Simple Htickel method

Figure 4.16. The n MOs and jt energy levels for an acyclic three-p-orbital system in the simple Htickel method. The MOs are composed of the basis functions (three p AOs) and the eigenvectors (the c s), while the energies of the MOs follow from the eigenvalues (Eq. (4.68)), In the drawings of the MOs, the relative sizes of the AOs in each MO suggest the relative contribution of each AO to that MO. This diagram is for the propenyl radical. The paired arrows represent a pair of electrons of opposite spin, in the fully-occupied lowest MO,, and the single arrow represents an unpaired electron in the nonbonding MO, 4 2> tlie highest n MO, fs, is empty in the radical.

In the simple Htickel method the Fock matrix elements Hij are not calculated, but are instead set equal to 0 or -1 according to simple rules based on atomic connectivity (section 4.3.3) in the EHM the Hij are calculated from the relative positions (through Sij) of the orbitals or basis functions and the ionization potentials of these orbitals (section 4.4.1) in neither case is Hij calculated from first principles. Section 4.3.3, Eqs (44) imply that Hij is ... 

In the simple Htickel method, Csi denotes the basis frmction coefficient for the contribution of atom number. y (in whatever numbering scheme we choose) to MO number In the ab initio method, Cgi still refers to MO number i, but the s does not necessarily denote atom number. y. Explain. 

In the simple and extended Htickel methods, the molecular orbitals are calculated and then filled from the bottom up with the available electrons. However, in ab initio calculations the occupancy of the orbitals is taken into account as they are being calculated. Explain. (Hint look at the expression for the Fock matrix elements in terms of the density matrix.)... 

The predictions of relative stability by the various approaches diverge more widely when nonbenzenoid systems are considered. The simple Htickel method using total TT delocalization energies relative to an isolated double-bond reference energy (a + 6) fails. This approach predicts stabilization of the same order of magnitude for such unstable systems as pentalene and fulvalene as it does for much more stable aromatics. The HMO , RE, and SCF-MO methods, which use polyene reference energies, do much better. All show drastically reduced stabilization for such systems and, in fact, indicate destabilization of systems such as butalene and pentalene (Scheme 8.2). 

Next we must choose the periodic basis set Uk x) for the tt MOs. The normal choice for this (and also the choice dictated by the simple Htickel method) is a 2pjr AO on each carbon. Furthermore, it is normal to assume that this basis does not change as we go from one MO to another, so we can dispense with the subscript k on U, (Hence, no variational calculation will be necessary in finding the wavefunctions for this system.)... 

The first of the zero differential overlap (ZDO) methods was the simple tt-electron method due to Htickel. Historically this method was very important in that it showed rather quickly that molecular orbital methods that... 

We assume here that the total electronic energy is equal to the sum of the individual electronic energies. This relationship, which has the advantage of being simple, is obtained when the electronic Hamiltonian is written as a sum of monoelectronic Hamiltonians, as in the Htickel and extended Hiickel methods. This approximate formula has, of course, limited application, but it is acceptable for a qualitative analysis of orbital interactions. Further details may be found in Structure ectronique des molecules, by Y. jean and F. Volatron, Volume 2, Chapter 13, Dunod,... 

The chemical shifts of the sp -hybridized carbons for methylbenzenium ions were compared with the jt-electron (q,) and the total (q = q, + q ) electron defkioM calculated by the Htickel LCAO MO and CNDO/2 methods. To judge by the values of root-mean-square deviations (s) and of correlation coefficients (r) the chemical shifts of the sp -hybridized carbons of the ions under consideration correlate better with the Tc-electron deficiency calculated by the simple Hiickel MO method... 

In analogy to extended Htickel theory, there are also charge iterative methods for simple Htickel theory. The equivalent of eq. (3.99) is given in eq. (3.104). 

This method is called Extended Htickel because it is in fact an extension, including all the a valence orbitals, of a much earlier method that considers only ir-electron systems in which each atom is represented by one electron in one pz orbital. Matrix elements are obtained as = a and = p, where a and P are empirical parameters for each atomic species. In the simple Htickel formulation butadiene is a four-orbital system and the secular equation can be solved by hand. 

Notice that, unlUce the case of the Henry equation, in the simple cases of Htickel and Smoluchowski equations the size of particles (radius) is not required. There are alternative graphical ways for correcting for the size of particles (Pashley and Karaman, 2004). Although the Htickel and Smoluchowski equations are very useful, it can be shown that they only cover a very small part of the colloidal domain. In most cases, corrections are needed, e.g. via the use of Henry equation or other graphical methods where the correction factor, f, can be estimated. Negative values of y/o can be obtained if p is negative. 

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