Block-diagonal matrix

Figure 5. Modified occurrence matrix (block diagonal form) for the system in Example 1...

If only zero-order states from the same polyad are conpled together, this constitutes a fantastic simplification in the Flamiltonian. Enonnons compntational economies result in fitting spectra, becanse the spectroscopic Flamiltonian is block diagonal in the polyad nnmber. That is, only zero-order states within blocks with the same polyad number are coupled the resulting small matrix diagonalization problem is vastly simpler than diagonalizing a matrix with all the zero-order states conpled to each other. 

That is, in the basis rj.F.irjjthe Hamiltonian matrix is block diagonal in Fand and we can rewrite (equation A1.4.8) as... 

The Hamiltonian matrix will be block diagonal in this basis set. There will be one block for each irreducible representation of the synnnetry group. 

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 ... 

The value of the vanishing integral rule is that it allows the matrix H to be block diagonalized. This occurs if... 

Th c Newton-Raph son block dingotial method is a second order optim izer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. I hese derivatives provide information ahont both the slope and curvature of lh e poten tial en ergy surface, Un like a full Newton -Raph son method, the block diagonal algorilh m calculates the second derivative matrix for one atom at a lime, avoiding the second derivatives with respect to two atoms. 

A diva It MM3 wilh Ihe cumrnand mm3. Answer questions file etheiie.mm3, parameter file Enter (default) line number 1, option 2. The defaull parameter sel is Ihe MM3 parameler sel don t ehange il. The line number starts Ihe system reading on the first line of your input file, and option 2 is the block diagonal followed by full matrix minimi7 ation mentioned at the end of the section on the Hessian matrix. You will see intermediate atomic coordinates as the system minimises the geometry, followed by a final steiic eireigy, Kird with 0, output Enter, cooidinates Enter,... 

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

Most ah initio calculations use symmetry-adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal. This means that every molecular orbital will have the symmetry properties of one of the irreducible representations of the point group. No orbitals will be described by mixing dilferent irreducible representations. 

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. 

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. 

Each of the atomic blocks in the density matrix is diagonalized to produce a set of non-orthogonal NAOs, often denoted pre-NAOs . 

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... 

The parity matrix commutes with the first entropy matrix, gS = Sg, because there is no coupling between variables of opposite parity at equilibrium, (xiXj)0 = 0 if e,e - = — 1. If variables of the same parity are grouped together, the first entropy matrix is block diagonal. 

The odd expansion coefficients are block-adiagonal and hence c j I c I [g.3+ k3] = 0. This means that the coefficient of x on the right hand side is identically zero. (Later it will be shown that 0 and that could be nonzero.) Since the parity matrix commutes with the block-diagonal even coefficients, the reduction condition gives... 

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. 

This idea can easily be extended to more than two matrices to yield a matrix with non-vanishing elements in square blocks along the main diagonal and zeros elsewhere. Such a block-diagonal matrix (e.g. D = A B C) has the self-evident important properties ... 

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... 

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... 

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