Water, equivalent transformation orbital

A sjfmmetry atomic orbital (SAO) represents such linear combination of equivalent-by-sjunmetry AOs that transforms according to one of the irreducible representations of the S) mmetry group of the Hamiltonian. Then, when molecular orbitals (MOs) are formed in the LCAO MO procedure, any given MO is a linear combination of the S AOs belonging to a particular irreducible representation. For example, the water molecule exhibits a sjnnmetry plane a) that is perpendicular to the plane of the molecule. A MO, which is S)nnmetric with respect to a does contain the SAO ISa + Isj, but does not contain the SAO ISa — l fc. 

To show how the matrix approach works, we will go over the (Zp c, 2py) example more formally. We have used an equivalent set of basis vectors for the water problem before. Figure 2.2 shows a set of vectors labelled x,y and z on the O atom of H2O along with the transformation that occurs after each of the symmetry operations in Cav is applied. These vectors have exactly the same symmetry properties as the p-orbital set on the O atom, since they are their functional forms. The paper models from Appendix 1 can also be used to follow the transformations discussed with this basis. If we consider the x and y vectors together, the C2 transformation can be written as. 

In equations (l7)-(20), A is the antisymmetrizer and U is a unitary matrix. Since the wavefiinctions in the different molecular orbital bases differ at most by their signs, the electron density and all molecular properties are invariant under the transformation of equation (18). Pople made use of this relationship to transform the canonical orbitals (the eigenfunctions of the Fock operator) of water to a set of equivalent orbitals , consisting of two equivalent O-H bond orbitals and two equivalent oxygen lone pair orbitals. Pople also noted that if one writes the total closed shell energy as the sum of one- and two-electron terms. 

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