Equal-probability contour, orbitals

How do we depict a probability function One way would be to draw contours connecting regions where there is an equal probability of finding the electron. If F2 for a Is orbital is plotted, a three-dimensional plot emerges. Of course, this is a two-dimensional representation of a three-dimensional plot—the contours are really spherical like the different layers of an onion. These circles are rather like the contour lines on a map except that they represent areas of equal probability of finding the electron instead of areas of equal altitude. 

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.

The square of the sum of (1/n/2)(Xj + X2) is a measure of the total electron probability (not the radial probability used on p. 2 ) and is here represented schematically both in cross section and from above with contour lines connected between points of equal probability as shown on the next page. It will be seen that the electron will have a considerable probability between the nuclei and will act to overcome the internuclear repulsion. While we cannot be sure without more detailed calculation whether or not the overall result will be net binding, at least the orbital might be classed as a bonding orbital because of the character of its electron distribution. On this basis, p must be a negative number. 

FIGURE 5.4 Four representations of hydrogen s orbitals, (a) A contour plot of the wave function amplitude for a hydrogen atom in its Is, 2s, and 3s states. The contours identify points at which i//takes on 0.05, 0.1, 0.3, 0.5, 0.7, and 0.9 of its maximum value. Contours with positive phase are shown in red those with negative phase are shown in blue. Nodal contours, where the amplitude of the wave function is zero, are shown in black. They are connected to the nodes in the lower plots by the vertical green lines, (b) The radial wave functions plotted against distance from the nucleus, r. (c) The radial probability density, equal to the square of the radial wave function multiplied by 1. (d) The "size" of the orbitals, as represented by spheres whose radius is the distance at which the probability falls to 0.05 of its maximum value. 

The boundary surfaces in Figure 3.11 are for the electron density, the probability of an electron being at any point in space. The electron density is given by the square of the wavefunction. Points with the same electron density will have the same numerical value for the wavefunction, but the wavefunction may be positive or negative. For example, if the probability of finding an electron at a particular point was one-quarter, 0.25, then the wavefunction at that point would have the value plus one half, + 0.5, or minus one half, -0.5, since both (0.5)2 and (-0.5)2 are equal to 0.25. The sign of the wavefunction gives its phase. To represent the wavefunction itself we can use the same contours as for electron density, but we also need to indicate the phase of the wavefunction. In atomic and molecular orbital representations, we shall use colour to show differences in phase. Is orbitals are all one phase and so are shown in one colour. 2p orbitals have two lobes, which are out... 

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