Simple Hiickel Assumptions

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—... 

Various refinements of the above model have been proposed for example, using alternative spherical potentials or allowing for nonspherical perturbations,and these can improve the agreement of the model with the abundance peaks observed in different experimental spectra. For small alkali metal clusters, the results are essentially equivalent to those obtained by TSH theory, for the simple reason that both approaches start from an assumption of zeroth-order spherical symmetry. This connection has been emphasized in two reviews,and also holds to some extent when considerations of symmetry breaking are applied. This aspect is discussed further below. The same shell structure is also observed in simple Hiickel calculations for alkali metals, again basically due to the symmetry of the systems considered. However, the developments of TSH theory, below, and the assumptions made in the jellium model itself, should make it clear that the latter approach is only likely to be successful for alkali and perhaps alkali earth metals. For example, recent results for aluminium clusters have led to the suggestion that symmetry-breaking effects are more important in these systems. ... 

We see therefore that no part of the Coulson-Rushbrooke Theorem on alternant hydrocarbons depends on having all non-zero Hamiltonian matrix-elements, Hrs, equal. In order for the reader to be quite clear which assumptions, in the context of the simple Hiickel-method, are necessary for the Theorem to hold, we summarise them again below. We require... 

The extended Hiickel (EH) method is much like the simple Hiickel method in many of its assumptions and limitations. However, it is of more general applicability since it takes account of all valence electrons, o and n, and it is of more recent vintage because it can only be carried out on a practical basis with the aid of a computer. The basic methods of extended Hiickel calculations have been proposed at several times by various people. We will describe the method of Hoffmann [1], which, because of its systematic development and application, is the EHMO method in common use. 

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... 

In Eqs. (3) and (4), H stands for the effective one-electron Hamiltonian the integration is over the whole space. In the simple method, a number of simplifying assumptions about the values of the integrals ftjk, j, and Sjk are introduced in the case of molecules containing no heteroatoms these are known as the Hiickel approximations. It seems useful to use this designation also for molecules with heteroatoms, and in the present review this method will be referred to as the Hiickel molecular orbital (HMO) method according to Streitwieser s suggestion.4... 

For this calculation some arbitrary assumption must be made, such as 7h3o+ = y the value of 7 can be estimated from the Debye-Hiickel equation. Alternatively, the known pH of an NBS primary standard buffer can be used in Eq. (5.30). This latter procedure probably is the most practical and, if the pH of the solutions lies between 2 and 12 and they contain only simple ions in concentrations less than 0.2 M, good constancy of Ej is found. This value of ( + Ej) consequently relates specifically to solutions that are similarly constituted. 

In the first instance, and as a first approximation valid for very dilute solutions, one may ignore all types of ion-ion interactions except those deriving from simple Coulombic" forces. Thus, short-range interactions (e.g dispersion interactions) are excluded. This is a fundamental assumption of the Debye-Hiickel theory. Then the potential of average force U simply becomes the Coulombic potential energy of an ion of charge z, q in the volume element dV, i.e., the charge on the ion times the electrostatic potential in the volume element dV. That is,... 

We also make another assumption, at the moment simply for convenience. It is an assumption which Hiickel made and one that has been made by a very large number of people subsequently. We invoke it partly on historical grounds and partly because it is simple afterwards ( 2.3) we shall go back and see what difference it has made. The assumption is that atomic orbitals on different centres are orthogonal—i.e. 

E. Hiickel introduced a simple quantum mechanical model for the description of the electronic structure of planar unsaturated molecules with the bonding connectivity as input. This model has been widely used. Although today s computing power and quantum chemistry software available for all chemists have made the assumptions of the Hiickel model unnecessarily simplistic, the model is still used to make estimates of molecular energies and has established itself as a useful teaching tool. 

The MSA is fundamentally connected to the Debye-Hiickel (DH) theory [7, 8], in which the linearized Poisson-Boltzmann equation is solved for a central ion surrounded by a neutralizing ionic cloud. In the DH framework, the main simplifying assumption is that the ions in the cloud are point ions. These ions are supposed to be able to approach the central ion to some minimum distance, the distance of closest approach. The MSA is the solution of the same linearized Poisson-Boltzmann equation but with finite size for all ions. The mathematical solution of the proper boundary conditions of this problem is more complex than for the DH theory. However, it is tractable and the MSA leads to analytical expressions. The latter shares with the DH theory the remarkable simplicity of being a function of a single screening parameter, generally denoted by r. For an arbitrary (neutral) mixture of ions, this parameter satisfies a simple equation which can be easily solved numerically by iterations. Its expression is explicit in the case of equisized ions (restricted case) [12]. One has... 

It is also important to understand why ionic solutions behave the way they do. A few simple assumptions lead us to the Debye-Hiickel theory for the description of ionic solutions. Even brief descriptions of these ideas will help us recognize why we devote an entire chapter to the interaction and chemistry of charged solutes. 

The Hiickel method is simple and has been in use for decades (Hiickel, 1931a,b). It is based on the ct-ti separation approximation while accounting for the pi-electrons only, i.e., the atomic orbitals involved refer to those 2p for Carbon atoms as well to the 2p and 3p orbitals for the second and third period elements as (N,0, F) and (S, Cl) respectively further discussion on the d-orbitals involvement may be also undertaken, yet the method essence reside in non explicitly counting on the electronic repulsion with an effective, not-defmed, mono-electronic Hamiltonian, as the most simple semi-empirical approximation. In these conditions, for the mono-electronic Hamiltonian matrix elements two basic assumptions are advanced the forthcoming discussion follows (Putz, 201 Id) ... 

The polyelectrolyte chains exist in the effective medium created by the counterions and ions from the simple salt. The charges on the polymer chains interact among themselves mediated by the neutralizing plasma that makes up the system except the polymer molecules. We further assume that the mediation by the background is adequately described by the Debye-Hiickel theory. This assumption allows us to deduce the key features, without resorting to heavy numerical work that will be needed to solve the very complex Poisson-Boltzmann equations for such topologically correlated objects as flexible polymers. 

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