Angular location probability

Figure 117 The angular location probability pattern for a 2p,-orbital. The pattern indicates that the electron is most likely to be found on either side of the nucleus along the r -axis. There is no chance of finding the...

Figure 118 The angular location probability patterns for the d- bitals. The lobes of the d,-orbital are located between the x- and y-axes the lobes of the d, -orbital are between the y- and z-axes perpendicular to the x-axis and the lobes of the d -orbital ate between the x- and z-axes perpendicular to the y-axis.

Because of the variation in potential energy with the angle of rotation, not all locations on the rim of the cones in Fig. 1.7b are equally favored. The probability of a particular angular position depends on the potential energy at that location, V, and an averaging procedure which considers this angular variation must be used to modify Eq. (1.60). The result of this procedure is... 

Let the minimum of the potential U(angular coordinate value

probability density fl co, (p) for a particle to be located at a point with the angular coordinate (p and to have the angular velocity co under thermodynamic equilibrium with a thermostat is given by the Gibbs distribution ... 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

Another measure of the size of an orbital is the most probable distance of the electron from the nucleus in that orbital. Figure 5.4c shows that the most probable location of the electron is progressively farther from the nucleus in ns orbitals for larger n. Nonetheless, there is a finite probability for finding the electron at the nucleus in both 2s and 3s orbitals. This happens because electrons in s orbitals have no angular momentum ( = 0), and thus can approach the nucleus along the radial direction. The ability of electrons in s orbitals to penetrate close to the nucleus has important consequences in the structure of many-electron atoms and molecules (see later). 

There are always 2n2 possible combinations of quantum numbers. We divide these into orbitals. Orbitals are maps of the probability of the electron being located at a certain region in space. They are designated by their angular momentum quantum numbers. The values of magnetic and spin quantum numbers define the electrons within an orbital. 

An electronic molecular transition is labeled by the main attributes of the initial and the final electronic state. For diatomic molecules, which play a major role in AAS, the labeling usually includes the following information, given in order of appearance (i) the energetic order of the electronic state, (ii) the total spin of the electrons, (iii) the total angular momentum of the electrons with respect to the molecular axis, and (iv) symmetry properties of the electronic wave function, whose squared value describes the probability of finding the electrons at any location within the molecule. A typical electronic state may therefore be labeled by ... 

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