Mathematical modeling, molecular orbitals

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

Finally, the present guide is much less academic and much more practical than ""Ab Initio Molecular Orbital Theory . Focus is not on the underlying elements of the theory or in the details of how the theory is actually implemented, but rather on providing an overview of how different theoretical models fit into the overall scheme. Mathematics has been kept to a minimum and for the most part, references are to monographs and reviews rather than to the primary literature. 

A wavefunction ip and its eigenvalue E define an orbital. The orbital is therefore an energy level available for electrons and it implies the relevant electron distribution. In mathematical models, these distributions extend to infinity, but in a pictorial representation it is sufficient to draw the volume in which the probability of presence of the electron is rather arbitrarily around 90%. The spatial distribution of atomic and molecular orbitals have implications for processes of electron tunneling (section 4.2.1). 

The relative importance of a and r contributions to the overall bonding is unclear, but several different combinations of relative strengths lead to limiting case models. When there are 2 electrons in the forward (T-bond and 2 electrons in the ir-backbond, there are 2 bonding electrons for each metal-carbon bond. This is mathematically equivalent to 2tr-bonds and a metallocyclopropane structure (72). This model does not necessitate strict sp3 hybridization at the carbon atoms. Molecular orbital calculations for cyclopropane (15) indicate that the C—C bonds have higher carbon atom p character than do the C—H bonds. Thus, the metallocyclopropane model allows it interactions with substituent groups on the olefin (68). 

The choice of topics is largely governed by the author s interests. Following a brief introduction the crystal field model is described non-mathematically in chapter 2. This treatment is extended to chapter 3, which outlines the theory of crystal field spectra of transition elements. Chapter 4 describes the information that can be obtained from measurements of absorption spectra of minerals, and chapter 5 describes the electronic spectra of suites of common, rock-forming silicates. The crystal chemistry of transition metal compounds and minerals is reviewed in chapter 6, while chapter 7 discusses thermodynamic properties of minerals using data derived from the spectra in chapter 5. Applications of crystal field theory to the distribution of transition elements in the crust are described in chapter 8, and properties of the mantle are considered in chapter 9. The final chapter is devoted to a brief outline of the molecular orbital theory, which is used to interpret some aspects of the sulphide mineralogy of transition elements. 

In the case of the HMO model the success of the search for structure-property relations is much enhanced by the possibility of applying the powerful mathematical apparatus of graph-spectral theory [18, 19]. There is hardly any application of graph (spectral) theory in more sophisticated molecular orbital models. 

This highly successful qualitative model parallels the most convenient quantum mechanical approach to molecular orbitals the method of linear combination of atomic orbitals (LCAO). We have assumed that the shapes and dispositions of bond orbitals are related, in a simple way to the shapes and dispositions of atomic orbitals. The LCAO method makes the same assumption mathematically to... 

As an alternative to ab initio methods, the semi-empirical quantum-chemical methods are fast and applicable for the calculation of molecular descriptors of long series of structurally complex and large molecules. Most of these methods have been developed within the mathematical framework of the molecular orbital theory (SCF MO), but use a number of simplifications and approximations in the computational procedure that reduce dramatically the computer time [6]. The most popular semi-empirical methods are Austin Model 1 (AMI) [7] and Parametric Model 3 (PM3) [8]. The results produced by different semi-empirical methods are generally not comparable, but they often do reproduce similar trends. For example, the electronic net charges calculated by the AMI, MNDO (modified neglect of diatomic overlap), and INDO (intermediate neglect of diatomic overlap) methods were found to be quite different in their absolute values, but were consistent in their trends. Intermediate between the ab initio and semi-empirical methods in terms of the demand in computational resources are algorithms based on density functional theory (DFT) [9]. 

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