Reference no: EM132174894

Assignment - Ground State of Helium Atom by Variational Method

What you need to do:

Choose a trial wave function and calculate E[ψ] and minimise it with respect to [ψ]. You need to do some preliminaries.

The ground state is a spin-singlet state. So the spatial part of ψ must be symmetric in r₁^→ and r₂^→. Note that the Hamiltonian does not contain any spin-dependent terms, so when you determine E[ψ] using Eq.(2), the spinors will give us a factor 1 because of normalization. So we can just assume that

Ψ = ψ(r₁^→, r₂^→)

The two electrons are expected to be in the s-orbital. So we can choose the trial wave function as

Ψ(r₁^→, r₂^→) = φ(r₁^→) φ(r₂^→)

[You need to justify this form in the light of discussions above.]

Suggestion:

Use X_p(r^→) = e^{-α_pr^2} (Gaussian)

with

α₁ = 0.297104

α₂ = 1.236745

α₃ = 5.749982

α₄ = 38.216677

Don't worry about how these were obtained.

So now your job is the following:

Step 1: Obtain expressions for T_pq, h_pq, S_pq and Q_pqrs.

Step 2: Choose initial values for C_p(p = 1, 2, 3, 4). You can choose any values. But not all of them should be zero.

Step 3: Normalise C_p, by using equation (10).

Step 4: Calculate all the elements (4x4) of the matrix H using equation (9).

Step 5: Solve the generalised eigenvalue problem: HC = E'SC

Choose the lower eigenvalue to be E' and find its corresponding eigenvector C^new.

Step 6: Calculate the error:

Step 7: Normalise C_p by using equation (10). Accept E' as correct value, and use equation (6) to calculate E, using E' and C_p.

Try to find an expression for E, using E' from equation (6).

Attachment:- Assignment File.rar