Symmetry-adapted LCAO method

Despite the huge increase in computational effort, this direct symmetry-adapted LCAO method was used to study ozone [22], tetrahedral Ni4 [23], and D5h-symmetric ferrocene (Fe(C5H5)2) [24] using molecular orbital (MO) contraction coefficients in the linear-combination-of-Gaussian-type orbital (LCGTO) computer code of [25]. Obviously, symmetry-adapted calculations are important enough to pay an order-TV computational price. The reasons are first, and foremost, that the calculations converge, and second that the wavefunction and one-electron orbitals can be used to address experiment, which typically must first determine the symmetry of the molecule. 

These same programs can also be used to transform any localized atomic basis into a symmetry-adapted linear-combination-of-atomic-orbitals (LCAO) basis. For methods that begin with an atomic basis set and include multi-center overlap, however, symmetry adaption can actually increase the complexity of the calculation as will be shown. Nevertheless, even with localized basis functions, symmetry can be used to significantly reduce the cost of the calculation if it is carefully applied. How to do so is the subject of the next section. On the other hand, sometimes one chooses not to symmetrize 14c to approximate real physical effects that cannot yet be treated exactly within density functional theory. 

For a proof of this theorem, we refer to the literature [19, 20]. The theorem is not only applicable to molecular vibrations but is also directly in line with the LCAO method in molecular quantum chemistry. In this method the molecular orbitals (MOs) are constructed from atomic basis sets that are defined on the constituent atoms. An atomic basis set, such as or 4/, corresponds to a fibre, emanating, as it were, from the atomic centre. Usually, such basis sets obey spherical symmetry, since they are defined for the isolated atoms. As such, they are also invariant under the molecular point group [21]. As an example, a set of 4/ polarisation functions on a chlorine ligand in a RhClg complex is itself adapted to octahedral symmetry as 2 + tiu + tiu This representation thus corresponds to V. In the C4 site symmetry these irreps subduce ai-ybi- -b2- -2e. According to the theorem, theLCAOs based on the 4/ orbitals thus will transform as ... 

To solve Eq.(9.57), one uses the SCF LCAO MO technique as applied for molecules (Chapter 8) and now adapted for crystals. This particular method will be called SCF LCAO CO. because the linear combinations (LC) of the symmetry AOs are used as the expansion fimctions for the... 

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