Charges and bond orders

Table I-l lists the various theoretical treatments published on the thiazole molecule for each the type of approximation, the mode of parametrization. and, eventually, the geometry employed are given net charges and bond orders for various theoretical calculations are listed in Tables 1-2 and 1-3.

T system, the PPP a—-tt method (133). The -tt and a net charges and bond orders of thiophene and thiazole are compared in Table 1-5. Whatever the method considered the variation of the indices occurs in the same sense when passing from thiophene to thiazole the replacement in the 3-position of a carbon atom by a nitrogen induces... 

The Yukawa-Tsuno r values have been measured for the solvolysis reactions fonning benzyl cations and several a-substituted derivatives, 6-3IG charges and bond orders have been calculated for the presumed cationic intermediates. Analyze the data for relationships between r and the structural parameters. (Hint. Plot r versus the bond orders and the charges at C-1, C-2, C-3, and C-4.)... 

The AIM facility in Gaussian can be used to predict a variety of atomic properties based on this theory. We will use it to compute atomic charges and bond order for the ally cation. 

Run an AIM=BondOrders calculation for allyl cation at the MP2/6-31G(d) model chemistry. What are the predicted atomic charges and bond orders for this molecule ... 

The tables below give some examples of atomic charges and bond orders calculated by various methods as a function of the basis set at the HF level of theory. It is evident that the Mulliken and Ldwdin methods do not converge as the basis set is increased, and the values in general behave unpredictably. Especially the presence of diffuse functions lead to absurd behaviours, as the aug-cc-pVXZ basis sets illustrate for CH4. Note also that... 

When one considers data available, such as energies, charges and bond orders, the following statements can be obtained from the simple potential energy surface shown in Fig. 2 (see part 2.3.2). The corresponding data from EHT calculations 16) for the two points represented by (O) are shown in Table 6. 

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

The charges and bond orders thus indicate the response of the system to infinitesimal changes of its coulomb and resonance integrals, respectively. 

It would be out of place to give a detailed review of the approximate SCF theory, as developed for 7r-electron systems, at this point. It is, however, necessary to explain the basic equations, and convenient to use a form in which only the charges and bond orders appear (McWeeny, 1956, 1964). The total 7r-electron energy E may then be written... 

Here Zg is the number of tt electrons provided by atom is essentially an ionization potential for an electron extracted from in the presence of the part of the framework associated with atom r alone (a somewhat hypothetical quantity), is a framework resonance integral, and is the coulomb interaction between electrons in orbitals < >, and <(>,. The essential parameters, in the semi-empirical form of the theory, are cug, and and from their definition these quantities are expected to be characteristic of atom r or bond r—s, not of the particular molecule in which they occur (for a discussion see McWeeny, 1964). In the SCF calculation, solution of (95) leads to MO s from which charges and bond orders are calculated using (97) these are used in setting up a revised Hamiltonian according to (98) and (99) and this is put back into (95) which is solved again to get new MO s, the process being continued until self-consistency is achieved. It is now clear that prediction of the variation of the self-consistent E with respect to the parameters is a matter of considerable difficulty. 

The approximate polarizabilities calculated by the steepest descent method are useful mainly because they are easily calculated from the charge-and-bond-order matrix of the imperturbed molecule but their... 

A recent re-examination of the perturbation problem (McWeeny, 1961 Diercksen and McWeeny, 1965), leads to a method of determining, to any prescribed order, the perturbation of an electronic system due to any applied field. This method involves the direct calculation of the first-, second-, third-order corrections to the charge-and-bond-order matrix... 

To summarize exactly the same self-consistent polarizabilities may be used in discussing the effect of heteroatoms as in discussing framework perturbations, but the perturbation parameter 8a , is replaced by given in (120). The changes in charges and bond orders are given by... 

The new charge-and-bond-order matrix may be used as a basis for discussion of the reactivity of the heterocyclic system. 

On this basis, a division of oxygen species into electron rich (O2-) and electron deficient (e.g., 03) is limited since it assumes that the species can be considered as entities separate from their environment. This approach is contrary to the picture that has emerged from the preceding discussion, where it became increasingly obvious that the nature of the species depends very much on its environment at the surface and that the formal description of a species as 03 does not necessarily accurately reflect its actual charge and bond order when adsorbed on the surface. In addition, it seems that many oxides can form a variety of different oxygen species on the surface depending on the reaction conditions. 

CHARGE AND BOND ORDER IN SHMO THEORY (SAB = 0, ONE ORBITAL PER ATOM)... 

Assigning atom charges and bond orders involves calculating the number of electrons belonging to an atom or shared between two atoms, i.e. the population of electrons on or between atoms hence such calculations are said to involve population analysis. Earlier schemes for population analysis bypassed the problem of defining the space occupied by atoms in molecules, and the space occupied by bonding electrons, by partitioning electron density in a somewhat arbitrary way. The earliest such schemes were utilized in the simple Hiickel or similar methods [256], and related these quantities to the basis functions (which in these methods are essentially valence, or even just p, atomic orbitals see Section 4.3.4). The simplest scheme used in ab initio calculations is Mulliken population analysis [257]. 

The quantities nr and nr s are used to calculate atom charges and bond orders. The Mulliken gross population in the basis function (fir is defined as the Mulliken net population nr (Eq. (5.211)) plus one half of all those Mulliken overlap populations nrh (Eq. (5.212)) which involve separated atoms Srs is very small) ... 

Since the formulas for calculating Mulliken charges and bond orders (Eqs. (5.211)-(5.218)) involve summing basis function coefficients and overlap integrals, it is not too surprising that they can be expressed neatly in terms of the density matrix (Section 5.2.3.6.4) P and the overlap matrix S (Section 4.3.3). The elements of the density matrix P are (cf. Eqs. 5.208 = 5.81)... 

As a simple illustration of the calculation of atom charges and bond orders, consider H-He+. From our Hartree-Fock calculations on this molecule (Section 5.2.3.6.5) we have... 

The discussion in Section 5.5.4 on dipole moments, charges and bond orders applies in a general way to the calculation of these quantities by semiempirical methods too. Electrostatic potentials, whether visualized as regions of space or mapped onto van der Waals surfaces, are usually qualitatively the same for AMI and PM3 as for ab initio methods. Atoms-in-molecules calculations are not viable for semiempirical methods, because the core orbitals, lacking in these methods, are important for AIM calculations. 

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