INDEX symmetry

Kier symmetry index - symmetry descriptors Kz index -> Hosoya Z matrix... 

Merrifield-Simmons bond order -> symmetry descriptors (O Merrifield-Simmons index) Merrifield-Simmons index -> symmetry descriptors mesomeric effect -> electronic substituent constants... 

As a result of index symmetry, a factor of one-half arises from the crossterm, with the identities of the two species being clearly distinguished. Note... 

The latter result (82) yields a quantum probability amplitude that, under Hermitian conjugation and time reversal, correctly equates to the corresponding amplitude for the time-inverse process of degenerate downconversion. To see this, we note that the matrix element for SHG invokes the tensor product Py (—2co co, ) p([/lC., where the brackets embracing two of the subscripts (jk) in the radiation tensor denote index symmetry, reflecting the equivalence of the two input photons. As shown previously [1], this allows the tensor product to be written without loss of generality as ( 2co co, co), entailing an index-symmetrized form of the molecular response tensor,... 

Before further developing the theory to a form more directly suited to a different kind of experimental application, we outline why SWM is a mechanism allowed for all possible molecular symmetries. By inspection of the index symmetry in the radiation tensor, it is clear that a harmonic signal can derive only from that part of the sixth-rank polarizability o4wm that is symmetric with respect to permutation among the four indices related to the absorbed pump... 

In this mechanism, two-photon transitions are forbidden and the excitation of the participating molecules occurs through one- and three-photon allowed transitions. Both the real (laser) photons are absorbed by one molecule, excitation of its partner resulting from the virtual photon coupling. Because of the difference in selection rules from the previous case, the first two terms of Eq. (5.13) are now zero, and contributions arise only from the third and fourth terms. It must also be noted that setting the two absorbed photon frequencies to be equal in Eq. (5.16) to produce zJy, (co,o>) introduces index symmetry into the tensor, as indicated by the brackets embracing the first two indices. A factor of j must then be introduced into the definition of this tensor in order to avoid over-counting contributions. The transition matrix... 

One may note that quasienergy derivatives dt listed above are symmetric with respect to the capital indices Ai... A . It must be emphasized that the indices must be thought of as a double index specifying a specific perturbation strength a, as well as a frequency cOa, and, as discussed previously, they must be permuted at the same time in order to retain symmetry. Further simplifications of the expressions for molecular properties are possible using the response equations, e.g. (48), (49) and (50), but generally at the expense of index symmetry and numerical precision [28,9]. As an example, using the response equation the expression for second-order molecular properties (52) can be rewritten as... 

Configurations equivalent to s, under figure index symmetry constitute the right coset of the isomorphic image of P with respect to s,... 

The expressions given above have complete sums over all indices, which can be exploited to simplify the implementation. However, we can reduce the operation count by restricting the summation ranges and exploiting the index symmetry. The relativistic expression reduces to... 

The general problem is now clear the quantities i /,. p are tensor components, with respect to the group U(m), and we want to find linear combinations of these components that will display particular symmetries under electron permutations and hence under index permutations. Each set of symmetrized products, with a particular index symmetry, will provide a basis for constructing spin-free CFs (as in Section 7.6) for states of given spin multiplicity and in this way the full-CI secular equations will be reduced into the desired block form, each block corresponding to an irreducible representation of U(m). It is therefore necessary to study both groups U(m), which describes possible orbital transformations, and which provides a route (via the Young tableaux of Chapter 4) to the construction of rank-N tensors of particular symmetry type with reject to index permutations. 

U(l), which involve the transformations of progressively fewer and fewer orbitals—only the first m — 1, then the first m — 2, and so on. For Sat the irreps were labelled by Young shapes and their basis vectors by the corresponding standard Young tableaux for U(m) they are again labelled by Young shapes (which indicate the index symmetry of the tensor bases), while their basis vectors refer to standard Weyl tableaux. 

The functions associated with these tableaux according to (10.3.4) provide a 20-dimensional irrep of U(4), the group of transformations that mix the orbitals 0,..., 4. To pass to the subgroup U(3) of transformations that mix only the first three orbitals 2. 3> we simply scratch out the cells containing orbital index 4. The 20 x 20 irrep of U(4) then reduces to a block form in which the blocks provide irreps of U(3), whose basis vectors are labelled by indicating the truncated tableaux that remain. There is an 8x8 block, with index symmetry... 

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