Boundary surfaces, atomic orbitals

FIGURE 3.8 Atomic orbitals. Boundary surface diagrams for electron densities of Is, 2s. 2p, 3s. 3p, and 3d orbitals. For the p orbitals, the subscript letter on the orbital notation (x, y, z) indicates the cartesian axis along which the orbital lies. 

Obviously, an orbital boundary surface defines an interior and an exterior. Outside the boundary, the function cp has very small values because its square, summed over all space from the boundary wall to infinity, has a value of only 0.1. Recognizing this fact allows the LCAO approximation to be interpreted in physical terms. When we say that a molecular orbital is a linear combination of AOs, we imply that it is almost indistinguishable from cpk in the neighbourhood of atom k. This is because we are then inside the boundary of cpk and outside the boundary oi(pfl = k), so that cpk has finite values and contributions from [Pg.24]

Optically pure (Section 7 4) Descnbing a chiral substance in which only a single enantiomer is present Orbital (Section 1 1) Strictly speaking a wave function i i It is convenient however to think of an orbital in terms of the probability i i of finding an electron at some point relative to the nucleus as the volume inside the boundary surface of an atom or the region in space where the probability of finding an electron is high... 

Tj FIGURE 1.33 The three s-orbitals of 5 lowest energy. The simplest way of drawing an atomic orbital is as a g boundary surface, a surface within which there is a high probability (typically 90%) of finding the electron. We shall use blue to denote s-orbitals, but that color is only an aid to their identification. The shading Jp within the boundary surfaces is an 9 approximate indication of the electron density at each point. 

FIGURE 3.10 A (T-bond is formed by the pairing ol electron spins in two 2p7-orbitals on neighboring atoms. At this stage, we are ignoring the interactions of any 2p,-(and 2p -) orbitals that also contain unpaired electrons, because they cannot form electron pair may be found anywhere within the boundary surface shown in the bottom diagram. Notice that the nodal plane of each p7-orbital survives in the tr-bond. 

The square of the wavefunction, T2, relates to the probability of finding the electron at a particular location in space, with atomic orbitals being conveniently pictured as boundary surfaces (regions of space where there is a 90% probability of finding the electron within the enclosed volume). 

The orbital angular-momentum quantum number, , defines the shape of the atomic orbital (for example, s-orbitals have a spherical boundary surface, while p-orbitals are represented by a two-lobed shaped boundary surface). can have integral values from 0 to (n - 1) for each value of n. The value of for a particular orbital is designated by the letters s, p, d and f, corresponding to values of 0, 1, 2 and 3 respectively (Table 1.2). 

Boundary surfaces and atomic orbital envelope diagrams... 

The solutions of the Schrodinger equation show how j/ is distributed in the space around the nucleus of the hydrogen atom. The solutions for v / are characterized by the values of three quantum numbers and every allowed set of values for the quantum numbers, together with the associated wave function, strictly defines that space which is termed an atomic orbital. Other representations are used for atomic orbitals, such as the boundary surface and orbital envelopes described later in the chapter. 

Although the formal method of describing orbitals is to use mathematical expressions, much understanding of orbital properties may be gained by the use of pictorial representations. The most useful pictorial representations of atomic orbitals are similar to boundary surfaces (which are based on V /2), but are based upon the distribution of jf values, with the sign of / being indicated in the various parts of the diagram. The shapes of these distributions are based upon the contours of jf within... 

This is the most stable orbital of a hydrogen-like atom—that is, the orbital with the lowest energy. Since a Is orbital has no angular dependency, the probability density 2 is spherically symmetrical. Furthermore, this is true for all s orbitals. We depict the boundary surface for an electron in an s orbital as a sphere (Figure 1-2). The radial function ensures that the probability for finding the particle goes to zero for r — °°. 

FIGURE 3.18 The valence-bond description of the bonding in an ethane molecule, CjHf,. Only two of the bonds are shown in terms of their boundary surfaces. Each pair of neighboring atoms is linked by a cr-bond formed by the pairing of electrons in either Hls-orbitals or C2sp3 hybrid orbitals. All the bond angles are close to 109.5° (the tetrahedral angle). 

Surfaces may be drawn to enclose the amplitude of the angular wave function. These boundary surfaces are the atomic orbitals, and lobes of each orbital have either positive or negative signs resulting as mathematical solutions to the Schrodinger wave equation. 

Figure 2.2 Boundary surface of atomic orbitals. The boundaries represent angular distribution probabilities for electrons in each orbital. The sign of each wave function is shown. The d orbitals have been classified into two groups, t2g and eg, on the basis of spatial configuration with respect to the cartesian axes. (Reproduced and modified from W. S. Fyfe, Geochemistry of Solids, McGraw-Hill, New York, 1964, figure 2.5, p. 19).

The most commonly used pictorial representation for a wavefunction is called the boundary surface and it is used to give a three-dimensional perspective of most of the electron density in an orbital. Usually these shapes are drawn so that their volume contains about 95% of the electron density in a molecular orbital. It is instmctive to draw them in a process that first sketches the atomic orbitals as separated functions and then brings them together, allowing mixing to occur via the secular determinant (see (14)). 

The combination of hydrogen Is atomic orbitals to form MOs. The phases of the orbitals are shown by signs inside the boundary surfaces. When the orbitals are added, the matching phases produce constructive interference, which give enhanced electron probability between the nuclei. This results in a bonding molecular orbital. When one orbital is subtracted from the other, destructive interference occurs between the opposite phases, leading to a node between the nuclei. This is an antibonding MO. 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

Because the boundary of an atomic orbital is fuzzy, the orbital does not have an exactly defined size. To overcome the inherent uncertainty about the electron s location, chemists arbitrarily draw an orbital s surface to contain 90% of the electron s total probability distribution. In other words, the electron spends 90% of the time within the volume defined by the surface, and 10% of the time somewhere outside the surface. The spherical surface shown in Figure 5-13b encloses 90% of the lowest-energy orbital of hydrogen. 

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