The 50 Probability Ellipsoid

The three-dimensional Gaussian distribution function is, in tensor notation, given by 

For an ellipsoidal surface defined by uTo u = c2, the probability of the atom being inside the surface is a function of c. 

Without loss of generality, we may change the metric of space such that erf = cr == of, which reduces the equations to those of the equivalent isotropic distribution with a2 = a = a — rf, or, with Eq. (C.l)  

on the surface of the sphere defined by Eq. (C.l), r = m1v/3 = ca. The probability Pc that the atom is within the sphere is given by the integral over the isotropic Gaussian distribution  

The integral can be related to the error function by partial integration, which gives 

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Fig. 1. ORTEP drawing for [Mn(L)Cl2], showing the labeling scheme and the 50% probability ellipsoids for the non-hydrogen atoms 34).

FIG. 10.1 3 Diagram of the bis(dicarbonyl-7t-cyclopentadienyl iron) molecule at 74 K. The 50% probability ellipsoids are shown. Source Mitschler et al. (1978). 

Figure 2. Packing diagram of Rh2H4(P(iso-Pr)3)4 with hydride ligands omitted. The 50% probability ellipsoids are shown, except for alkyl hydrogen atoms which are drawn artifically small. The partially occupied positions of the THF molecules are shown.

Fig. 5. A drawing of the CoH(PF3)4 molecule, with the H atom omitted. The 50% probability ellipsoids are shown. [Reproduced from reference (US) with permission.]...

Figure 12 (a) Drawings of the [Fe(phen)2(NCS)2] (left) and [Fe(btz)2(NCS)2] (right) units showing the 50% probability ellipsoids. Flydrogen atoms have been omitted for clarity, (b) Projection along the a axis of the crystal structures of [Fe(phen)2(NCS)2] (left) and [Fe(btz)2(NCS)2] (right). Reproduced with permission from ref. 48. 

FIGURE 3.2 X-ray crystallographic results for the product of Eq. 3.24 showing the 50% probability ellipsoid. for each atom. Hydrogen atoms, poorly located by X-ray methods, are omitted for clarity. Inset shows the unusual distortion in more detail. [Reproduced from Ref. 14a with permission.]... 

Figure 5.18 (a) Drawing of the [Au(S2CN (C5Hii)2)]2 molecules in DMSO with two repeating units of chains in the structure viewed perpendicular to the stacking axis. Thermal ellipsoids are drawn at the 50% probability level. The pentyl moieties have been omitted forclarity. 

Figure 3. ORTEP [7] drawing of the benzoylacetone molecule showing 50% probability ellipsoids.

Fig. 9. Composition (5a, X12 = H 5d, X12 = OCH3) and ORTEP (5a) structures of the black p-oxo-bridged diiron(IV) complex rapidly formed in a high yield from la,d and 02 in CH2C12 or other weakly coordinating solvents. The ellipsoids are drawn at the 50% probability level. From Ref. (38).

The compilation of bond length and angle information, all of which must agree with known chemical principles, enables one to deduce the molecular conformation of the molecule in question. Such illustrations are usually provided in the form of ORTEP drawings, where atomic sizes are depicted in the form of ellipsoids at the 50% probability level. The molecular conformation of theophylline in the anhydrate and monohydrate phase is shown in Fig. 7.4. As would be expected from a rigid and planar molecule, these conformations do not differ significantly. 

Fig. 9. Molecular structure of the anion [FeCbdmpzalCla] of 7b NEtJ cation omitted for better view thermal ellipsoids are drawn at the 50% probability level 49).

Fig. 23. Molecular structure of [Ru(bdmpza)Cl( = C = CHTolXPPhs)] (33b) with thermal ellipsoids drawn at the 50% probability level. Most hydrogen atoms and solvent molecules are omitted for clarity (67).

Figure I. ORTEP drawing of i,3-(Ph2PNPPh2)S2N3 showing bond lengths (A) and angles (deg). All ORTEP figures are drawn with thermal ellipsoids at the 50% probability level, and only a-C atoms of the phenyl rings are shown.

Figure 6.5 ORTEP view of the cation of complex 21 (50% probability ellipsoids, reproduced with permission from Elsevier). Selected bond lengths (A) and angles (°) Ir-Pl 2.349(2), Ir-Cl 2.208(7), lr-P2 2.3007(18), lr-C2 2.194(7), lr-P3 2.3022(18),...

Figure 14 ORTEP representations (a) the core and (b) complete complex [Mn3o024(OH)8(02CCH2Bu )32-(H20)2(CH3N02)4] (78) showing 50% probability ellipsoids.

Fig. 3a, b Crystal structures of 2 and poly(2) viewed down along the crystallographic a- and h-axes at the top and bottom, respectively. Hydrogen atoms are omitted for clarity, c ORTEP drawing for 2 (-120 °C) and repeating unit of poly(2). Thermal ellipsoids are plotted at the 50% probability level [56]... 

Figure 1. Face and edge ellipsoid plot of the complex 2a from the salt [Fe(Cp )2][2a] at the 50% probability level.

Figure 12. Ellipsoid plots of the anion 11 and its neutral analogue 12 at the 50% probability level. 

Figure 1. ORTEP diagram of diphenylguanidinium nicotinate hydrate. The thermal ellipsoids were drawn at the 50% probability level.

Figure 29 The 50% probability thermal ellipsoids of Co4(CO) 2Sb4. The molecule of idealized T -Aint geometry has...

Figure 1. Sketch of the Rh2H4(P(iso-Pr)3)4 molecule. Hydrogen atoms are omitted. The other atoms are drawn as 50% probability ellipsoids.

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