Symmetry as a Computational Tool

The use of symmetry—at least the translational subgroup—is essential to modem first-principles calculations on crystalline solids. Group theory is simplest for Abelian groups such as the translational subgroup of a crystal or the six-fold-rotational subgroup of the benzene molecule. For such simple cyclic groups, the irreducible representations are characterized by a phase, exp(ifc), associated with each step in a direction of periodicity. For one-dimensional (or cyclic) periodicity, 

If these are symmetry-adapted basis functions, then the local potential must transform according to the invariant representation. Thus for use with localized basis sets it must be expressible as a superposition of identical single-center terms, 

A straight-forward LCAO calculation requires N3 matrix elements of the form 

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. 

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