Block-diagonal structure

The disappearance of matrix elements between the HF reference and singly excited states is known as Brillouins theorem. The HF reference state therefore only has nonzero matrix elements with doubly excited determinants, and the full Cl matrix acquires a block diagonal structure. 

The conditions determined by the selection rules for Clebsch-Gordan coefficients and 6j symbols provide the following block-diagonal structure of the operator... 

Because many physical systems possess certain types of symmetry, its adaptation has become an important issue in theoretical studies of molecules. For example, symmetry facilitates the assignment of energy levels and determines selection rules in optical transitions. In direct diagonalization, symmetry adaptation, often performed on a symmetrized basis, significantly reduces the numerical costs in diagonalizing the Hamiltonian matrix because the resulting block-diagonal structure of the Hamiltonian matrix allows for the separate... 

Again, the 2-RDM has a very simple block-diagonal structure of only two blocks, of which one is diagonal ... 

Table 4.1 shows that the MRs T(T) of the symmetry operators T e C3v for the basis (ei e2 e3 all have the same block-diagonal structure so that T = Tj T3. We shall soon deduce a simple rule for deciding whether or not a given representation is reducible, and we shall see then that T3 is in fact irreducible. 

Thus the matrix D has been transformed into one with a block-diagonal structure.The diagonal blocks labeled DCT(q) are matrices of dimensions c T / c T, with elements... 

Finite difference approximation of the boundary-value problem In this case, matrix D has a septa-block-diagonal structure ... 

These matrix elements are equivalent to those of Eq. (3.124), apart from anharmonic contributions of the order of v/N. So we see that the extended Majorana operator has the required effect on the states involved in the resonance mechanism. At the same time > SB does not preserve the coupled 65 (2) symmetry in other words, + Vg is not conserved anymore. Consequently, the block-diagonal structure of the Hamiltonian operator is destroyed and the numerical diagonalization of... 

As a second step, symmetry type 2 can be applied to the set of the s k points, which allows one to further reduce the Fock matrix into a block-diagonal form. By transforming the basis set into an equivalent set of symmetry adapted basis functions, every block of the transformed matrix in Figure 4, which corresponds to one particular point ky, reveals, in turn, a block-diagonal structure, for example, of the kind depicted in Figure 20. 

Since V has the same decoupled block-diagonal structure as in Eq. (14.14), only the scalar term K VK is needed to construct V. For the special relativistic potential matrix W, four real matrices,... 

If the UPSR matrix is rearranged in a block diagonal structure then the matrix is always structurally - but not numerically - symmetric, because the UPSR measures feedback strengths. If state Xi affects state Xj, then for this interaction to appear in the UPSR the feedback loop,must be closed and state Xj must affect state X. ... 

The block diagonal structure of a process within the UPSR can reveal when dynamic boundaries do not correspond to process unit boundaries. Some boundaries will be the result of modelling choices, and the physical structure and chosen model structure can both be different from the dynamic structure. A knowledge of when dynamic structure crosses process unit boundaries can be very important. For example, Samyudia [9], found that the key to decentralised control of multi-unit process plants was determining a plant decomposition that took interaction between units into account. Strong interaction between units required a plant decomposition that crossed unit boundaries. 

If a transformation can be found that reduces all the matrices of one representation to the same block diagonal structure (e.g a set of 5 x 5 matrices are reduced to 2x2- -2x2- -lxl diagonal blocks), the representation is reducible, otherwise it is irreducible. We will use the abbreviation irrep for irreducible representation, and rep for a representation in general. 

Fig. 14.12. The block-diagonal structure of the zero-order CASPT Hamiltonian matrix.

For macroscopically isotropic media, the orientation-averaged scattering matrix has sixteen nonzero elements (cf. (1.113)) but only ten of them are independent. For macroscopically isotropic and mirror-symmetric media, the orientation-averaged scattering matrix has a block-diagonal structure (cf. (1.114)), so that only eight elements are nonzero and only six of them are independent. In this case we determine the six quantities Sei3 9) ),... 

Because of the block-diagonal structure of W, these conditions are equivalent to the matrix equation (IOE.2.9). 

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