Orbitals radial distribution function

Likewise, a basis set can be improved by uncontracting some of the outer basis function primitives (individual GTO orbitals). This will always lower the total energy slightly. It will improve the accuracy of chemical predictions if the primitives being uncontracted are those describing the wave function in the middle of a chemical bond. The distance from the nucleus at which a basis function has the most significant effect on the wave function is the distance at which there is a peak in the radial distribution function for that GTO primitive. The formula for a normalized radial GTO primitive in atomic units is... 

FIGURE 1.32 The radial distribution function tells us the probability density for finding an electron at a given radius summed over all directions. The graph shows the radial distribution function for the 1s-, 2s-, and 3s-orbitals in hydrogen. Note how the most probable radius icorresponding to the greatest maximum) increases as n increases. 

FIGURE 1.42 The radial distribution functions for s-, p-, and cf-orbitals in the first three shells of a hydrogen atom. Note that the probability maxima for orbitals of the same shell are close to each other however, note that an electron in an ns-orbital has a higher probability of being found close to the nucleus than does an electron in an np-orbital or an nd-orbital. 

Show thar, if the radial distribution function is defined as P = r2R2, then the expression for P for an s-orbital is... 

This plot shows the radial distribution function of the 3s and 3p orbitals of a hydrogen atom. Identify each curve and explain how you made your decision. 

In Fig. 5.12, the radial distribution functions for the neutral iron-atom are plotted. It is evident that the orbitals with the same main quantum number occupy similar regions in space and are relatively well separated from the next higher and next lower shell. In particular, the 4s orbital is rather diffuse and shows its maximum close to typical bonding distances while the 3d orbitals are much more compact. 

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... 

There are n — / — 1 nodes in the radial distribution functions of all orbitals, Tor example, the 3s orbital has two nodes, the 4d orbitals each have one. 

As we have seen from the radial distribution functions, the most probable radius tends to increase with increasing n. Counteracting this tendency is the effect of increasing effective nuclear charge, which tends to contract tlie orbitals. From these opposing forces we obtain the following results ... 

By analogy with the functions plotted in Fig. 4.7, give rough sketches of radial wavefunctions and radial distribution functions for 4r, 4p, 4d, and 4/orbitals. 

Fig. 5.4 Radial distribution functions for sodium, showing the inner shells, and orbitals with n=3. Note the different degrees of penetration, 3s being most penetrating, and 3d least so.

The radial distribution functions of 3s, 3p, and 3d orbitals together with that of the sodium core (ls22s22p6). 

In general, the FEUDAL model appears to answer many of the fundamental questions regarding the bonding within f-electron systems however, certain discrepancies exist within the physical data of the actinide systems and the theoretical understanding of the radial distribution functions of the actinides. Evidence that the f orbitals are accessible for covalent bonding continues to be found. 

The square ot the radial part of th(e wavefunction of an orbital provides information about how tho electron density within the orbital varies as a function of distance from the nucleus. These radial distribution functions show that, in a given principle shell, the maximum electron density is reached nearer to the nucleus as the quantum number I increases. However, the proportion of the total electron density which is near to the nucleus is larger for an electron in an s orbital than in a p orbital. 

Figure 7.6 Density p r) and radial distribution function D r) for a hydrogen Ij orbital.

The Hartree-Fock or self-consistent field (SCF) method is a procedure for optimizing the orbital functions in the Slater determinant (9.1), so as to minimize the energy (9.4). SCF computations have been carried out for all the atoms of the periodic table, with predictions of total energies and ionization energies generally accurate in the 1-2% range. Fig. 9.2 shows the electronic radial distribution function in the argon atom, obtained from a Hartree-Fock computation. The shell structure of the electron cloud is readily apparent. 

The radial distribution function is P(r) = r R (for the s orbitals this expression is the same as 47tr ). The plot of vs. r for a Is orbital in Figure 1.10 is a radial distribution function. Figure 1.12 provides plots of the radial... 

Fig. 1. Radial distribution functions at 2rz for the two natural orbitals with largest occupation numbers in the lowest triplet state of helium, the ls2s 3S state. Values of the radius are measured in atomic units.

Fig. 2. This figure shows the radial distribution functions of the two most highly occupied natural orbitals in the the ls3s 3S state of helium.

Fig. 1.7 Radial distribution functions, Anr R(r4, for the li, 2s and 3 atomic orbitals of the hydrogen atom.

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