Bond order operator

One should note that when only nearest-neighbor interactions are considered for j3rs the atomic sites r and s belong to different classes and the associated bond order operator product satisfies the relation... 

The expectation value of the bond-order operator is a measure of the strength of that bond, as illustrated by the simple example of ethylene. Modelling this by two TT-orbitals with two electrons shared between them it is easily shown that the bonding molecular orbital has a bond-order value of +1, while the antibonding molecular orbital has a bond-order value of —1. Thus, a larger bond-order value implies a stronger bond. 

We conclude this section by remarking on the character of the broken symmetry ground state. The staggered dimerization of the ground state, represented by eqn (4.9), together with eqn (4.21), implies that there is an alternating deviation of the expectation value of the bond order operator from its average value. This... 

We start with the alternating Huckel chain in Eq. (1), which simply consists of the bond order operators pd,s for nearest-neighbor hopping between sites 2n, 2n -1 and 2n, 2n + 1. Following SSH [10,11] and LHS [8], we discuss the dimerized ground state and write the linear e-ph Hamiltonian, Eq. (11), in terms of the symmetry coordinates in Eq. (16) for single- and double-bond stretches. In the = 0 subspace, we have... 

Draw bond order and free valency index diagrams for the butadienyl system. Write a counter into program MOBAS to detemiine how many iterations are executed in solving for the allyl system. The number is not the same for all computers or operating systems. Change the convergence criterion (statement 300) to several different values and determine the number of iterations for each. 

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. 

Orientation effects in benzene derivatives operate in two ways. If the substituent is inductive there are large first order charge displacements at the ortho and para positions, and these can be estimated approximately using the atom polarizabilities (which is very small at the meta position). The changes of bond order, however, and consequently of free valence, vanish in first order and hence depend on Sa. The charge g g at position s therefore increases or decreases from the value unity in the... 

The bond polarity is given by eq. (3.16) even if the second-order correction is considered (i.e., the contribution to the bond polarity proportional to p m is absent). The second-order corrected expectation values of the pseudospin operators defining the bond ionicity and bond order by contrast are not vanishing and have the following forms ... 

Here, P is the Dyson time-ordering operator [57], Q (t)IP is the coordinate in the interaction picture with respect to the thermal bath and to the diagonal part of the Hamiltonian of the H-bond bridge, and the notation (( )e)siow has the meaning of a partial trace on the thermal bath and on the H-bond bridge coordinates. 

Besides P, which showed not be confused with the momentum operator P of the H-bond bridge, is the Dyson time-ordering operator [57] acting on the Taylor expansion terms of the exponential operator in such a way so that the time arguments involved in the different integrals will be t > t > t". 

The bond orders and the resonance-integrals of the Core Hamilton Operators for PF5 are shown in Table 11. The negative product of these two is the stabilizing contribution of the particular ligand orbital with the phosphorus d-orbitals. 

Bond Orders (Upper Number) and Resonance-integrals of the Core Hamilton Operators (Lower Number in Italics) for PFj. Total Phosphorus Electron Density = 3-8890. 

The present position is this that for reliable quantitative calculations the M.O. method is used throughout. In the interpretation of the results one operates, however, with concepts such as bond order, free valency, etc., which actually correspond to ideas of the V.B. method. Again for qualitative inter-... 

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