Block-diagonal form

If we start with an t -dimensional representation of A consisting of the matrices M, M2, M3,. .., it may be that we can find a matrix V such that when it is used with ( equation A1.4.34) it produces an equivalent representation M, M 2, M 3,. .. each of whose matrices is in the same block diagonal form. For example, the nonvanishing elements of each of the matrices could fonn an upper-left-comer ... 

Note that every matrix in the four dimensional group representation labeled DN) has the so-called block diagonal form... 

The distance matrix A, which holds the relative distances (by whatever similarity measure) between the individual confonnations, is rarely informative by itself. For example, when sampling along a molecular dynamics trajectory, the A matrix can have a block diagonal form, indicating that the trajectory has moved from one conformational basin to another. Nonetheless, even in this case, the matrix in itself does not give reliable information about the size and shape of the respective basins. In general, the distance matrix requires further processing. 

Finally, we consider the contribution from the last term, j d>g A j. To find the eigenvalues in this case, we note that we can relabel the submatrix d>g so that it appears in block diagonal form, with each block corresponding to a particular cycle, Ci, i = 1,2,..., / ... 

In addition one can always find a transformation leading to a symmetry adapted basis [4] e, so that T is brought to the block diagonal form T via the associated similarity transformation. The T matrix can be written as a direct sum... 

As indicated above there may be many equivalent matrix representations for a given operation in a point group. Although the form depends on the choice of basis coordinates, the character is Independent of such a choice. However, for each application there exists a particular set of basis coordinates in terms of which the representation matrix is reduced to block-diagonal form. This result is shown symbolically in Fig. 4. ft can be expressed mathematically by the relation... 

Rfl. 4 Block-diagonal form of a representation matrix The tedueed representation. 

Coordinates such as these, which have the symmetry properties of the point group are known as symmetry coordinates. As they transform in the same manner as the IRs when used as basis coordinates, they factor the secular determinant into block-diagonal form. Thus, while normal coordinates most be found to diagonalize the secular determinant, the factorization resulting horn the use of symmetry coordinates often provides considerable simplification of the vibrational problem. Furthermore, symmetry coordinates can be chosen a priori by a simple analysis of the molecular structure. 

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

Of course for large systems, it is impractical to ascertain by inspection which row and column permutations will yield a block diagonal form or to... 

In addition, G and F matrix elements have been tabulated (see Appendix VII in Nakamoto 1997) for many simple molecular structure types (including bent triatomic, pyramidal and planar tetratomic, tetrahedral and square-planar 5-atom, and octahedral 7-atom molecules) in block-diagonalized form. MUBFF G and F matrices for tetrahedral XY4 and octahedral XY molecules are reproduced in Table 1. Tabulated matrices greatly facilitate calculations, and can easily be applied to vibrational modeling of isotopically substituted molecules. Matrix elements change, however, if the symmetry of the substituted molecule is lowered by isotopic substitution, and the tabulated matrices will not work in these circumstances. For instance, C Cl4, and all share full XY4 tetrahedral symmetry (point group Tj), but... 

We have used for the row vectors of the respective entities, while we denote by ( ) and O the orbitals and many-electron functions, and by O and T(0) the two corresponding linear transformations, respectively. Various types of many-electron space for which such transformations may be carried out have been described by Malmqvist [34], In general, O may be non-unitary, possibly with subsidiary conditions imposed for ensuring that the corresponding transformation of the V-electron space exists e.g. a block-diagonal form according to orbital subsets or irreducible representations). 

By assuming and maintaining the orbital coefficient variation matrix in block diagonal form. 

This second-order reduced density matrix for SOAGP has a block diagonal form—one dense block for each geminal and one diagonal block for the mixing between geminals. 

Since in the FHCF(L) the effective crystal field is given in terms of the l-system Green s function, the natural way to go further with this technique is to apply the perturbation theory to obtain estimates of the /-system Green s function entering Fqs. (22) and/or (25). It was assumed and reasoned in [29] that the bare Green s function for the /-system has a block-diagonal form ... 

The secular determinant is in block-diagonal form and factors into three determinants. Two of the roots of the secular equation are... 

Aside from the infinite number of representations equivalent to (9.25), there are other representations of C3 >. The matrices (9.25) are each in block-diagonal form. If we partition them into submatrices of the form (2.30), then the submatrices S12 and S2i will be column and row matrices whose elements are all zero. The product of any two of these partitioned... 

The convention is to use lowercase letters for the symmetry species of one-electron functions.) Since each irreducible representation occurs only once in (9.72), these symmetry-adapted orbitals are also the (unnormalized) MOs. As a check, using (9.73), (9.74), and (9.65), we find for the matrix A of the similarity transformation that reduces the matrices of TAO to block-diagonal form... 

Beveridge, D. L., 380 Binomial coefficients, 351,374 Biochemical applications of ESR, 380-381 of IR spectroscopy, 268 of NMR, 357, 363-364 of Raman spectroscopy, 270-271 Birge-Sponer extrapolation, 304-305 Blackbody radiation, 121-122 Bloch, F 328 Block-diagonal form, 15 Bohr (unit), 23 Bohr magneton, 51, 368 Bohr radius, 42 Bolometer, 260... 

When the secular determinant is in block-diagonal form, the secular equation (1.207) splits up into several equations. For example, if we have a five-fold degenerate unperturbed level and the secular determinant happens to consist of a 2x2 block and a 3x3 block [as in (1.79)], then the secular equation splits into two separate equations moreover, two of the... 

Interchange of rows 2 and 4 followed by interchange of columns 2 and 4 puts the secular determinant in block-diagonal form. The roots are found to be... 

Suppose r is reducible. Then the matrices R S T,... either are in the same block-diagonal form or can be put in the same block-diagonal form by a similarity transformation. To see the significance of this, we write... 

Let Rao be the matrix representing the symmetry operation R in the representation rAG. Equation (9.64) means that there is a similarity transformation that transforms RAO to RAO, where RAO is in block-diagonal form, the blocks being the matrices R],R2,...,Rfc of the irreducible representations ri,r2,...,ri. Let rAO denote the block-diagonal representation equivalent to TAO and let A be the matrix of the similarity transformation that converts the matrices of I AO to TAO Rao = A, RaoA. We form the following linear combinations of the AOs ... 

Since HSF is one example of such an operator X, different permutational symmetries are not mixed by HSF and the secular determinant in quantum mechanical calculations is factored into block diagonal form. We will also find in Section IV that eq. (2-18) leads to selection rules and the concept of spin conservation. 

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