Angular nodal surfaces

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. 

Describe the angular nodal surfaces for a d. orbital, whose angular wave function is... 

The angular nodal surfaces for a d orbital are the planes where xz = 0, which means that either r or z must be zero. The yz and xy planes satisfy this requirement. 

Because there are two solutions to the equation T = 0 (setting x - y = 0, the solutions are X = y and x = - y), the planes defined by these equations are the angular nodal surfaces. They are planes containing the z axis and making 45° angles with the X and y axes (see Table 2.5). The function is positive where x > y and negative where x < y. In addition, a 3dy2 y2 orbital has no radial nodes, a has one radial node, and so on. 

The nodal surfaces require 2z — — yr = 0, so the angular nodal surface for a orbital... 

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). 

There are 1 nodal surfaces in the angular distributional functions of all orbitals, for example, s orbitals have none, d orbitals have two. 

Specifically, the number of nodal surfaces in the radial function depends on the value of n. For n = 1, the total number of nodal surfaces is 0, for n = 2 the total number is 1, for n = 3, the total number is 2 and so on. Hence, the radial function for a 3s orbital has two nodal surfaces and the angular function has none whereas the radial function for a 3d orbital has none but the angular functions have two. A rule of thumb that also applies to molecules is that the greater the total number of nodal surfaces, the higher the orbital energy. 

The cutoff radii, r , are not adjustable pseudo-potential parameters. The choice of a given set of cutoff radii establishes only the region where the pseudo and true wave-functions coincide. Therefore, the cutoff radii can be considered as a measure of the quality of the pseudo-potential. Their smallest possible value is determined by the location of the outermost nodal surface of the true wave-functions. For cutoff radii close to this minimum, the pseudopotential is very realistic, but also very strong. If very large cutoff radii are chosen, the pseudo-potentials will be smooth and almost angular momentum independent, but also very unrealistic. A smooth potential leads to a fast convergence of plane-wave basis calculations [58]. The choice of the ideal cutoff radii is then the result of a balance between basis-set size and pseudopotential accuracy. 

The nuclear contribution Bintemai is called the Fermi contact interaction. Mjj is the quantum number for the z component of the nuclear spin angular moment of nucleus number j and the constant aj is called the coupling constant for that nucleus. The coupling constants for nuclei in many molecules have values near 1 gauss (1 x 10 T), but the coupling constant is appreciably nonzero only if the electron approaches closely to the nucleus. If an unpaired electron occupies an orbital that has a nodal surface at... 

Fig. 1.9 Boundary surfaces for the angular parts of the li and Ip atomic orbitals of the hydrogen atom. The nodal plane shown in grey for the Ip atomic orbital lies in the xy plane.

The real part of the term is equal to cosw/0, and it would contribute w/ additional nodal planes. Consequently, the total number of angular nodes in the real part of if/ is equal to /. Each angular node corresponds to a surface (a plane for nodes in (f), a cone for nodes in 6) on which the electron wavefunction vanishes. For example, the do wavefunction has nodes at 6 = 54.7° and 125.3°, and all the points at each value of 6 form a cone centered on the z axis. On the other hand, the real part of the pi wavefunction has a node at 0 = 90°, which defines the yz plane. The nodal planes along cf) vanish when Yf is multiplied by its complex conjugate, and therefore the probability density has no nodes along 4>, but retains the /— m/ ... 

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