“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 product theorem, with numbers

Exercises

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

Practice Problem 5 of 8

The product theorem, with numbers

Problem

Two 1s Gaussians sit on the atoms of

H_{2}

at separation

R = 1.4

bohr:

g_{A}

(

α = 0.5

) at

A = (0, 0, 0)

and

g_{B}

(

β = 0.5

) at

B = (0, 0, 1.4)

. Their product is a single Gaussian. Find its center

P_{z}

and the prefactor

K = exp (- \frac{α β}{α + β} ∣ A - B ∣^{2})

Solution

P = \frac{α A + βB}{α + β} \Rightarrow P_{z} = \frac{0.5 \cdot 0 + 0.5 \cdot 1.4}{1.0} = 0.7 bohr (the midpoint)

K = exp (- \frac{0.5 \cdot 0.5}{1.0} (1.4)^{2}) = exp (- 0.25 \cdot 1.96) = e^{- 0.49} = 0.6126

Equal exponents put $P$ at the midpoint; the prefactor $K < 1$ measures how much the two clouds overlap. Collapsing a two-center product onto one center is exactly why Gaussian integrals are fast — the whole reason we tolerate the wrong shape.