CLASS EXAMPLE – PROBABILITIES:  For a H atom in the ground electronic state find the total probability that the electron and the nucleus (proton) occupy the same volume if the radius of the nucleus is 1.0 x m. How can (a) numerical integration and (b) approximations be used to simplify the calculation?

CLASS EXAMPLE – THE 1s H WAVE FUNCTION: 

H ATOM S ORBITALS – GRAPHS: 

 The corresponding radial part of the 2s orbital wave function, R 2,0 (r) looks very similar (see text).  Class example: At what point does the probability density for an electron in the 2s orbital of a H atom have its maximum value?

H ATOM S ORBITALS – GRAPHS:  Graphs will be constructed/drawn in class showing two routes to the desired information:  Route 1: Plot either ψ 2,0,0 (r) or R 2,0 (r) vs r  Route 2: Plot either ψ 2 2,0,0 (r) or R 2 2,0 (r) vs r  Aside: One might wish to multiply both ψ 2,0,0 (r) and R 2,0 (r)! Is “Route 2” better?

H ATOM WAVE FUNCTIONS:  Plots of R n,l (r) vs r are much less regular than we saw earlier for the one dimensional PIAB. For the H atom, for example, nodes are not equally spaced and R n,l (r) amplitudes do not show the “regular” variations seen for the PIAB (where all ψ n (x) maxima and minima had the same absolute value).

S ORBITALS ARE SPHERICAL!!  Although hardly news, the form of all s orbital wave functions – having no angular dependence - mandates that the smallest volume containing a given % of the atom’s electron density is a sphere.

NODAL SURFACES:  While not immediately obvious for the H atom we can easily/often calculate/describe the position of a nodal surface. The simplest example of a nodal surface occurs for the H atom with the single electron in a 2s orbital (an electronically excited state!).

NODAL SURFACES – THE H 2s ORBITAL: 

NODAL SURFACES – OTHER 1 e - SPECIES:  One electron “atoms” with higher nuclear charges (He +, Li 2+ …) have a spherical nodal surface (for the 2s orbital) with a smaller radius (and a much smaller associated volume). This follows from the form of the relevant wave functions and/or coulombic considerations.

R n,l and R 2 n,l PLOTS FOR s ORBITALS:  Class examples: Use the R n,o (r) functions to draw R n,o (r) vs r plots and R 2 n,o (r) plots for the 1s, 2s and 3s H atoms orbitals. Can both sets of plots be used, for example, to locate radial nodes?

THE RADIAL DISTRIBUTION FUNCTION: 

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Radial probability distributions  FIGURE 8-35 COPYRIGHT © 2011 PEARSON CANADA INC. GENERAL CHEMISTRY: CHAPTER 8 Slide 14 of 50

H ATOM – RADIAL AND ANGULAR NODES: 

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