Symmetry, arbitrary

Recently, a unitarily invariant decomposition of Hermitian second-order matrices of arbitrary symmetry under permutation of the indices within the row or column subsets of indices has been reported by Alcoba [77]. This decomposition, which generalizes that of Coleman, also presents three components that are mutually orthogonal with respect to the trace scalar product [77] ... 

It must be noted that, due to the arbitrary symmetry under permutation of indices of this second-order matrix, a larger set of contractions into the 0- and 1 -body space must be taken into account. 

In order for the induced dipole moment, ft, to transform as a vector, the spherical harmonics describing the various orientations have to be coupled in an appropriate way. We write the induced dipole components of a system of two molecules of arbitrary symmetry, according to [141]... 

It is an unusual problem to construct a homogeneously distorted two-dimensional domain structure which allows us to cover the range of the hexagonal closely packed crystal up to a two-dimensional gas. For example, the concept of the paracrystal is based on the philosophy that every state of condensed matter has at least a micro-paracrystalline arrangement of segments within a distorted lattice of arbitrary symmetry with a defined coordination number. 

Inserting the expansion eq. (4.40) rewritten in terms of matrices V in the energy expression eq. (4.31) with the perturbed Fock operator eq. (4.39) yields a DMM model of the CC of an arbitrary symmetry since the transition densities V take account of all possible perturbations of the electronic structure, keeping the CLS a separate entity. The series eq. (4.40) in fact appears by expanding the closed expression for the projection operator ... 

Leavitt RP (1980) Erratum An irreducible tensor method of deriving the longrange anisotropic interactions between molecules of arbitrary symmetry [J. Chem. Phys. 72, 3472 (1980)]. J Chem Phys 73 2017-2017... 

Symmetry is a property we find in objects with at least one dimension (D) (1-D symmetry of beads on a string 2-D symmetry of objects in a plane 3-D symmetry of objects in space). Empty space has the most symmetry. In zero dimensions, any symmetry is allowed. An object that does not have to fill space can have any arbitrary symmetry (e.g., no symmetry, or a sevenfold rotation axis). However, if this object must fill 2-D or 3-D space, it must meet certain local symmetry requirements, which, coupled with translational symmetry operators, allows the space to be completely filled. 

Amorphous solids and polycrystalline substances composed of crystals of arbitrary symmetry arranged with a perfectly disordered or random orientation are elastically isotropic macroscopically (taken as a whole). They may be described by nine elastic constants, which may be reduced to two independent (effective) elastic constants. 

S.L. Price, A.J. Stone and M. Alderton, Explicit formulae for the electrostatic energy, forces and torques between a pair of molecules of arbitrary symmetry, Mol.Phys., 52 (1984) 987-1001. 

S. L. Price, A. J. Stone, and M. Alderton, Mol. Phys., 52, 987 (1984). Explicit Formulas for the Electrostatic Energy, Forces and Torques Between a Pair of Molecules of Arbitrary Symmetry. 

The preferred numerical method is then a combination of Newton-Raphson and Picard schemes first proposed by Gillan, together with the Ng method for handling long-range Coulomb potentials. It is a completely general technique and can be used with any closure or potential model to solve any integral equation for molecules of arbitrary symmetry. We will demonstrate the application of the method to the SSOZ equation. For this purpose, it is convenient to write the SSOZ equation in the form... 

This form of the general Jacobian element allows for the straightforward solution of the SSOZ equation for molecules of arbitrary symmetry. However, in the numerical solution using Gillan s methods, most of the computation time is involved in calculating the elements of the Jacobian matrix, rather than in the calculation of its inverse or in the calculation of transforms. Indeed, as the forward and backward Fourier transforms can be carried out using a fast Fourier transform routine, the time-limiting step is the double summation over / and j in Eq. (4.3.36). With this restriction in mind, it is... 

Hunt KLC. Dispersion dipoles and dispersion forces proof of Feynman s conjecture and generalization to interacting molecules of arbitrary symmetry. J Chem Phys 1990 92 1180-1187. 

What are the full generalizations of the formally exact expressions for e we have exhibited in terms of correlation functions when one generalizes the models we have been considering to simultaneously include quadrupoles and higher multipoles, as well as short-range anisotropy of arbitrary symmetry along with a dipolar term and polarizability ... 

Quite probably the answer to the second question will look not too much different from the expressions for the models that have been thoroughly analyzed here, but the establishment of this result may turn out to be tedious. We have seen in Section II how to handle rigid nonpolarizable particles of arbitrary symmetry using the formalism of Hoye and Stell. The addition of fluctuating polarizability has been considered by those authors only for molecules of cylindrical symmetry, but its extension to molecules of arbitrary symmetry is unlikely to raise fundamental problems. On the other hand, particles lacking cylindrical symmetry even in the nonpolarizable case are substantially more awkward to deal with than cylindrically symmetric particles. In treating the constant-polarizability case, Wertheim excludes all permanent multipoles beyond the dipole clearly the quadrupole at least must also be included to provide a realistic model for many real fluids of interest. 

Tensors and are the forth-rank tensors of flexomagnetic and flexoelecttic couplings respectively. Flexoelectric effect exists for arbitrary symmetry. 

MkMi exist in the materials of arbitrary symmetry, since the flexoelectric... 

The two wave functions change phase during certain symmetry operations. We call an arbitrary symmetry operation S. Operating on a wave function ip, we have S and we call X the eigen-... 

The occurrence of electron degeneracy brings two effects. First, the molecular Hamiltonian is invariant with respect to arbitrary symmetry operations, so that the adiabatic potential belongs to the totally symmetric representation (A-type or 2-type). Therefore, only certain combinations of QjQj or QjQjQk are allowed for symmetry coordinates and they must span the totally symmetric representation. Secondly, the interaction matrix U... 

Morphology with Arbitrary Symmetry and Distribution of Length Scales... 

Without going into further details the conclusion from the analysis is that one can indeed obtain both 2 and i correlation functions quite directly from the two responses at least for the simple molecular anisotropy and simple field pulses assumed. Further generalization to molecules of arbitrary symmetry presents problems discussed by Rosato and Williams (75) while other pulse shapes require more complicated treatments than for linear effects with simple superposition principles (from unpublished calculations of the writer). There is also the question of local field effects. 

Voigt [Voigt 1889, 1910] has shown that, under the assumption that the strain inside the material is uniform (isostrain assumption), the effective elastic moduli of a dense (i.e. pore-free) polycrystalline material, e g. a densely sintered ceramic, composed of crystallites of arbitrary symmetry can be ealeulated from the 9 elastic constants (stiffnesses) C, C22, C33,... 

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