence characterized by X and parity...

Tables 1 and 2 assume the strong-field limit for configurations of d electrons in ligand fields (16). In Table 1, all ground states have been considered which are likely to occur in tetrahedral and octahedral symmetries. Table 2 was originally intended for D 5 symmetries appropriate to the metallocenes, but it can also be used for Dn where n > 5, and (with some change of notation) for linear symmetries.

Systems of molecules with linear symmetry can be viewed as a special application of spherical-polar coordinates. Placing the polar axis along the molecular axis of symmetry allows one to average over the angle 0. The spatial distribution function then becomes a two-dimensional function, g(r,6), which can be much more readily calculated and visualized (for an example, see Figure 1). 

In the present paper AOM is reparameterized for the case when the central ion-to-ligand bonds have linear symmetry and a comparison with the conventional perturbation model is made. It is shown that an electrostatic contribution to the semiempirical parameters can never be evaluated from experiments determining energy parameters. Further, some of the problems associated with the zero points for the energy (72) are discussed [Ref. [13) p. 130]. 

In mesophase C with its linear symmetry the conditions are to some extent similar to those in phase D, but they present well defined differences. If it is assumed that, as in the two other linear phases, there is a coherent double layer of amphiphilic molecules, this will result in a decrease in the layer thickness and in a rapid increase in the area per hydrophilic group when the water content of the phase rises. In addition, the interplanar distance increases with the water content only one half as rapidly as in the D phase. This indicates a difference in the arrangement of... 

It is apparent that the symmetry of a sphere or cylinder is such that any angle will make Eqs. 3.23 a linear symmetry transformation. The matrix of this transformation is ... 

THE APPLICATION OF THE ONE-DIMENSIONAL OR LINEAR SYMMETRY GROUPS TO THE SYSTEMATIZATION OF LINEAR STRUCTURE SERIES AND THEIR REPRESENTATIVES... 

Figure 1. Examples of zigzag lines, numeric codes and linear symmetry groups for structures with f2 T (left) and Cl = 2 (right) a) Example of a homogeneous structure b - d) Inhomogeneous structures of different construction.

Appendix Short review of linear symmetry groups... 

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