Irreps definition

The definitions of the perturbation corrections to the interaction energy, as given by Eqs. (8), (9), and (11), involve nonsymmetric operators, like H0, V, and Rq. These operators do not include all electrons in a fully symmetric way, so H0, V, and Rq must act in a larger space them the dimer Hilbert space Hab adapted to a specific irrep of 4. To use these operators we have to consider the space Ha Hb, the tensor product of Hilbert spaces Ha and Hb for the monomers A and B. For the interaction of two closed-shell two-electron systems this tensor product space should be adapted to the irreps of the 2 2 group. In most of the quantum chemical applications the Hilbert spaces Hx, X = A and B, are constructed from one-electron spaces of finite dimension Cx - In the particular case of two-electron systems in singlet states we have Hx — x x, where the symbol denotes the symmetrized tensor product. We define the one-electron space as the space spanned by the union of two atomic bases associated with the monomers in the dimer. The basis of the space includes functions centered on all atoms in the dimer and, consequently, will be referred to as the dimer-centered basis. We assume that the same one-electron space is used to construct the Hilbert spaces Hx 1 X = A and B, i.e., Hx = In such case Ha )Hb can be represented as a direct sum of Hilbert spaces H adapted to the irreps entering the induced product [2] [2] 4, i.e., Ha ( Hb = H[2i] f[3i] f[4]- This means that every function from H , v = [22], [31], or [4], can be expanded in terms of functions from Ha Hb-... 

An important consequence of the matrix element theorem concerns the definition of selection rules. An interaction will be forbidden if the corresponding coupling coefficient in the Wigner-Eckart theorem is zero. The conditions that control the zero values of the coupling coefficients are called triangular conditions, since they involve the combination of three irreps. Two kinds of triangular conditions must be taken into account ... 

In our examples, aU the induced irreps are simple, excluding the BR corresponding to the 6-sheeted lower valence subband (see Fig. 3.3). This band representation is a composite one as it is formed by two simple band representations d,aig) and (6, t u) induced by 0 2s- and Sr 4pstates, respectively. Analysis of the space symmetry of crystalline orbitals is used to consider the possible centers of localization of chemical bonding in crystals. This task requires the Wannier-function definition and is considered in the next section. 

WTAOs are defined as the Wannier functions that are constructed from a set of specially chosen occupied and vacant bands and have a definite symmetry (they are centered on atoms and transform via irreducible representations (irreps) of the corresponding site groups). Thus, the index t in (3.114) may be substituted by several indices - - that reflect the symmetry properties of the WTAOs —... 

It is particularly usefiil to make this definition so that real irreps reduce to complex irreps, where the rows and the multiplication table are well-defined. 

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