CamCASP/Programming/3 - Revision history

import>Am592: /* Why do we need the robust form? */

2009-06-18T17:08:36Z

Why do we need the robust form?

← Older revision		Revision as of 17:08, 18 June 2009
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	So our 2-electron Coulomb integrals are already robust. See Fred Manby's paper in J. Chem. Phys. 119, 4607 (2003) for more details.		So our 2-electron Coulomb integrals are already robust. See Fred Manby's paper in J. Chem. Phys. 119, 4607 (2003) for more details.

	==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.

import>Am592: /* Implementation issues */

2009-04-30T13:50:43Z

Implementation issues

← Older revision		Revision as of 13:50, 30 April 2009
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	==Why do we need the robust form?==		==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.		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).

import>Am592: /* Outline */

2009-04-30T13:50:18Z

Outline

← Older revision		Revision as of 13:50, 30 April 2009
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	== Outline ==		== Outline ==
	~~At present, we~~ calculate our 4-index 2-electron integrals in the following way (chemical notation):		We calculate our 4-index 2-electron integrals in the following way (chemical notation):

	<math>		<math>
Line 8:		Line 8:
	</math>		</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:		where we have assumed <math>p,q ~ (r,s)</math> belong to molecule A (B). These integrals have errors that are second order in the error in the density. Here's why:

			Consider the difference
	We have:

	<math>		<math>
	\|pq) = \|\tilde{pq}) + \~~delta_~~{pq}		(pq\|rs) - (\tilde{pq}\|\tilde{rs}) = (pq-\tilde{pq}\|rs-\tilde{rs}) + (\tilde{pq}\|rs-\tilde{rs}) + (ps-\tilde{pq}\|\tilde{rs})
	</math>		</math>

			Since <math>\|pq-\tilde{pq})</math> is the error made in fitting transition density <math>\|pq)</math>, it appears that the error in this expression is ''linear'' in the error in the (transition)-density. But this is not the case, and Dunlap (B.I. Dunlap, Int. J. Quantum Chem. 64, 193 (1997) and Phys. Chem. Chem. Phys. 2, 2113 (2000)) has shown that the terms with linear errors actually vanish. To see this, take one of these terms,
	which implicitly defines the error <math>\delta_{pq}</math>. Using this, our approximation to the 4-index integral above can be written as

	<math>		<math>
	(\tilde{pq}\|~~\tilde{~~rs~~}) = (pq\|rs)~~ - (\~~delta_{pq}\|rs) - (pq\|\delta_~~{rs}) ~~+ (\delta_{pq}\|\delta_{rs})~~		(\tilde{pq}\|rs-\tilde{rs})
	</math>		</math>

			together with the expansion <math>\|\tilde{pq}) = D_{pq,k} \|k)</math> and the density-fitting equations
	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>		<math>
	R(pq\|rs) ~~= (~~pq~~\|\tilde{rs~~}) + (~~\tilde{pq}~~\|~~rs) - (\tilde{~~pq~~}\|\tilde{rs}~~)		(k\|k') D_{pq,k'} = (k\|pq).
	</math>		</math>

			So our 2-electron Coulomb integrals are already robust. See Fred Manby's paper in J. Chem. Phys. 119, 4607 (2003) for more details.
	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?==		==Why do we need the robust form?==

import>Am592: /* Outline */

2009-02-26T16:38:32Z

Outline

← Older revision		Revision as of 16:38, 26 February 2009
Line 37:		Line 37:

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

import>Am592: /* Outline */

2009-02-26T16:34:26Z

Outline

← Older revision		Revision as of 16:34, 26 February 2009
Line 21:		Line 21:
	(\tilde{pq}\|\tilde{rs}) = (pq\|rs) - (\delta_{pq}\|rs) - (pq\|\delta_{rs}) + (\delta_{pq}\|\delta_{rs})		(\tilde{pq}\|\tilde{rs}) = (pq\|rs) - (\delta_{pq}\|rs) - (pq\|\delta_{rs}) + (\delta_{pq}\|\delta_{rs})
	</math>		</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.

import>Am592: /* Outline */

2009-02-26T16:27:15Z

Outline

← Older revision		Revision as of 16:27, 26 February 2009
Line 8:		Line 8:
	</math>		</math>

	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:

			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>

import>Am592 at 16:10, 26 February 2009

2009-02-26T16:10:41Z

New page

[[CamCASP]] => [[CamCASP/Programming | 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).