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

Basis Sets by Hand — The cusp a Gaussian can never make

Exercises

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

Practice Problem 3 of 8

The cusp a Gaussian can never make

Problem

Compare the radial slope at the nucleus,

d χ / d r

r = 0

, for a 1s STO

e^{- ζ r}

and a 1s Gaussian

e^{- α r^{2}}

. The Kato cusp condition requires

χ^{'} (0) / χ (0) = - Z

at a nucleus of charge

Z

. Which function can satisfy it?

Solution

\frac{d}{d r} e^{- ζ r}_{0} = - ζ, \frac{d}{d r} e^{- α r^{2}}_{0} = (- 2 α r e^{- α r^{2}})_{0} = 0

The STO has a finite, nonzero slope at the origin — a genuine cusp — and setting $- ζ = - Z$ satisfies the condition exactly for a hydrogenic atom. The Gaussian has zero slope at $r = 0$ : it is smooth there, no cusp, so a single Gaussian can never meet the cusp condition. This is the one place STOs are simply right and Gaussians are simply wrong.