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

Basis Sets by Hand — Derive the Gaussian product theorem

Exercises

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

Check Problem 7 of 8

Derive the Gaussian product theorem

Problem

Show that two s-Gaussians multiply to one:

e^{- α ∣ r - A ∣^{2}} e^{- β ∣ r - B ∣^{2}} = K e^{- (α + β) ∣ r - P ∣^{2}}

. Identify

P

and

K

Solution

Add the exponents and complete the square. With $p = α + β$ , collect the quadratic and linear terms in $r$ :

- α ∣ r - A ∣^{2} - β ∣ r - B ∣^{2} = - p ∣ r - P ∣^{2} - \frac{α β}{p} ∣ A - B ∣^{2}, P = \frac{α A + β B}{p}

The first term is one Gaussian centered at the exponent-weighted mean $P$ ; the second is constant in $r$ and pulls out as $K = exp (- \frac{α β}{p} ∣ A - B ∣^{2})$ . Two centers became one — the identity that makes molecular integrals tractable.