Questions: The Variational Principle and Trial Wavefunctions

Question 1

You apply the variational method to helium using two different trial wavefunctions. Trial A gives an energy of −77.5 eV, and Trial B gives −79.0 eV. What can you conclude?

• A. Trial A is a better approximation because it gives a higher energy
• B. Trial B is a better approximation because it gives a lower energy, closer to the true ground state
• C. Trial B must give the exact ground-state energy since it is the lower value
• D. Neither can be a valid approximation because both may be below the exact energy

Answer: (B)

The variational principle guarantees that both trial energies are upper bounds on the true ground-state energy. A lower energy means the trial wavefunction is closer to the true ground state — so Trial B is the better approximation. However, the true energy could still be lower than −79.0 eV, so Trial B need not be exact. Option C confuses 'closer to' with 'equal to' the true energy.

Question 2

A chemist introduces an additional variational parameter into a trial wavefunction for a molecule. After re-optimization, the energy is identical to before. What does this indicate?

• A. The variational method has broken down and the result is unreliable
• B. The additional parameter is redundant — the original trial function already captured everything the new parameter could improve
• C. The true ground-state energy has been reached exactly
• D. The new parameter was incorrectly defined and the calculation must be redone

Answer: (B)

The variational principle guarantees energies can only decrease (or stay the same) as you add more flexibility to the trial function. If adding a parameter doesn't change the energy, the original trial function already captured whatever improvement was possible in that direction. It does NOT mean the exact energy has been reached — there may still be a gap to the true energy that this type of parameter cannot bridge.

Question 3

The variational principle states that the energy calculated from any trial wavefunction is generally less than or equal to the true ground-state energy.

• T. True
• F. False

Answer: False

This reverses the inequality. The variational principle states that the expectation value of the energy for any trial wavefunction is always GREATER THAN OR EQUAL TO the true ground-state energy. The true ground state is a lower bound that no trial function can beat — every approximate wavefunction overshoots. Improving the trial function brings the energy down toward the true value, but always from above.

Question 4

Expanding a trial wavefunction as a linear combination of basis functions and minimizing the energy with respect to the coefficients leads to a matrix eigenvalue problem.

• T. True
• F. False

Answer: True

When the trial function is φ = Σ cᵢχᵢ, minimizing the energy with respect to the coefficients leads to the secular equation |H − ES| = 0, where H and S are the Hamiltonian and overlap matrices. Solving this eigenvalue equation gives the variational energies as eigenvalues and the optimal coefficients as eigenvectors. This connection to linear algebra is what makes the variational method computationally tractable for large molecular systems.

Question 5

Why can the variational principle transform the problem of solving the many-electron Schrödinger equation into an optimization problem?

Model answer: The variational principle guarantees that the energy expectation value of any trial wavefunction is an upper bound on the true ground-state energy. This means 'lower energy = better wavefunction,' so you can improve your approximation by minimizing the energy with respect to adjustable parameters. The intractable task of solving a multi-body differential equation becomes the tractable task of minimizing a function — and the minimum is always pushing toward, not past, the correct answer.

The key is the one-way guarantee: since trial energies can only be too high, minimization is a reliable guide toward the truth. Without this guarantee, minimizing would have no physical meaning. The principle converts a problem with no known exact solution into a systematic approximation where each step provably improves the result.