Bond order matrix

Platt number total edge adjacency index -> edge adjacency matrix PLS-based variable selection - variable selection P matrix -> bond order indices (O graphical bond order)... 

P-matrix bond order indices (0 graphical bond order)... 

TABLE 2.4. Total Energy per C2H2 Unit Relative Energy Measured from the Equidistant Symmetry-Adapted State (AE), the Single-Particle Gap AE, and Off-Diagonal r-Electron Density Matrix (Bond Order) Elements Obtained for Various HF Solutions of trans-FA ... 

With such a matrix representation, the storage space is dependent only on the number of nodc.s (atoms) and independent of the number of bonds. As Figure 2-14 dcmon.stratcs, all the e.sscntial information in an adjacency matrix can also be lound in the much smaller non-rediindant matrix. But the adjacency matrix is unsuitable for reconstructing the constitution of a molecule, because it does not provide any information about the bond orders. 

Both the adjacency and distance matrices provide information about the connections in the molceular structure, but no additional information such as atom type or bond order. One type of matrix which includes more information, the Atom Connectivity Matrix (ACM), was introduced by Spialtcr and is discussed in Ref, [38]. This approach was eventually abandoned but is listed here because it was quite a unique approach. 

The bond matrix is related to the adjacency matrix but gives information also on the bond order of the connected atoms. Elements of the matrix obtain the value of 2 if there is a double bond between the atoms, c.g, between atoms 2 and 3... 

Adjactney matrix describes connections of atoms contains only 0 and 1 (bits) no bond types and bond orders no number of free electrons... 

Distunct matrix describes geometric distances no bond types or bond orders no number of free electrons cannot be represented by bits... 

P is the total spinless density matrix (P = P + P ) and P is the spin density matrix (P = p" + P ). For a closed-shell system Mayer s definition of the bond order reduces to ... 

The total it electron energy is the sum of occupied orbital energies multiplied by two if. as is usually the ease, the orbital is doubly occupied. The charge densities and free valency indices were treated in separate sections above. The bond order output should be read as a lower triangular serni matrix. The bond order semi matrix for the butadiene output is shown in Fig. 7-7. 

The full bond order matrix is a symmetric tridiagonal matrix (Chapter 2). It is symmetric because the bond order Cji Y Na,ack is the same as tiie bond order Na,i, a,j. Elements off the tridiagonal (-.4472 in the butadiene example) are artifacts of the minimization atid should be disi egarded. The full bond order matrix for butadietie is... 

We have the makings of an iterative computer method. Start by assuming values for the matr ix elements and calculate electron densities (charge densities and bond orders). Modify the matr ix elements according to the results of the electron density calculations, rediagonalize using the new matrix elements to get new densities, and so on. When the results of one iteration are not different from those of the last by more than some specified small amount, the results are self-consistent. 

In the Huckel theory of simple hydrocarbons, one assumes that the election density on a carbon atom and the order of bonds connected to it (which is an election density between atoms) are uninfluenced by election densities and bond orders elsewhere in the molecule. In PPP-SCF theory, exchange and electrostatic repulsion among electrons are specifically built into the method by including exchange and electrostatic terms in the elements of the F matrix. A simple example is the 1,3 element of the matrix for the allyl anion, which is zero in the Huckel method but is 1.44 eV due to election repulsion between the 1 and 3 carbon atoms in one implementation of the PPP-SCF method. 

The elements of the F matr ix depend on either the charge densities q or the bond orders p, which in turn depend on the elements of the F matrix. This circular dependence means that we must start with some initial F matrix, calculate eigenvectors, use the eigenvectors to calculate q and p, which lead to new elements in the F matr ix, calculate new eigenvectors leading to a new F matrix, and so on, until repeated iteration brings about no change in the results. The job now is to fill in the elements of the F matr ix. 

Internally, molecules can be represented several different ways. One possibility is to use a bond-order matrix representation. A second possibility is to use a list of bonds. Matrices are convenient for carrying out mathematical operations, but they waste memory due to many zero entries corresponding to pairs of atoms that are not bonded. For this reason, bond lists are the more widely used technique. 

The density matrix can also be used to generate information about bond strengths. A quantitative measure is given by the Bond Order (BO). It was originally defined" from bond distances as... 

Defining ethane, ethylene and acetylene to have bond orders of 1, 2 and 3, the constant a-is found to have a value of approximately 0.3 A. For bond orders less than 1 (i.e. breaking and fonning single bonds) it appears that 0.6 A is a more appropriate proportionality constant. A Mulliken style measure of the bond strength between atoms A and B can be defined from the density matrix as (note that this involves the elements of the product of the D and S matrices). 

For k = J and for degenerate eigenvalues the elements of f are taken equal to zero. Let us next consider the variation of the population-bond-order matrix, which, in the orthogonal case of eqn 1, is just ... 

Huber et al. 189) have investigated cocondensation (4.2—10 °K) reactions between Ni, Pd and Pt and molecular oxygen in pure O2 and Oj/Ar matrices. These reactions were studied by matrix isolation infra-red spectroscopy, including isotopic and diffusion controlled warm-up studies. They established that both M(02) and (0j)M(02) species were present. The 0-0 bond order suggested significant back-bonding, and this led them to reject the monodentate stmcture. Fig. 8 (b). 

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

D is the charge and bond order matrix 4>v(r) and r) are the basis functions... 

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