gaussian_basis_integrals.md

List of Gaussian Basis Integrals

Basic integrals

$$ \int_{0}^{u} e^{-x^2}\mathrm{d}x = \frac{\sqrt{\pi}}{2}\mathrm{erf}(u) $$

$$ \int_{0}^{u} e^{-\alpha x^2}\mathrm{d}x = \frac{\sqrt{\pi}}{2\sqrt{\alpha}}\mathrm{erf}(\sqrt{\alpha}u) $$

Gaussian basis integrals

1st order

$$ \int_{0}^{u} xe^{-\alpha x^2}\mathrm{d}x = \frac{1}{2\alpha}\left[1-e^{-\alpha u^2} \right] $$

$$ \int_{0}^{\infty} xe^{-\alpha x^2}\mathrm{d}x = \frac{1}{2\alpha} $$ $$ \int_{\infty}^{\infty} xe^{-\alpha x^2}\mathrm{d}x = 0 $$

2nd order

$$ \int_{0}^{u} x^2e^{-\alpha x^2}\mathrm{d}x = \frac{1}{2\alpha\sqrt{\alpha}} \left[ \frac{\sqrt{\pi}}{2}\mathrm{erf}(\sqrt{\alpha}u)-\sqrt{\alpha}ue^{\alpha u^2} \right] $$

$$ \int_{0}^{\infty} x^2e^{-\alpha x^2}\mathrm{d}x = \frac{\sqrt{\pi}}{4\alpha\sqrt{\alpha}} $$ $$ \int_{-\infty}^{\infty} x^2e^{-\alpha x^2}\mathrm{d}x = \frac{\sqrt{\pi}}{2\alpha\sqrt{\alpha}} $$

3rd order

$$ \int_{0}^{u} x^3e^{-\alpha x^2}\mathrm{d}x = \frac{1}{2\alpha\sqrt{\alpha}} \left[ 1 - (1+\alpha u^2)e^{-\alpha u^2} \right] $$

$$ \int_{0}^{\infty} x^3e^{-\alpha x^2}\mathrm{d}x = \frac{1}{2\alpha^2} $$ $$ \int_{-\infty}^{\infty} x^3e^{-\alpha x^2}\mathrm{d}x = 0 $$