Radial distribution functions hydrogen atomic orbitals

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. 

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. 

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

An important point that must not be forgotten is that d-block metal atoms are, of course, many-electron species, and when we discuss, for example, radial distribution functions of the nd atomic orbitals, we refer to hydrogen-... 

Figure 1.7 The radial distribution function, P(r), for the Is atomic orbital in hydrogen. Note the maximum occurs at r = 1 Bohr unit.

Radial distribution function for the 2s atomic orbital in the hydrogen atom. 

The clear suggestions from this analysis are that the atomic orbitals for many-electron atoms are similar to the equivalent hydrogenic ones and so it might be possible to discover mathematical functions to represent analytically and continuously the essential details of the radial distribution functions. Our appreciation of how to achieve this becomes clearer if we construct the radial distribution functions as in the last diagram of Figure 1.10. 

The radial function / (r) (a) and the radial distribution function (b) for several types of orbitals in the hydrogen atom. The y-scale varies from one orbital to the next. 

Figure 3.2 shows the radial distribution functions for the hydrogen 2s and 2p orbitals, from which it can be seen that the 2s orbital has a considerably larger probability near the nucleus than the 2p orbital. When an electron in a polyelectronic atom occupies the n = 2 level, it would be more stable in the 2s orbital than in the 2p orbital. In the 2s orbital it would be nearer the nucleus and be more strongly attracted than if it were to occupy the 2p sub-set. 

Fig. 1.13 Radial distribution functions, A-KrR(rY, for the Is, 2s and 2p atomic orbitals of the hydrogen atran.

Our individual one-electron HF or KS wavefunctions represent the individual molecular orbitals, and the square of the wavefunction gives us the probability distribution of each electron within the molecule. We do not know the form of the real multi-electron wavefunction a priori, nor the individual one-electron HF or KS functions, but we can use the mathematical principle that any unknown function can be modeled by a linear combination of known functions. A natural choice for chemists would be to use a set of functions that are similar in shape to individual atomic orbitals. To do this, we need to consider atomic radial distribution functions, such as the ones shown in Figure 3.2 for hydrogen. These are plots of how the electron density varies at any given distance away from the nucleus. 

Fig. 2.4 The hydrogen-atom and 2s orbitals. (A, B) Contour plots (lines of constant amplitude) of the wavefunctions in the plane of the nucleus, with so/id lines for positive amplitudes and dashed lines for negative amplitudes. The Cartesian coordinates x and y are expressed as dimensionless multiples of the Bohr radius (Aq = 0.529 A). The contour intervals for the amplitude are in (A) and 0.05a<, in (B). (C, D) The amplitudes of the wavefunctions as functions of the X coordinate. (E, F) The radial distribution functions...

The maximum in the radial distribution function, 52.9 pm, turns out to be the very same radius that Bohr had predicted for the innermost orbit of the hydrogen atom. However, there is a significant conceptnal difference between the two radii. In the Bohr model, every time you probe the atom (in its lowest energy state), yon would find the electron at a radius of 52.9 pm. In the quantum-mechanical model, yon would generally find the electron at various radii, with 52.9 pm having the greatest probability. 

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