Question

Ignoring electron spin, the Hamiltonian for the two electrons of helium atom, whose positions relative to the nucleus are r_i (i = 1, 2), can be written as

(a) Show that there are 8 orbital wave functions that are eigenfunctions of H - V with one electron in the hydrogenic ground state and the others in the first excited state.

(b) Using symmetry arguments show that all the matrix elements of V among these 8 states can be expressed in terms of four of them. [It may be helpful to use linear combinations of 1 = 1 spherical harmonics proportional to

\(\frac x{[r]}, \frac y{[r]}\) and \(\frac z{[r]}\).

(c) Show that the variational principle leads to a determinants1 equation for the energies of the 8 excited states if a linear combination of the 8 eigenfunctions of H - V is used as a trial function. Express the energy splitting in terms of the four independent matrix elements of V.

(d) Discuss the degeneracies of the levels due to the Pauli principle.

Answer 1

Treating V as perturbation, the zero-order wave function is a product of two eigenfunctions \(|n, l, m\rangle\) of a hydrogen-like atom. Thus the 8 eigenfunctions for H₀ = H - V with one electron in the hydrogen ground state can be written as

with l = 0, 1, m = -1,. . .l, where the subscripts 1 and 2 refer to the two electrons. The corresponding energies are

To take account of the perturbation we have to calculate the matrix elements

\(\langle l'm'\pm|V|lm\pm\rangle\)

As V is rotation-invariant and symmetric in the two electrons and \(|lm\pm\rangle\) are spatial rotation eigenstates, we have

Because the wave functions were formed taking into account the symmetry with respect to the interchange of the two electrons, the perturbation matrix is diagonal, whence the four discrete energy levels follow:

The first levels \(|1m+\rangle\) have energy E_b + A₁ + B₁, second levels \(|1m-\rangle\) have energy E_b + A₁ - B₁, the third level \(|00+\rangle\) has energy E_b + A₀ + B₀, the fourth level \(|00-\rangle\) has energy E_b + A₀ - B₀. Note that the levels \(|1m+\rangle\) and \(|1m-\rangle\) are each three-fold degenerate (m = \(\pm1\), 0).

According to Pauli's principle, we must also consider the spin wave functions. Neglecting spin-orbit coupling, the total spin wave functions are xc, antisymmetric, a singlet state; \(x_{1s_z}\), symmetric, a triplet state.

Since the total electron wave function must be antisymmetric for interchange of the electrons, we must take combinations as follows,

\(|lm+\rangle x_0,\)

\(|lm-\rangle x_{1s_z},\)

Hence the degeneracies of the energy levels are