Hydrogen atom radial functions

Figure 1.3 presents the graph for the Is hydrogen atom radial function. All the spreadsheets on the CD are named fig - . xls to correspond to the diagrams in the text. If you wish and I recommend that you do, it is good practice to generate this... 

Consider the radial probability density function, D(r), for the ground state of the hydrogen atom. This function describes the probability per unit length of finding an electron at a radial distance between r and r + dr (see Figure 6.5). 

Hydrogen Atom Wave Functions Radial Functions... 

Fig. 2.2 Radial density functions for = 2 for the hydrogen atom. These functions give the relative electron density (e pm ) as a function of distance from the nucleus. They were prepared by squaring the wave functions given in Fig. 2.1.

Fig. 2.4 Radial probability functions for n = 1,2, 3 for the hydrogen atom. The function gives the probability of finding the electron in a spherical shell of thickness dr at a distance r from the nucleus. [From Herzberg, G. Atomic Spectra and Atomic Structure Dover. New York, 1944. Reproduced with permission.]...

Tables 1.7 and 1.8 list Huzinaga s (41,45) basis sets for the Is and 2p Slater functions in hydrogen. These basis sets are interesting particularly because they were the first basis sets subjeeted to the double-zeta procedure of Slater theory. This procedure has the effect of reducing the number of terms for variation in calculations, but, more particularly, philosophically, provides for the better representation of details of the atomic radial functions, such as the cusp near the origin in s-functions, since the components subject to optimization separately, can be included in the linear combination.

The Radial Factor in the Hydrogen Atom Wave Function. The Energy Levels of the Hydrogen Atom... 

Figure 8-6. Comparison of the radial distribution function of the ctiair, boat, and twist conformations of cyclohexane (hydrogen atoms are not considered).

FIGURE 13.1 Graphs that have a one-dimensional data space, (a) Radial portion of the wave function for the hydrogen atom in the l.v ground state and 2p excited state. (A) Hypothetical salary chart. 

Table Al.l Normalized radial functions / ,/(r) for hydrogen-like atoms...

Figure Al.l Radial functions for a hydrogen atom. (Note that the horizontal scale is the same in each graph but the vertical scale varies by as much as a factor of 100. The Bohr radius Oq = 52.9 pm.)...

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. 

The function R(r) is called the radial wavefunction it tells us how the wavefunction varies as we move away from the nucleus in any direction. The function Y(0,c[>) is called the angular wavefunction it tells us how the wavefunction varies as the angles 0 and c > change. For example, the wavefunction corresponding to the ground state of the hydrogen atom ( = 1) is... 

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. 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

At first sight, we are foreed to solve this equation numerically, but its overall form allows a qualitative insight into the number of solutions and their approximate values. For example, one easily see that S represents a sum of two identical quasi-atomic (onedimensional) functions each centered on the corresponding hydrogen nucleus. The functions are quite similar to 2pz Gaussian functions, but they differ by their one-dimensionality and by a different radial dependence. Indeed, instead of the usual... 

Table 6.1. Radial functions R i for the hydrogen-like atom for n = to 6. The variable p is given by p = 2 Zrj na ...

Figure 6.4 The radial functions for the hydrogen-like atom.

As the integers if and / both begin at zero, y = 1,2,3... can of course be identified as the principal quantum number n for the hydrogen atom (see Section 6.6.1). Thus, the quantization of the energy is due to the termination of the series, a condition imposed to obtain an acceptable solution. The associated Laguerre polynomials provide quantitative descriptions of the radial part of the wave functions for the hydrogen atom, as described in Appendix IV. 

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