Atomic orbitals radial nodes

So called Hydrogenic atomic orbitals (exact solutions for the hydrogen atom) have radial nodes (values of the distance r where the orbital s value goes to zero) that make them somewhat inconvenient for computation. Results are not sensitive to these nodes and most simple calculations use Slater atomic orbitals of the form... 

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. 

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. 

Fig.4- Radial part / (r) of three 2p type orbitals (/= , one radial node) of the Carbon atom corresponding to three different energy values. The full line corresponds to the RHF energy and the other ones to the RHF energy plus or minus 0.2 H. The radius r is given in Bohr units.

Fig.5. Same as fig.4 for 3p type orbitals (1=1, two radial nodes) of the Silicon atom.

Fig.6. Same as fig.4 for 3d type orbitals (1=2, one radial node) of the Scandium atom.

The radial functions Rniir) for the Is, 2s, 2p, 3s, 3p, and 3d atomic orbitals are shown in Figure 6.4. For states with / 0, the radial functions vanish at the origin. For states with no angular momentum (/ = 0), however, the radial function Rno r) has a non-zero value at the origin. The function Rniir) has ( — /— ) nodes between 0 and oo, i.e., the function crosses the r-axis in — I — ) times, not counting the origin. 

Atomic Size The associated Laguerre polynomial (x) is a polynomial of degree nr = n — l — 1, which has nr radial nodes (zeros). The radial distribution function therefore exhibits n — l maxima. Whenever n = l + 1 and the orbital quantum number, l has its largest value, there is only one maximum. In this case nT = 0 and from (14) follows... 

H2) because the H atom has no electronic core underlying the valence shell, its valence orbital also has no radial nodes. 

The unique absence of angular and radial nodes in the H-atom valence orbital has two important consequences for the efficacy of nB—oah interaction at the H-terminus. 

In summary, we can say that, because of the unique absence of angular and radial nodes in the H-atom valence shell, the hydride oah orbital is uniquely suited to strong n-a donor-acceptor interactions with Lewis bases. In turn, the unique energetic and angular features of nB-aAH interactions (or equivalently, of B H—A <—> B—H+ A covalent-ionic resonance) can be directly associated with the distinctive structural and spectroscopic properties of B - H—A hydrogen bonding. 

A second example is the minimal-basis-set (MBS) Hartree-Fock wave function for the diatomic molecule hydrogen fluoride, HF (Ransil 1960). The basis orbitals are six Slater-type (i.e., single exponential) functions, one for each inner and valence shell orbital of the two atoms. They are the Is function on the hydrogen atom, and the Is, 2s, 2per, and two 2pn functions on the fluorine atom. The 2sF function is an exponential function to which a term is added that introduces the radial node, and ensures orthogonality with the Is function on fluorine. To indicate the orthogonality, it is labeled 2s F. The orbital is described by... 

Calculate the r value in pm at which a radial node will appear for the 2s orbital oT ihe hydrogen atom. 

In order to locate the nodes in the radial part of the hydrogen 4s atomic orbital ... 

Figure 4.7 shows the forms of other radial wavefunctions for orbitals with n up to three. It can be seen that functions for r orbitals are non-zero at the nucleus, whereas the other orbitals are zero there because of the centrifugal repulsion term. The other obvious feature is that the number of nodes in these functions increases with n. In fact, the number of radial nodes is equal to n - / — 1. If we recall that the angular functions can also have nodes, the number being /, then it is apparent that the total number of nodes in any atomic orbital is equal to n -1 this increases with the energy, as we have found with the wavefunctions for other systems. 

The quantum content of current theories of chemical cohesion is, in reality, close to nil. The conceptual model of covalent bonding still amounts to one or more pairs of electrons, situated between two atomic nuclei, with paired spins, and confined to the region in which hybrid orbitals of the two atoms overlap. The bond strength depends on the degree of overlap. This model is simply a paraphrase of the 19th century concept of atomic valencies, with the incorporation of the electron-pair conjectures of Lewis and Langmuir. Hybrid orbitals came to be introduced to substitute for spatially oriented elliptic orbits, but in fact, these one-electron orbits are spin-free. The orbitals are next interpreted as if they were atomic wave functions with non-radial nodes at the nuclear position. Both assumptions are misleading. 

Orbitals (GTO). Slater type orbitals have the functional form e, if) = NYi, d, e- -- (5.1) is a normalization constant and T 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. centre of a bond. 5.2 Classification of Basis Sets Having decided on the type of function (STO/GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smallest number of functions... 

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