Representations, completely reduced equivalent

If each of the blocks in the matrices comprising the matrix system A cannot be reduced ftirther, the matrix system has been reduced completely and each of the matrix systems A1, A2, . .. in the direct sum is said to be irreducible. Matrix systems that are isomorphous to a group G are called matrix representations (Chapter 4). Irreducible representations (IRs) are of great importance in applications of group theory in physics and chemistry. A matrix representation in which the matrices are unitary matrices is called a unitary representation. Matrix representations are not necessarily unitary, but any representation of a finite group that consists of non-singular matrices is equivalent to a unitary representation, as will be demonstrated in Section A1.5. 

At this point we may abandon the Heisenberg picture and proceed to the Schrodinger representation. Of course, both representations are equivalent, as soon as the field problem has been reduced to studying a finite-dimensional quantum system. However, the most of numerous investigations of the time-dependent quantum oscillator, since Husimi s paper [285], were performed in the Schrodinger picture. So it is natural to use the known results. According to several studies [279,285,286], all the characteristics of the quantum oscillator are determined completely by the complex solution of the classical oscillator equation of motion... 

So far, we focused on conventional quantum chemical approaches that approximate the FCI wave function by truncating the complete N-particle Hilbert space based on predefined configuration selection procedures. In a different approach, the number of independent Cf coefficients can be reduced without pruning the FCI space. This is equivalent to seeking a more efficient parameterization of the wave function expansion, where the Cl coeflBcients are approximated by a smaller set of variational parameters that allow for an optimal representation of the quantum state of interest. Different approaches, which we will call modern solely to distinguish them from the standard quantum chemical methods, have emerged from solid-state physics. 

Such terms are equivalent to those obtained by Satten et al. (1983) in their study of the excitations of the T,u vibrational mode of the octahedral complex UCI5 at 260 cm L They were aware that their approach would work in a similar way for lanthanide complexes their choice of an actinide was partly determined by their familiarity with the UClj complex. Another important consideration was the high symmetry of the uranium site, which greatly reduces the number of intensity parameters required in the analysis. To assess the situation we note first that the possible pairs (k,t) in expression (94) are (1,2), (3,2), (3,4), (5,4), (5,6) and (7,6). The number of Ti representations of 0(, that occur in the representations of 0(3) are 1, 1,2 and 2 for /c = 1, 3, 5 and 7, so we expect 9 parameters in all. Satten et al. (1983) reduced that number to 6 by carrying out a complete closure over all virtual electronic states, which has the effect of fixing the ratio of the two operators (94) for which t = /c 1. They were able to account quite well for the relative intensities of some 20 or so transitions of the type Aig( H4)-> T, ( Lj), where is a... 

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