Atomic orbitals theoretical terms

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

The concepts of hybridisation and resonance are the cornerstones of VB theory. Unfortunately, they are often misunderstood and have consequently suffered from much unjust criticism. Hybridisation is not a phenomenon, nor a physical process. It is essentially a mathematical manipulation of atomic wave functions which is often necessary if we are to describe electron-pair bonds in terms of orbital overlap. This manipulation is justified by a theorem of quantum mechanics which states that, given a set of n respectable wave functions for a chemical system which turn out to be inconvenient or unsuitable, it is permissible to transform these into a new set of n functions which are linear combinations of the old ones, subject to the constraint that the functions are all mutually orthogonal, i.e. the overlap integral J p/ip dT between any pair of functions ip, and op, (i = j) is always zero. This theorem is exploited in a great many theoretical arguments it forms the basis for the construction of molecular orbitals as linear combinations of atomic orbitals (see below and Section 7.1). 

Hybridization is the mixing of orbitals on an atom to produce new, hybridized (in the spirit of the biological use of the term), atomic orbitals. This is done mathematically but can be appreciated pictorially (Fig. 4.5). One way to justify the procedure theoretically is to recognize that atomic orbitals are vectors in the generalized... 

Despite the quantitative victory of molecular orbital (MO) theory, much of our qualitative understanding of electronic structure is still couched in terms of local bonds and lone pairs, that are key conceptual elements of the valence bond (VB) picture. VB theory is essentially the quantum chemical formulation of the Lewis concept of the chemical bond [1,2]. Thus, a chemical bond involves spin-pairing of electrons which occupy valence atomic orbitals or hybrids of adjacent atoms that are bonded in the Lewis structure. In this manner, each term of a VB wave function corresponds to a specific chemical structure, and the isomorphism of the theoretical elements with the chemical elements creates an intimate relationship between the abstract theory and the nature of the... 

Complex molecules may not possess any symmetry elements, or if they do, the localizations of the electrons can so distort the electron cloud that its symmetry bears little relation to the molecular symmetry. In such cases it may be best to revert to a description of states in terms of the individual orbitals. As an example, we will consider formaldehyde, although a molecule as simple as this is probably best described by the group-theoretical term symbol of the last paragraph. The last filled orbitals in H2CO can easily be shown to be. ..(jtco)2 (no)2, where no represents the nonbonding orbital on the O atom and the two electrons in it are the lone pair. The first unfilled orbitals in formaldehyde are the tt 0 and rr o antibonding orbitals. Promotion of one... 

The topological or graph-theoretical approaches attempt to define the bonding and antibonding orbitals available to a cluster in terms of a sort of valence-bond description, where hybridized atomic orbitals are directed in space to form either localized or multicenter functions. (See also Hybridization) The analysis of a complex structure involving delocalized bonding may, however, require some seemingly subjective decisions about how the atomic orbitals overlap, where the multicenter bonds should be formed, and so forth. 

It was, therefore, clear in 1974 that electronic-structure methods were not sufficiently advanced to reproduce experimental data accurately for even a simple ionic oxide such as MgO. The emphasis at the time was on the determination of the effects of different approximations upon the calculated results. Comparison was usually made between one calculation and another rather than between calculation and experiment. The theoretical papers reported the quantities arising directly from the calculations, such as orbital eigenvalues and atomic-orbital charge decompositions, and spectral properties were interpreted primarily in terms of orbital energies. No attempt was made to evaluate equilibrium structural or energetic properties. 

Although the use of strokes to represent bonds between atoms in molecules comes from the nineteenth century, the electron pair concept as necessary for the understanding of chemical bonding was introduced by G.N. Lewis (1875-1946) in 1916 (ref. 90) following Bohr s, then recently proposed, model of the atom. Indeed, the Lewis model still lies at the basis of much of present-day chemical thinking, although it was advanced before both the development of quantum mechanics and the introduction of the concept of electron spin. In a more quantitative way, it found a natural theoretical extension in the valence-bond approximation to the molecular wavefunction, as expressed in terms of the overlap of (pure or hybridized) atomic orbitals to describe the pairing of electrons, coupled with the concept of electron spin. 

In theoretical terms, the total electron density in a molecule is easily expressed in terms of the occupied molecular orbitals. Additional information is gained from the m.o. approach especially regarding the electronic energy for ground and excited states and the detailed features (e.g. phase) of individual m.o.s. Molecular orbitals are mathematical functions that can be constructed as linear combinations of orbitals of the contributing atoms, in a process where the atoms lose their individuality, except for the respective nuclei and, perhaps, the core electrons. The valence electrons are described by functions which, in general, extend to several atoms or even to the whole molecule. 

Given a specific atomic orbital basis, it is always possible to calculate all the matrix elements by integration, and to introduce the theoretical values found in that way as corrections to the terms Wp Spq of the ordinary GMS potential 31>. Unfortunately, these corrections are very sensitive to the choice of the basis, and it is difficult to give them a definite meaning. 

Unlike the s orbitals, the atomic p orbitals possess one angular node. At each cluster vertex, linear combinations of px, py, and pz may be taken, which generate one radial (p°) and two tangential (p31) orbitals. In terms of TSH theory these orbitals form the basis of the cluster Lp and Lnp/Lp MO s. In group theoretical terms, Tp can be divided into radial and tangential parts as follows ... 

The large cross-sections can be explained in terms of an electron jump mechanism [233, 265]. The chlorine atom p-orbital which is out of the plane of the three atoms ( P) participates in the reaction and this then must lead to an excited sdkali metal atom ( P). Theoretical studies [266, 267] indicate that such reactions involving out-of-plane p- or 7r-orbitals will have large cross-sections. 

The discussion presented above has demonstrated that non-bonding orbitals play an important role in determining the stereochemistries and closed shell requirements of both co-ordination and cluster compounds. In group theoretical terms, whenever there are an odd number of irreducible representations, a non-bonding orbital is generated which is either localised exclusively on the central atom, or the peripheral atoms and has nodal characteristics which lead to a zero or minimal overlap with the remaining atoms in the structure. 

It was immediately recognized by M.F. Hawthorne and coworkers that the open face of either B9C2Hj ion was essentially identical to the cyclopentadienide ion C5HJ in terms of electrons and orbitals. The theoretical description of the open face of the B9C2Hj ion has 6 electrons occupying 5 atomic orbitals directed toward the missing vertex of the icosahedron. The cyclopentadienide ion has six electrons occupying five atomic orbitals oriented perpendicular to the plane of the ion and is well known for its ability to sandwich a metal ion between two cyclopentadienide... 

Theoretical studies (MNDO-SCF) of commo-silicarbaboranes including commo- A -Si( 1,2,3-SiC2B4H6)2 have been carried out. The stabilities and structures of the commo-clusters are discussed in terms of the interactions of the silicon atomic orbitals with molecular orbitals of ic-symmctry of the open face of the carbaborane.25... 

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