Zero electron density

Node (Section 1.2) A surface of zero electron density within an orbital. For example, a p orbital has a nodal plane passing through the cenler of the nucleus, perpendicular to the axis of the orbital. 

As Figure 10-19 shows, bonds that form from the side-by-side overlap of atomic p orbitals have different electron density profdes than a bonds. A p orbital has zero electron density—a node—in a plane passing through the nucleus, so bonds that form from side-by-side overlap have no electron density directly on the bond axis. High electron density exists between the bonded atoms, but it is concentrated above and below the bond axis. A bond of this type is called a pi ( r) bond, and a bonding orbital that describes a ttbond is a tt orbital. 

Fig. 31. Approximation of van der Waals cross-sections of inclusion channels in 1 alcohol clathrates21 (dimensions are in A hatched regions represent O atoms of the host matrix continous solid lines indicate surfaces of apolar attribute) (a) 1 MeOH (1 2) (approximately parallel to the 0(I -Cul vectors, cf. Fig. 17a) (b) 1 2-PrOH (1 2) (orientation as before) (c) 1 2-BuOH (1 1) (through a center of symmetry at 1,1/2,1/2, cf. Fig. 30c non-zero electron density contours) (d) 1 ethylene glycol (1 1) (in the plane of the C—C single bonds of a guest molecule, indicated by projected stick models non-zero electron density contours)...

One simple case of disordered structure involves many of the long charged side chains exposed to solvent, particularly lysines. For example, 16 of the 19 lysines in myoglobin are listed as uncertain past C8 and 5 of them for all atoms past C/J (Watson, 1969) for ribonuclease S Wyckoff et al. (1970) report 6 of the 10 lysine side chains in zero electron density in trypsin the ends of 9 of the 13 lysines refined to the maximum allowed temperature factor of 40 (R. Stroud and J. Chambers, personal communication) and in rubredoxin refined at 1.2 A resolution the average temperature factor for the last 4 atoms in the side chain is 9.2 for one of the four lysines versus 43.6, 74.4, and 79.3 for the others. Figure 57 shows the refined electron density for the well-ordered lysine and for the best of the disordered ones in ru-... 

Two parallel p orbitals overlap side-by-side to form a pi (tt) bond. Fig. 2-3(u), or a n bond. Fig. 2-3(6). The bond axis lies in a nodal plane (plane of zero electronic density) perpendicular to the cross-sectional plane of the tt bond. 

Here, M is taken to be a member of the 3d series. The 3d, 4s and 4p orbitals are deemed to constitute its valence orbitals. L is a ligand which has available for bonding to M only a lone pair, oriented such that o overlap is possible with M orbitals having non-zero electron density along the M-L axes. Fig. 8.1 shows the labelling of the Cartesian axes, with respect to which the M atomic orbitals are labelled. The L lone pairs are... 

A region in an orbital with zero electron density, (p. 4)... 

The structure of the nonbonding orbital (ir2) may seem strange because there is zero electron density on the center p orbital (C2). This is the case because tt2 must have one node, and the only symmetrical position for one node is in the center of the molecule, crossing C2. We can tell from its structure that tt2 must be nonbonding, because Cl and C3 both have zero overlap with C2. The total is zero bonding, implying a nonbonding orbital. 

The right-hand column of Figure 15-11 shows the electronic structure for the allyl radical, with three pi electrons in the lowest available molecular orbitals. Two electrons are in the all-bonding MO (iri), representing the pi bond shared between the Cl—C2 bond and the C2—C3 bond. The unpaired electron goes into tt2 with zero electron density on the center carbon atom (C2). This MO representation agrees with the resonance picture showing the radical electron shared equally by Cl and C3, but not C2. Both... 

The second electron shell consists of the 2s and 2p orbitals. The 2s orbital is spherically symmetrical like the Is orbital, but its electron density is not a simple exponential function. The 2s orbital has a smaller amount of electron density close to the nucleus. Most of the electron density is farther away, beyond a region of zero electron density called a node. Because most of the 2s electron density is farther from the nucleus than that of the b, the 2s orbital is higher in energy. Figure 1-3 shows a graph of the 2s orbital. 

Graph and diagram of the 2s atomic orbital. The 2s orbital has a small region of high electron density close to the nucleus, but most of the electron density is farther from the nucleus, beyond a node, or region of zero electron density. 

Equation (10) shows that the isomer shift IS is a direct measure of the total electronic density at the probe nucleus. This density derives almost exclusively from 5-type orbitals, which have non-zero electron densities at the nucleus. Band electrons, which have non-zero occurrence probabilities at the nucleus and 5-type conduction electrons in metals may also contribute, but to a lesser extent. Figure 3 shows the linear correlation that is observed between the experimental values of Sb Mossbauer isomer shift and the calculated values of the valence electron density at the nucleus p (0). The total electron density at the nucleus p C ) (Eq. 10) is the sum of the valence electron density p (0) and the core electron density p (0), which is assumed to be constant. This density is not only determined by the 5-electrons themselves but also by the screening by other outer electrons p-, d-, or /-electrons) and consequently by the ionicity or covalency and length of the chemical bonds. IS is thus a probe of the formal oxidation state of the isotope under investigation and of the crystal field around it (high- and low-spin Fe may be differentiated). The variation of IS with temperature can be used to determine the Debye temperature of a compound (see Eq. (13)). 

In molecular orbitals, as in atomic orbitals, nodes are regions of zero electron density that divide an orbital into lobes with amplitudes of opposite sign. When a node coincides with a nuclear position, there are no lobes depicted on that atom. In the following diagram, we see that the bonding v molecular orbital for ethylene has no nodes perpendicular to the bond axis, whereas the antibonding tt orbital has one node perpendicular to the bond axis. 

At large distances from the nucleus, the electron density, or probability of finding the electron, falls off rapidly. The 2s orbital also has a nodal surface, a surface with zero electron density, in this case a sphere with r = 2uq where the probability is zero. Nodes appear naturally as a result of the wave nature of the electron they occur in the functions that result from solving the wave equation for 4. A node is a surface where the wave function is zero as it changes sign (as at r = 2aQ, in the 2s orbital) this requires that = 0, and the probability of finding the electron at that point is also zero. 

As in the case of the orbitals, the overlap of two regions with the same sign leads to an increased concentration of electrons, and the overlap of two regions of opposite sign leads to a node of zero electron density. In addition, the nodes of the atomic orbitals become the nodes of the resulting molecular orbitals. In the it antibonding case, four lobes result that are similar in appearance to an expanded d orbital (Figure 5-2(c)). 

Note that the plane passing between the two lobes of each p orbital in Figure 1.5 is in a region of zero electron density, called a node. As we ll see, nodes have important consequences with respect to chemical reactivity. 

The difference electron density distribution also finds an interesting application when the fully refined model of the crystal structure is used to compute a Fourier transformation. Although it may seem that such a Fourier map should result in zero electron density throughout the unit cell (since the differences in Eq. 2.135 are expected to approach zero), this is true only if electron shells of atoms in the crystal structure were not deformed. In reality, atoms do interact and form chemical bonds with their neighbors. This causes a redistribution of the electron density when compared to isolated atoms, for which atomic scattering functions are known. 

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