CamCASP/Programming/5 - Revision history

import>Ajs1 at 09:53, 25 March 2010

2010-03-25T09:53:49Z

← Older revision		Revision as of 09:53, 25 March 2010
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	* Spherical: Now the auxiliary basis consists of the functions:		* Spherical: Now the auxiliary basis consists of the functions:
	<math>		<math>
	s \ \ x \ \ y \ \ z \ \ \sqrt{3}xy \ \ \sqrt{3}yz \ \ z^2-1/2(x^2+y^2) \ \ \sqrt{3}xz \ \ \sqrt{3}/2(x^2+y^2)		s \ \ x \ \ y \ \ z \ \ \sqrt{3}xy \ \ \sqrt{3}yz \ \ z^2-1/2(x^2+y^2) \ \ \sqrt{3}xz \ \ \sqrt{3}/2(x^2-y^2)
	</math>		</math>

import>Am592: /* Simple example: 1 */

2009-05-12T16:08:39Z

Simple example: 1

← Older revision		Revision as of 16:08, 12 May 2009
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	* Spherical: 2.53362		* Spherical: 2.53362
	I.e., a 2.8% error in the spherical case. This error is reflected in the normalization constraints. These are satisfied exactly with the Cartesian auxiliary basis, but the spherical basis results in a normalization constraint of 1.1227 for the occupied x occupied orbital product. This should be exactly one.		I.e., a 2.8% error in the spherical case. This error is reflected in the normalization constraints. These are satisfied exactly with the Cartesian auxiliary basis, but the spherical basis results in a normalization constraint of 1.1227 for the occupied x occupied orbital product. This should be exactly one.
			--[[User:am592\|alston]] 17:08, 12 May 2009 (BST)

import>Am592: /* Simple example: 1 */

2009-05-12T16:04:05Z

Simple example: 1

← Older revision		Revision as of 16:04, 12 May 2009
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	[[ CamCASP/Programming/5/example1\| Here]] are the input files for a simple CamCASP calculation that demonstrates this.		[[ CamCASP/Programming/5/example1\| Here]] are the input files for a simple CamCASP calculation that demonstrates this.
			The example calculates the nuclear integral of the occupied orbital with itself. The exact integral is <math>2.60588</math>, and what we get is:
			* Cartesian: 2.60588
			* Spherical: 2.53362
			I.e., a 2.8% error in the spherical case. This error is reflected in the normalization constraints. These are satisfied exactly with the Cartesian auxiliary basis, but the spherical basis results in a normalization constraint of 1.1227 for the occupied x occupied orbital product. This should be exactly one.

import>Am592: /* Simple example: 1 */

2009-05-12T15:54:31Z

Simple example: 1

← Older revision		Revision as of 15:54, 12 May 2009
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	Amazingly, the density <math>x^2</math> cannot be exactly written as a linear combination of these functions.		Amazingly, the density <math>x^2</math> cannot be exactly written as a linear combination of these functions.

	Here are the input files for a simple CamCASP calculation that demonstrates this.		[[ CamCASP/Programming/5/example1\| Here]] are the input files for a simple CamCASP calculation that demonstrates this.

import>Am592: /* Simple example: 1 */

2009-05-12T15:53:43Z

Simple example: 1

← Older revision		Revision as of 15:53, 12 May 2009
Line 40:		Line 40:
	Amazingly, the density <math>x^2</math> cannot be exactly written as a linear combination of these functions.		Amazingly, the density <math>x^2</math> cannot be exactly written as a linear combination of these functions.

			Here are the input files for a simple CamCASP calculation that demonstrates this.
	Here is

import>Am592: /* Simple example: 1 */

2009-05-12T15:51:17Z

Simple example: 1

← Older revision		Revision as of 15:51, 12 May 2009
Line 39:		Line 39:

	Amazingly, the density <math>x^2</math> cannot be exactly written as a linear combination of these functions.		Amazingly, the density <math>x^2</math> cannot be exactly written as a linear combination of these functions.

			Here is

import>Am592: /* Simple example: 1 */

2009-05-12T15:49:56Z

Simple example: 1

← Older revision		Revision as of 15:49, 12 May 2009
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	s \ \ x \ \ y \ \ z \ \ \sqrt{3}xy \ \ \sqrt{3}yz \ \ z^2-1/2(x^2+y^2) \ \ \sqrt{3}xz \ \ \sqrt{3}/2(x^2+y^2)		s \ \ x \ \ y \ \ z \ \ \sqrt{3}xy \ \ \sqrt{3}yz \ \ z^2-1/2(x^2+y^2) \ \ \sqrt{3}xz \ \ \sqrt{3}/2(x^2+y^2)
	</math>		</math>

			Amazingly, the density <math>x^2</math> cannot be exactly written as a linear combination of these functions.

import>Am592: /* Simple example: 1 */

2009-05-12T15:49:01Z

Simple example: 1

← Older revision		Revision as of 15:49, 12 May 2009
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	===Simple example: 1===		===Simple example: 1===
	Here's an example that demonstrates the inadequacy of the spherical GTOs. Consider a ''1p1s'' (yes, in this order) main basis and a single occupied orbital. This would be the ''px'' orbital. The space the auxiliary basis needs to span is ''1p1s x 1p1s'' = ''1s1p1d''. So let us construct this auxiliary basis and perform the DF. ~~Here's the question: '''Can the density be exactly fitted using this auxiliary basis?'''~~		Here's an example that demonstrates the inadequacy of the spherical GTOs. Consider a ''1p1s'' (yes, in this order) main basis and a single occupied orbital. This would be the ''px'' orbital. The space the auxiliary basis needs to span is ''1p1s x 1p1s'' = ''1s1p1d''. So let us construct this auxiliary basis and perform the DF.

			Here's the question: '''Can the density be exactly fitted using this auxiliary basis?'''

	The answer depends on which basis type is used for the auxiliary basis.		The answer depends on which basis type is used for the auxiliary basis.
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	* Spherical: Now the auxiliary basis consists of the functions:		* Spherical: Now the auxiliary basis consists of the functions:
	<math>		<math>
			s \ \ x \ \ y \ \ z \ \ \sqrt{3}xy \ \ \sqrt{3}yz \ \ z^2-1/2(x^2+y^2) \ \ \sqrt{3}xz \ \ \sqrt{3}/2(x^2+y^2)
	s x y z
	</math>		</math>

import>Am592: /* Spherical versus Cartesian GTOs */

2009-05-12T15:47:19Z

Spherical versus Cartesian GTOs

← Older revision		Revision as of 15:47, 12 May 2009
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	# More consistent behaviour of aug-cc-pV''n''Z series.		# More consistent behaviour of aug-cc-pV''n''Z series.
	# Don't have rotation matrices coded, but these have been derived.		# Don't have rotation matrices coded, but these have been derived.

			===Simple example: 1===
			Here's an example that demonstrates the inadequacy of the spherical GTOs. Consider a ''1p1s'' (yes, in this order) main basis and a single occupied orbital. This would be the ''px'' orbital. The space the auxiliary basis needs to span is ''1p1s x 1p1s'' = ''1s1p1d''. So let us construct this auxiliary basis and perform the DF. Here's the question: '''Can the density be exactly fitted using this auxiliary basis?'''

			The answer depends on which basis type is used for the auxiliary basis.

			* Cartesian: The auxiliary basis consists of the following combinations:
			<math>
			s \ \ x \ \ y \ \ z \ \ x^2 \ \ xy \ \ xz \ \ y^2 \ \ yz \ \ z^2
			</math>

			So the density, <math>x^2</math>, can be exactly described.

			* Spherical: Now the auxiliary basis consists of the functions:
			<math>
			s x y z
			</math>

import>Am592 at 15:37, 12 May 2009

2009-05-12T15:37:43Z

New page

[[CamCASP]] => [[CamCASP/Programming | Programming]] => Auxiliary basis sets

==Spherical versus Cartesian GTOs==
* What are the advantages of the two basis types: Spherical and Cartesian?
* Which one results in more accurate energies and why?

Spherical GTOs:
# Smaller in size
# More well-defined set of linear equations during DF
# Lower accuracy.
# Seemingly erratic behaviour in aug-cc-pV''n''Z series.
# Rotations made easy as we have the Wigner rotation matrices.

Cartesian GTOs:
# Larger in size
# DF equations seem OK.
# Higher accuracy of energies
# More consistent behaviour of aug-cc-pV''n''Z series.
# Don't have rotation matrices coded, but these have been derived.