The Resonance Integral

In the zeroth approximation, H- amounts to the energy of an electron in the fields of atoms i and j involving the wave functions Xj and Xj. It is usually called the resonance inte-gral. is a function of atomic number, orbital types, and 

Methods for calculating and for such situations will be discussed later. In the zeroth approximation neglected 

Mulliken has provided data for the following graph of semi-empirical p against for carbon 2 —ir overlap relative to that of the isolated carbon—carbon double bond 

when i and j are not nearest neighbors, is set to zero, then our matrix becomes very simple since most of the H.. 

Thus we find two possible energy levels for the hydrogen molecule ion. Our problem now is to determine the wave functions corresponding to each so that we can find out which calculated energy corresponds to the more stable state. Remembering (see p. 28) that 

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One convention (Dickson. 1968) for oxygen heterocycles sets the coulomb integral at z 2f) and the resonance integral at Eor the oxirane moiety,... 

Secondly, the use of a value of the resonance integral yS derived from empirical resonance energies in other contexts is not justifiable. 

The two-center one-electron integral Hj y, sometimes called the resonance integral, is approximated in MINDO/3 by using the overlap integral, Sj y, in a related but slightly different manner to... 

In the MNDO method the resonance integral, is proportional to the overlap integral,... 

The developers of ZINDO found that the parameters required to reproduce orbital energy orderings and UV spectra are different from those required to reproduce accurate structures by geometry optimization. They introduced anew pair of parameters, called the overlap weighting factors, to account for this. These parameters are provided in HyperChem in the Semi-empirical Options dialog box. Their effect is to modify the resonance integrals for the off-diagonal elements of the Fock matrix. 

Hi2 is the resonance integral, usually symbolized by p. In a homonuclear diatomic molecule Hi I = H22 = a, which is known as the Coulomb integral, and the secular determinant becomes... 

There is, in principle, no reason why linear combinations should not be made between AOs which have the correct symmetry but very different energies, such as the lx orbital on the oxygen atom and the lx orbital on the phosphorus atom. The result would be that the resonance integral /i (see Figure 7.12) would be extremely small so that the MOs would be virtually unchanged from the AOs and the linear combination would be ineffective. 

When m 7 n the resonance integral is assumed to be the same for any pair of directly bonded atoms and is given the symbol /i ... 

The resonance integral of the 7r-bond between the heteroatom and carbon is another possible parameter in the treatment of heteroatomic molecules. However, for nitrogen compounds more detailed calculations have suggested that this resonance integral is similar to that for a C—C bond and moreover the relative values of the reactivity Indices at different positions are not very sensitive to change in this parameter. 

In contrast, when ot,P-unsaturated multiple bond systems act as dienophiles in concerted [4+2] cycloaddition reactions, they react across the C=C double bond Periselectivity as well as regiochemistry are explained on the basis of the size of the orbital coefficients and the resonance integrals [25S]... 

It is possible that the explanation of these discrepancies is to be found in the fact that the resonance integral, may vary with the row and group of the periodic table. Such a variation must almost certainly exist, but it can be taken into account only with difficulty. Furthermore, the introduction of the large number of additional arbitrary parameters would deprive the whole procedure of much of its significance. A second possible explanation is that, with phenol for ex-... 

Now the magnitude of the resonance integral, which determines the resonance energy and the resonance frequency, depends on the nature of the structures involved. In benzene it is large (about 36 kcal./mole) but it might have been much smaller. Let us consider what the benzene molecule would be like if the value of the resonance integral were very small, so that the resonance frequency were less than the frequency of nuclear... 

The number 3-14 is a measure of the coefficient of the resonance integral for synchronous resonance. 

If Li+ and Li- ions (the latter bicovalent) are also present, their a priori probabilities in class A are 1 and 28, respectively, with geometrical mean 2-7 (the ions must be present in pairs), which corresponds to 8 for neutral atoms. A calculation similar to that above, on the assumption that there is no energy difference between Li Li and Li+ Li-, leads to (77/86)" (1 + 74/2)" for the number of ways of placing the bonds and hence to the number (77/86) (1 + 74/2) = 3-14 X 2 32 as the measure of the coefficient of the resonance integral for uninhibited resonance. This result, containing the factor 2-32, indicates the importance of uninhibited resonance. 

The Coulomb repulsion integrals are evaluated using the Mataga-Nishimoto formula The resonance integral is assumed to be of exponential form p=Be , the value of exponent a being taken as 1.7 A... 

In order to discuss the geometrical structures of electronically excited states, the same procedure as described above is used, except for the use of a different value 3.3 for exponent a in the exponential form of the resonance integral This value of a was determined so that the predicted fluorescence energy from the lowest singlet excited state CB2J in benzene may fit the experimental value. 

The anisotropy of the magnetic susceptibility of a cyclic conjugated system, attributable to induced ring currents in its rc-electron network, is one of the important quantities indicative of 7t-electron delocalization. The method used for the calculation of the magnetic susceptibilities of nonalternant hydrocarbons is the London-Hoarau method taken together with the Wheland-Mann SCF technique . The resonance integral is assumed again to be of exponential form but... 

The electron is delocalized in the metal. Therefore, the resonance integral... 

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