Hydrogen atom electron probability density

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 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. 

Each state of the hydrogen atom, 1a, 2a> etc., has a corresponding function from which the electron probability density and energy can be calculated. 

FIGURE 3.7 The representations of the electronic probability density of existence (wave-functions) for Hydrogenic atoms, for the superior (excited) levels (or shells, quantified by the number n) with the respective sub-sheUs (or orbitals, quantified by the mixed numbers nl), employing the derived radial wave functions in terms of respective Laguerre polynomials of Table 3.1. 

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 1.34 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 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. 

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. 

Electron density probability graphs for the lowest energy level in the hydrogen atom. These diagrams represent the probability of finding an electron at any point in this energy level. 

Fig. S-S8. Electron levels of dehydrated redox particles, H ld + bh /h = H,d, adsorbed on an interface of metal electrodes D = state density (electron level density) 6 = adsorption coverage shVi - most probable vacant electron level of adsorbed protons (oxidants) eH(d = most probable occupied electron level of adsorbed hydrogen atoms (reductants) RO.d = adsorbed redox particles.

The hydrogen atom orbitals are functions of three variables the coordinates of the electron. Their physical interpretation is that the square of the amplitude of the wave function at any point is proportional to the probability of finding a particle at that point. Mathematically, the electron density distribution is equal to the square of the absolute value of the wave function ... 

Fig. JL5 (a) V-a and ifo for mfividual hydrogen atoms (cf. Fig. 2.1). (bj = if>A + V b-(c) Probability function for the bending orbital.. (d) 4> - tfiA 4>b fe> Probability function for the antibonding orbital.

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. 

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