Difference between revisions of "CamCASP/Programming/3"
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import>Am592 |
import>Am592 |
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| − | where we have assumed <math>p,q (r,s)</math> belong to molecule A (B). |
+ | 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: |
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| + | We have: |
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| + | |||
| + | <math> |
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| + | |pq) = |\tilde{pq}) + \delta_{pq} |
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| + | </math> |
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| + | |||
| + | which implicitly defines the error <math>\delta_{pq}</math>. Using this, our approximation to the 4-index integral above can be written as |
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| + | |||
| + | <math> |
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| + | (\tilde{pq}|\tilde{rs}) = (pq|rs) - (\delta_{pq}|rs) - (pq|\delta_{rs}) + (\delta_{pq}|\delta_{rs}) |
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| + | </math> |
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Revision as of 16:27, 26 February 2009
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>