Angular node

For all orbitals except s there are regions in space where 0, ) = 0 because either Yimt = 0 or = 0. In these regions the electron density is zero and we call them nodal surfaces or, simply, nodes. For example, the 2p orbital has a nodal plane, while each of the 3d orbitals has two nodal planes. In general, there are I such angular nodes where = 0. The 2s orbital has one spherical nodal plane, or radial node, as Figure 1.7 shows. In general, there are (n — 1) radial nodes for an ns orbital (or n if we count the one at infinity). 

However, the authors soon inform the reader that that they are adopting a rather exotic sense of the term radial node, as well as treating the angular nodes in an unconventional manner. In addition to the radial nodes, given by the well-known equation of n — Z — 1, the authors include an additional radial node because of the existence of a node at infinity. The result of this change is to produce a total of n — i radial nodes. 

Next, instead of counting the normal number of angular nodes, or i, the authors consider twice the value of i. The net result of both of these changes is to give n - ( + 2f or a total of n + (. Not... 

HI) the only available hydrogen valence orbital is an isotropic s-type orbital with no angular nodes and,... 

This is equivalent to saying that the smaller number of radial nodes (rather than angular nodes) tends to dictate the favored low-energy orbital in orbital Aufbau, i.e., the orbital of... 

Hew many radial nodes do 3s, 4p, 3d and 5if orbitals exhibit How many angular nodes ... 

The closest forerunner to TSH theory was due to Hoffmann, Ruedenberg, and Verkade. These authors used spherical harmonics at the center of the cluster to generate linear combinations of orbitals with particular symmetry and nodal characteristics. The energy of these orbitals could then be estimated by the number of angular nodes in the parent spherical harmonic. For aUcah metal clusters, one could actually produce the same answers for the symmetries of... 

The energies of the depend principally upon the value of L, because this defines the number of angular nodes. The nodeless orbital always lies lowest, followed by... 

Table 2-5 summarizes the nodes for several orbitals. Note that the total number of nodes in any orbital is n — 1 if the conical nodes of some d and / orbitals count as 2. Angular nodes result when 7 = 0 and are planar or conical. Angular nodes can be determined in terms of 0 and < ), but may be easier to visualize if 7 is expressed in... 

Cartesian x, y, z) coordinates (see Table 2-3). In addition, the regions where the wave function is positive and where it is negative can be found. This information will be useful in working with molecular orbitals in later chapters. There are / angular nodes in any orbital, with the conical surface in the and similar orbitals counted as two nodes. 

This orbital is designated because z appears in the Y expression. For an angular node, Y must equal zero, which is true only if z = 0. Therefore, z = 0 (the xy plane) is an angular nodal surface for the orbital as shown in Table 2-5 and Figure 2-8. The wave function is positive where z > 0 and negative where z < 0. In addition, a 2p orbital has no spherical nodes, a3p. orbital has one spherical node, and so on. 

Based on your knowledge of the first few hydrogenic eigenfunctions, deduce general formulas, in terms of n and , for (i) the number of radial nodes in an atomic orbital, (ii) the number of angular nodes, (iii) the total number of nodes. 

You can count n —f — 1 radial nodes, t angular nodes, n -1 total nodes. 

A contour plot is shown in Fig. 7.9. This function is also cylindrically symmetric about the z-axis with two angular nodes—conical surfaces with 3z — r = 0. The remaining four 3d orbitals are complex functions containing the spherical harmonics Y2.t and Y2 2 pictiiied in Fig. 6.4. We can again construct real... 

It is clear that there is overlap not only in each pair zj, Z2 and Z3, Z4 but also between Z2 and Z3. Recalling the study of a linear H3 molecule (page 141), we can relate the successive energies of the four tt m.o.s to the number of nodes (besides the angular nodes of the p orbitals) in various combinations. Thus, the 7T orbital of minimum energy must be bonding for each pair of adjacent atoms (no new nodes) ... 

An orbital with quantum numbers n and has angular nodes and n — — I radial nodes, giving a total of m — 1 nodes. An angular node is defined by a plane. A radial node is defined by a spherical surface. For a one-electron atom or ion, the energy depends only on the number of nodes—that is, on n but not on or m. The energy increases as the number of nodes increases. 

How many radial nodes and how many angular nodes does each of the orbitals in Problem 3 have ... 

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