Atomic orbitals pictorial representation

Fig. 1. Simplified pictorial representation of the transmission paths for cis and tram effects in metalloporphyrins according to Phillips and Williams (2). A 3 d orbital is shown for the metal, two p-orbitals of opposite N-atoms representative of the empty, antibonding porphyrin (rr ) orbitals, and a p-orbital for each of the axial ligands. Lobes containing arrows are occupied by electrons the arrows indicate the direction of charge delocalization. The types A-F are explained in Sect. 5...

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... 

A wavefunction ip and its eigenvalue E define an orbital. The orbital is therefore an energy level available for electrons and it implies the relevant electron distribution. In mathematical models, these distributions extend to infinity, but in a pictorial representation it is sufficient to draw the volume in which the probability of presence of the electron is rather arbitrarily around 90%. The spatial distribution of atomic and molecular orbitals have implications for processes of electron tunneling (section 4.2.1). 

Integral Sab is called the overlap integral. Physically it represents the common volume of the two atomic orbitals. A pictorial representation of Sab is given in Fig. 3.1.2. It is clear that Sab varies with rab, the internuclear distance when rab = 0, Sab = 1 when rab -> oo, Sab 0. Mathematically, it can be shown that, in a.u.,... 

The quadratic functions x + y% x2 — y2, z2 (more correctly 37 — 1), xy, xz, and yz. The x2 + y2, function cannot represent a d orbital because it does not have a node. The remaining five are taken as pictorial representations of the five d orbitals of central atom A in the AH2 molecule. Note that xy, xz, yz, and x2 — y2 each has two nodal planes, whereas die nodes of z2 are the curved surfaces of two co-axial cones sharing a common vertex. 

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)). 

Fig. 2. A pictorial representation of the possible interactions on spacecraft surfaces in low Earth orbit that involve atomic-oxygen.

Figure 7.5 Orbital alignment in the glucopyranosyl radical. The top shows the unoccupied, singly occupied and doubly occupied atomic orbitals involved, the bottom a pictorial valence bond representation of the same phenomenon.

Fig. 1.26 The relative energies of atomic orbitals of X and Y will dictate whether an interaction (formally allowed by symmetry) will lead to efficient overlap or not. Here, an interaction occurs but the contribution made by 4 y to V mo is greater than that made by ipx> while ipx contributes more than ->pY to the antibonding MO. The diagrams on the right give pictorial representations of the bonding and antibonding MOs.

Figure 4.4 gives a pictorial representation of the way in which the three sp hybrid orbitals are constructed. Remember that a change in sign for the atomic wavefunc-tion means a change in phase. The resultant directions of the lower two hybrid orbitals in Figure 4.4 are determined by resolving the vectors associated with the and 2py atomic orbitals. 

Table 4.6 gives the results of a Fenske-Hall self-consistent field (SCF) quantum chemical calculation for H2O using an orbital basis set of the atomic orbitals of O and the LGOs of an H—H fragment. The axis set is as defined in Figure 4.15. (a) Use the data to construct pictorial representations of the MOs of H2O and confirm that... 

These coefficients give rise to the pictorial representation of the 1,3,5-hexatriene molecular orbitals shown in Figure 1.8. Note in particular the increase in the energy of the orbital as the number of nodes goes from 0 to 5. The magnitude of each atomic coefficient indicates the relative contribution at that atom to the MO. In i ji, for example, the central atoms C(3) and C(4) have larger coefficients than the terminal atoms C(l) and C(6), whereas for ijij the terminal carbons have the largest coefficients. 

Although we must not lose sight of the fact that wave-functions are mathematical in origin, most chemists find such functions hard to visualize and prefer pictorial representations of orbitals. The boundary surfaces of the s and three p atomic orbitals are shown in Figure 1.9. The... 

Let us apply this pictorial representation of bonding to one of the fundamental concepts in organic chemistry—the bonding of methane. We usually begin with the atomic orbitals of the carbon atom and then consider... 

Pictorial representations of the molecular orbitals of ethylene are given in Fig. 1.P17 (pp. 36-37). Show qualitatively how these orbitals are derived from the atomic orbitals of the constituent atoms. 

We can also represent the electron configuration of the hydrogen atom using an orbital diagram, which is a pictorial representation of the electron configuration that shows the spin of the electron (see Figure 1.30) ... 

A plot of RDF,j against r is shown in Figure 2.4, and indicates that the maximum probability of finding the electron is at a distance (= 52.917726 pm) from the nucleus. One common pictorial representation of atomic orbitals is the solid figure, or boundary surface, in which there is a 95% chance (a probability of 0.95) of finding the electron. The 95% boundary surface for the Is atomic orbital of hydrogen is shown in Figure 2.5. It has a radius of 160 pm. 

Figure 1.25. Pictorial representations of the carbonate anion (CO3") showing that the carbon lies in the plane defined by the three oxygen atoms, (a) Orbitals (p) are shown for purposes of illustration only and the charges have been omitted, (b) One of the occupied molecular orbitals of the carbonate anion (charges are omitted).

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