Probability contour

Finally, it is important to be clear about what orbitals are when you view diagrams such as those in Figure 3.16. Orbitals, remember, are solutions to mathematical equations. Those solutions, when manipulated, describe the motion and position of the electron in terms of probabilities. Contour diagrams, such as those shown here and in numerous print and electronic resources, appear solid. It therefore becomes easy to begin thinking about orbitals as physical containers that are occupied by electrons. In some ways, this is unavoidable. Try to remind yourself, now and then, of the following ... 

The term in brackets gives the sum of all probable contours that are not in contact with the nodes of the path Pt — xn, xt, . . ., xik. This means that the denominators of the contour weights are the sums... 

The two-dimensional drawing on the left shows that the electron spends most of its time fairly close to the nucleus. The circle around the cloud in the centre encompasses 95 percent of the two-dimensional cloud. The diagram on the right shows the 95 percent probability contour in three dimensions. 

Fig. 11a, b. Two ORTEP views of ester 43 (a) and 44 (b). Thermal ellipsoids represent 35% probability contours. Taken from Ref. 53... 

Figure 8 Bivariate normal distributions as probability contour plots for data having different covariance relationships...

The 95% probability contour for a Is electron in a hydrogen atom in 3-D, a boundary surface. 

How far away from the nucleus can we find the electron This is the same as asking How large is the atom Recall from Figure 7.16B that the probability of finding the electron far from the nucleus is not zero. Therefore, we cannot assign a definite volume to an atom. However, we often visualize atoms with a 90% probability contour, such as in Figure 7.16E, which shows the volume within which the electron of the hydrogen atom spends 90% of its time. 

So far we have discussed the electron density for the ground state of the H atom. When the atom absorbs energy, it exists in an excited state and the region of space occupied by the electron is described by a different atomic orbital (wave function). As you ll see, each atomic orbital has a distinctive radial probability distribution and 90% probability contour. 

Figure 7.19 The 2p orbitals. A, A radial probability distribution plot of the 2p orbital shows a single peak. It lies at nearly the same distance from the nucleus as the larger peak in the 2s plot (shown in Figure 7.18B). B, A cross section shows an electron cloud representation of the 90% probability contour of the 2p orbital. An electron occupies both regions of a 2p orbital equally and spends 90% of its time within this volume. Note the nodal plane at the nucleus. C, An accurate representation of the 2pj probability contour. The 2p and 2py orbitals lie along the x and y axes, respectively. D, The stylized depiction of the 2p probability contour used throughout the text. E, In an atom, the three 2p orbitals occupy mutually perpendicular regions of space, contributing to the atom s overall spherical shape.

Fig. 7. The B4F6-PFa molecule, which is bisected by a crystallographic mirror plane which contains F1(P), P, B(l), B(2), Fl(2), and F2(2). The thermal elipsoids have been reduced to 30% probability contours for the sake of clarity. From B. G. De Boer et al., Inorg. Chem. 8, 836 (1969) with permission.

One of the unsolved problems in the interaction of low-energy ions with surfaces is the mechanism of charge transfer and prediction of the charge composition of the flux of scattered, recoiled and sputtered atoms. The ability to collect spectra of neutrals plus ions and only neutrals provides a direct measure of scattered and recoiled ion fractions. SARIS images can provide electronic transition probability contour maps which are related to surface electron density and reactivity along the various azimuths. 

Fig. 8. A SNOOPI diagram (E. K. Davies, plotting routine, 1984, Chemical Crystallography Laboratory, 9 Parks Road, Oxford, England) of the hemin chloride of 202a (50% probability contours for all atoms hydrogen atoms have been omitted for clarity the dashed bonds are used to distinguish between the strap and the porphyrin skeleton)...

Fig. 5 Probability map for sodium cations. The figure is centered in the middle of a cage, with six D8Rs surrounding it. The framework is shown in black with the A1 atoms in green, a The probability scale is such that even locations that are very infrequendy visited are shown in blue. These blue regions reveal that the cations explore within their site and are thus to some extent mobile, b Most visited locations (the 0.1 max probability contour is shown)... 

As you U see later in this section, each atomic orbital has a distinctive radial probability distribution and 90% probability contour. 

For each of the s orbitals, a plot of probability density vs. distance (top, with the relief map, inset, showing the plot in three dimensions) lies above a quarter section of an electron cloud depiction of the 90% probability contour (middle), which lies above a radial probability distribution plot (bottom). A,The Is orbital. B,The 2s orbital. C,The 3s orbital. 

Figure 7.18 The 3rf orbitals. A, A radial probability distribution plot. B. Cross section of an electron cloud depiction of the orbital probability contour shows two mutually perpendicular nodal planes and lobes lying between the axes. C. An accurate representation of the orbital probability contour. D,The stylized depiction of the 3dy orbital used in the text. E,The 3d orbital. F.The 3d orbital. G,The lobes of the d 2 2 orbital lie a/ongr the x and y axes. H.The d orbital has two lobes and a central, donut-shaped region. I, A composite of the five 3d (stylized) orbitals shows how they contribute to the atom s spherical shape.

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