Antisymmetric direct product

Analogously, the characters of antisymmetric direct product [F ] of dimension n(n — l)/2 is defined as follows [10]... 

The antisymmetrized direct product of the two-dimensional representations with themselves is A for C and A2 for independent of whether n is even or odd. The basic rule (p. 210) for the choice of standard basis functions for the irreducible representations naturally proposes the fimction as a standard function for this representation. It is noted that Cn in this way requires two standard basis functions for A, namely further the function representing the symmetrized direct product of the most reduced representations with themselves. This is due to the fact that the group Cn with a real basis is not a simply reducible group since in... 

Scheme of direct multiplication of the irreducible representations of Da>. where A > A. From this table the multiplication schemes for the other point groups can be derived using the relationships discussed in Sect. 7. The antisymmetrized direct product (E ) is always A2 and this is true also of all the dihedral sub-groups of -Doo... 

In the three-dimensional rotation group Rs every other irreducible representation occurs in the antisymmetrized direct products [Eq. (28)], whereas in the dihedral group Dm only one particular irreducible representation occurs in these direct products [Table 12]. For the octahedral rotation group 0, one has an intermediate situation. 

The representations are called the symmetric and antisymmetric direct product representations. The relation of Eq. 7.53 is extremely useful since it provides a recipe for extracting symmetric and antisymmetric reps from the reducible direct product rep. The dimensions of the representations [C X Cl " " and [C X C) are not equal, and the sum of their dimensions equals the dimension of the direct product representation X C , namely /x. ... 

The underlined modes are contained in symmetrized as well as in antisymmetrized direct products of irreducible representations of electronic states. There must be at least two sets of coordinates belonging to these modes and at least one of them is vibronic active. 

The antisymmetric direct product, which we use to generate the triplet levels of 4.15, is defined by... 

It was explained in Section 4.3.2 that the direct product of two identical degenerate symmetry species contains a symmetric part and an antisymmetric part. The antisymmetric part is an A (or 2") species and, where possible, not the totally symmetric species. Therefore, in the product in Equation (7.79), Ig is the antisymmetric and Ig + Ag the symmetric part. 

Suppose now that A) and B) belong to an electronic representation I ,. Since H is totally symmetric, Eq. (6) implies that the matrix elements (A II TB) belong to the representation of symmetrized or anti-symmetrized products of the bras (A with the kets 7 A). However, the set TA) is, however, simply a reordering of the set ( A). Hence, the symmetry of the matrix elements in the even- and odd-electron cases is given, respectively, by the symmetrized [Ye x Te] and antisymmetrized Ff x I parts of the direct product of I , with itself. A final consideration is that coordinates belonging to the totally symmetric representation, To, cannot break any symmetry determined degeneracy. The symmetries of the Jahn-Teller active modes are therefore given by... 

A tableau may be used to define certain subgroups of which are themselves direct products of smaller permutation groups the symmetrizing and antisymmetrizing operators for these subgroups lead, as we shall see, to projection operators on irreducible representations of... 

The antisymmetrizer -d merely selects a certain portion of the total Hilbert space of H = HSF + fl this portion is antisymmetric under the group S r 0 Sn". Here the inner, direct product group S F 0 SN contains elements P Q, Pe SfF, Q e SN such that P — Q. The space W is invariant with respect to the group S F 0 Ss . Since the only physically observed electronic states are of antisymmetric symmetry, this symmetry is always conserved, and hence in a certain sense less interesting. We will always assume that electronic kets for H are of this symmetry, though we will not label the kets with this symmetry. 

The irrep matrices are duplicated for the additional operations for those direct product group irreps derived from the symmetric irrep of S2 or C while they are duplicated with a change of sign for irreps derived from the antisymmetric irrep of S2 or Ca. 

Independent of the choice of the full set of electron-vibrational functions with which the average values in Eqs. (85a)-(85c) are calculated, owing to the preceding assumption the final results are always expressed by matrix elements of electronic operators /ir , ajl, and calculated within the basis functions of the initial electronic term I Thus, following Eq. (87), the matrix elements of these operators are nonzero if in the case of the ystem with an even (odd) number of electrons the direct products [P] T and P rm or [P] P ( P T and [P] Tm or P p) contain the totally symmetric representation. The brackets and braces in these expressions denote, respectively, symmetric and antisymmetric products of the representation T by itself. 

The (iV ) factor included in these expressions ensures that the determinants are normalized when the orbitals are normalized. Eq. (41) gives an explicit representation of the antisymmetrizer. This summation is over the N permutations of electron coordinates for a fixed orbital order, or equivalently, over the permutations of spin-orbital labels for a fixed order of electrons. The exponent Pp is the number of interchanges required to bring a particular permuted order of electron coordinates, or of spin-orbital labels, back to the original order. Different expansion terms are generated when different spin orbitals are employed in the determinant. For convenience, we will choose this spin-orbital basis to be the direct product of the set of n spatial orbitals and the set of spin factors a, / . A particular spin orbital of this form may be written as where r (= 1 to n) labels the spatial orbital and spin factor, or simply as (j), where the combined index r (= 1 to 2n) labels both the spatial and spin components. The notation used will be clear from the context. 

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