Orbit, concept

The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept. 

Color from Transition-Metal Compounds and Impurities. The energy levels of the excited states of the unpaked electrons of transition-metal ions in crystals are controlled by the field of the surrounding cations or cationic groups. Erom a purely ionic point of view, this is explained by the electrostatic interactions of crystal field theory ligand field theory is a more advanced approach also incorporating molecular orbital concepts. 

When both the 1,3-dipoIe and the dipolarophile are unsymmetrical, there are two possible orientations for addition. Both steric and electronic factors play a role in determining the regioselectivity of the addition. The most generally satisfactory interpretation of the regiochemistry of dipolar cycloadditions is based on frontier orbital concepts. As with the Diels-Alder reaction, the most favorable orientation is that which involves complementary interaction between the frontier orbitals of the 1,3-dipole and the dipolarophile. Although most dipolar cycloadditions are of the type in which the LUMO of the dipolarophile interacts with the HOMO of the 1,3-dipole, there are a significant number of systems in which the relationship is reversed. There are also some in which the two possible HOMO-LUMO interactions are of comparable magnitude. 

This chapter will try to cover some developments in the theoretical understanding of metal-catalyzed cycloaddition reactions. The reactions to be discussed below are related to the other chapters in this book in an attempt to obtain a coherent picture of the metal-catalyzed reactions discussed. The intention with this chapter is not to go into details of the theoretical methods used for the calculations - the reader must go to the original literature to obtain this information. The examples chosen are related to the different chapters, i.e. this chapter will cover carbo-Diels-Alder, hetero-Diels-Alder and 1,3-dipolar cycloaddition reactions. Each section will start with a description of the reactions considered, based on the frontier molecular orbital approach, in an attempt for the reader to understand the basis molecular orbital concepts for the reaction. 

The most advanced among today s projects is the future American reusable launch system known as Venture Star. It utilizes two new concepts that are expected to boost efficiency and cut down the cost of a payload. The first is the single stage-to-orbit concept—the launch vehicle carries the entire propellant load and does not have any expendable parts. The latter is planned to be achieved by a revolutionary rocket engine design called Aerospike. The bell-shaped noz-... 

In the cases other than [case A] and [case B), so called "open-shell SCF methods are employed. The orbital concept becomes not quite certain. The methods are divided into classes which are "restricted 18> and "unrestricted 19> Hartree-Fock procedures. In the latter case the wave function obtained is no longer a spin eigenfunction. 

H2, N2, or CO dissociates on a surface, we need to take two orbitals of the molecule into account, the highest occupied and the lowest unoccupied molecular orbital (the HOMO and LUMO of the so-called frontier orbital concept). Let us take a simple case to start with the molecule A2 with occupied bonding level a and unoccupied anti-bonding level a. We use jellium as the substrate metal and discuss the chemisorption of A2 in the resonant level model. What happens is that the two levels broaden because of the rather weak interaction with the free electron cloud of the metal. 

Given the ubiquitous character of molecular orbital concepts in contemporary discourse on electronic structure, ionization energies and electron affinities provide valuable parameters for one-electron models of chemical bonding and spectra. Electron binding energies may be assigned to delocalized molecular orbitals and thereby provide measures of chemical reactivity. Notions of hardness and softness, electronegativity,... 

This system has its merits and uses - indeed, we shall employ the line notation almost exclusively - but to understand how bonding occurs, and to explain molecular shape and chemical reactivity, we need to use orbital concepts. 

It is then quite understandable why, without the necessary mathematical machinery, the relevant concepts cannot be properly grasped. On the other hand, the mathematical disguise that is characteristic of quantum-chemistry courses makes both teachers and students pay more attention to the complexities of the mathematics (the tools, the trees ) and lose the physics (the actual world, the forest ). Although mathematics is essential for a deep understanding of quantum chemistry, the underlying physical picture and its connection with mathematics are equally important. AOs, MOs and related concepts derive from SchrOdinger s wave mechanics, which is an approximation to nature. According to Simons (96), "orbital concepts are merely aspects of the best presently available model they are not real in the same sense that experimental observations are. ... 

Quantum chemical methods may be divided into two classes wave function-based techniques and functionals of the density and its derivatives. In the former, a simple Hamiltonian describes the interactions while a hierarchy of wave functions of increasing complexity is used to improve the calculation. With this approach it is in principle possible to come arbitrarily close to the correct solution, but at the expense of interpretability of the wave function the molecular orbital concept loses meaning for correlated wave functions. In DFT on the other hand, the complexity is built into the energy expression, rather than in the wave function which can still be written similar to a simple single-determinant Hartree-Fock wave function. We can thus still interpret our results in terms of a simple molecular orbital picture when using a cluster model of the metal substrate, i.e., the surface represented by a suitable number of metal atoms. 

The description of the bonding of unsaturated hydrocarbons to metals was originally developed by Dewar, Chatt and Duncanson and is now known as the well-established DCD model based on a frontier-orbital concept [82]. In this model, the interaction is viewed in terms of a donation of charge from the highest occupied -orbital into the metal and a subsequent backdonation from filled metal-states into the lowest unoccupied -orbital, see Figure 2.33. Contrary to the case of the standard Blyholder model for CO and N2 the DCD frontier-orbital model is supported by experimental XES measurements [83]. In the present section, we will show how we can experimentally identify and quantify the contributions of the different -orbitals involved in the interaction with the surface. The DCD model will be shown to very well describe the chemical bonding of ethylene on Cu and Ni surfaces. Furthermore, the differences in bonding of benzene to Cu and Ni will be discussed. 

The bond length data, although not satisfactory, do suggest that the P—C bond has multiple character with a bond order of about 2. Other physical measurements which support this and provide further insight are detailed below. Note that many of the original studies which reported these data discuss them in terms of dn-pn overlap on the basis that it is evidence for such overlap. Since the d orbital concept is now discarded, these discussions are obsolete, but the experimental results still give useful information. 

The molecular orbital (MO) is the most fundamental quantity in contemporary quantum chemistry. Almost all of our understanding of "what the electrons are doing in molecules" is based on the molecular orbital concept. Also most of the computational methods used today start by a calculation of the MO s of the system. 

This description of a covalent bond in terms of overlapping atomic orbitals stems from the desire to retain and use the atomic orbital concept. Atomic orbitals were designed to describe electron distributions in isolated atoms however, one might expect distortions of the isolated atom electron cloud when it is in close proximity to the positive charge of another atom. A more detailed description of the covalent bond would have to include this distortion. 

Parr and collaborators [8-12] showed how Fukui s frontier-orbital concept could be grounded in a rigorous many-electron theory, density-functional theory (DFT) [8,16-18], They used the ensemble formulation of DFT to introduce the expectation value Jf of the total electron number as a continuous variable. They then defined the Fukui functions... 

The Ambiguity of the Frontier-Orbital Concept and the Complexity of Chemical Responses... 

We have now seen that the effort of Parr and collaborators [8-12] to put Fukui s frontier-orbital concept of chemical reactivity on sound footing in density-functional theory through the definition of the Fukui function and the local and global softness works only for extended systems. This restriction to extended systems raises a sixth issue. In both the local softness and the Fukui function, Eqs. (54) and (53a), the orbitals at the chemical potential represent both the LUMO and the HOMO in the Fukui sense. However, there is a continuum of unoccupied KS states above the chemical potential accessible even to weak chemical perturbations any linear combination of which could in principle be selected as the LUMO, and similarly for states below fi and the HOMO. This ambiguity in the frontier-orbital concept obviously applies as well to localized systems when there is more than one KS state significantly affected by a chemical perturbation. 

This chapter has presented an overview of several important aspects of the chemistry of coordination compounds. In addition to the elementary ideas related to bonding presented here, there is an extensive application of molecular orbital concepts to coordination chemistry. However, most aspects of the chemistry of coordination compounds treated in this book do not require this approach, so it is left to more advanced texts. The references at the end of this chapter should be consulted for more details on bonding in complexes. 

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