“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

The Variational Principle by Hand — An energy below exact

Exercises

Quantum Chemistry I › Unit 2 · The variational game › The Variational Principle by Hand › all problems

Check Problem 7 of 8

An energy below exact

Problem

A classmate computes a variational energy for helium and reports

- 2.95

hartree — below the exact

- 2.9037

. Without redoing a single integral, what can you conclude, and what are the two most likely mistakes?

Solution

A correct variational energy can never fall below the exact ground state, so the number is wrong — only the arithmetic needs checking, not the physics.

The two classic culprits: (1) they forgot to divide by $⟨ ψ ∣ ψ ⟩$ — an unnormalized trial breaks the bound; or (2) a sign error in an expectation value, most often dropping or flipping the positive $⟨ V_{ee} ⟩$ , which pushes the energy artificially low. The principle is itself the diagnostic.