Screening and Energy Dependence on ℓ: In Section 8.4, we claimed that the effects of scree...

Screening and Energy Dependence on ℓ: In Section 8.4, we claimed that the effects of screening by inner electrons were such that for outer electrons, a lower ℓ would have lower energy, given the same n. Here we investigate the claim. We begin with a form of the Schrödinger equation already adapted to the task. Computational Exercise 7.92 lays out the steps that transform the hydrogen atom radial equation (7-31) to the following:

The function f(x) is actually rR(r), and the transformation gives us a system of units where x = 1 is one Bohr- radius and an energy Ẽ = -1 is the hydrogen ground-state energy - 13.6 eV, Computational Exercise 7.92 solves the equation numerically and verifies that physically acceptable solutions occur when (Ẽ,ℓ) = (-1.0),  ,

that is, the 1s, 2s, 2p, 3s, 3p, and 3d. Notably, energy depends only on n. A common way of attacking multielectron atoms is to assume that a given electron orbits the nucleus plus a rather inert cloud of other electrons. The second term in brackets in the above differential equation is the electrostatic potential energy term. To handle a diffuse charge cloud, we need only insert a Z(x) in this term, accounting for the potential energy "felt" by an electron at a certain distance from the origin. Actually, we insert it in a form obtained later in Exercise 7.92 that further adapts the differential equation to a numerical solution. f(x+∆x)=2f(x)-f(x)+

We study lithium's lone n = 2 valence electron. To a good approximation, the effective Z produced by its three nuclear protons and the roughly fixed cloud from its two n = 1 electrons is given by

Z(x) - 7.5cxp(-1.26,x) - 5.5 exp(-1.1 1x) + 1

(a) Plot Z(x) from x = 0 to 10, Does it make sense at the limits? (b) Lithium's valence electron is not allowed in the full 1s level, whose two electrons have energies of roughly —100 eV, or about -7 in the units of this exercise. Following the procedures laid out in Exercise 7.92 and using Figure 7.15, which plots the square of rR(r), as a guide, try values of Ẽ and ℓ until you find physically acceptable solutions whose shape is what we expect of 2s and 2p states. (Plotting out to x = 15 is sufficient.) Do these n = 2 states have equal energy? Can you explain the first ionization energy shown for lithium in Figure 8.16?