Symmetry-adapted orbitals

In general, the produa of two functions of the same symmetry is either a basis for the totally symmetric representation of the group, or is a basis for a reducible representation that contains it. In contrast, the product of two functions whose symmetries are different never contains the totally symmetric representation. 

Once the decomposition of a reducible representation into a sum of irreducible representations has been achieved, the following step consists of finding the linear combination of orbitals that are bases for these irreducible representations. These are often referred to as symmetry-adapted linear combinations of orbitals (SALCO). 

---

An example will help illustrate these ideas. Consider the formaldehyde molecule H2CO in C2v symmetry. The configuration which dominates the ground-state waveflinction has doubly occupied O and C 1 s orbitals, two CH bonds, a CO a bond, a CO n bond, and two 0-centered lone pairs this configuration is described in terms of symmetry adapted orbitals as follows (Iai22ai23ai2lb2 ... 

To generate the above Ai and E symmetry-adapted orbitals, we make use of so-ealled symmetry projeetion operators Pe and Pa These operators are given in terms of... 

These equations state that the three Ish orbitals can be combined to give one Ai orbital and, since E is degenerate, one pair of E orbitals, as established above. With knowledge of the ni, the symmetry-adapted orbitals can be formed by allowing the projectors... 

Molecular orbitals are not unique. The same exact wave function could be expressed an infinite number of ways with different, but equivalent orbitals. Two commonly used sets of orbitals are localized orbitals and symmetry-adapted orbitals (also called canonical orbitals). Localized orbitals are sometimes used because they look very much like a chemist s qualitative models of molecular bonds, lone-pair electrons, core electrons, and the like. Symmetry-adapted orbitals are more commonly used because they allow the calculation to be executed much more quickly for high-symmetry molecules. Localized orbitals can give the fastest calculations for very large molecules without symmetry due to many long-distance interactions becoming negligible. 

An orbital correlation diagram can be constructed by examining the symmetry of the reactant and product orbitals with respect to this plane. The orbitals are classified by symmetry with respect to this plane in Fig. 11.9. For the reactants ethylene and butadiene, the classifications are the same as for the consideration of electrocyclic reactions on p. 610. An additional feature must be taken into account in the case of cyclohexene. The cyclohexene orbitals tr, t72. < i> and are called symmetry-adapted orbitals. We might be inclined to think of the a and a orbitals as localized between specific pairs of carbon... 

Multipole analysis with high-resolution X-ray data for [Ni(thmbtacn)]2+ was carried out to determine the electron configuration in the C3 symmetry-adapted orbitals of the Ni ion, confirming a higher occupancy of the crystal field-stabilized t2g orbitals relative to the destabilized eg orbitals. This is interpreted in terms of a predominantly ionic metal-ligand interaction.1424... 

Finally, it must be mentioned that localized orbitals are not always simply related to symmetry. There are cases where the localized orbitals form neither a set of symmetry adapted orbitals, belonging to irreducible representations, nor a set of equivalent orbitals, permuting under symmetry operations, but a set of orbitals with little or no apparent relationship to the molecular symmetry group. This can occur, for example, when the symmetry is such that sev-... 

For the point group D4h, the atomic d-orbital functions belong to four different group-theoretical representations. When the same representation occurs more than once, as it does, for example, in trigonal point groups, M-1 will contain cross terms between orbitals of the same symmetry, as shown in Table 10.3(a). In this case, we are interested in the population of the symmetry-adapted orbitals ysk, such as defined for the trigonal case by expression (10.2). The symmetry-adapted orbitals are linear combinations of the original functions, that is,... 

The populations Pkl of the symmetry-adapted orbital products follow, in analogy to Eq. (10.7), from... 

The original set of 19 orbitals would have led to a 1 x 19 determinant in eqn (12-2.1), but now instead, by using the equally valid set of symmetry adapted orbitals, we have ... 

The convention is to use lowercase letters for the symmetry species of one-electron functions.) Since each irreducible representation occurs only once in (9.72), these symmetry-adapted orbitals are also the (unnormalized) MOs. As a check, using (9.73), (9.74), and (9.65), we find for the matrix A of the similarity transformation that reduces the matrices of TAO to block-diagonal form... 

The z axis is the internuclear axis, and the z axes on atoms a and b point toward each other. Without using the full group-theory formalism, we can write down by inspection the following symmetry-adapted orbitals, which... 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

A number of basis functions were generated and utilized for two steps of the continuum calculations. First, an extended basis set with continuum wavefunctions of limited spatial extent and the minimal set of atomic orbitals corresponding to the states filled with electrons were used to obtain self-consistent charge density. Then a further extended basis set was used to represent the states in the continuum through the orthogonalization to the wavefunctions for the electron charge density. To reduce the number of matrix elements, the basis functions were transformed to the symmetry-adapted orbitals before the orthogonalization. 

When the basis set was enlarged, almost linearly dependent basis functions appeared. They were akin to some orbitals generated on different atoms in the moleeule beeause the linear eombination of many basis functions at multi-centers spanned the space repeatedly. The linear dependence among the symmetry-adapted orbitals was removed before the orthogonalization. 

In order to solve the Kohn-Sham equations (Eqn. (2)) we used the molecular orbital-linear combination of atomic orbitals (MO-LCAO) approach. The molecular wave functions 0j are expanded the symmetry adapted orbitals Xj) which are also expanded in terms of the atomic orbitals... 

---