Radial nodes

In all-electron calculations, the number of radial nodes of an atomic orbital (AO) increases by one as the principal quantum increases by one. Accordingly, while a Is atomic orbital is nodeless (in this and the following discussion, nodes at r = 0 and are ignored), the 2s, 3s, 4s, and higher s orbitals contain one, two, three, and so forth radial nodes. Radial nodes are required to ensure that the radial portions of the atomic wavefunctions remain orthogonal. With replacement of core electrons and orbitals by a potential, one must remove the appropriate number of nodes in the valence orbitals to ensure that the s, p, d, f, etc., orbital with the lowest principal quantum number not replaced by the ECP is nodeless, as is the Is, 2p, 3d, 4f, etc., atomic orbital in an all-electron calculation. Wavefunctions derived from relativistic calculations should be referred to as spinors (to denote their j dependence, j = V2).i We will use the terms spinor and orbital interchangeably. 

So called Ilydrogenic atomic orbitals (exact solutions for the hydrogen atom) h ave radial nodes (values of th e distance r where the orbital s value goes to zero) that make them somewhat inconvenient for computation. Results are n ot sensitive to these nodes and most simple calculation s use Slater atom ic orbitals ofthe form... 

Radial nodes provide a means by whieh an orbital aequires density eloser to the nueleus... 

The presenee of radial nodes also indieates that the eleetron has radial kinetie energy. The 3s orbital with 2 radial nodes has more radial kinetie energy than does the 3p whieh, in turn, has more than the 3d. On the other hand, the 3d orbital has the most angular energy... 

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). 

Normalized radial functions for a hydrogenlike atom are given in Table A 1.1 and plotted graphically in Fig. A 1.1 for the first ten combinations of n and /. It will be seen that the radial functions for Is, 2p, 3d, and 4f orbitals have no nodes and are everywhere of... 

The Extended Hiickel model treats all valence electrons within the spirit of the TT-electron model. Each molecular orbital is written as an LCAO expansion of the valence orbitals, which can be thought of as being Slater-type orbitals (to look ahead to Chapter 9). Slater-type orbitals are very similar to hydrogenic ones except that they do not have radial nodes. Once again we can understand the model best by considering the HF-LCAO equations... 

It is instructive to look at the form of the Is, 2s and 3s orbitals (Table 9.1). By convention, we use the dimensionless variable p = Zrjaa rather than r. Here 2 is the nuclear charge number and oq the first Bohr radius (approximately 52.9 pm). The quantity Z/n is usually called the orbital exponent, written These exponents have an increasing number of radial nodes, and they are orthonormal. 

A is a normalization constant and T/.m are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. 

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. 

FIGURE 1.34 The radial wavefunctions of the first three s-orbitals of a hydrogen atom. Note that the number of radial nodes increases (as n 1), as does the average distance of the electron from the nucleus (compare with Fig. 1.32). Because the probability density is given by ip3, all s-orbitals correspond to a nonzero probability density at the nucleus. 

The Relation between the Shell Model and Layers of Spherons.—In the customary nomenclature for nucleon orbitals the principal quantum number n is taken to be nr + 1, where nr> the radial quantum number, is the number of nodes in the radial wave function. (For electrons n is taken to be nT + l + 1.) The nucleon distribution function for n = 1 corresponds to a single shell (for Is a ball) about the origin. For n = 2 the wave function has a small negative value inside the nodal surface, that is, in the region where the wave function for n = 1 and the same value of l is large, and a large value in the region just beyond this surface. 

The numerical experiment started at a steady-state value of 200 C for both temperature nodes with an output of 16.89% for both heaters output number 1 was then stepped to 19.00%. If both outputs had been stepped to 19%, then both nodes would have gone to 220 C. The temperature of node 5 does not go as high, and the temperature of node 55 goes too high. In the reduced order model, the time constant x represents the effect of radial heat conduction, while the time constant X2 represents the effect of axial heat conduction. SimuSolv estimates these two parameters of the dynamic model as ... 

The shared memory OpenMP library is used for parallelization within each node. The evaluation of the action of potential energy, rotational kinetic energy, and T2 kinetic energy are local to each node. These local calculations are performed with the help of a task farm. Each thread dynamically obtains a triple (/2, il, ir) of radial indices and performs evaluation of first the kinetic energy and then the potential energy contribution to hps local( , i2, il, ir) for all rotational indices. 

Fig.l. Radial part /,(r) of three Is type orbitals (/ = 0, no node) of the Hydrogen atom corresponding to three different energy values. The full line corresponds to the RIIF energy and the other ones to the RHF energy plus or minus 0.2 II. The radius r is given in Bohr units. 

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