Zero node

Using the nomenclature of Dewar and Zimmerman, the transition state for the 2, + 2S cycloaddition is a 4n Hiickel system (zero nodes) and is antiaromatic in the ground state and aromatic in the excited state. The transition state for the 2S + 20 cycloaddition is a 4n Mobius system (one node) and is aromatic in the ground state and antiaromatic in the excited state (see Chapter 8). The general cycloaddition rules are given in Table 9.5. 

The Self-Consistent-Field (SCF) procedure can be initiated with hydrogenic wave functions and Thomas-Fermi potentials. It leads to a set of solutions w(fj), each with k nodes between 0 and oo, with zero nodes for the lowest energy and increasing by one for each higher energy level. The quantum number n can now be defined asn = / + l + A to give rise to Is, 2s, 2p, etc. orbitals. 

Figure 6. Solid-support-based protocol for the synthesis of a quadrilateral. Beginning with the support containing a closed junction, alternate cycles of restriction and ligation are performed, always at the position indicated as 1 . Selection of the target product (triangle, quadrilateral, pentalateral,...) is determined by the point at which one chooses to restrict at site 2, exposing a sticky end complementary to that exposed by restriction at site 1. This action corresponds to a strand switch (eliminating a zero node), of the sort shown in Figure 8, below. This is emphasized by the lines of different thickness with which the square catenane is drawn.

Figure 8. Zero node operations, (a) Strand switch to remove nodes in knots and catenanes. On the left is a 5-noded knot (50, with its polarity indicated by arrowheads. Passing to the middle, one strand switch has been performed, converting the knot to a catenane, drawn with lines of two different thicknesses. On the right, another strand switch has been performed, making a new 3-noded knot, (b) A strand switch in a DNA context. Backbones are indicated by thick arrows, held together by three base pairs on each side. The helix axis is horizontal, and the dyad axis is vertical. The strand switch reconnects the strands, but maintains polarity. The reaction symbol replaces the right directional in (a), (c) View down the dyad axis. The view in (b) has been rotated 90° about the horizontal axis. The hairpin nature of the product is clear here. It should be clear that the leftward reaction shown here is identical to the ligation shown in the last step of Figure 6.

While it is obvious that the principal quantum number n must be a positive integer (it is impossible to have less than zero nodes in a wavefunction), the values of the other quantum numbers can be negative when they are defined in polar coordinates. The allowed values of the various quantum numbers are... 

Figure 3.7 The probability density of the reactive resonance at c = 0.52 kcal/mol. In the top panel, the F-H-D collinear subspace is shown using the Jacobi coordinates (R, r). In the bottom panel, the probability density is sliced r = 2 Bohr and is shown in the [R, y) coordinates. The plot clearly shows a state with three nodes along the asymmetric stretch and zero nodes in the symmetric stretch and bend.

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. 

The HOMO of NH3 is slightly bonding, because it contains an electron pair in an orbital resulting from interaction of the 2p orbital of nitrogen with the H orbitals of the hydrogens (from the zero-node group orbital). This is the lone pair of the electron-dot and VSEPR models. It is also the pair donated by ammonia when it functions as a Lewis base (discussed in Chapter 6). 

The overall bonding in ferrocene can now be summarized. The occupied orbitals of the cyclopentadienyl ligands—the zero-node and one-node group orbitals—are stabilized by their interactions with iron. In addition, six electrons occupy molecular orbitals that are largely derived from iron d orbitals (as one would expect for d6 iron(II)), but these occupied orbitals, displayed in Figure 5-10, have significant ligand character too. The molecular orbital picture in this... 

Basis set orbitals for supra,supra [jt2 + Jt4] cycloaddition. Six electrons, zero nodes aromatic... 

Figure 1 gives an example of an eight node network that represents a distribution system between factories and retailers. Nodes 1 and 2 are factories whose production capacities (net supplies) are 6 and 9. Nodes 3, 4, and 5 are warehouses (intermediate nodes) whose net supplies are zero. Nodes 6, 7, and 8 correspond to the retailers, whose net supplies are —3, —5, and —7. Each arc is labeled with its unit transportation cost and its upper bound. In this example, there are only arcs between source nodes and intermediate nodes and tetween intermediate nodes and sink nodes. In a general minimum cost flow model there can be arcs between any two nodes in the network. 

Thus, the 02u MO will have zero nodes and lie lowest in energy, the Cjg set of MOs will each have one nodal plane and lie just beneath the barycenter, the C2u MOs will have two nodal planes and lie right above the barycenter, and the b2g MO that has three nodal planes will lie highest in energy. Using Equation (10.49), where n = 6 (even), the following energies are obtained ... 

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