Matrix elements symmetry reduction

In order to apply the direct product representation to the derivation of selection rules, recognize that a matrix element of the form ipi, O lpj) will be equal to zero for symmetry reasons if there is even one symmetry operation that takes the integrand into its negative. The argument follows exactly the course of that of section 10.2. Thus the matrix element will vanish unless the direct product representation is totally symmetric (Ai), or contains A upon reduction. 

We have shown in Ref. [19] that if the systems in question have three levels, one can completely eliminate decoherence and disentanglement by imposing a special symmetry using the appropriate modulation. Thus, even if drastic reduction of all the decoherence matrix elements is not possible, then by using local modulations, one may equate the intraparticle elements, eliminate the interparficle elements, and code the QI in the ground and antisymmetric dark state of the two excited levels, and consequently completely preserve coherence and entanglement. 

In Section 8.5.13 we have seen that the matrix elements of the spin-orbit coupling can interconnect certain atomic terms. In real metal complexes, however, the matrix elements of the spin-orbit coupling operator have to be evaluated between the strong-field kets. In the crystal field of cubic (Oh) symmetry the following reduction applies... 

Since (a) and (d) are identically zero, the matrix elements of the (a) and (d) rows of the direct product will vanish. The function (b) is the negative of (c), so the set of functions of E

antisymmetric product is only one-dimensional. If equals the symmetric product yields three independent functions and the antisymmetric product yields none. It is important to use symmetric direct products when examining the symmetry of products of partner functions. We now outline the general method of obtaining the characters of these direct product matrices. Once the characters are known the reduction to constituent reps can be carried out in the usual way. 

Here we have, for the first time, a formula with a triad of irreps. This will form the basis for the symmetry evaluation of general matrix elements. The cp coefficients are obtained by performing product manipulations on the character tables. As an example. Table 6.1 illustrates the reduction of the Eg x T2g product in Oh, as given in Eq. (6.9). Product tables are given in Appendix E. 

In the preceding chapter, we have shown how the use of time-reversal symmetry can lead to considerable reduction in the number of unique matrix elements that appear in the operator expressions. However, we are also interested in the overall structure of the matrices of the operators. In particular, we are interested in possible block structures, where classes of matrix elements may be set to zero a priori. If the matrices can be cast in block diagonal form, we may save on storage as well as computational effort in solving eigenvalue problems, for example. 

Similar relations can be written for the odd-bar integrals. Relations like these can be used to reduce the number of unique integrals by a factor of 2 and the number of density matrix elements by a factor of 2. However, the odd-bar integrals are not necessarily zero because they are not integrals between symmetry functions, and all the terms in the expressions for the Fock matrix elements remain. The reduction in work comes from the fact that the sums are restricted to sums over symmetry-unique basis functions, and that only the symmetry-unique parts of the Fock matrix are constmcted. For matrix elements between functions on the symmetry axis, the reductions follow from the fact that they are symmetry functions, and hence an overall factor of 4 reduction in the work for the two-electron integrals is obtained. 

To quantify the resulting reduction in computational cost, we compare the number of matrix element evaluations for independent basis functions to that required when only Ng of these are independent. The total number of basis functions is Ninn in both cases. For simplicity, we ignore the symmetry of the Hamiltonian matrix— we assume that all elements are evaluated even though H is Hermitian. When aU iVMt basis functions are independent, traveling on A init distinct paths in phase space, TV , matrix element evaluations are required at each time step. When the total number of basis functions is but only Ng distinct paths are traversed by these basis functions, the number of matrix element evaluations reduces to 0 Nf + The number of matrix... 

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