Residual electron density - Revision history

GervaisChapuis at 20:32, 15 August 2026

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GervaisChapuis at 20:15, 15 August 2026

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GervaisChapuis at 20:04, 15 August 2026

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GervaisChapuis at 19:52, 15 August 2026

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GervaisChapuis at 19:49, 15 August 2026

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GervaisChapuis at 18:07, 15 August 2026

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GervaisChapuis: Created page with "(work in progress) In the classical i.e. kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (r..."

2026-08-15T17:07:20Z

Created page with "(work in progress) In the classical i.e. kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (r..."

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(work in progress)

In the classical i.e. kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the density model. The calculated electron density if based on spherical electron densities of individual atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. For example, we can observe e.g. pi-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.

← Older revision		Revision as of 20:32, 15 August 2026
Line 12:		Line 12:
	== Note ==		== Note ==

−	In the classical ''i.e.'' [[kinematical theory]] approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density is based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, ~~We can compare~~ the two models by calculating the residual densities, which ~~give~~ the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. We can observe ''e.g.'' <math>\pi</math>-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.	+	In the classical ''i.e.'' [[kinematical theory]] approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density is based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, the two models can be compared by calculating the residual densities, which gives the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. We can observe ''e.g.'' <math>\pi</math>-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.

← Older revision		Revision as of 20:26, 15 August 2026
Line 12:		Line 12:
	== Note ==		== Note ==

−	In the classical ''i.e.'' [[kinematical theory]~~kinematical~~] approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density is based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. We can observe ''e.g.'' <math>\pi</math>-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.	+	In the classical ''i.e.'' [[kinematical theory]] approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density is based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. We can observe ''e.g.'' <math>\pi</math>-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.

← Older revision		Revision as of 20:15, 15 August 2026
Line 12:		Line 12:
	== Note ==		== Note ==

−	In the classical ''i.e.'' kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density is based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. We can observe ''e.g.'' <math>\pi</math>-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.	+	In the classical ''i.e.'' [[kinematical theory]kinematical] approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density is based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. We can observe ''e.g.'' <math>\pi</math>-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.

@@ Line 2: / Line 2: @@
 The residual density is defined as
 <math>
 \Delta\rho(\textbf x)=
 \frac{1}{V}\sum_{\mathbf{h}} (|F_o(\textbf{h})|-|F_c(\textbf{h})|)\exp(i\phi_c(\textbf{h})\exp (-2\pi\mathrm i\mathbf{h}\cdot\mathbf{x})
 </math>
 Here the subscripts o respectively c refer to the observed and calculated structure factors

← Older revision		Revision as of 19:52, 15 August 2026
Line 10:		Line 10:
	== Note ==		== Note ==

−	In the classical ''i.e.'' kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density if based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. ~~For example, we~~ can observe ''e.g.'' <math>\pi</math>-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.	+	In the classical ''i.e.'' kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density is based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. We can observe ''e.g.'' <math>\pi</math>-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.

@@ Line 1: / Line 1: @@
 == Definition ==
+Here the subscripts o respectively c refer to the observed and calculated structure factors
 == Note ==
-In the classical ''i.e.'' kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the density model. The calculated electron density if based on spherical electron densities of individual atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. For example, we can observe ''e.g.'' pi-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.
+In the classical ''i.e.'' kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density if based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. For example, we can observe ''e.g.'' <math>\pi</math>-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.

← Older revision		Revision as of 18:07, 15 August 2026
Line 1:		Line 1:
−	~~(work in progress)~~	+	== Definition ==
		+
		+	== Note ==

	In the classical i.e. kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the density model. The calculated electron density if based on spherical electron densities of individual atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. For example, we can observe e.g. pi-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.		In the classical i.e. kinematical approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the density model. The calculated electron density if based on spherical electron densities of individual atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, We can compare the two models by calculating the residual densities, which give the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. For example, we can observe e.g. pi-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.