Hydrogen probability density

FIGURE 1.31 The three-dimensional electron cloud corresponding to an electron in a Ij-orbital of hydrogen. The density of shading represents the probability of finding the electron at any point. The superimposed graph shows how the probability varies with the distance of the point from the nucleus along any radius. 

All s-orbitals are independent of the angles 0 and c[>, so we say that they are spherically symmetrical (Fig. 1.31). The probability density of an electron at the point (r,0,ct>) when it is in a ls-orbital is found from the wavefunction for the ground state of the hydrogen atom. When we square the wavefunction (which was given earlier, but can also be constructed as RY from the entries for R and V in Tables 1.2a and 1.2b) we find that... 

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 1.34 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 radial parts of the wavefunctions for the hydrogen atom can be constructed from the general form of the associated Laguerre polynomials, as developed in Section 5.5.3. However, in applications in physics and chemistry it is often the probability density that is more important (see Section 5.4.1). This quantity in this case represents the probability of finding the electron in the appropriate three-dimensional volume element. 

Fig. 8 Radial probability density for the hydrogen atom in ns states.

Because of interelectronic effects this Hamiltonian is not separable. Only when these effects are ignored may the total probability density ip ip be assumed to be a product of one-electron probability densities and the wave function a product of hydrogenic atomic wave functions... 

The simplest physical picture for the tunneling of a hydrogen nucleus during a hydrogen-transfer reaction takes note of the nuclear probability-density function for... 

Fig. 6-S8. Probability density for the energy level of interfadal redox electrons in adsorbed redox particles of proton-hydrogen and hydroxyl-hydroxide on the electrode interface of semiconductor ADS = adsorption > ost probable...

Figure 2.1 Radial probability density plots for Is and 2s orbitals of hydrogen atom...

Scheme 4.2 Bond energy as a function of hydrogen position (black solid line), assuming identical pff, values for the donor and acceptor, relative to the lowest vibrational energy level of the hydrogen atom (highlighted by a dotted line), (a) A standard, symmetric hydrogen bond (b) the corresponding low-barrier hydrogen bond (LBHB). The red line represents the probability density function [27, 28].

Fig. 2.12 The radial function, / n/ (dashed lines) and the probability density. Pm (solid lines) as a function of r for the 1s, 2s and 2p states of hydrogen.

Fig. 6.77. Calculations done using the statistical mechanical theory of electrolyte solutions. Probability density p(6,r) for molecular orientations of water molecules (tetrahedral symmetry) as a function of distance rfrom a neutral surface (distances are given in units of solvent diameter d = 0.28 nm) (a) 60H OH bond orientation and (b) dipolar orientation, (c) Ice-like arrangement found to dominate the liquid structure of water models at uncharged surfaces. The arrows point from oxygen to hydrogen of the same molecule. The peaks at 180 and 70° in p(0OH,r) for the contact layer correspond to the one hydrogen bond directed into the surface and the three directed to the adjacent solvent layer, respectively, in (c). (Reprinted from G. M. Tome and G. N. Patey, ElectrocNm. Acta 36 1677, copyright 1991, Figs. 1 and 2, with permission from Elsevier Science.

The quantity pv C is the unpaired w-electron spin density at the carbon atom to which the hydrogen atom in question is bonded p c is defined as 1 times the fractional number of unpaired it electrons on the carbon atom, with the sign being determined by whether the net unpaired spin at the carbon atom is in the same or opposite direction as the spin vector of the molecule. The term -electron spin density is somewhat misleading in that pn is not an electron probability density (which is measured in electrons/cm3), but rather is a pure number. The semiempirical constant Q (no connection with nuclear quadrupole moments) is approximately —23 G. 

Fig, 7. Probability density distributions as a function ol radial distance from the nucleus for several stales of a hydrogen mom. The dashed lines are proportional in Ihe probability of finding Ihe electron in an incremental volume Jr ai ihc indicated radial distance. The solid lines are proportional to the probability for finding the electron in an incremental shell of votume 4nr2dr at the indicated radius... 

The probability of finding the electron in the ground state of the hydrogen atom between radii r and r + dr is given by D(r)Ar, where D(r) is the radial probability density function shown in Figure 4.5. The most probable distance of the electron from the nucleus is found by locating the maximum in D(r) (see Problem 4.12 below). It should come as no surprise to discover that this maximum occurs at the value r = ao, the Bohr radius. 

Figure 4.5 The radial probability density function for the 1s atomic orbital of the hydrogen atom...

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). 

This is the most stable orbital of a hydrogen-like atom—that is, the orbital with the lowest energy. Since a Is orbital has no angular dependency, the probability density 2 is spherically symmetrical. Furthermore, this is true for all s orbitals. We depict the boundary surface for an electron in an s orbital as a sphere (Figure 1-2). The radial function ensures that the probability for finding the particle goes to zero for r — °°. 

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