Symmetry operator belongs

These are the characters of the so-called representation of the Oh group. It should be noted that symmetry operations belonging to the same class have the same character, for a given basis function. Here, we obtain a second important simplification it is only necessary to work with classes, instead of invoking all of the symmetry operations (48 in the case of the Oh group). 

The character table of C2v (Table 5) is easily constructed considering that in this group the barred and unbarred symmetry operations belong to the same class, with the exception of E and E, which always represent a class of their own. 

Along with the primitive translations, and glide-reflections when appropriate, there axe other symmetry operations belonging to the space group. In bipartite systems, it is relevant to classify any symmetry operation according to whether it leaves each sublattice invariant or transforms one into each other (see, for instance, Fig. 2). 

The set of symmetry operations belonging to a given molecule has the following four properties ... 

The four properties are the defining properties of a group A group is a set of elements of whatever kind for which we can define the product of a pair of elements and which has the properties (i)-(iv) above. Hence it follows that the set of all the symmetry operations belonging to a molecule forms a group, the symmetry group of the molecule, and that the methods of group theory can be used to discuss the symmetry of a molecule. 

During all normal vibrations the moving atoms deform a molecule or a crystal such that the deformed object is either symmetric or antisymmetric with respect to the different symmetry operations belonging to its point group. Symmetric means that the symmetry operation transforms one atom to an equivalent atom which is moving in the same direction. Antisymmetric means that the equivalent atom is moving in the opposite direction. 

The symmetry operators belonging to a symmetrical object such as the equilibrium nuclear framework of a molecule form a group. 

A uniform spherical object is the most highly symmetrical object. If the center of the sphere is at the origin, every mirror plane, every rotation axis, every improper rotation axis, and the inversion center at the origin are symmetry elements of symmetry operators belonging to the sphere. 

Group theory is a branch of mathematics that involves elements with defined properties and a single method to combine two elements called multiplication. The symmetry operators belonging to any symmetrical object form a group. The theorems of group theory can provide useful information about electronic wave functions for symmetrical molecules, spectroscopic transitions, and so forth. 

In certain point groups, some of the symmetry operations belong to the same class and will therefore have identical characters for their matrices. Mathematically, a class is a set of operations that are conjugate to each other, or related by a similarity transform. In the world of matrix math, two matrices (A and 6) will be conjugate with each other if there exists a third matrix Q, which is also a member of the group, such that 6 and A are related to each other by a similarity transform, given by Equation (8.16) ... 

For nonmerohedral twins, only certain zones of data are affected by overlap. The diffraction patterns of nonmero-hedrally twinned crystals are difficult to index since the individual reflections may result from different domains of the twin. The twin law is usually a symmetry operation belonging to a higher symmetry supercell. An example of nonmerohedral twinning is an orthorhombic crystal where a 2b. A metrically tetragonal supercell can be obtained by donbling the length of b so that there is a pseudofourfold axis about c. 

Each molecular electronic term of H2O is designated by giving the symmetry species of the electronic wave functions of the term, with the spin multiplicity 25 -I- 1 as a left superscript. For example, an electronic state of H2O with two electrons unpaired and with the electronic wave function unchanged by all four symmetry operators belongs to a Ai term. (The subscript 1 is not an angular-momentum eigenvalue but is part of the symmetry-species label.)... 

Symmetry operators move points in space relative to a symmetry element. If a symmetry operator belongs to a symmetrical object, it leaves that object in the same conformation after operating on aU particles of the object. 

The symmetry operators belonging to a symmetrical object form a group. 

The set of symmetry operators belonging to a symmetrical object forms a group if we define operator multiplication to be the method of combining two members of the group. We illustrate this fact for the ammonia... 

In the Bom-Oppenheimer approximation a symmetry operator operates on all of the electron coordinates but does not affect the nuclear coordinates. If a symmetry operator leaves the potential energy unchanged, it must move every electfon to a position where it either is at the same distance from each nucleus as in its original position, or is at the same distance from another nucleus of the same kind. We can test whether a symmetry operator belongs to a molecule by applying it to all electrons and seeing if each electron is now at the same distance from each nucleus as it originally was from a nucleus of the same kind. 

There is a second test to determine whether a symmetry operator belongs to a molecule in a particular nuclear conformation. We apply it to all of the nuclei of the molecule instead of applying it to the electrons. If a symmetry operator moves every nucleus to a location previously occupied by that nucleus or by a nucleus of the same... 

In order to discuss molecules beyond Bc2, we construct additional LCAOMOs, following the same policies that we applied earlier (1) Each LCAOMO is a combination of two atomic orbitals centered on different nuclei (2) each LCAOMO is an eigenfunction of the symmetry operators belonging to the molecule. From each pair of atomic orbitals, a bonding orbital and an antibonding orbital can be constructed. We construct six space orbitals from the real atomic orbitals of the 2p subshells ... 

Our treatment of polyatomic molecules thus far has not exploited the symmetry properties of the molecules. We have largely restricted our descriptions of chemical bonding to orbitals made from no more than two atomic orbitals, and have included hybrid orbitals in our basis functions to achieve this goal. The approximate LCAO molecular orbitals that we have created are not necessarily eigenfunctions of any symmetry operators belonging to the molecule. 

If all the members of the group commute with each other, the group is called abelian. It is a fact that the symmetry operators belonging to any symmetrical object form a mathematical group. 

Show that the symmetry operations belonging to the H2O molecule form a group. 

Once a molecule has been assigned to a point group we can draw some conclusions about it. If a molecule has a dipole moment, any symmetry operation belonging to the... 

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