Q, bond order

A. Partial Derivatives and Polarizability Coefficients Expansion of (8) yields a polynomial, the characteristic or secular polynomial, whose roots are determined by the values of the parameters , vw- The ground state energy (12) is likewise a function of the (a,j3) parameter values, as are all quantities such as AO coefficients in the MO s, charges q bond orders p t, etc. It is possible, therefore, to specify the h partial derivative with respect to any or at an arbitrary point defined by a set of values (a,j8) in the parameter space, and to make expansions such as... 

The sum of all these factors for a specific problem is equal to the trace of the matrix of the corresponding characteristic problem4 the sum of their squares equals the sum of the squares of all matrix elements of the corresponding characteristic problem.64 These relations are useful for checking the values of orbital energies. By an equally simple procedure1-4 the values of expansion coefficients yield the values of ir-electron densities (q), bond orders (p), and free valences (F) these quantities are usually presented in the form of a molecular diagram ... 

Recently, a series of models of 16 polynuclear pyridine-like heterocycles (Fig. 9 shows formulas of eleven of these) were treated using the HMO approximation7 (SN = 0.6, inductive effect not allowed for) and the following reactivity indices calculated 77-electron densities (q), bond orders (p), free valences (F, N x = Wheland s atom localization energies (A(,Ar,An), and superdelocalizabilities, both exact (Se,Sr,Sn) and approximate (S e,S r,S n). Atom-atom polarizabilities150 (773) had been calculated earlier.151 Some of the indices calculated are presented in Section VI, B. 

From tabulated values of rc-electron densities (q,), bond orders (pij, atom-atom (7Cj,i). bond-bond nij.jJ, and bond-atom polarizabilities itij-k), AX, has been calculated. These values (Table I) resemble those obtained by direct solution of the Hiickel matrices. 

Selected atom charges q, bond orders p and distances do-Mo. (A) of the M015O56H22 cluster representing MoOslOlO) surface. For a definition of the two sets of data, bulk termination and optimized cluster , see text. Results refer to DFT calculations using the RPBE approximation. 

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. 

Table 6. Atomic charges (a.u.) and bond orders for two special points of the potential energy surface shown in Fig. 2 as (Q)...

Table 2. Heats offormation, H (kcal), magnitude of electron density, Q (Of), displace from the cluster to adsorbed oxygen molecules, equilibrium interatomic distances R (A-B) and bond orders P (A-B), corresponding to them, in PANI-O2 adsorption complexes.

Table 3.33. NBO/NRT descriptors of molecules in Table 3.32, including atomic charges (Q), central atom d-orbital occupancy (c/m )> NRT bond orders, and central-atom valency (with percentage ionic character)...

Table 3.35. Natural atomic charges (Q), d-orbital occupancies (d f), and NRT bond orders and valencies (with percentage ionic character) for ArF species...

This consideration is corroborated by calculations at different levels of theory [11-13], showing that 1,2-additions to C q occur preferentially at a and (3 bonds between C(l) and C(2) and C(5) and C(6), respectively (Figure 13.2). Also, Mulliken charges on the fullerene core have been used in predicting the regiochemistry of nucleophilic additions to C70 [11]. Finally, 7t-bond orders have also been employed to evaluate the reactivity of different types of bonds [14-16]. 

Q The carbide ion, C22, is a constituent of some compounds of electropositive metals, such as in CaC2. Write down the electronic configuration of the carbide ion and determine the bond order. The C-C distance in CaC2 is 119 pm. Is this distance compatible with your value for the bond order ... 

Q The S-S distance in the molecule S2 is 189 pm. Determine the electronic configuration of this molecule and calculate the bond order. Would you expect the molecule to be paramagnetic ... 

Q The molecule Ne2 does not exist. Determine the bond order of the Ne2+ ion and comment upon its possible existence. Under which conditions might the ion be produced ... 

Q Estimate the bond order for the molecule CN, and compare the value with that for the CN ion. Should the bond in CN be shorter or longer than that in CN ... 

Using a similar approach, Evans and co-workers obtained values of hsi = -1.50, kcsi = 0.55, 8 = 0.15 for a series of trimethylsilyl-substituted naphthalene radical anions based on a Q value of -26.1 49). For phenyltrimethylsilylacetylene radical anion, the values of the heteroatom parameters that gave the best fit of the Huckel calculated spin densities with experimental values, using a Q value of 28, were hs, = -1.3, and kCSi = 0.65 when a 8 value of 0.1 was assumed. A resulting C—Si 7r-bond order of about 0.3 is obtained 43). 

B. Indices of Chemical Reactivity Table X presents the following data for models of molecules XV-XXV 7r-electron densities (q), atom-atom polarizabilities (77 ), free valences (F, Nmhx = /3), and exact superdelocalizabilities (Se,8r, and Sn). Table XI gives Wheland s atom-localization energies for a few molecules. Bond orders for molecules XV-XXV are compiled in Table XII. 

There are four nonequivalent types of C—C bonds in naphthalene, these being represented by C,—C2, C2—C3, C4—C10, and Q—C,(). By using the MO expressions in Table 7.2, the n bond order of each type may be computed. As before, in treating tetramethylenecyclobutane, the bond order, pmtn of the bond between atoms m and n is defined as the sum of contributions from each occupied MO, each contribution being given by twice (for two electrons) the product of the coefficients of 0OT and 0 in that MO. For p,2 in naphthalene we have... 

The bond orders in the polymethine chain are equalized in the ground and excited states. If 4>0 = 45°, the bond equalization is maximum. This is the ideal polymethine state (1) of the polymethine chain. Any deviation from this state (ie, Q is greater than or less than 45°) causes the bond to alternate from the polymethine chain center to its ends. The alternation amplitude is found to be proportional to the absolute value 45° — 01. 

Here g refers to electron densities on the atomic orbitals, Q is the total valence-shell electron density for an atom, and P represents the bond order matrix. 

Here Aq = qv (X) — q (CeHgF) is the -electron charge density on the fluorine atom in X relative to fluorobenzene Aq c is the corresponding difference in jt-electron bond order for the FC bond Aqc is the corre-... 

Table 6. The values of bond orders for Pq4(q = 0, — 1, -2) in the nuclear configuration with Td symmetry at / PP = 2.21 A...

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