Symmetry operations, group

The symmetry operations, grouped in classes, appear on the first line. However, in this point group, there is only one operation per class. 

Table B.l includes character tables for a number of molecular point groups. The headings for the columns are the symmetry operators grouped into classes. The horizontal rows are the characters of the representations with their designations in the leftmost column. The designation for the point group is in the upper left comer.

CO, CO, co, and o, respectively. The integrals in Eqs. (E.9) and (E.IO) will then be different from zero only if the integrands are invariant under all symmetry operations allowed by the symmetry point group, in particular under C3. It is readily seen that the linear terms in Q+ and Q- vanish in and H In turn. 

SymApps converts 2D structures From the ChemWindow drawing program into 3D representations with the help of a modified MM2 force field (see Section 7.2). Besides basic visualization tools such as display styles, perspective views, and light source adjustments, the module additionally provides calculations of bond lengths, angles, etc, Moreover, point groups and character tables can be determined. Animations of spinning movements and symmetry operations can also he created and saved as movie files (. avi). 

Periodic boundary conditions can also be used to simulate solid state con dition s although TlyperChem has few specific tools to assist in setting up specific crystal symmetry space groups. The group operation s In vert, Reflect, and Rotate can, however, be used to set up a unit cell manually, provided it is rectangular. 

Operators that eommute with the Hamiltonian and with one another form a partieularly important elass beeause eaeh sueh operator permits eaeh of the energy eigenstates of the system to be labelled with a eorresponding quantum number. These operators are ealled symmetry operators. As will be seen later, they inelude angular momenta (e.g., L2,Lz, S, Sz, for atoms) and point group symmetries (e.g., planes and rotations about axes). Every operator that qualifies as a symmetry operator provides a quantum number with whieh the energy levels of the system ean be labeled. 

Because symmetry operators eommute with the eleetronie Hamiltonian, the wavefunetions that are eigenstates of H ean be labeled by the symmetry of the point group of the moleeule (i.e., those operators that leave H invariant). It is for this reason that one eonstruets symmetry-adapted atomie basis orbitals to use in forming moleeular orbitals. 

Point groups in whieh degenerate orbital symmetries appear ean be treated in like fashion but require more analysis beeause a symmetry operation R aeting on a degenerate... 

Here g is the order of the group (the number of symmetry operations in the group- 6 in this ease) and Xr(R) is the eharaeter for the partieular symmetry T whose eomponent in the direet produet is being ealeulated. 

These veetors form the basis for a redueible representation. Evaluate the eharaeters for this redueible representation under the symmetry operations of the D h group. 

The ammonia moleeule NH3 belongs, in its ground-state equilibrium geometry, to the C3v point group. Its symmetry operations eonsist of two C3 rotations, C3, 3 ... 

These six symmetry operations form a mathematieal group. A group is defined as a set of objeets satisfying four properties. 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

We ean likewise write matrix representations for eaeh of the symmetry operations of the C3v point group ... 

In faet, one finds that the six matriees, Df4)(R), when multiplied together in all 36 possible ways obey the same multiplieation table as did the six symmetry operations. We say the matriees form a representation of the group beeause the matriees have all the properties of the group. 

These six matrices form another representation of the group. In this basis, each character is equal to unity. The representation formed by allowing the six symmetry operations to act on the Is N-atom orbital is clearly not the same as that formed when the same six operations acted on the (8]s[,S 1,82,83) basis. We now need to learn how to further analyze the information content of a specific representation of the group formed when the symmetry operations act on any specific set of objects. 

These six matrices can be verified to multiply just as the symmetry operations do thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Tj functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. 

Another one-dimensional representation of the group ean be obtained by taking rotation about the Z-axis (the C3 axis) as the objeet on whieh the symmetry operations aet ... 

These one-dimensional matriees ean be shown to multiply together just like the symmetry operations of the C3V group. They form an irredueible representation of the group (beeause it is one-dimensional, it ean not be further redueed). Note that this one-dimensional representation is not identieal to that found above for the Is N-atom orbital, or the Ti funetion. 

For a function to transform according to a specific irreducible representation means that the function, when operated upon by a point-group symmetry operator, yields a linear combination of the functions that transform according to that irreducible representation. For example, a 2pz orbital (z is the C3 axis of NH3) on the nitrogen atom... 

We now return to the symmetry analysis of orbital produets. Sueh knowledge is important beeause one is routinely faeed with eonstrueting symmetry-adapted N-eleetron eonfigurations that eonsist of produets of N individual orbitals. A point-group symmetry operator S, when aeting on sueh a produet of orbitals, gives the produet of S aeting on eaeh of the individual orbitals... 

All molecules possess the identity element of symmetry, for which the symbol is / (some authors use E, but this may cause confusion with the E symmetry species see Section 4.3.2). The symmetry operation / consists of doing nothing to the molecule, so that it may seem too trivial to be of importance but it is a necessary element required by the mles of group theory. Since the C operation is a rotation by 2n radians, Ci = I and the symbol is not used. 

Except for the multiplication of by we follow the rules for forming direct products used in non-degenerate point groups the characters under the various symmetry operations are obtained by multiplying the characters of the species being multiplied, giving... 

The dipole moment vector /i must be totally symmetric, and therefore symmetric to all operations of the point group to which the molecule belongs otherwise the direction of the dipole moment could be reversed by carrying out a symmetry operation, and this clearly cannot happen. The vector /i has components fiy and along the cartesian axes of the molecule. In the examples of NH3 and NF3, shown in Figures 4.18(b) and 4.18(e), respectively, if the C3 axis is the z-axis, 7 0 but = 0. Similarly in H2O and cis-... 

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

The symmetry groups for the chiral tubules are Abelian groups. The corresponding space groups are non-symmorphic and the basic symmetry operations... 

From the symmetry operations R = (xP t) for tubules the non-symmorphic symmetry group of... 

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