Hiickel assumption

The Hiickel assumption posits a total wave function of the type... 

One might reasonably observe that the Hiickel assumption of negligible overlap between neighboringpn orbitals seems physically unlikely, since just such overlap would be required to allow formation of multicenter molecular orbitals. This point has been considered in more detail by Mulliken, Rieke, and Brown,t and we summarize here their discussion. 

We now prove the second part of the Coulson-Rushbrooke Theorem that in pairs of complementary orbitals the LCAO-coefficients of orbitals centred on the starred atoms are the same, while those of orbitals centred on the unstarred atoms are the same in magnitude but opposite in sign. Let us consider the 7th orbital of energy e, (let it be, for argument s sake, a bonding orbital) and its associated set of LCAO-coefficients c/rh r = 1, 2,. .., n. These will satisfy a series of secular equations, the rth one of which, on the simple Hiickel assumptions, is... 

In what follows we make, to begin with, the simple Hiickel-assumptions—... 

The Debye-Hiickel assumption, discussed earlier in this chapter, is often used when we make simplifications - approximations in the theory of the electrical double layer as many equations are largely simplified if we use this assumption. This approximation means that the surface potential should be much lower than 25 mV. The surface potential can, however, in reality for colloidal systems of practical interest be much higher (see also Figure 10.22). 

In a formal sense, isoindole can be regarde,d as a IOtt- electron system and, as such, complies vith the Hiickel (4w- -2) rule for aromatic stabilization, with the usual implicit assumption that the crossing bond (8, 9 in 1) represents a relatively small perturbation of the monocyclic, conjugated system. The question in more explicit terms is whether isoindole possesses aromatic stabilization in excess of that exhibited by pyrrole. 

Turning now to the experimental data, we find that the mobility of the (NH4)+ ion in liquid ammonia does not have any high value that would indicate a contribution from proton jumps.1 Nor docs the (CH,0) ion in methanol or the (C2H5O)- ion in ethanol solution.2 These experimental data do not force us to accept the Hiickel mechanism but if we do not accept the mechanism, we shall have to make some ad hoc assumptions to explain these experimental results. 

The first assumption of the Debye-Hiickel theory is that is spherically symmetric. With the elimination of any angular dependence, the Poisson equation (expressed in spherical-polar coordinates) reduces to... 

We refer those who are interested in the details of the Debye Hiickel derivation to the following sources R. A. Robinson and R. H. Stokes. "Electrolyte Solutions", Academic Press, Inc., New York (1955). The Robinson/Stokes reference does an especially good job of summarizing and evaluating the assumptions made in the derivation H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions." Rcinhold Publishing Corporation, New York (1958) K. S. Pitzer. "Thermodynamics." Third Edition, McGraw Hill, Inc., New York (1995). 

In the second approximation, Debye and Hiickel introduced the idea that the centers of the ions cannot come closer than a certain minimum distance a, which depends on ion size the ions were now treated as entities with a finite radius. The mathematical result of this assumption are charge densities Qy, which are zero for r[Pg.120]

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

The theory of Debye and Hiickel started from the assumption that strong electrolytes are completely dissociated into ions, which results, however, in electrical interactions between the ions in such a manner that a given ion is surrounded by a spherically symmetrical distribution of other ions mainly of opposite charges, the ionic atmosphere. The nearer to the central ions the higher will be the potential U and the charge density the limit of approach to the central ion is its radius r = a. 

The Debye-Hiickel limiting law is the least accurate approximation to the actual situation, analogous to the ideal gas law. It is based on the assumption that the ions are material points and that the potential of the ionic atmosphere is distributed from r = 0 to r->oo. Within these limits the last equation is integrated by parts yielding, for constant k, the value ezk/Aite. Potential pk is given by the expression... 

An extended Hiickel MO calculation supports the assumptions made in the above analysis in that the three t2g orbitals are indeed close together in energy and remain nearly nonbonding metal-based d-orbitals. The detailed agreement is less satisfactory the SOMO is predicted to be primarily dx2 y2 with a small dxz admixture (hybrid 6 of Table 4.13), a result that can be ruled out from our analysis of the ESR results. The EHMO overlap matrix based on the X-ray structure suggests that the molecule is much closer to C2 symmetry than to Cs. If we accept that conclusion, then dxzjdxy hybridization is less likely than dyjdxy, as we tacitly assumed above. 

This section concerns unsubstituted planar polyenes C H, , n even and m = n+2(l—r), r being the number of rings, i.e. 7r-sy sterns with all atoms, C and H, in a common plane. Low man on the totem pole of quantum-chemical models adequate for such polyenes (and, of course, for aromatic 7r-sy sterns) is the Hiickel (HMO) treatment2 which assumes strict orthogonality between the molecular [Pg.199]

In the early years of quantum theory, Hiickel developed a remarkably simple form of MO theory that retains great influence on the concepts of organic chemistry to this day. The Hiickel molecular orbital (HMO) picture for a planar conjugated pi network is based on the assumption of a minimal basis of orthonormal p-type AOs pr and an effective pi-Hamiltonian h(ctT) with matrix elements... 

Equations (87)-(89) apply in aqueous solutions of two electrolytes in which the interaction potentials are conformal. For example, the assumptions utilized in the extensions of the Debye-Hiickel theory (e.g. water is considered as a continuous dielectric medium of dielectric constant D, that the cation-anion repulsive potential is that of hard spheres, and that all the... 

With these assumptions, it is readily verified that each term in (301) is finite, and we may thus restrict ourselves to the lowest-order term. Taking for the Debye-Hiickel result (168)... 

Edwards et al. (6) made the assumption that was equal to 4>pure a at the same pressure and temperature. Further theyused the virial equation, truncated after the second term to estimate pUre a These assumptions are satisfactory when the total pressure is low or when the mole fraction of the solute in the vapor phase is near unity. For the water, the assumption was made that <(>w, , aw and the exponential term were unity. These assumptions are valid when the solution consists mostly of water and the total pressure is low. The activity coefficient of the electrolyte was calculated using the extended Debye-Hiickel theory ... 

At least a partial solution to this problem is attained by the conventional activity scale method [5, 6, 7, 9, 10, 11]. This procedure was first used by Bates and Guggenheim [8] when formulating the operational definition of pH (see [86a], chapter 1), on the basis of which the National Bureau of Standards in the USA developed a method for determining conventional hydrogen ion activities. The basic assumption is the use of the Debye-Hiickel relationship for the individual activity of chloride ions ... 

This serious limitation of the utility of C-nmr has been recognized (Olah et al., 1970), but dismissed by recourse to the results of extended Hiickel M.O. calculations by Hoffmann (1964). These calculations indicated that the charge on the cationic centre of t-butyl is, in fact, more positive than in the isopropyl ion. Hoffmann noted that his results are at odds with the widely held assumption that a methyl group is a better electron donor than hydrogen but showed that the correct order of carbonium ion stability could be predicted even if a methyl group is electron withdrawing with respect to hydrogen. 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

The Debye Hiickel Limiting Law. Although beyond the scope of this Handbook, the derivation of this quantitative relationship rests on the following simplifying assumptions (a) Electrolytes are assumed to be completely... 

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