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

Basis Sets by Hand — Normalize a primitive 1s Gaussian

Exercises

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

Worked example Problem 1 of 8

Normalize a primitive 1s Gaussian

Problem

Find the normalization constant

N

for a primitive 1s Gaussian

g (r) = N e^{- α r^{2}}

— the

N

making

\int ∣ g ∣^{2} d^{3} r = 1

. Then evaluate it for the tightest STO-3G hydrogen primitive,

α = 3.42525

Solution

Set up the normalization integral. For an s-function the angular part is trivial; the work is one Gaussian integral, which factorizes in Cartesian coordinates because $r^{2} = x^{2} + y^{2} + z^{2}$ .

\int ∣ g ∣^{2} d^{3} r = N^{2} \int e^{- 2 α r^{2}} d^{3} r = N^{2} (\int_{- \infty}^{\infty} e^{- 2 α x^{2}} d x)^{3}

The one-dimensional Gaussian integral is $\int_{- \infty}^{\infty} e^{- 2 α x^{2}} d x = π /2 α$ , so the cube is:

\int ∣ g ∣^{2} d^{3} r = N^{2} (\frac{π}{2 α})^{3/2}

Set it to 1 and solve for $N$ :

N = (\frac{2 α}{π})^{3/4}

Evaluate for $α = 3.42525$ : $N = (2 \cdot 3.42525/ π)^{3/4} = (2.1805)^{3/4} \approx 1.794$ .

Check. Tighter Gaussians (larger $α$ ) are more sharply peaked, so they need a larger normalization constant — $N$ grows as $α^{3/4}$ , exactly as the formula shows.

Result

N = (\frac{2 α}{π})^{3/4}