“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 — A basis of the wrong shape

Exercises

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

Practice Problem 3 of 8

A basis of the wrong shape

Problem

Try a single Gaussian trial on hydrogen,

e^{- α r^{2}}

, whose energy is

E (α) = \frac{3}{2} α - 2 2 α / π

. Minimize and compare with

- 0.5

Solution

Predict before reading on. Same single-parameter recipe — but the minimum will stall above $- 0.5$ . Which feature of $e^{- α r^{2}}$ at the origin can a Gaussian never reproduce?

\frac{d E}{d α} = \frac{3}{2} - \frac{2}{π α} = 0 \Rightarrow α = \frac{8}{9 π}

E_{m i n} = - \frac{4}{3 π} = - 0.4244 hartree

A full 15% above exact. The Gaussian has zero slope at $r = 0$ and cannot make the nuclear cusp; that missing kink is energy left on the table — the same trade you meet again in basis sets.