The quantum numbers n, I and

An atomic orbital is usually described in terms of three integral quantum numbers. We have already encountered the principal quantum number, n, in the Bohr model of the hydrogen atom. The principal quantum number is a positive integer with values lying between the limits 1 oo. Allowed values of n arise when the radial part of the wavefunctirai is solved. 

Two more quantum numbers, I and m/, appear when the angular part of the wavefunction is solved. The quantum number / is called the orbital quantum number and has allowed values of 0,1,2... (ra 1). The value of / determines the shape of the atomic orbital, and the orbital angular momentum of the electron. The value of the magnetic quantum number, mi, gives information about the directionality of an atomic orbital and has integral values between +l and /. 

Worked example 1.2 Quantum numbers atomic orbitals 

Given that the principal quantum number, n, is 2, write down the allowed values of / and mi, and determine the number of atomic orbitals possible for n = 3. 

For 1=1, allowed values of m = —1,0, -1-1 Each set of three quantum numbers defines a particular atomic orbital, and, therefore, for n = 2, there are four atomic orbitals with the sets of quantum numbers  

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Concerning the hydrogen atom, what information do we get from the values of the quantum numbers n, i, and... 

We can now summarize the relation between the quantum numbers n, I, and m, and the physical pictures of electron distribution in orbitals by a few simple rules. It... 

In writing this equation, we have made use of Van Vleck s pure precession hypofiiesis [12], in which the molecular orbital A.) is approximated by an atomic orbital with well-defined values for the quantum numbers n, I and A.. Such an orbital implies a spherically symmetric potential and its use is most appropriate when the electronic distribution is nearly spherical. Examples of this situation occur quite often in the description of Rydberg states. It is also appropriate for hydrides like OH where the molecule is essentially an oxygen atom with a small pimple, the hydrogen atom, on its side. Accepting the pure precession hypothesis allows the matrix elements of the orbital operators to be evaluated since... 

The forms taken by this wavefunction for various values of the quantum numbers n, I and (up to n = 2) are shown in Figure 6.7. The function e " is the same one that we had for a particle moving around a circle, and here it represents the motion of the electron in the xy plane. The allowed values of the quantum number are as follows ... 

The quantum rules are statements of the permitted values of the quantum numbers, n, I and m. 

Identify the quantum numbers n, I, and the absolute value of m for the wavefunction of the He ion that has exactly one radial node, at 6oq, and with exactly one angular node in 6. 

We have now twice as many possible states of electronic motion in a hydrogen-like atom as we did before, because for every set of values of the quantum numbers n, I, and m, ms can equal either 1/2 or -1/2. There are two ways to include spin in our notation. The first is to attach the value of ms as another subscript on the orbital symbol, replacing nlm by nlm ms. There is no need to include the value of since it is fixed at 5 = 1/2. The orbital is now called a spin orbital. The second way, which we will usually use, is to multiply the original orbital by a spin function that is called a for ms = +1/2 and f for ms = -1/2. The original orbital is now called a space orbital, and the product of the space orbital and the spin function represents the spin orbital. The two ways of writing a spin orbital are equivalent ... 

Table 2.1 Summary of the Relationships among the Quantum Numbers n, I, and Numbers of Orbitals...

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