Hiickel

The Hiickel description of aromaticity was based in part on benzene, a cyclic fully conjugated hydrocarbon having (4n -l- 2) -electrons (ff = I) in the closed shell (ring). 

Debye-Hiickel theory The activity coefficient of an electrolyte depends markedly upon concentration. Jn dilute solutions, due to the Coulombic forces of attraction and repulsion, the ions tend to surround themselves with an atmosphere of oppositely charged ions. Debye and Hiickel showed that it was possible to explain the abnormal activity coefficients at least for very dilute solutions of electrolytes. 

Debye-Hiickel equation Debye-length Condenser capacity... 

At this point an interesting simplification can be made if it is assumed that r, as representing the depth in which the ion discrimination occurs, is taken to be just equal to 1/x, the ion atmosphere thickness given by Debye-Hiickel theory (see Section V-2). In the present case of a 1 1 electrolyte, k = (8ire V/1000eitr) / c /, and on making the substitution into Eq. XV-7 and inserting numbers (for the case of water at 20°C), one obtains, for t/ o in millivolts ... 

A more general classification considers the phase of the total electronic wave function [13]. We have treated the case of cyclic polyenes in detail [28,48,49] and showed that for Hiickel systems the ground state may be considered as the combination of two Kekule structures. If the number of electron pairs in the system is odd, the ground state is the in-phase combination, and the system is aromatic. If the number of electron pairs is even (as in cyclobutadiene, pentalene, etc.), the ground state is the out-of-phase combination, and the system is antiaromatic. These ideas are in line with previous work on specific systems [40,50]. 

Finally, the distinction between Huckel and Mobius systems is considered. The above definitions are valid for Hiickel-type reactions. For aromatic Mobius-type reations, the reverse holds An ATS is formed when an even number of electron pairs is re-paired. 

Hiickel-type systems (such as Hilcfcel pericyclic reactions and suprafacial sigmatropic shifts) obey the same rules as for sigma electron. The rationale for this observation is clear If the overlap between adjacent p-electron orbitals is positive along the reaction coordinate, only the peraiutational mechanism can... 

Electi ocyclic reactions are examples of cases where ic-electiDn bonds transform to sigma ones [32,49,55]. A prototype is the cyclization of butadiene to cyclobutene (Fig. 8, lower panel). In this four electron system, phase inversion occurs if no new nodes are fomred along the reaction coordinate. Therefore, when the ring closure is disrotatory, the system is Hiickel type, and the reaction a phase-inverting one. If, however, the motion is conrotatory, a new node is formed along the reaction coordinate just as in the HCl + H system. The reaction is now Mdbius type, and phase preserving. This result, which is in line with the Woodward-Hoffmann rules and with Zimmerman s Mdbius-Huckel model [20], was obtained without consideration of nuclear symmetry. This conclusion was previously reached by Goddard [22,39]. 

In this section, the systematic search for conical intersections based on the Longuet-Higgins phase-change rule is described. For conciseness sake, we limit the present discussion to Hiickel-type systems only, unless specifically noted otherwise. The first step in the antilysis is the determination of the LH loops containing a conical intersection for the reaction of interest. 

To become familar with the algorithm for charge calculation by partial equalization of orbital electronegativity (PEOE) and by a modified Hiickel Molecular Orbital method... 

In the non-linear differential equation Eq. (43), k is related to the inverse Debye-Hiickel length. The method briefly outlined above is implemented, e.g., in the pro-... 

The theoretical methods used commonly can be divided into three main categories, semi-empirical MO theory, DFT and ab-initio MO theory. Although it is no longer applied often, Hiickel molecular orbital (HMO) theory will be employed to introduce some of the principles used by the more modem techniques. 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

We can now assign the four carbon p-orbitals, one to each carbon. For simplicity, we will label them with the subscript corresponding to the number of the carbon atom to which the AO belongs. We will use the symbol p to denote AOs and P for MOs. We can now write the Hiickel matrix as a square matrix involving the AOs as shown in Figure 7-20. 

The diagonal elements of the Hiickel matrix represent the energies of the contributing AOs, which in this case are all a. Each of the bonds (in this case tpi-Wi> arid 3- 4) is assigned the overlap energy and all other elements of... 

Figure 7-20. The Hiickel matrix (above) and the eigenvalues and eigenvectors for 1,3-butadiene.

The PEOE method in conjunction with a modified Hiickel Molecular Orbital (HMO) method allows charge calculation in conjugated r-systems. 

I h c Hiickel eon stari t (k) scales Ih e in teraetiou energy between two atomic orbitals (see Kxtended Hiickel Method on page 125). HyperChem uses the defatill value of 1.75 (see the second part of th is book. Theory and Methods). You shoti Id tise th is value for m ost eases, A suggested ran ge for experimcri tal adjiistmen L of th is eon -stant is 1,6 2,0.- ... 

Note You cannot use the Hxicndcd Hiickel method or any one of th e-SCH m eth ods with theCI option being turn ed on forgeometry optim i/.ation s, m olecular dynam ics sim ulation s or vibrational cal-culations, in the ctirrcrit version ofHypcrChem. 

You can use any ah initio SCT calciilalion and all Ihe semi-empiri-cal methods, except Extended Hiickel. for molecular dynamics simulations. The procedures and considerations are similar for sim u lation s using molecular mech anics m eihods (see Molecular Dynamics" on page 69). 

Extended Hiickel is the simplest and fastest senii-empirical method included m IlyperC hem, but it isalso the least accurate. It Is particularly simple in its treatment of electron-electron interactions it has no explicit treatment of these interactions, although it may include some of their effects by parameteri/.aiioii. 

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