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

Basis Sets by Hand — Counting the degrees of freedom

Exercises

Quantum Chemistry I › Unit 3 · Basis sets › Basis Sets by Hand › all problems

Practice Problem 6 of 8

Counting the degrees of freedom

Problem

STO-3G contracts three primitive Gaussians into one basis function with fixed coefficients. In an STO-3G calculation on

H_{2}

, how many molecular-orbital coefficients are varied? How many if the three primitives on each atom were left uncontracted?

Solution

STO-3G H $_{2}$ : one contracted 1s on each atom → 2 basis functions. Each MO is $ψ = \sum_{μ} C_{μ} χ_{μ}$ , so 2 coefficients per MO are varied — the three contraction coefficients inside each $χ$ are frozen.

Uncontracted: 3 primitives per atom → 6 basis functions, so 6 MO coefficients are varied. The extra freedom lowers the energy (variational principle), but at higher cost — and the optimizer would largely re-discover the contraction the atom prefers. Contraction freezes the combination atoms almost always want, trading a little accuracy for a large cost saving.