Symmetry orbitals unnormalized

There are 14 minimal-basis-set AOs. It is easy to set up symmetry orbitals by trial and error. The reader will readily verify the symmetry species of the (unnormalized) symmetry functions listed in Table 15.4. 

One can use the character table to generate symmetry orbitals. This is done by picking any one of the AOs, say 1 s 1, and operating on it with each synunetry operation times the character for each operation in the representation of interest. The sum of all these operations is an unnormalized symmetry orbital. Thus, for the a representation in NH3,... 

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... 

A set of a-type atomic orbitals transform amongst themselves under point group operations like scalar quantities. Hence, to consttuct descent-in-symmetry-type LCAO cluster orbitals, we simply follow the same reasoning as for the linear and cychc polyenes, above, and define (unnormalized) cr-type cluster orbitals as... 

Since each methane symmetry operator permutes the hydrogen Is orbitals among themselves, (15.42) is sent into itself by each symmetry operation and belongs to the totally symmetric species Aj. We need three more symmetry functions. The construction of these is not obvious without the use of group theory, and we shall simply write down the results. The remaining three orthogonal (unnormalized) symmetry-adapted basis functions can be taken as... 

Then we set up the six simultaneous equations for the coefficients and the corresponding 6x6 secular determinant and apply the Hiickel approximations. Its form resembles that for cyclobutadiene in Exercise 10.33, but with six rows and six columns. Full-frontal attack on it to determine the six values of E is rather tedious, especially as there are procedures that make use of symmetry that greatly simplifies the solution. As should be verified, the energies and the corresponding (unnormalized) molecular orbitals obtained are as follows (Fig. 10.39) ... 

In atom/ions with N = 5 due to the presence of angular orbitals in the wave function, L — S there are many kinds of configurations to be constructed, whose contribution to the nonrelativistic energy is important. In particular, the ground states of B and C atoms are of P-symmetry. The Hy-CI program is general for any type of orbital, although in Hy-CI we use unnormalized s-, p- and d-orbitals. 

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