Self-consistent Hiickel methods

The u) parameter determines the weight of the charge on the diagonal elements. Since Ga is calculated from the results (MO coefficients, eq. (3.90)), but enters the Hiickel matrix which produces the results (by diagonalization), such schemes become iterative. Methods where the matrix elements are modified by the calculated charge are often called charge iteration or self-consistent (Hiickel) methods. 

The charge distributions and electron energies have been calculated by a self-consistent Hiickel method for phosphazenes of the type 0-X-CeH4)3-P=N-C6H4-p-N02 (X = H, Cl, or NMeg) and (15) and have been compared with their u.v. spectra. Hiickel-type MO calculations have also been carried out on other iV-nitrophenylphosphazenes, and the... 

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

These methods appear rather simple, yet they were the starting point of a long evolution. Gilles Klopman, whose research interests at Case-Western Reserve University later turned to modeling bioactive molecules, was the first to use Sandorfy s methods. Kenichi Fukui made extensive use of them in his well-known work on the structures and reactions of saturated hydrocarbons and their derivatives. Fukui added his frontier orbital considerations. Around 1959 the milieu of developments in quantum chemistry contributed to inspire William N. Lipscomb to conceive the extended Hiickel method, which was subsequently implemented by Lawrence L. Lohr and Roald Hoffmann.83 Soon thereafter, John Pople and his coworkers introduced self-consistent field methods based on the zero-differential overlap approximation.815... 

Leading references to the self-consistent field method employed here are cited by Pople, J. A., /. Phys. Chem. 61, 6 (1957). Calculations of the Hiickel and SCF orbitals were executed on an IBM-7090 computer using an entirely automatic program written by Dr. Bessis of the Centre de Mecanique Ondulatoire Appliqufee, Paris, France. 

The Hiickel molecular orbital (HMO) model of pi electrons goes back to the early days of quantum mechanics [7], and is a standard tool of the organic chemist for predicting orbital symmetries and degeneracies, chemical reactivity, and rough energetics. It represents the ultimate uncorrelated picture of electrons in that electron-electron repulsion is not explicitly included at all, not even in an average way as in the Hartree Fock self consistent field method. As a result, each electron moves independently in a fully delocalized molecular orbital, subject only to the Pauli Exclusion Principle limitation to one electron of each spin in each molecular orbital. 

H was the matrix-component of the Hiickel effective-Hamiltonian operator, effective between two basis atomic-orbitals, 4>r and 4>s, Srs was the overlap integral between 4>r and s, and H was set equal to a, H to / . This is how we developed the simple HMO-approach in Chapter Two. What Roothaan did was to show that a formally similar determinant is obtained in a full treatment of the re-electrons, but that it involves a somewhat more complicated expression for the matrix-elements, H . Furthermore, he showed that this more-complicated expression somehow had to take into account interactions between any one re-electron and all the other re-electrons. We do not go into the details of this here, except to say that, in order to find the LCAO-MO coefficients for one molecular orbital, it is necessary to know all the others, because all the others appear in the expressions for the equivalent terms, Hrs. This is a very familiar situation which mathematicians have long known how to deal with and which we encountered during our discussion of the self-consistent" Huckel-methods in 7.2—7.5 it is necessary to use an iterative scheme. An initial guess is made of all the orbitals except one and these are used to calculate the H -terms for the one orbital which has not yet... 

The self-consistent field method and the Hiickel approximation... 

The standard way of treating conjugated systems using the o-n separation approximation in the 1930s and into the 1960s was the so-called Hiickel methody In this method, the electron-electron interactions are not explicitly considered. Rather, the positions of the nuclei are fixed, and the electrons move in the field of the nuclei. Much of the error that results from neglecting the interactions of the electrons with one another can be circumvented with proper adjustment of empirical parameters. These parameters are also adjusted to allow (approximately) for the interaction from the o system, which is thus taken into account without specific calculations. This method is crude but does often give qualitative results that enable rationalization of many chemical phenomena of interest. It was a powerful and useful tool in its time. A better approximation is the Hartree-Fock or self-consistent field (SCF) method in which the electron-electron interactions are explicitly considered (Self-Consistent Field Method in Chapter 3). The quantum mechanical calculations on the n system in this case are carried out in a... 

The electronic spectrum of aniline has been analyzed, using wave-mechanical methods, by various authors - . The self-consistent field method which has been presented in section I and used in section II to describe the ground state of this molecule can also be used to represent its electronically excited states. For practical reasons it is necessary to introduce the lcao approximation and either the Hiickel or the Pariser and Parr method is used. Furthermore, the virtual orbitals approximation is added. Let us consider the eigenfunctions 77j of the self-consistent field operator A " corresponding to the ground state of aniline in the lcao approximation. They obey the equation... 

The Pariser-Pople-Parr method is the simplest implementation of the self-consistent field method. Like the Hiickel method, it treats only the electrons in delocalized pi orbitals in planar molecules and represents these orbitals as linear combinations of basis functions that include only the unhybridized p orbitals. All overlap integrals are assumed to vanish, as in the Hiickel method, but not all of the integrals representing electron-electron repulsions are assumed to vanish. Two notations are used for these integrals ... 

This situation arises when the geometries of the molecules A and 6 are identical with the corresponding fragments of the composite molecule AB, if MO calculations are carried out by a non-self-consistent-field method such as the extended Hiickel method. In such a case, the diagonal blocks of the matrices and S° are zero (Eq. 20). Consequently,... 

Conjugated polymers have been studied theoretically since 1937. Quantum chemical calculations applied to conjugated chains include (i) Hiickel methods, for the rapid assessment of band structures " (ii) the Paiser-Parr-Pople configuration interaction method " (iii) complete neglect of differential overlap (CNDO) self-consistent field methods " (iv) ab initio Hartree-Fock calculations, which have given very close agreement with experimental results " " and (v) the valence effective Hamiltonian (VEH) method, which offers a considerable reduction in computational effort. ... 

For planar unsaturated and aromatic molecules, many MO calculations have been made by treating the a and n electrons separately. It is assumed that the o orbitals can be treated as localized bonds and the calculations involve only the tt electrons. The first such calculations were made by Hiickel such calculations are often called Hiickel molecular orbital (HMO) calculations Because electron-electron repulsions are either neglected or averaged out in the HMO method, another approach, the self-consistent field (SCF), or Hartree-Fock (HF), method, was devised. Although these methods give many useful results for planar unsaturated and aromatic molecules, they are often unsuccessful for other molecules it would obviously be better if all electrons, both a and it, could be included in the calculations. The development of modem computers has now made this possible. Many such calculations have been made" using a number of methods, among them an extension of the Hiickel method (EHMO) and the application of the SCF method to all valence electrons. ... 

Most of these developments may be applied most directly within the framework of the isolated molecule method, in which the reactivity indices are the charges and self-polarizabilities of the unperturbed ground state of a given molecule calculations based on the localization model (e.g. Nesbet, 1962) have made less progress, and will not be considered. It is therefore natural to enquire whether indices similar to and tt,, in Hiickel theory can still be defined, and calculated more precisely, in self-consistent field theory. The obvious questions are... 

The second-order changes, in terms of which polarizability coefficients may be defined, are much more difficult to discuss because they involve essentially a change in the wave function (made in such a way as to preserve self-consistency)—unlike the first-order changes, which involve the Mwperturbed wave function only. Approximate formulae for the polarizabilities were first obtained (McWeeny, 1956) using a steepest descent method to minimize the energy, a useful result being the establishment of a connection between tt,, and F, valid for systems of any kind (non-alternant or heteroaromatic included) and applicable either in Hiickel theory or in a more complete theory. 

Finally, if we abandon Hiickel s topological approach altogether and consider more elaborate quantum-mechanical approaches, the concept of aromaticity derived purely from a consideration of -electrons becomes blurred and tends to disappear completely. In fact, allelectron methods allow the calculation of aromatic properties (Section V,B) of a given substance without introducing explicitly the concept of aromaticity. Certain authors, notably Dewar,19 have published resonance energies derived from self-consistent field molecular-orbital (SCF-MO) calculations, and these could be used as a measure of aromaticity. 

Stochastic aggregation does not emerge for oppositely charged particles, when electroneutrality holds due to conditions nk(t) = nB(f) = n(f), particle charge ea = — eB = e. Let us introduce, following the Debye-Hiickel method, the self-consistent potential (J> through Poisson equation... 

Over the past 20 years and, in particular since 1955, many theoretical studies of the electronic structure of pyrrole using the molecular orbital approach with varying degrees of refinement have been reported. The 7r-electronic structure of pyrrole has been extensively discussed in terms of both the simple Hiickel molecular orbital (LCAO) theory37- 41-55-65 and the more sophisticated self-consistent field molecular orbital method (SCFMO method).18- 66-77 Extended... 

The somewhat arbitrary use of the auxiliary inductive parameter in the Hiickel MO calculations has been questioned and the effect of the nonuniform distribution of a-electron densities, particularly in the CN bond,90 upon the 77-electron distribution has been discussed.65 Variable electronegativity self-consistent field (VESCF) molecular orbital calculations, which are an elaboration of the conventional SCF method and allow for the variation in electronegativity of an... 

Hiickel and extended Huckel methods are termed semi-empirical because they rely on experimental data for the quantification of parameters. There are other semi-empirical methods, such as CNDO, MINDO, INDO, in which experimental data are still used, but more care is taken in evaluating the Htj. These methods are self-consistent field procedures based on 3 SCF. They are discussed in various works on molecular orbital theory.4... 

In the first place the Hiickel approximation for the tt electrons has been replaced by a self-consistent field (SCF) procedure, generally in a semiempirical approximation of the Pariser-Parr-Pople type completed with some limited configuration interaction (PPP-CI method).59 00 Second, the a skeleton of the molecules has been treated by the Del Re procedure61 for saturated systems (which is the counterpart for the a electrons of the Hiickel method for 77 electrons) as refined for the a skeletons of conjugated heterocycles by... 

In this class of compounds, there are many interesting properties whose solutions, using approximate wave functions, may yield sufficiently accurate results to permit interpretation of the desired phenomena. For this reason we have performed calculations using the semiempirical extended Hiickel method, modified to include a seemingly more justifiable physical interpretation of the matrix elements as well as iterative processes which introduce a measure of self-consistency. We shall discuss this method in detail later, present some results of the calculations, and show their good agreement with experimental results. 

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