Quantum number combinations, Table

Table 2.1 Quantum number combinations and atomic orbitals...

Paper four first appeared in the Journal of Chemical Education and aimed to highlight one of the important ways in which the periodic table is not fully explained by quantum mechanics. The orbital model and the four quantum number description of electrons, as described earlier, is generally taken as the explanation of the periodic table but there is an important and often neglected limitation in this explanation. This is the fact that the possible combinations of four quantum numbers, which are strictly deduced from the theory, explain the closing of electron shells but not the closing of the periods. That is to say the deductive explanation only shows why successive electron shells can contain 2, 8, 18 and 32 electrons respectively. 

The hierarchy of shells, subshells, and orbitals is summarized in Fig. 1.30 and Table 1.3. Each possible combination of the three quantum numbers specifies an individual orbital. For example, an electron in the ground state of a hydrogen atom has the specification n = 1, / = 0, nij = 0. Because 1=0, the ground-state wavefunction is an example of an s-orbital and is denoted Is. Each... 

Table 2.1 shows the possible combinations of quantum numbers for n = 1 to 3. 

Using the above definitions for the four quantum numbers, we can list what combinations of quantum numbers are possible. A basic rule when working with quantum numbers is that no two electrons in the same atom can have an identical set of quantum numbers. This rule is known as the Pauli Exclusion Principle named after Wolfgang Pauli (1900-1958). For example, when n = 1,1 and mj can be only 0 and m can be + / or -1/ This means the K shell can hold a maximum of two electrons. The two electrons would have quantum numbers of 1,0,0, + / and 1,0,0,- /, respectively. We see that the opposite spin of the two electrons in the K orbital means the electrons do not violate the Pauli Exclusion Principle. Possible values for quantum numbers and the maximum number of electrons each orbital can hold are given in Table 4.3 and shown in Figure 4.7. 

Explicit formulas and numerical tables for the overlap integral S between AOs (atomic orbitals) of two overlapping atoms a and b are given. These cover all the most important combinations of AO pairs involving ns, npSlater type, each containing two parameters i [equal to Z/( - 5)], and n — S, where n — S is an effective principal quantum number. The S formulas are given as functions of two parameters p and t, where p = p- + p,s)R/ao, R being the interatomic distance, and t = — Mb)/(ma + Mb)- Master tables of... 

A summary of the allowed combinations of quantum numbers for the first four shells is given in Table 5.2. 

PROBLEM 5.11 Extend Table 5.2 to show allowed combinations of quantum numbers when n = 5. How many orbitals are in the fifth shell ... 

Combination of Eqs. 15 and 16 with the expressions in Table 2 thus finally yields the symmetry adapted (t2g)3 multiplets AA2g, 2Eg + 2Tlg, 2T2g described by five quantum numbers SLTMsMry. These resulting states are identical -within a multiplet dependent phase factor - to the state functions published by Griffith [3],... 

Orbitals having the same azimuthal quantum number Z have the same shape all s orbitals have spherical symmetry and all p orbitals have cylindrical symmetry. The d orbital is drawn differently from the other d orbitals but, being a linear combination of d x2 and d orbitals, it is perfectly equivalent to them. (This statement may be checked, using Table 2.1). The whole field of stereochemistry is founded upon the directional character of p and d orbitals. 

We can see the relationship between these terms and quantum numbers by looking carefully at Table 4.3. First, each unique value of n represents an energy level. Each 7 value represents a specific sublevel within an energy level. Recall from the previous section that these sublevels are typically referred to using their common names s, p, d, and f. Each unique combination of n and 1 values corresponds to a different sublevel. For example, for n = 3 and 7=2, this corresponds to the 3dsublevel of the atom. The rn values tell us how many orbitals are found in a given sublevel. For instance, in the 3d sublevel there are 5 orbitals possible (for 3d, rn —2, — 1, 0, 1, 2). The spin quantum number tells us that there can be no more than 2 electrons in any orbital, which you will learn more about later in this chapter. Let s summarize what we know in a new chart. 

With lithium (Z = 3) there are the two electrons in a spherical cloud (as with helium) plus a third electron. In most cases, in considering the electronic configurations of the element, the last electron is the only one that need be considered, all the remaining having been present in the preceding atom in the periodic table (there are a few important exceptions, however). For the third lithium electron there are no more possible combinations of quantum numbers where w = 1 since neither l or m can exceed zero if n is only one. The last (outermost) electron of lithium has the quantum numbers ... 

Molecular orbitals (MOs) were constructed using linear combinations of basis functions of atomic orbitals. The MO eigenfunctions were obtained by solving the Schrodinger equations in numerical form, including Is— (n+l)p, that is to say, Is, 2s, 2p, -ns, np, nd, (n+l)s, (n+l)p orbitals for elements from n-th row in the periodic table and ls-2p orbitals for O, where n—1 corresponded to the principal quantum number of the valence shell. 

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. 

It is conventional to label specific states by replacing the angular momentum quantum number with a letter we signify = 0 with s, = 1 with p, = 2 with d, = 3 with f, (.= 4 with g, and on through the alphabet. Thus, a state with n = and = 0 is called a Is state, one with n = 3 and F = 1 is a 3p state, one with n = 4 and = 3 is a 4/state, and so forth. The letters s, p, d, and / derive from early (pre-quantum mechanics) spectroscopy, in which certain spectral lines were referred to as sharp, principal, diffuse, and fundamental. These terms are not used in modern spectroscopy, but the historical labels for the values of the quantum number are still followed. Table 5.1 summarizes the allowed combinations of quantum numbers. 

In the following table we have collected all the possible combinations for these four quantum numbers, omitting all those combinations which would contradict the exclusion principle. Further, we have not written down those combinations which arise from a given set merely by interchanging the quantum numbers of the two electrons for since we cannot distinguish one electron from the other these of course n.re identical and are associated with the same energy term. Hence we have the following table ... 

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