B.6 Fundamental ERI

The fundamental ERI ${[ss|ss]}^{(0)}\equiv {[\mathrm{𝟎}]}^{(0)}$, which is the basis of all ERI algorithms, is usually represented as³⁰⁰

$${[\mathrm{𝟎}]}^{(0)}=D_A{D}_B{D}_C{D}_D\int e^{-\alpha {|{𝐫}_1-𝐀|}^2}{e}^{-\beta {|{𝐫}_1-𝐁|}^2}\left(\frac{1}{r_{12}}\right)e^{-\gamma {|{𝐫}_2-𝐂|}^2}{e}^{-\delta {|{𝐫}_2-𝐃|}^2}𝑑{𝐫}_1𝑑{𝐫}_2$$

(B.3)

which can be reduced to a one-dimensional integral of the form

$${[\mathrm{𝟎}]}^{(0)}=U{(2{\vartheta }^2)}^{1/2}{\left(\frac{2}{\pi }\right)}^{1/2}{\int }_0^1{e}^{-T{u}^2}𝑑u$$

(B.4)

and can be efficiently computed using a modified Chebyshev interpolation scheme.²⁹⁵ Equation (B.4) can also be adapted for the general case ${[\mathrm{𝟎}]}^{(m)}$ integrals required for most calculations. Following the fundamental ERI, building up to the full bra-ket ERI (or intermediary matrix elements, see later) are the problems of angular momentum and contraction.

Note:  Square brackets denote primitive integrals and parentheses denote fully-contracted integrals.