B AOInts

B.8 Contraction Problem

The contraction problem may be described by considering a general contracted ERI of s-type functions derived from the STO-3G basis set. Each basis function has degree of contraction K=3. Thus, the ERI may be written

(ss|ss)=i=13j=13k=13=13DAiDBjDCkDD×e-αi|𝐫1-𝐀|2e-βj|𝐫1-𝐁|2(1r12)e-γk|𝐫2-𝐂|2e-δ|𝐫2-𝐃|2d𝐫1d𝐫2=i=13j=13k=13=13[sisj|sks] (B.5)

and requires 81 primitive integrals for the single ERI. The problem escalates dramatically for more highly contracted sets (STO-6G, 6-311G) and has been the motivation for the development of techniques for shell-pair modeling,11 in which a second shell-pair is constructed with fewer primitives that the first, but introduces no extra error relative to the integral threshold sought.

The Pople-Hehre axis-switch method776 is excellent for high contraction low angular momentum integral classes.