Symmetry, potential

Kozlowski, P. M., Rauhut, G., Pulay, P., 1995, Potential Symmetry Breaking, Structure and Definite Vibrational Assignment for Azulene Multiconfigurational and Density Functional Results , J. Chem. Phys., 103, 5650. 

Much more subtle are those molecules which exhibit only potential symmetry. Usnic acid (21), for instance, which does not exhibit any kind of symmetry, can be synthesised in two steps by the unsymmetrical coupling of two molecules of the same phenoxy radical 19 (Scheme 4.4) [17], which is prepared from readily accessible starting materials 13 and 14. 

Bertz [5] has drawn the attention to the relationship existing between the concept of potential symmetry and the symmetry present in the corresponding "synthesis tree" (or "synthesis graph") [18]. In the abovementioned synthesis of usnic acid both branches of the synthesis tree are identical, and collapse into a single path, a fact that is most easily visualised by comparing synthetic trees for labelled (a) and unlabelled (b) usnic acid (see Diagram 4.1). 

In this chapter, dielectric response of only isotropic medium is considered. However, in a local-order scale, such a medium is actually anisotropic. The anisotropy is characterized by a local axially symmetric potential. Spatial motion of a dipole in such a potential can be represented as a superposition of oscillations (librations) in a symmetry-axis plane and of a dipole s precession about this axis. In our theory this anisotropy is revealed as follows. The spectral function presents a linear combination of the transverse (K ) and the longitudinal (K ) spectral functions, which are found, respectively, for the parallel and the transverse orientations of the potential symmetry axis with... 

The first and second terms in the right-hand part are, respectively, the transverse and longitudinal components of the spectral function. In other words, these terms are stipulated by reorientation of the projections of a dipole moment, which are, respectively, normal and collinear to the potential symmetry axis. The potential under consideration comprises two wells with oppositely directed symmetry axes. Such is the cosine-squared potential... 

In this approach, the principal planes of the environment are taken as the best approximation of the potential symmetry planes, and the asymmetry of the environment as seen from the considered / th atom is defined as proportional to the distance from / to the symmetry plane. 

As we noted above, in fact calculations on stable molecules usually do generate canonical Hartree-Fock orbitals which have the symmetry of the molecular framework. It seems likely that this is a contingent, numerical fact rather than one of principle. If the one-electron terms in the Hartree-Fock Hamiltonian are so large as to dominate the form of the molecular orbitals, these MOs will, presumably, take up a distribution which optimises the energy due to the dominant terms in the Hamiltonian, and the electron-repulsion terms involving the potentially symmetry-breaking effects are too weak to chauige this situation. In molecules, radicals or (particularly) anions which have very weakly bound electrons, one would expect to see the effect of broken symmetry. 

With periodic boundary condition, /l(v) has the potential symmetry. Thus if the potential is symmetric the integrand in Eqn (h) is antisymmetric and the effective potential is periodic Wgff( /) = Wefi(0) for integer n. The motor therefore is flat on large scales and cannot generate motion. 

One-dimensional velocity distribution Specific conductivity, hard-sphere diameter for a collision, length parameter in Lennard-Jones potential, symmetry number of a molecule Intrinsic lifetime of a photoexcited state Azimuthal angular velocity in spherical polar coordinates, azimuthal angle in spherical polar coordinates, angle of deflection Quantum yield at wavelength A Fluorescence (phosphorescence) quantum efficiency... 

Table 9.5. Dependence of proton tunnelling frequency on conformation of the system and potential symmetry in Eqs. (9.11)-(9.13), according to Ref. [75]...

Scheme 2.60) (Luo and De Brabander 2015). These adducts 329 were transformed into chiral 3-suhstituted 3,4-dihydroquinoxalin-2(l//)-ones 330 (Scheme 2.61) upon trifluoroacetic acid-mediated Boc deprotection/in situ cyclodehydration. Importantiy, the entire two-step procedure occurs without racemization, even with racemization-prone arylglycine substrates. This approach uniquely avoids potential symmetry-related limitations in the substrate scope compared to previous approaches to enantiomerically pure 3-aryl-3,4-dihydroquinoxalin-2(lfl)-ones that emanate from symmetrical dianilines. 

The occurrence of resonances on potential ridges has obvious relevance for the "transition state" of chemical reactions. This is the state of a transition complex as it traverses the activation threshold (i.e., the ridge) that separates the initial and final stable configurations of the reactants (Fig. 5). Indeed studies of the complex wavefunction in coordinates appropriate to the potential symmetry on the ridge have attracted attention recently accounting, e.g., for a remarkable concentration of vibrational energy in the output of the reaction H2+F->-H+HF. ... 

In terms of the elemental potential, symmetry requires that U = U(t), whereas the reorientation induced asymmetry requires U = U(t,x)- Thus, it may be that the... 

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