Quantum numbers, atomic permitted values

When dealing with atoms possessing more than one electron it is necessary to introduce a fourth quantum number s, the spin quantum number. However, to take into account the intrinsic energy of an electron, the value of 5 is taken to be A. Essentially the intrinsic energy of the electron may interact in a quantized manner with that associated with the angular momentum represented by /, such that the only permitted interactions are l + s and / - s. For atoms possessing more than one electron it is necessary to specify the values of s with respect to an applied magnetic field these are expressed as values of ra of + A or - A. 

As discussed in Section 5.1, the structure of many-electron atoms can be understood only by assuming that no more than two electrons can occupy each separate orbital. Taking account of the electron spin allows a deeper interpretation of this fact. One way of expressing the Pauli exclusion principle is no two electrons can have the same values of all four quantum numbers, n, l, m, and ms. As only two values of ms are permitted, it follows that each orbital, specified by a given set of values of n, l, and m, can hold... 

A lithium atom has three electrons. The hrst two of these can have the same sets of quantum numbers as the two electrons of hehum. What should the set of quantum numbers for the third electron be We cannot choose the lowest permitted value for n, which is 1, because C and m would then both be 0. If we choose -j as the value of m, the third electron would have a set of quantum numbers exactly the same as that of one of the hrst two electrons, and if we choose the value = +5, the third electron would have the same set of quantum numbers as the other. Because neither of these situations is permitted by the Pauli principle, n cannot be 1 for the third electron. We must choose the next higher value, = 2. With = 2, the permitted values of are 0 and 1. Because = 0 wUl give a lower value for the sum + , we choose that value for . With = 0, must be 0, and we can choose either 5 or +5 for m. The quantum numbers for the three electrons of the Uthium atom can thus be as follows ... 

The 77 + f rule, the Pauli exclusion principle, and the permitted values of the quantum numbers enable us to determine the order of the electrons in an atom in inereasing energy. 

The second electron also can have n = 1, Z =0, and m = 0. Its value of can be either + or — but not the same as that for the first electron. If it were, this second electron would have the same set of four quantum numbers that the first electron has, which is not permitted by the Pauli principle. If we were to try to give the third electron the same values for the first three quantum numbers, we would be stuck when we came to assign the m, value. Both + j and — j have already been used, and we would have a duplicate set of quantum numbers for two electrons, which is not permitted. We cannot use any other values for Z or m with the value of n = 1, and so the third electron must have the next-higher n value, n = 2. The Z values could be 0 or 1, and since 0 will give a lower n + l sum, we choose that value for the third electron. Again the value of m must be 0 since Z = 0, and rris can have a value — (or + j). For the fourth electron, n = 2,1 = 0, nii = 0, and mj = + (or —j if the third were + ). The fifth electron can have n = 2 but not 1=0, since all combinations of n = 2 and 1=0 have been used. Therefore, n = 2,1 =, nii = —, and nis = — are assigned. The rest of the electrons in the aluminum atom are assigned quantum numbers somewhat arbitrarily as shown in Table 4-3. 

For the higher quantum numbers the relationship between the energy values of the orbitals is more complicated (see Figure 7). Thus for example, the 4J orbital ( == 4, / = m) is more stable than the orbital (n = 3, / = 2). This complexity docs not permit a construction of the electronic distribution of the elements on the basis only of the analogy to a hydrogen like atom and it is necessary for each element to use the spectroscopic data to determine the electronic states. The sequence of the distribution thus obtained can, with only a few exceptions, be expressed by the data given in Figure 7. 

Slater s method which has been described in Chapter i6 for the problem of three electrons may be applied to systems with any number of electrons. Each electron may exist in the field of any of the nuclei, and resonance among the different states, representing different electronic distributions, will occur. With four atoms having four valency electrons, 4 = 24 different arrangements of the electrons between the nuclei are possible, with 2 = 16 different spin states. In all there will therefore be 24 X 16 = 384 different complete wave functions. But as we have already seen, states with different values of the spin quantum number do not interact with one another. This permits a considerable reduction in the number of states to be considered. We are only concerned with stable configurations in which all the electrons in pairs neutralize their spin by the formation of a covalent bond, i.e. S—0 and S =0. 

These individual functions R, , and give rise to the three orbital quantum numbers n, i, and m. We have seen that solutions of the Schrodinger equation are possible only for certain values of the total energy E. For hydrogen-like atoms, the permitted total energy values are given by the equation... 

We saw in Table 1.1 the eigenfunctions for a p orbital, for which the permitted m values are 1, 0, and — 1, and Figure 1.6 shows their orientation in space. The case of d orbitals is shown in Figure 1.7. The magnetic properties of atoms are related to the magnetic quantum number. 

The parameter m is the component of angular momentum in a given direction it is also known as the magnetic quantum number because it is related to the components of the electron angular momentum when a magnetic field is applied to the atom. The permitted values are — / < m, < +/. 

The classical case corresponds to the vanishing de Broglie wave length. Formula (18.1.2) permits to visualize quite easily the factors which determine quantum effects. For heavy particles A is small and we may n lect the quantum effects. Also an increase in the interaction parameter e lowers A. If we consider for example the rare gases, both factors work in the same direction because with increasing number of electrons there is an increase in the polarisability of the molecule and consequently of e. For this reason the value of A drops rapidly if we go from He to A (cf. Table 18.1.1). Table 18.1.1 gives the values of the quantum mechanical parameter a for some atoms and molecules. 

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