Symmetry irreducible representations

Find the atomic orbitals of the central atom with the same symmetries (irreducible representations) as those found in Step 4. 

H 2 transforms according to the representation being the direet product of three irreducible representations that of, that of i/ 2 and that of Wg (the last is, however, fully symmetric, and therefore, does not count in this direct product). In order to have H12 5 0 this direct product, after decomposition into irreducible representations, has to contain a fully symmetrie irreducible representation. This, however, is possible only when ijji and i/(2 transform according to the same irreducible representation. 

To see how these SALCs conform to the symmetry irreducible representations, it is interesting to consider the affect of the C2V symmetry operations on the functions themselves. For example, after a C2 operation the two H atoms swap over and so do the basis vectors. However, the mode is a function of both vectors, and it is the symmetry of the overall function that determines the value of its character. Looking at the result for each function as a whole, there are still vectors at Hj and H2. For the symmetric mode, and ( 2 will have swapped but the function looks identical it is still bi + 2, so has a character of 1. In terms of the old basis, b has become 2 and vice versa. 

Let us now consider the necessary conditions for the appearance of phonons in impurity-ion electronic spectra. The presence of a substitutional defect in an otherwise perfect crystal removes the translational symmetry of the system and reduces the symmetry group of the system from the crystal space group to the point group of the lattice site. Loudon [26] has provided a table for the reduction of the space group representations of a face-centered cubic lattice into a sum of cubic point-group representations. A portion of that table is shown in Table 1 here. Consider an impurity ion that undergoes a vibronic electric-dipole allowed transition, with T and Tf the irreducible representations of the initial and final electronic states. Since the electric dipole operator transforms as Tj in the cubic point group, Oh, the selection rule for participation of a phonon is that one of its site symmetry irreducible representations is contained in the direct product T x Ti X Tf. 

We have described here one particular type of molecular synnnetry, rotational symmetry. On one hand, this example is complicated because the appropriate symmetry group, K (spatial), has infinitely many elements. On the other hand, it is simple because each irreducible representation of K (spatial) corresponds to a particular value of the quantum number F which is associated with a physically observable quantity, the angular momentum. Below we describe other types of molecular synnnetry, some of which give rise to finite synnnetry groups. 

The irreducible representations of a symmetry group of a molecule are used to label its energy levels. The way we label the energy levels follows from an examination of the effect of a synnnetry operation on the molecular Sclnodinger equation. 

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

When Cj symmetry is present, irreducible representations of the double group (see Table II) so that = 0 by symmetry. In this case, there are only... 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

Consider trans-C2H2Cl2. The vibrational normal modes of this molecule are shown below. What is the symmetry of the molecule Eabel each of the modes with the appropriate irreducible representation. 

The basic idea of symmetry analysis is that any basis of orbitals, displacements, rotations, etc. transforms either as one of the irreducible representations or as a direct sum (reducible) representation. Symmetry tools are used to first determine how the basis transforms under action of the symmetry operations. They are then used to decompose the resultant representations into their irreducible components. 

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... 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

In order to obtain this savings in the computational cost, orbitals are symmetry-adapted. As various positive and negative combinations of orbitals are used, there are a number of ways to break down the total wave function. These various orbital functions will obey different sets of symmetry constraints, such as having positive or negative values across a mirror plane of the molecule. These various symmetry sets are called irreducible representations. 

Most ah initio calculations use symmetry-adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal. This means that every molecular orbital will have the symmetry properties of one of the irreducible representations of the point group. No orbitals will be described by mixing dilferent irreducible representations. 

We have seen that any two of the C2, ( Jxz), (r Jyz) elements may be regarded as generating elements. There are four possible combinations of + 1 or — 1 characters with respect to these generating elements, + 1 and + 1, + 1 and -1,-1 and +1,-1 and —1, with respect to C2 and (tJxz). These combinations are entered in columns 3 and 4 of the C2 character table in Table A.l 1 in Appendix A. The character with respect to / must always be + 1 and, just as (r Jyz) is generated from C2 and (tJxz), the character with respect to (r Jyz) is the product of characters with respect to C2 and (tJxz). Each of the four rows of characters is called an irreducible representation of the group and, for convenience, each is represented by a symmetry species Aj, A2, or B2. The A] species is said to be totally symmetric since all the characters are + 1 the other three species are non-totally symmetric. 

R = (i/ r) require translations t in addition to rotations j/. The irreducible representations for all Abelian groups have a phase factor c, consistent with the requirement that all h symmetry elements of the symmetry group commute. These symmetry elements of the Abelian group are obtained by multiplication of the symmetry element./ = (i/ lr) by itself an appropriate number of times, since R = E, where E is the identity element, and h is the number of elements in the Abelian group. We note that N, the number of hexagons in the ID unit cell of the nanotube, is not always equal h, particularly when d 1 and dfi d. 

In the E irreducible representation, the character of any symmetry operation corresponding to a rotation by an angle is given by... 

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