In the LCAO treatment of

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · Molecular structure and spectra · Molecular structure and spectra

Deriving the Spectrum, Ch The Born-Oppenheimer separation and the chemical bond

Problem

In the LCAO treatment of H2+\mathrm{H}_2^+, J(R)J(R) is the potential energy that an electron in ϕA=(πa03)−1/2e−rA/a0\phi_A=(\pi a_0^3)^{-1/2}e^{-r_A/a_0} has in the field of proton BB, a distance RR from proton AA: J(R)=−e24πϵ0∫∣ϕA∣2rB d3rJ(R)=-\dfrac{e^2}{4\pi\epsilon_0}\displaystyle\int\frac{|\phi_A|^2}{r_B}\,\mathrm d^3r. ((a)) The electron's charge cloud is spherically symmetric about AA. Use the shell theorem (a thin spherical shell of charge qq and radius rr produces the potential q/4πϵ0dq/4\pi\epsilon_0d at a distance d>rd>r from its centre, and the constant potential q/4πϵ0rq/4\pi\epsilon_0r everywhere inside it) to show that J(R)=e24πϵ0a0[−a0R+e−2R/a0(1+a0R)].J(R)=\frac{e^2}{4\pi\epsilon_0a_0}\left[-\frac{a_0}{R}+e^{-2R/a_0}\left(1+\frac{a_0}{R}\right)\right]. ((b)) Add the repulsion of the two protons. Show that e2/4πϵ0R+J(R)e^2/4\pi\epsilon_0R+J(R) is positive at every RR and falls off exponentially at large RR, and explain each fact in one sentence. ((c)) Evaluate e2/4πϵ0R+J(R)e^2/4\pi\epsilon_0R+J(R) at R=2.49 a0R=2.49\,a_0, in eV. The LCAO bonding curve can be written E+(R)−E1s=[e24πϵ0R+J(R)]+K(R)−S(R)J(R)1+S(R).E_+(R)-E_{1s}=\left[\frac{e^2}{4\pi\epsilon_0R}+J(R)\right]+\frac{K(R)-S(R)J(R)}{1+S(R)}. Using the chapter's S(R)S(R) and K(R)K(R), evaluate the second term at the same RR, and say which of the two terms makes the bond.

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