Hiickel theory

Hiickel theory separates the tt system from the underlying a framework and constructs molecular orbitals into which the tt electrons are then fed in the usual way according to the Aufbau principle. The tt electrons are thus considered to be moving in a field created by the nuclei and the core of a electrons. The molecular orbitals are constructed from linear combinations of atomic orbitals and so the theory is an LCAO method. For our purposes it is most appropriate to consider Hiickel theory in terms of the CNDO approximation (in fact, Hiickel theory was the first ZDO molecular orbital theory to be developed). Let us examine the three types of Fock matrix element in Equations (2.252)-(2.254). First, In 

As with the other semi-empirical methods that we have considered so far, the overlap matrix is equal to the identity matrix. The following simple matrix equation must then be solved  

The equation can be solved by standard methods to give the basis set coefficients and the molecular orbital energies E. The orbital energies for benzene are Ei = a 2/3 E2, 3 = 0-1- / Ei, E5 = a — (i E(, = a — 2/3, and so the ground state places two electrons in 3 and two each in the two degenerate orbitals i/ij and i/)3. The lowest-energy orbital i/)i is a linear combination of the six carbon p orbitals. 

The results of their calculations were summarised in two rules. The first rule states that at least one isomer C with a properly closed p shell (i.e. bonding HOMO, antibonding LUMO) exists for all u = 60 - - 6fc (fc = 0,2,3. but not 1). Thus Csq, C72, C78, etc., are in this group. The second rule is for carbon cylinders and states that a closed-shell structure is found for n = 2p 7 + 3k) (for all k). C70 is the parent of this family. The calculations were extended to cover different types of structure and fullerenes doped with metals. 

Hiickel theory is clearly limited, in part because it is restricted to tt systems. The extended Huckel method is a molecular orbital theory that takes account of all the valence electrons in the molecule [Hoffmann 1963]. It is largely associated with R Hoffmann, who received the Nobel Prize for his contributions The equation to be solved is FC = SCE, with the 

In Section 12.2 we use simple Hiickel theory to quantify these results. 

The Huckel jt energy levels and coefficients of the (a) allyl and (b) butadiene systems. 

Simple functions describe both the energy levels and orbital coefficients for these acyclic systems. The energy of the f MO for a system with N p orbitals is given by 

The orbital coefficient for the r atomic orbital in molecular orbital 0 where j runs from I to N, is given by 

The corresponding functions for cyclic systems are described in Section 12.3. The energy levels and orbital coefficients for more complex systems are to be found in the mammoth compilation of Streitweiser and Coulson [4]. 

In this freimework It is clear that the electrolytes are structure breakers. At high concentrations the effects are considerable, reaching 13° for 15m LlCl, and about 100° for 10m CaClz. It Is interesting to note that the differences between NaCl, NaBr, Nal are not as marked as those between LiCl, NaCl, NaBr, Nal. -This Is 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

Comparlsion of Excess Enthalpies of Transfer AH. , In cal/mole. (H2O D2O) at 25°C and Selected Concentrations  

The Intercept corresponds to a predicted Isotope effect on the heats of fusion of the pure solvent of 71+2 cal/m. In good agreement with the best calorimetric value 65+4 cal/m (64) taking Tf(H) - T (D) = -3.82. The present result is somewhat more precise than the old calorimetric data. The limiting slope Is in good agreement with the value predicted from the extrapolated 

When using this expression, we must include all the ions present in the solution, not just those of interest. For instance, if you are calculating the ionic strength of a solution of silver chloride and potassium nitrate, there are contributions to the ionic strength from all four types of ion. When more than two ions contribute to the ionic strength, we write 

The sulfate ion, SO, is an important source of sulfur used in the synthesis of the amino acids cysteine and methionine in plants and bacteria. To estimate the mean activity coefficient for the ions in 0.0010 m Na2S04(aq) at 25 C, we begin by evaluating the ionic strength of the solution from eqn 5.5  

3 Estimate the mean activity coefficient of NaCl in a solution 

For a derivation of the Debye-Huckel limiting law, see our Physical chemistry (2010). 

As we have stressed, eqn 5.4 is a limiting law and is reliable only in very dilute solutions. For solutions more concentrated than about 10 M, it is better to use an empirical modification known as the extended Debye-Hiickel law  

---

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. 

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 ... 

Extended Hiickel theory Generalised valence bond model Hartree-Fock... 

In the Hiickel theory, the 7r-electron energy of a conjugated molecule can be expressed by the following equation ... 

The simplest approximation to the Schrodinger equation is an independent-electron approximation, such as the Hiickel method for Jt-electron systems, developed by E. Hiickel. Later, others, principally Roald Hoffmann of Cornell University, extended the Hiickel approximations to arbitrary systems having both n and a electrons—the Extended Hiickel Theory (EHT) approximation. This chapter describes some of the basics of molecular orbital theory with a view to later explaining the specifics of HyperChem EHT calculations. 

HyperChem currently supports one first-principle method ab initio theory), one independent-electron method (extended Hiickel theory), and eight semi-empirical SCFmethods (CNDO, INDO, MINDO/3, MNDO, AMI, PM3, ZINDO/1, and ZINDO/S). This section gives sufficient details on each method to serve as an introduction to approximate molecular orbital calculations. For further details, the original papers on each method should be consulted, as well as other research literature. References appear in the following sections. 

The off-diagonal elements of Extended Hiickel theory, (fi v) represent the effects of bonding between the atoms and are assumed to be proportional to the overlap, Sj y. An approximation for differential overlap referred to as the Mulliken approximation... 

Although it is not possible to measure an individual ionic activity coefficient,, it may be estimated from the following equation of the Debye-Hiickel theory ... 

In an early investigation (66T539) the two highest occupied and the two lowest unoccupied orbitals were calculated on the basis of an extended Hiickel theory to determine the electron transition responsible for the long wavelength UV absorption. An Ai- Bi, [Pg.197]

The pioneering calculations of Wolfsberg and Helmholtz on Mn04, Cr04 and CIO4 are usually cited as the first applications of extended Hiickel theory... 

The systematic application of extended Hiickel theory to organic molecules comes with the work of Hoffmann. 

An Extended Hiickel Theory. I Hydrocarbons Roald Hoffmann... 

Prior to 1965, all we had in our armoury were the a and it Hiickel theories, and a very small number of rigorous calculations designated ab initio (to be discussed later). The aims of quantum chemistry in those days were to give total energies and charge distributions for real molecules, and the seventh decimal place in the calculated properties of LiH. Practical chemists wanted things like reliable enthalpy changes for reactions, reaction paths, and so on. It should come as no surprise to learn that the practical chemists therefore treated theoreticians with scepticism. 

Theoreticians did little to improve their case by proposing yet more complicated and obviously unreUable parameter schemes. For example, it is usual to call the C2 axis of the water molecule the z-axis. The molecule doesn t care, it must have the same energy, electric dipole moment and enthalpy of formation no matter how we label the axes. I have to tell you that some of the more esoteric versions of extended Hiickel theory did not satisfy this simple criterion. It proved possible to calculate different physical properties depending on the arbitrary choice of coordinate system. 

---