Hiickel theory defined

For compounds containing X = N, O, or S in place of C, some trivial extensions of Hiickel theory define the on-site and off-site integrals c/.x and (3XY in terms of the corresponding integrals a,c and (3Cc for carbon (called a and (3 further above), through numerical "fudge" (i.e., adjustable) parameters hx and kXy (both typically between 0.5 and 2)... 

An estimate of the magnitude of the salt effect can be obtained from the Debye-Hiickel theory. Defining... 

Most of these developments may be applied most directly within the framework of the isolated molecule method, in which the reactivity indices are the charges and self-polarizabilities of the unperturbed ground state of a given molecule calculations based on the localization model (e.g. Nesbet, 1962) have made less progress, and will not be considered. It is therefore natural to enquire whether indices similar to and tt,, in Hiickel theory can still be defined, and calculated more precisely, in self-consistent field theory. The obvious questions are... 

The second-order changes, in terms of which polarizability coefficients may be defined, are much more difficult to discuss because they involve essentially a change in the wave function (made in such a way as to preserve self-consistency)—unlike the first-order changes, which involve the Mwperturbed wave function only. Approximate formulae for the polarizabilities were first obtained (McWeeny, 1956) using a steepest descent method to minimize the energy, a useful result being the establishment of a connection between tt,, and F, valid for systems of any kind (non-alternant or heteroaromatic included) and applicable either in Hiickel theory or in a more complete theory. 

In the PPP model, each first-row atom such as carbon and nitrogen contributes a single basis function to the n system. Just as in Hiickel theory, the orbitals y,- are not rigorously defined but we can visualize them as 2pff atomic orbitals. Each first-row atom contributes a certain number of jr-electrons—in the pyridine case, one electron per atom just as in Hiickel 7r-electron theory. 

Having defined various activities and activity coefficients in solutions made up from strong electrolytes, we now turn to the determination of y . For this purpose we briefly discuss some aspects of the Debye-Hiickel Theory. 

Hiickel theory [or the Giintelberg or Davies equation (Table 3.3)] may be used to convert the solubility equilibrium constant given at infinite dilution or at a specified / to an operational constant, valid for the ionic strength of interest. In seawater solubility equilibrium constants, experimentally determined in seawater, may be used. For example, the CaC03 calcite solubility in seawater of specified salinity may be defined by = [Ca " ] [CO f ], where [Caj ] and [C03f ] are the total concentrations of calcium and carbonate ions, for example,... 

For the RPM, the first term cancels, and the second is dominant. In principle, c (r ) defined in this way may also vary locally. Third, one may also calculate C2g by working backward from the Laplace transformation of the pair correlation function, as done by Lee and Fisher [94] for the pair correlation function of their generalized Debye-Hiickel theory (GDH). 

The dominant parameter in the Debye-Hiickel theory is the ionic strength I, defined as... 

Spin densities (p) are theoretical quantities, defined as the sum of the squared atomic orbital coefficients in the nonbonding semi-occupied molecular orbital (SOMO) of the radical species (Hiickel theory). For monoradical species, the spin density is connected to the experimental EPR hyperfine coupling constant a through the McConnell equation [38]. This relation provides the opportunity to test the spin density dependence of the D parameter [Eq. (8)] for the cyclopentane-1,3-diyl triplet diradicals 10 by comparing them with the known experimental hyperfine coupling constants (ap) of the corresponding substituted cumyl radicals 14 [39]. The good semiquadratic correlation (Fig. 9) between these two EPR spectral quantities demonstrates unequivocally that the localized triplet 1,3-diradicals 9-11 constitute an excellent model system to assess electronic substituent effects on the spin density in cumyl-type monoradicals. 

The concept of the ionic strength was introduced into electrolyte solution studies some time prior to the development of the Debye-Hiickel theory, but it rises to prominence in this theory. The ionic strength is a special type of summation over all charged species in solution (see Section 8.21.1) and is defined as ... 

The identiheation of the coulomb integrals with the electronegativity has some very interesting implications. It allows to redefine the reactivity indices, introduced earlier [45] through the parameters of Hiickel theory, in terms of density functional theory [40]. Indeed, the self and mutual polarizabilities, defined [45] respectively as ... 

The Debye-Hiickel theory gives, strictly, a relation for/ in terms of the ionic strength (/) defined on the molar scale... 

In an electrolyte, the activity coefficient of the metal ions cannot be neglected and deviates considerably from one. The activity coefficient depends on the ionic strength /, defined in Eq. (1.12). Eor diluted solutions (c<10 mol kg i), it can be calculated by the Debye-Hiickel theory in the first approximation (Eq. (1.15)). In practice, if working with an excess of supporting electrolytes, the activity coefficient is approximately constant. The variation of the equilibrium potential with the concentration is determined by the Nemst factor 2.303 RTIF=59.2 mV at 25 °C for z+ = l (decadal logarithm). 

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