The SCF MO Treatment of

Now for the hydrogen Is AOs. Reflection in the yz plane leaves each of them unchanged. However, rotation by 180 about the z axis sends Hjls into H2IS and vice versa (Fig. 15.3) the result is the same for reflection in the xz plane. The H,ls and H2IS 

As a preliminary step in finding the MOs of a molecule, it is helpful (but not essential) to construct linear combinations of the original basis AOs such that each linear combination does transform according to one of the molecular symmetry species. Such linear combinations are called symmetry oihitals or symmetry-adapted basis functions. The symmetry orbitals are used as the basis functions Xs n the expansions l i = c iXs [Eq. (13.156)] of the MOs The use of basis functions that transform according to the molecular symmetry species simplifies the calculation by putting the secular determiruint in block-diagonal form this will be illustrated below. 

Each oxygen AO transforms according to one of the symmetry species of H2O and can serve as a symmetry orbital. However, neither of the two hydrogen Is AOs belongs to a symmetry species of HjO, and we must construct two symmetry orbitals from these AOs. Consider the following linear combinations  

Thus the first and second functions in (15.14) are eigenfunctions of Ci z) with eigenvalues +1 and -1, respectively. Examination of the effects of the other three symmetry operators shows the functions (15.14) to belong to the symmetry species ai and 2 respectively. We shall not bother to normalize the symmetry orbitals (15.14). The seven basis symmetry functions and their symmetry species are then 

The set of Roothaan simultaneous equations (13.157) then breaks up into one set of four simultaneous equations, one set of two simultaneous equations, and one set of one simultaneous equation (Section 9.6). The first set contains matrix elements involving only the four syrmnetry orbitals. Therefore, four of the lowest seven H2O MOs are linear combinations of the four ai symmetry orbitals these four MOs must have a, symmetry. Similarly, we have two MOs of 62 symmetry and one MO of h, symmetry. The symmetry orbitals are not (in general) the MOs, but each MO must be a linear combination of those symmetry orbitals having the same symmetry species as the MO. The forms of the lowest MOs of H2O are then 

To describe a quantum-mechanical calculation, one specifies the method and the basis set. The letters HF (for Hartree-Fock) denote any ab initio SCF MO calculation, whether or not the basis set is large enough to come close to the Hartree-Fock limit. Thus the notation HF/3-21G denotes an ab initio SCF MO calculation that uses the 3-21G basis set. 

For a minimal-basis-set MO treatment of H2O (Fig. 15.1 in Section 15.2), we start with the Oli, 02s, 02p 02py, and 02p inner-shell and valence oxygen AOs, and the Hi Is and H2IS valence AOs of the hydrogen atoms. Linear combinations of these seven basis AOs give LCAO approximations to the seven lowest MOs of water. As stated in Section 15.3, the MOs can be chosen so that upon application of the molecular symmetry operators each 

MO transforms according to one of the symmetry species of the molecular point group. To aid in choosing the right linear combinations of AOs, we examine the symmetry behavior of the AOs. 

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