Molecular symmetry reducible representations

If there is a molecular symmetry group whose elements leave the hamiltonian 36 invariant, then the closed-shell wavefunction belongs to the totally symmetric representation of both the spin and symmetry groups.8 It is further true that under these symmetry operations the molecular orbitals transform among each other by means of an orthogonal transformation, such as mentioned in Eq. (5) 9) and, therefore, span a representation of the molecular symmetry group. In general, this representation is reducible. 

In this case, it can be proved that the canonical SCF orbitals, being solutions of Eq. (26), are symmetry orbitals, i.e. that they belong to irreducible representations of the symmetry group. 12) If the number of molecular orbitals is larger than the dimension of the largest irreducible representation of the symmetry group, it must then be concluded that the set of all N molecular orbitals form a reducible representation of the group which is the direct sum of all the irreducible representations spanned by the CMO s. 

Except for this change, we find hyb(l ) in the same way as before. We note, however, that this time the direction of a vector may be reversed as the result of a symmetry operation and in such a case there will be a contribution of — 1 to the character of that operation. Furthermore, we immediately see that in carrying out the different symmetry operations, no vector perpendicular to the molecular plane is ever interchanged with one in the molecular plane and vice versa. This implies two things the representation rhyb is at once in a partially reduced form (the matrices are already in block form, each consisting of two blocks) and the vectors perpendicular to the molecular plane on their own form a basis for a reducible representation of (which we will call rby ) and the vectors in the molecular plane on their own also form a basis for a reducible representation of 9 (which we will call iy.T) necessarily... 

Since each MO belongs to some irreducible representation of the molecular point group, we must find linear combinations of the AOs that transform according to the irreducible representations group theory enables us to do this. A symmetry operation sends each nucleus either into itself or into an atom of the same type a symmetry operation will thus transform each AO into some linear combination of the AOs (9.63). Therefore (Section 9.6), the AOs form a basis for some representation TAO of the point group of the molecule. This representation (as any reducible representation) will be the direct sum of certain of the irreducible representations r r2,...,r (not necessarily all different) of the molecular point group ... 

In a quantum chemical calculation on a molecule we may wish to classify the symmetries spanned by our atomic orbitals, and perhaps to symmetry-adapt them. Since simple arguments can usually give us a qualitative MO description of the molecule, we will also be interested to classify the symmetries of the possible MOs. The formal methods required to accomplish these tasks were given in Chapters 1 and 2. That is, by determining the (generally reducible) representation spanned by the atomic basis functions and reducing it, we can identify which atomic basis functions contribute to which symmetries. A similar procedure can be followed for localized molecular orbitals, for example. Finally, if we wish to obtain explicit symmetry-adapted functions, we can apply projection and shift operators. 

While we have chosen to proceed here by reducing representations for the full group D3h, it would have been simpler to take advantage of the fact that D3h is the direct product of C3u and C where the plane in the latter is perpendicular to the principal axis of the former. The behaviour of any atomic basis functions with respect to the C3 subgroup is trivial to determine, and there are only two classes of non-trivial operations in C3v. In more general cases, it is often worthwhile to look for such simplifications. It is seldom useful, for instance, to employ the full character table for a group that contains the inversion, or a unique horizontal plane, since the symmetry with respect to these operations can be determined by inspection. With these observations and the transformation properties of spherical harmonics given in the Supplementary Notes, it should be possible to determine the symmetries spanned by sets of atomic basis functions for any molecular system. Finally, with access to the appropriate literature the labour can be eliminated entirely for some cases, since... 

This is a reducible representation of the D point group which reduces to ug + uu. Two molecular orbitals must be generated, one with ag and the other with antibonding orbitals which can be formed from the two 1. s atomic orbitals. 

The symmetry of the normal mode of vibration that can take the molecule out of the degenerate electronic state will have to be such as to satisfy Eq. (6-7). The direct product of E with itself (see Table 6-11) reduces to A + A 2 + E. The molecule has three normal modes of vibration [(3 x 3) - 6 = 3], and their symmetry species are A + E. A totally symmetric normal mode, A, does not reduce the molecular symmetry (this is the symmetric stretching mode), and thus the only possibility is a vibration of E symmetry. This matches one of the irreducible representations of the direct product E E therefore, this normal mode of vibration is capable of reducing th eZ)3/, symmetry of the H3 molecule. These types of vibrations are called Jahn-Teller active vibrations. 

Since p and E are vector quantities, a, P, 7, etc., are tensors. For example, the electric field vector in the first term will have three components in the molecular coordinate system. Each electric field component can contribute to polarization along each of the three directions in the molecular coordinate system. This triple contribution of electric field components leads to a total of nine elements to the second rank polarizability tensor. Similarly, there are 27 components to the P tensor and 81 components to 7. Molecular symmetry generally reduces these tensors to only a few independent elements. Unless the molecular coordinate system lacks an inversion center, the form of the odd-rank tensors such as P will lead to zero induced polarization in this representation of optical nonlinearities. For molecules such as benzene and polymers such as poly[bis(p-toluenesulfonate)diacetylene]... 

Examining each of the D4 , symmetry operations in turn generates the following reducible representation for all the molecular motions of XeF4 ... 

The C—0 o--bond is of type ai, and the remaining two orbitals are constructed from linear combinations of C and H orbitals from the ai and f>2 sets. Since the a orbital assigned to the C—O bond can contain contributions from the other orbitals that have symmetry it is best to consider all the atomic orbitals of ai type in each of the ai molecular orbitals. However, chemical intuition should indicate that one of the Oi orbitals is principally a C—O orbital, and that the other is principally a CH> orbital. The reducible representation formed by the set 0 px) and C(p ) is ... 

The T-molecular orbitals constructed from the 2pi orbitals on each carbon atom of butadiene provide an example for this case. The symmetry is Cih and the reducible representation r(0) has the characters... 

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