Orbital wave characteristics

The following presentation is limited to closed-shell molecular orbital wave-functions. The first section discusses the unique ability of molecular orbital theory to make chemical comparisons. The second section contains a discussion of the underlying basic concepts. The next two sections describe characteristics of canonical and localized orbitals. The fifth section examines illustrative examples from the field of diatomic molecules, and the last section demonstrates how the approach can be valuable even for the delocalized electrons in aromatic ir-systems. All localized orbitals considered here are based on the self-energy criterion, since only for these do the authors possess detailed information of the type illustrated. We plan to give elsewhere a survey of work involving other types of localization criteria. 

It has not proved mathematically feasible to calculate the electron-electron repulsion that causes this change in orbital-energies for many-electron molecules. It is even difficult to rationalize the qualitative changes in sequence on the basis of the shapes of the 11orbitals. Greater success has been achieved by an approximate method which begins with orbitals characteristic of the isolated atoms present in the molecule, and assumes that molecular orbital wave functions can be obtained by taking linear combinations of atomic orbital wave functions (abbreviated L.C.A.O.). For... 

Just as atomic orbitals are solutions to the quantum mechanical treatment of atoms, molecular orbitals (MOs) are solutions to the molecular problem. MOs have many of the same characteristics as atomic orbitals. Two of the most important are (1) they can hold two electrons with opposite spins and (2) the square of the molecular orbital wave function indicates the electron probability. 

In the bonding molecular orbital the electron density is greatest between the nuclei of the bonding atoms. In the antibonding molecular orbital, on the other hand, the electron density decreases to zero between the nuclei. We can understand this distinction if we recall that electrons in orbitals have wave characteristics. A property unique to waves allows waves of the same type to interact in such a way that the resultant wave has either an enhanced amplitude or a diminished amphtude. In the former case, we call the interaction constructive interference in the latter case, it is destructive interference (Figure 10.21). 

The projection operator method has applications beyond the deduction of group orbital SALCs. Deduce the wave function equations for the six ir molecular orbitals of benzene, using the labels specified for each 2p orbital. First, derive initial SALCs using each representation of the D /, point group some combinations wiU afford zero. Using sketches of the deduced orbitals, symmetry characteristics of the representations, and a coefficient table like that in Section 5.4.4, deduce the SALCs not derived initially from the character table analysis. Provide normalized equations and a sketch for each ir molecular orbital. 

Transition metals owe their location in the periodic table to the filling of the d subshells, as you Saw in Figure 6.31. Many of the chemical and physical properties of transition metals result from the unique characteristics of the d orbitals. For a given transition-metal atom, the valence (n — l)d orbitals are smaller than the corresponding valence ns and np orbitals. In quantum mechanical terms, the (n — l)d orbital wave functions drop off more rapidly as we move away from the nucleus than do the ns and np orbital wave functions. This characteristic feature of the d orbitals limits their interaction with orbitals on neighboring atoms, but not so much that they are insensitive to... 

The author of [73] deduced a physical model of the hydrogen bond from ah initio molecular orbital wave functions. The characteristic of the model are as follows the dipole moment of the A-H bond /Ta-h ) the difference between the first ionization potential of the electron donor and the noble gas in its row of the Mendeleev Table A / the length R of the hydrogen bonding lone pair. A number of characteristic of the intermolecular strength can be described in terms of these quantities. 

Atomic orbital (AO) (Sections 1.10, 1.11, and 1.15) A volume of space about the nucleus of an atom where there is a high probability of finding an electron. An atomic orbital can be described mathematically by its wave function. Atomic orbitals have characteristic quantum numbers the principal quantum number, n, is related to the energy of the electron in an atomic orbital and can have the... 

Atomic orbital (AO) (Sections 1.10, 1.11, and 1.15) A volume of space about the nucleus of an atom where there is a high probability of finding an electron. An atomic orbital can be described mathematically by its wave function. Atomic orbitals have characteristic quantum numbers the principal quantum number, n, is related to the energy of the electron in an atomic orbital and can have the values 1, 2, 3,. The azimuthal quantum number I, determines the angular momentum of the electron that results from its motion around the nucleus, and can have the values 0, 1, 2,. .., (n — 1). The magnetic quantum number, m, determines the orientation in space of the angular momentum and can have values from 3-/ to /. The spin quantum number, s, specifies the intrinsic angular... 

X-ray Electromagnetic radiation of wave length c. 1 k. X-rays are generated in various ways, including the bombarding of solids with electrons, when they are emitted as a result of electron transitions in the inner orbits of the atoms bombarded. Each element has a characteristic X-ray spectrum. 

Molecular properties and reactions are controlled by electrons in the molecules. Electrons had been thonght to be particles. Quantum mechanics showed that electrons have properties not only as particles but also as waves. A chemical theory is required to think abont the wave properties of electrons in molecules. These properties are well represented by orbitals, which contain the amplitude and phase characteristics of waves. This volume is a result of our attempt to establish a theory of chemistry in terms of orbitals — A Chemical Orbital Theory. 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

Independently of each other, Pauling and Slater worked out a quantum mechanical explanation of the directional valences characteristic of chemical molecules. They did this by proposing directional properties for the p wave functions and for the sp3 wave functions resulting from "hybridization" of electron wave functions, or orbitals.73... 

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