Atomic oibitals

Fig. 1.19. Atomic oibitals of caibon relative to methane in a cubic fiame of reference.

Rg. 5.28 Atomic oibitals (lefi)and resulting molecular orbatals (nght) in the mtriie ion (a) bonding and (b) aniibonding. 

The atomic orbitals listed in Table 7.1 are illustrated in Fig. 7.10. Blue and yellow indieate, respeetively. positive and negative regions of the wavefunetions (the radial nodes of the 2s and 3s orbitals are obscured). These pictures are intended as stylized representations of atomic oibitals and should not be interpreted as quantitatively aeeurate. 

Fig. 12.8. A mDlecule in a homogeneous electric field (al. In (bl. i is a parameter describing the shift of the Gaussian atomic oibitals along the electric field, with ij = 0 showing the centering on the nucleL The total energy E(S, x) is a function of the electric field intensity and the basis set shift parameter r. Optimization of q gives a result close to the Sadlej value >7 = 4- Laiger absolute t] values first lead to an increase of E, but then end up in a decrease toward a catastrophe linix —ooE( ,x) = —00.

Table 6.2 Atomic oibital energy differences in the outermost p orbitals of magnetic quantum numbers mi = and 0 (E(mi = 1) — E(mi = 0)) in kcal/mol...

As is well known, conventional hydrogenoid spherical oibitals are strictly linked to tetradimensional harmonics when the atomic oibitals for the tridimensional hydrogen atom are considered in momentum space. We have therefore studied an alternative representation, providing the Stark and Zeeman basis sets, related to the spherical one by orthogonal transformation, see eqs. (12) and (15). The latter can also be interpreted as suitable timber coefficients relating different tree stmctures of hyperspherical harmonics for (Fig. 1). 

Let us analyze chemical bonding as viewed by the poor version of the MO method (only two Is hydrogen atom oibitals are used in the LCAO expansion, see Appendix R on p. 1009). Much can be seen thanks to such a poor version. The mean kinetic energy of the (only) electron of H, residing on the bonding MO = [2(1 -F S)] / (a + b), is given as (a and b denote the atomic Is orbitals centred, respectively, on the a and b nuclei)... 

As a consequence of this rotation, the orientation of the original atomic oibitals x changes. If we now compare the nodal structure of these rotated orbitals with the corresponding orbitals in the product it is possible to see that whereas the orientation of the orbitals at the center Cj is the same, for the orbitals at the center C4 it is just opposite. This result can be formally rewritten in the form of relation (13). 

In quantum mechanics, three quantum numbers are required to describe the distribution of electrons in hydrogen and other atoms. These numbeis arc derived from the mathematical solution of the Schrodinger equation for the hydrogen atom. They are called the principal quantum number, the angular momentum quantum number, and the magnetic quantum number. These quantum numbeis will be used to describe atomic oibitals and to label electrons that reside in them. A fourth quantum number— the spin quantum number—describes the behavior of a specific electron and completes the description of electrons in atoms. 

Gradient optimization of a two-quartet stracture (i.e. G-octet) can result in BSSE error originating from the incompleteness of the basis set of atomic oibitals and causing an artefactual stabilization of complexes. As has been already mentioned, this error can be corrected for single-point calculations by employing the standard counterpoise method [101]. It should be mentioned that empirical dispersion corrections are also able to absorb small BSSE effects [120]. 

We can do similar analyses of the molecules Li2 and Bc2. (The Li and Be atoms have ground-state electron configurations of [He]2s and [He]2s, respectively.) The 2s atomic oibitals also combine to form the corresponding cr and tr molecular orbitals. Figure 9.14 shows the molecular orbital diagrams and bond orders for Li2 and Be2. 

Rg. 6.19 Directional propcfties of hybrid oibitals from s. p. and d atomic oibitals. [From Kasha. M. adapted ftoro Kimbal. C. Am. Hev. Phys. Chem. I9 l. 2.177. Reproduced with peimissioa.l... 

Fig. 4.1 Charge distribution and atomic oibitals in the ehains consisting of five C atoms...

An important new development of Ae 1980 s is the use of oibitally aligned atoms to study inelastic [45] and reactive [46] atom-molecule collisions. Here an atomic beam is excited by polarized laser radiation, which produces a laboratory alignment of the appropriate atomic oibital. The beam of excited, aligned atoms is crossed by a molecular beam and the angle between the polarization axis and the relative velocity vector Vj varied. For example, the cross section for the inelastic energy-transfer collision... 

This is the first reference in which the reinterpretation of the GIAO acronym for gauge-including-atomic-oibitals has been proposed, see footnote 6, p. 5047... 

Strictly speaking, an atomic oibital does not have a well-defined shape because the wave function characterizing the orbital extends from the nucleus to infinity. In that sense, it is difficult to say what an orbital looks like. On the other hand, it is certainly useful to think of orbitals as having specific shapes. Being able to visualize atomic orbitals is essential to understanding the formation of chemical bortds and molectrlar geometry, which are discussed in Chapters 8 and 9. In this section, we will look at each type of orbital separately. 

In Chapter 6 we learned that althongh an election is a paiticle with a known mass, it exhibits wavelike properties. The quantum mechanical model of the atom, which gives rise to the familiar shapes of 5 and p atomic oibitals, treats elections in atoms as waves, rather than particles. Therefore, rather than use arrows to denote the locations and spins of electrons, we will adopt a convention whereby a singly occupied oibital will appear as a Ught color and a doubly occupied... 

The Lewis theoiy of chemical bonding provides a relatively simple way for us to visuaUze the arrangement of electrons in molecules. It is insufficient, however, to ejqilain the differences between the covalent bonds in compounds such as H2, F2, and HF. Although Lewis theory describes the bonds in these three molecules in exactly the same way, they really are quite different from one another, as evidenced by their bond lengths and bond enthalpies listed in Table 9.3. Understanding these differences and why eovalent bonds form in the first place requires a bonding model that combines Lewis s notion of atoms sharing electron pairs and the quantum mechanical descriptions of atomic oibitals. 

According to valence bond theory, atoms share electrons when an atomic orbital on one atom overlaps with an atomic oibital on the other. Each of the overlapping atomic oibitals must... 

To explain these and other observations, we need to extend our discussion of orbital overlap to include the concept of hybridization or mixing of atomic oibitals. 

For the solution of equation [10.1.54], the molecular wavefunction can be presented as a proper spin-projected antisymmetrized product of molecular (or atomic) oibitals ... 

Several ceramic materials are oxides. The valence band, essentially built from the 2p atomic oibitals of oxygen, is 2/3 filled with oxygen electrons (atomic configuration 2p" ) and is completed by the electrons provided by cations. In case of MgO, for example, the band 2pO is saturated, the immediate higher eneigy band 3sMg is empty and we can legitimately speak of Mg + and 0 ions. This is an insulator. 

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