Group from basis functions

It would appear that identical particle permutation groups are not of help in providing distinguishing symmetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very useful restrictions on the way we can build up the complete molecular wavefimction from basis functions. Molecular wavefunctions are usually built up from basis functions that are products of electronic and nuclear parts. Each of these parts is further built up from products of separate uncoupled coordinate (or orbital) and spin basis functions. When we combine these separate functions, the final overall product states must conform to the permutation symmetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis functions. 

Basis functions for the irreducible representations of Dj may be obtained directly from basis functions in 0. A table showing the correspondence between the two groups is given in Table 8 in which the signs that have been chosen constitute one of several phase combinations which are consistent with the transformation properties. 

For Abelian point groups, the use of symmetry is particularly simple. Under each operation R in the group, each basis function transforms entirely into another, apart from a possible change of sign, i.e. 

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. 

Projection operators are a technique for constructing linear combinations of basis functions that transform according to irreducible representations of a group. Projection operators can be used to form molecular orbitals from a basis set of atomic orbitals, or to form normal modes of vibration from a basis of displacement vectors. With projection operators we can revisit a number of topics considered previously but which can now be treated in a uniform way. 

The principle action of the DMI fungicides is to inhibit the action of the cytochrome 450-dependent removal of the 14a-methyl group from 24-methylene dihydrolanosterol. The resulting accumulation of the substrate 24-methylene dihydrolanosterol, obtusifoliol and 14a-methyl-fecosterol, together with the consequent reduction in ergosterol synthesis, disrupts the normal functioning of cell membranes and was thought to be the basis of DMI activity. 

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. 

We end this chapter with an example of the determination of the irreducible representations produced by certain basis functions using the rules and theorems which we have developed. It is at this point that we are ready, at last, to produce results of genuine chemical interest from the sole knowledge of the point group to which a molecule belongs. 

Let us now consider the n-dimensional reducible representation r d which is produced from the function space whose basis functions are Qi> 9i> - Qn> d let us assume that in the reduction of I 1 no irreducible representation of the point group occurs more than once. One way of looking at the reduction is to see it as a change of basis functions from gl9 9i> gn to... 

It is always possible to form a new, and in general reducible, representation r of a given point group from any two given representations T and r of the group. This is done by forming a new function space for which the basis functions are all possible products of the basis functions of T and T Let the basis functions of T and t9 be... 

The atomic orbitals suitable for combination into hybrid orbitals in a given molecule or ion will he those that meet certain symmetry criteria. The relevant symmetry properties of orbitals can be extracted from character tables by simple inspection. We have already pointed out (page 60) that the p, orbital transforms in a particular point group in the same manner as an x vector. In other words, a px orbital can serve as a basis function for any irreducible representation that has "x" listed among its basis functions in a character table. Likewise, the pr and p. orbitals transform as y and vectors. The d orbitals—d d dy, d >, t, and d ,—transform as the binary products xy, xz, yr, x2 — y2, and z2, respectively. Recall that degenerate groups of vectors, orbitals, etc, are denoted in character tables by inclusion within parentheses. 

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