The angular momentum quantum number

The angular momentum quantum number / describes the shape of the orbital, and the shape is limited by the principal quantum number n The angular momentum quantum number I can have positive integer values from 0 to n - 1. For exeunple, if the n value is 3, three values are allowed for 1 0, 1, and 2. 

The value of I defines the shape of the orbital, and the value of n defines the size. 

Orbitals that have the same value of n but different values of / are called subshells. These subshells eire given different letters to help chemists distinguish them from each other. Table 2-3 shows the letters corresponding to the different values of /. 

When chemists describe one particular subshell in an atom, they can use both the n value and the subshell letter — 2p, 3d, and so on. Normally, a subshell value of 4 is the largest needed to describe a peirticular subshell. If chemists ever need a larger value, they can create subshell numbers and letters. 

The magnetic quantum number m, describes how the Vcirious orbitals are oriented in space. The value of m, depends on the value of 1. The values allowed are integers from -/ to 0 to +/. For example, if the value of / = 1 (p orbital — see Table 3-4), you can write three values for m, -1, 0, and +1. This means that there are three different p subshells for a pcirticular orbital. The subshells have the same energy but different orientations in space. 

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Consider now the solutions of the spherical potential well with a barrier at the center. Figure 14 shows how the energies of the subshells vary as a function of the ratio between the radius of the C o barrier Rc and the outer radius of the metal layer R ui- The subshells are labeled with n and /, where n is the principal quantum number used in nuclear physics denoting the number of extrema in the radial wave function, and / is the angular momentum quantum number. 

If we consider the angular momentum quantum number of each of these orbitals, s = 0, p = 1, d = 2, f = 3, etc., we obtain the following sequence of numbers for the order of filling. Each sequence shown on consecutive lines, is repeated just once. 

The second quantum number is called the angular momentum quantum number. It is designated by the letter f and can be thought of as representing a subshell within a principal energy... 

The angular momentum quantum number is denoted /. It also affects the energy of the electron, but in general not as much as the principal quantum number does. In the absence of an electric or magnetic field around the atom, only these two quantum numbers have any effect on the energy of the electron. The value of / can be 0 or any positive integer up to, but not including, the value of n for that electron. 

The angular momentum quantum numbers are often given letter designations, so that when they are stated along with principal quantum numbers, less confusion results. The letter designations of importance in the ground states of atoms are the following ... 

Ans. (a) 2. (h) 10, and (c) 6. Note that the principal quantum number docs not affect the number of orbitals and thus the maximum number of electrons. The angular momentum quantum number is the only criterion of that. 

The angular momentum quantum number, , for a particular energy level as defined by the principle quantum number, n, depends on the value of n. I can take integral values from 0 up to and including (n - 1). 

Fig. 2.5. Dependence of total reaction cross-section on the angular momentum quantum number.

Each shell contains one or more subshells, each with one or more orbitals. The second quantum number is the angular momentum quantum number (/) that describes the shape of the orbitals. Its value is related to the principle quantum number and has allowed values of 0 to (n-1). For example, if n = 4, then the possible values of / would be 0,1, 2, and 3 (= 4-1). 

The third quantum number is the magnetic quantum number ( /). It describes the orientation of the orbital around the nucleus. The possible values of m1 depend on the value of the angular momentum quantum number, /. The allowed values for m/ are —/ through zero to +/. For example, for /= 2 the possible values of mi would be —2, —1, 0, +1, +2. This is why, for example, if / = 1 (a p orbital), then there are three p orbitals corresponding to m/ values of—1, 0, +1. This is also shown in Figure 10.3. 

This empirical result is consistent with the theoretical analysis of the partial-wave expansion (where the truncation of the FCI expansion is based on the angular-momentum quantum number rather than on the principal quantum number n), for which it has been proved that the truncation error is proportional to L-3 when all STOs up to l = L are included in the FCI wavefunction [49, 50],... 

We return to the simple example of the angular momentum algebra, SO(3). Its tensor representations are characterized by one integer (Table A.4), that is, the angular momentum quantum number J. Similarly, the representations of SO(2) are characterized by one integer (Table A.4) that is, M the projection of the angular momentum on the z axis. The complete chain of algebras is... 

In Equation 12.6 p, is the permanent dipole moment, h is Planck s constant, I the moment of inertia, j the angular momentum quantum number, and M and K the projection of the angular momentum on the electric field vector or axis of symmetry of the molecule, respectively. Obviously if the electric field strength is known, and the j state is reliably identified (this can be done using the Stark shift itself) it is possible to determine the dipole moment precisely. The high sensitivity of the method enables one to measure differences in dipole moments between isotopes and/or between ground and excited vibrational states (and in favorable cases dipole differences between rotational states). Dipole measurements precise to 0.001 D, or better, for moments in the range 0.5-2D are typical (Table 12.1). 

The second quantum number describes an orbital s shape, and is a positive integer that ranges in value from 0 to (n - 1). Chemists use a variety of names for the second quantum number. For example, you may see it referred to as the angular momentum quantum number, the azimuthal quantum number, the secondary quantum number, or the orbital-shape quantum number. 

All electrons in an atom can be defined in terms of four quantum numbers. The four quantum numbers are the principal quantum number, n, the angular momentum quantum number, /, the magnetic quantum number, m, and the spin quantum number, s. 

Values of /, the angular momentum quantum number, lie between 0 and (n- 1). The value of / determines the type of orbital. The values of m are from -I through 0 to +/. This can be seen in the table. 

The first shell or energy level out from the nucleus is called the K shell or energy level and contains a maximum of two electrons in the s orbital— that is, K = s2, where the K represents the shell number (or principle quantum number), the s describes the orbital shape of the angular momentum quantum number, and the 2 is the maximum number of electrons that the s orbital can contain. This particular sequence is K = s2, which means K shell contains 2 electrons in the s orbital. This is the sequence for the element helium. Look up helium in the text for more information. 

Quantum mechanical considerations show that, like many other atomic quantities, this angular momentum is quantized and depends on I, which is the angular momentum quantum number, commonly referred to as nuclear spin. The nuclear spins of / = 0, 1/2, 1, 3/2, 2. .. up to 6 have been observed (see also Table 1). Neither the values of I nor those of L (see below) can yet be predicted from theory. 

Then the momental densities n (p) = (4ti) u t p) p for various values of the angular momentum quantum number have the following expansions [187] ... 

The last equation is formally identical to the radial Schrodinger equation with a non-integer value of the angular momentum quantum number. Its spectrum is bounded from below and the discrete eigenvalues are given by... 

The y>Ee(R) are the radial free-state wavefunctions (see Chapter 5 for details). The free state energies E are positive and the bound state energies E(v,S) are negative v and ( are vibrational and rotational dimer quantum numbers t is also the angular momentum quantum number of the fth partial wave. The g( are nuclear weights. We will occasionally refer to a third partition sum, that of pre-dissociating (sometimes called metastable ) dimer states,... 

When the wave equation for a hydrogen-like atom is solved in the most direct way for orbitals with the angular momentum quantum number / = 3, the following results are obtained for the purely angular parts (i.e., omitting all numerical factors) ... 

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