The helium-like ion has nuclear charge and two electrons.

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Many-electron atoms and the periodic table

Deriving the Spectrum, Ch Helium: exchange and the variational method

Problem

The helium-like ion

\mathrm{Ne}^{8+}

has nuclear charge

Z = 10

and two electrons. ((a)) Using the hydrogenic one-electron energy formula

E_n = -\dfrac{Z^2}{2n^2}E_{\rm h}

(Chapter 1), find the independent-electron ground-state energy

E_0

of the

1s^2

configuration. Add the first-order repulsion

\langle 1/r_{12}\rangle_{1s^2} = \tfrac58 Z\,E_{\rm h}

to obtain

E_1

, and use the one-electron energy of the remaining ion

\mathrm{Ne}^{9+}

to find the first ionisation energy that

E_1

predicts. Give all energies in eV, with

E_{\rm h} = 27.211\,\mathrm{eV}

. ((b)) Two electrons each carry spin

\tfrac12

. Using the rule for combining two angular momenta into a total (the same rule used in Chapter 2 to combine

\vec L

and

\vec S

into

\vec J

), list the possible values of the total spin quantum number

S

for this pair, and for each value evaluate

\vec s_1\cdot\vec s_2 = \tfrac{\hbar^2}{2}\left[S(S+1)-\tfrac32\right]

. For the

1s2s

configuration of

\mathrm{Ne}^{8+}

, unscreened hydrogenic orbitals give

J_{1s2s} = \tfrac{17}{81}Z\,E_{\rm h}

and

K_{1s2s} = \tfrac{16}{729}Z\,E_{\rm h}

. Using the effective form

E = E_a+E_b+J_{ab}-\tfrac12K_{ab}(1+4\vec s_1\cdot\vec s_2/\hbar^2)

, state which value of

S

gives the lower energy and give the unscreened singlet-triplet splitting

2K_{1s2s}

in eV.

Related theory

Quantum Mechanics

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