Coordinate symmetry properties

The most significant symmetry property for the second-order nonlinear optics is inversion synnnetry. A material possessing inversion synnnetry (or centrosymmetry) is one that, for an appropriate origin, remains unchanged when all spatial coordinates are inverted via / —> - r. For such materials, the second-order nonlmear response vanishes. This fact is of sufficient importance that we shall explain its origm briefly. For a... 

Let us discuss further the pemrutational symmetry properties of the nuclei subsystem. Since the elechonic spatial wave function t / (r,s Ro) depends parameti ically on the nuclear coordinates, and the electronic spacial and spin coordinates are defined in the BF, it follows that one must take into account the effects of the nuclei under the permutations of the identical nuclei. Of course. 

The symmetry properties of a fundamental vibrational wave function are the same as those of the corresponding normal coordinate Q. For example, when the C3 operation is carried out on Qi, the normal coordinate for Vj, it is transformed into Q[, where... 

Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. 

The definitions are here given under the assumption that the wave function XP is either antisymmetric or symmetric for a trial function without symmetry property, one has to replace the binomial factor NCV before the integrand by a factor l/p and sum over the N(N—l). . . (N—p+l) possible integrals which are obtained by placing the fixed coordinates x, x 2,. . ., x P in various ways in the N places of the first factor W and the fixed coordinates xv x2,. . xv similarly in the second factor W. By using Eq. II.8 we then obtain... 

When one of the cartesian coordinates (i.e. x, y, or z) of a centrosymmetric molecule is inverted through the center of symmetry it is transformed into the negative of itself. On the other hand, a binary product of coordinates (i.e. xx, yy, zz, xz, yz, zx) does not change sign on inversion since each coordinate changes sign separately. Hence for a centrosymmetric molecule every vibration which is infrared active has different symmetry properties with respect to the center of symmetry than does any Raman active mode. Therefore, for a centrosymmetric molecule no single vibration can be active in both the Raman and infrared spectrum. 

Coordinates such as these, which have the symmetry properties of the point group are known as symmetry coordinates. As they transform in the same manner as the IRs when used as basis coordinates, they factor the secular determinant into block-diagonal form. Thus, while normal coordinates most be found to diagonalize the secular determinant, the factorization resulting horn the use of symmetry coordinates often provides considerable simplification of the vibrational problem. Furthermore, symmetry coordinates can be chosen a priori by a simple analysis of the molecular structure. 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

This is not as useful as Eq. (19.4) because products of different coordinates appear in the second term. However, the symmetry properties of this term ensure the existence of a coordinate system in which the cross-terms can be eliminated and the nuclear Hamiltonian reduces to a sum of harmonic oscillator terms ... 

In the first term, Uc, usually called the Coulombic term, the initially excited electron on D returns to the ground state orbital while an electron on A is simultaneously promoted to the excited state. In the second term, called the exchange term, Liex, there is an exchange of two electrons on D and A. The exchange interaction is a quantum-mechanical effect arising from the symmetry properties of the wavefunctions with respect to exchange of spin and space coordinates of two electrons. 

Table 11.3 General classification of nuclides significance of parity is related to symmetry properties of nnclear wave fnnctions. A nuclide is said to have odd or even parity if the sign of the wave fnnction of the system respectively changes or not with changing sign in all spatial coordinates (see Friedlander et ah, 1981 for more detailed treatment). The value assigned to dxA is appropriate for A > 80. For < 60, a value of 65 is more appropriate.

The following is a summary of the local group molecular orbitals from which one may select the primary building blocks for interaction diagrams. The orbitals are classified according to the coordination number of the central atom. The local symmetry properties and relative energies are independent of the atomic number of the central atom itself. 

Parity, as used in nuclear science, refers to the symmetry properties of the wave function for a particle or a system of particles. If the wave function that specifies the state of the system is Tir,. v) where r represents the position coordinates of the system (x, y, z) and s represents the spin orientation, then (r,. v) is said to have positive or even parity when... 

The parity of a system is related to the symmetry properties of the spatial portion of the wave function. Another important quantum mechanical property of a system of two or more identical particles is the effect on the wave function of exchanging the coordinates of two particles. If no change in the wave function occurs when the spatial and spin coordinates are exchanged, we say the wave function is symmetric... 

Properties related to angular momentum follow. The spin coordinates are implicit in the whole argument. Insofar as symmetry properties are concerned, it is... 

Variational principles have turned out to be of great practical use in modem theory. They often provide a compact and general statement of theory, invariant or covariant under transformations of coordinates or functions, and can be used to formulate internally consistent computational algorithms. Symmetry properties are often most easily derived in a variational formalism. 

The valence and coordination symmetry of a transition metal ion in a crystal structure govern the relative energies and energy separations of its 3d orbitals and, hence, influence the positions of absorption bands in a crystal field spectrum. The intensities of the absorption bands depend on the valences and spin states of each cation, the centrosymmetric properties of the coordination sites, the covalency of cation-anion bonds, and next-nearest-neighbour interactions with adjacent cations. These factors may produce characteristic spectra for most transition metal ions, particularly when the cation occurs alone in a simple oxide structure. Conversely, it is sometimes possible to identify the valence of a transition metal ion and the symmetry of its coordination site from the absorption spectrum of a mineral. 

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