The Characteristics of Hydrogen Orbitals

1 A = 10 ° m the angstrom is often used as the unit for atomic radius because of its convenient size. Another convenient unit is the picometer (1 pm = 10 m). 

We are also interested in knowing the total probability of finding the electron in the hydrogen atom at a particular distance from the nucleus. Imagine that the space around the hydrogen nucleus is made up of a series of thin spherical shells (rather like layers in an onion), as shown in Fig. 12.17(a). When the total probability of finding the electron in each spherical shell is plotted versus the distance from the nucleus, the plot in Fig. 12.17(b) is obtained. This graph is called the radial probability distribution, which is a plot of Atrr R versus r, where R represents the radial part of the wave function. 

The maximum in the curve occurs because of two opposing effects. The probability of finding an electron at a particular position is greatest near the nucleus, but the volume of the spherical shell increases with the distance from the nucleus. Therefore, as we move away from the nucleus, the probability of finding the electron at a given position decreases. However, we are summing more positions. Thus the total probability increases to a certain radius and then decreases as the electron probability at each position becomes very small. Mathematically, the maximum occurs because in the function in- 

One more characteristic of the hydrogen Is orbital that we must consider is its size. As we can see from Fig. 12.16, the size of this orbital cannot be precisely defined, since the probability never becomes zero (although it drops to an extremely small value at large values of r). Therefore, the hydrogen Is orbital has no distinct size. However, it is useful to have a definition of relative orbital size. The normally accepted arbitrary definition of the size of the hydrogen Is orbital is the radius of the sphere that encloses 90% of the total electron probability. That is, 90% of the time the electron is found inside this sphere. Application of this rule to the hydrogen atom Is orbital gives a sphere with radius 2.6 ao, or 1.4 X 10 ° m (140 pm). 

So far we have described only the lowest-energy wave function in the hydrogen atom, the Is orbital. Hydrogen has many other orbitals, which are described in the next section. 

As we have seen, when we solve the Schrodinger equation for the hydrogen atom, we find many wave functions (orbitals) that satisfy it. Each of these orbitals is characterized by a set of quantum numbers that arise when the boundary conditions are applied. Now we will systematically describe these quantum numbers in terms of the values they can assume and their physical meanings. 

TABLE 12.2 The Angular Momentum Quantum Numbers and Corresponding Letter Symbols 

The labels s, p, d, and f are used for historical reasons. They originally referred to characteristics of lines observed in the atomic spectra s (sharp), p (principal), d (diffuse), and f (fundamental). Beyond f the letters become alphabetic g, h. skipping , which is reserved as a symbol for angular momentum. 

For principal quantum level n = 5, determine the number of subshells (different values of ) and give the designation of each. 

The first four levels of orbitals in the hydrogen atom are listed with their quantum numbers in Table 12.3. Note that each set of orbitals with a given value of (sometimes called a subshell) is designated by giving the value of 

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The propylene double bond consists of a (7-bond formed by two ovedapping orbitals, and a 7t-bond formed above and below the plane by the side overlap of two p orbitals. The 7t-bond is responsible for many of the reactions that ate characteristic of alkenes. It serves as a source of electrons for electrophilic reactions such as addition reactions. Simple examples are the addition of hydrogen or a halogen, eg, chlorine ... 

There are totally six molecular orbitals (cts, a, az, a, tt", and tt") formed by the six atomic orbitals (2s and 2p orbitals on Be and 1 s orbitals on the hydrogens). Note that the a molecular orbitals have cylindrical symmetry around the molecular axis, while the nonbonding n orbitals do not. Another important characteristic of these orbitals is that they are delocalized" in nature. For example, an electron occupying the ers orbital has its density spread over all three atoms. Table 3.4.1 summarizes the way the molecular orbitals of BeH2 are formed by the atomic orbitals on Be and H, where the linear combinations of H orbitals are normalized. 

A complete review of the characteristics of various types of basis sets has been given recently by Schaefer.44 The radial form of STO s is similar to the (nodeless) hydrogenic atomic orbitals, rn -1e - r where n is the principal quantum number and C is a variable exponent.45 Their angular dependence is described by multiplication by a spherical harmonic The use of (jTO s in molecular calculations was... 

We have seen that the meaning of an orbital is illustrated most clearly by a probability distribution. Each orbital in the hydrogen atom has a unique probability distribution. We also have seen that another means of representing an orbital is by the surface that surrounds 90% of the total electron probability. These three types of representations for the hydrogen Is, 2s, and 3s orbitals are shown in Fig. 12.18. Note the characteristic spherical shape of each of the s orbitals. Note also that the 2s and 3s orbitals contain areas of high probability separated by areas of zero probability. These latter areas are called nodal surfaces, or simply nodes. The number of nodes increases as n increases. For s orbitals the number of nodes is given by n — 1. For our purposes, however, we will think of s orbitals only in terms of their overall spherical shape, which becomes larger as the value of n increases. 

The examples of H2O and NH3 (Table 6.4) illustrate a characteristic property of the F h representations constituted by the set of Ishi orbitals on the hydrogen atoms of a molecule. An orbital of this type can only be transformed either into itself (contribution to the character equal to +1) or into an orbital located on another hydrogen atom (contribution to the character of 0). To determine the character associated with a particular symmetry operation, it is therefore sufficient to count the number of hydrogen atoms that are left unchanged by this operation. This comment is also applicable to other orbitals that possess this property (e.g. the s orbitals on heavy atoms that are equivalent by symmetry). 

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