CamCASP/Programming/3

CamCASP => Programming => Robust integrals

Outline

At present, we calculate our 4-index 2-electron integrals in the following way (chemical notation):

<math>

 (pq|rs) \approx (\tilde{pq}|\tilde{rs}) = D^A_{pq,k} (k|l) D^B_{rs,l}

</math>

where we have assumed <math>p,q ~ (r,s)</math> belong to molecule A (B). This method is not good enough as the integrals have errors that are first order in the error in the density. Here's why:

We have:

<math>

 |pq) = |\tilde{pq}) + \delta_{pq} 

</math>

which implicitly defines the error <math>\delta_{pq}</math>. Using this, our approximation to the 4-index integral above can be written as

<math>

 (\tilde{pq}|\tilde{rs}) = (pq|rs) - (\delta_{pq}|rs) - (pq|\delta_{rs}) + (\delta_{pq}|\delta_{rs})

</math>

The error in this expression is linear in the error in the (transition)-density.

To remove the linear error we need an alternative expression for the integral. This has been derived long ago, probably by Dulap (see Manby's 2003 JCP paper on density-fitting) and is

<math>

 R(pq|rs) = (pq|\tilde{rs}) + (\tilde{pq}|rs) - (\tilde{pq}|\tilde{rs}) 

</math>

where the '<math>R</math>' has been used to indicate ROBUSTNESS. This robust form can be re-written as

<math>

 R(pq|rs) = (pq|rs) - (\delta_{pq}|\delta_{rs}).

</math>

Therefore the error is now quadratic in the error made in the density.