B AOInts

B.6 Fundamental ERI

The fundamental ERI [ss|ss](0)[𝟎](0), which is the basis of all ERI algorithms, is usually represented as303

[𝟎](0)=DADBDCDDe-α|𝐫1-𝐀|2e-β|𝐫1-𝐁|2(1r12)e-γ|𝐫2-𝐂|2e-δ|𝐫2-𝐃|2𝑑𝐫1𝑑𝐫2 (B.3)

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

[𝟎](0)=U(2ϑ2)1/2(2π)1/201e-Tu2𝑑u (B.4)

and can be efficiently computed using a modified Chebyshev interpolation scheme.298 Equation (B.4) can also be adapted for the general case [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.