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.

Why do we need the robust form?

The main, and probably only reason is that the simple form of these 4-index integrals results in too large an error in the first-order electrostatic energy.

Implementation issues

The robust form requires T-integrals of the form OOX,AAB. That is, occupied MOs of A with the auxiliary basis of B (and OOX,BBA). These are not calculated in the present code (well they are if the dimer basis is used).