Symmetry-adapted combinations

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. 

The phrase symmetry adapted basis functions refers to those linear combinations of basis functions (on several atoms) that transform like the particular irreducible representation of the appropriate point group. Molecular symmetry is used at various points in these calculations twenty years ago I would have had to write several chapters on molecular symmetry, point groups, constructing symmetry-adapted combinations of basis functions, factoring a Hamiltonian matrix using symmetry and related topics. The point is that twenty... 

FIGURE 5. Symmetry-adapted combinations of two Is hydrogen AOs (labelled symmetric or asymmetric with respect to the plane perpendicular to the HSH plane) and of the p orbitals of oxygens, where the labels are with respect to planes CSC and... 

Form symmetry-adapted combinations (SAC) of POs. This is accomplished by combining couples into in-phase and out-of-phase pairs across symmetry elements. Figure 5 shows examples. 

FIGURE 8. (a) Symmetry-adapted combinations of sp hybrids and of p orbitals on the oxygens of an OXO group. Symmetry labels are with respect to the 0X0 plane and to the plane perpendicular to it. (b) Combinations of these SACs with two sp hybrids and two p orbitals on sulphur to give the MOs of S02. 

Symmetry-adapted combinations 6 Symmetry coordinates 13 Synthons 767, 788... 

Essentially, the n ag, b u) and n b g, b2u) orbitals are the four symmetry-adapted combinations of the in-plane n orbitals of the CN groups. It is important to distinguish between the two n orthogonal orbitals of the CN group. They are degenerate for CN itself because of the cylindrical symmetry but become nondegenerate in TCNQ. Because of the symmetry plane of the molecule, the two formally degenerate orbitals lead to one in-plane tt orbital, denoted as a (nr), which is symmetrical with respect to the molecular plane even if locally it is a nr-type orbital, and one out-of-plane nr orbital, denoted as n n), which is anti-symmetrical with respect to the molecular plane and is locally a nr-type orbital. 

The four a (nr) orbitals lead to four symmetry-adapted combinations which are the main components of the four molecular orbitals. Two of them are lower in... 

The 7t (jt(b3g, Ou)) orbitals are two of the symmetry-adapted combinations of the orbitals. By being symmetrical with respect to the C2 axis along the long molecular axis, the it orbital of the central carbon atom of the C(CN)2 substituent cannot mix into these symmetry-adapted combinations, and thus, in practice, the orbitals do not delocalize toward the CeHe ring. 

We have now developed the analogy between the decomposition of a vector into components along coordinate axes, and the decomposition of a set of objects into symmetry-adapted combinations. 

In the derivation of normal modes of vibration we started with a set of displacements of individual atoms. By determining the reducible representation Ltot and decomposing it, we calculated the number of normal modes of each symmetry species. We could determine what these modes are by solving a secular equation. We could alternatively have used projection operators to determine the symmetry-adapted combinations. 

One more quantum number, that relating to the inversion (i) symmetry operator can be used in atomic cases because the total potential energy V is unchanged when ah of the electrons have their position vectors subjected to inversion (i r = -r). This quantum number is straightforward to determine. Because each L, S, Ml, Ms, H state discussed above consist of a few (or, in the case of configuration interaction several) symmetry adapted combinations of Slater determinant functions, the effect of the inversion operator on such a wavefunction P can be determined by ... 

Individual molecular orbitals, which in symmetric systems may be expressed as symmetry-adapted combinations of atomic orbital basis functions, may be assigned to individual irreps. The many-electron wave function is an antisymmetrized product of these orbitals, and thus the assignment of the wave function to an irrep requires us to have defined mathematics for taking the product between two irreps, e.g., a 0 a" in the Q point group. These product relationships may be determined from so-called character tables found in standard textbooks on group theory. Tables B.l through B.5 list the product rules for the simple point groups G, C, C2, C2/, and C2 , respectively. 

These symmetry adapted combinations are found by using the projection operators ... 

To find the symmetry adapted combinations we first consider the application of the transformation operators Om to the 33 atomic orbitals. We see at once that for all symmetry operations R of the... 

Six lxl blocks, three corresponding to I 1 and three corresponding to TV These will provide one triply-degenerate level of each type tlg and tiu) and the final MOs will only involve ligand 77-type p orbitals and will simply be the symmetry adapted combinations themselves y>Tuu) and y>rt%u)> = 1,2, 3). 

A very similar approach can be used to interpret the UPS of bis-chelate complexes of acac, hfa, and related monothio and bisthio analogues (45, 51). Monomeric Ni(acac)2 has been found to possess D2h symmetry in the vapor phase (258), and this geometry is assumed here for the other bis-chelates. In D2d symmetry the rr3, +, and n- MOs of each 0-diketonate ligand combine to afford the symmetry-adapted combinations shown in Fig. 44. Introduction of the metal d orbitals, which transform as ag(d2 ), ag(dx> yz), blg(dxy), bigidxz), and b2g(dyz), completes the scheme. By analogy with the tris-chelates, the strongest interaction anticipated is that between the dxy orbital and the antisymmetric lone pair combination of b g symmetry. 

Scheme 38. (a) Four Kekule Structures for Anthracene and (b) Their Symmetry Adapted Combinations, (c) Generation of the Twin States 0 o, I ) from These Combinations... 

It is possible to use as the basis functions symmetry-adapted combinations of primitive basis functions. This affords a decomposition of the orbital interaction energy of Eq. [21] according to irreducible representations of the point group... 

To find the symmetry-adapted combinations Ffy that represent the tunneling multiplet, we can apply symmetry projection ... 

Figure 7.7c shows how the K12B and Kij2a structures spread into the symmetry adapted combinations in each subset the lowest combination is the positive combination, which lies below the corresponding negative... 

FIGURE 7.7 (a) Kekule structures of anthracene, (b) Symmetry adapted combinations... 

Naphthalene and anthracene are archetypes of the even and odd members of the polyacene series. In each subseries, one can start by classifying the classical Kekule structures by using the symmetry operations i, C2, and point group. Then one can form symmetry-adapted linear combinations of the mutually transformable Kekule structures and deduce their bonding characteristics. Finally, these 1 Ag and 1 B2u symmetry-adapted combinations are allowed to mix and form the states of interest, the ground and first covalent excited states (16). 

FIGURE 7.10 Covalent resonance structures for hexatriene. (b) Generation of symmetry adapted combinations of R2 and R3. (c) A VB mixing diagram of the symmetry adapted VB combinations yielding the covalent ground and excited states. 

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