Atoms and Quantum Numbers

Main energy levels corresponding to electron shells discussed earlier in this chapter are designated by a principal quantum number n. Both the size of orbitals 

Within each main energy level (shell) represented by a principal quantum number, there are sublevels (subshells). Each sublevel is denoted by an azimuthal quantum number I For any shell with a principal quantum number n, the possible values of Z are 0,1,2, 3. n - 1. This gives the following  

From the above, it can be seen that the maximum number of sublevels within a principal energy level designated by n is equal to n. Within a main energy level, sub-levels denoted by different values of Z have slightly different energies. Furthermore, Z designates different shapes of orbitals. 

The letters s, p, d, and/, corresponding to values of Z = 0,1,2, and 3, respectively, are conventionally used to designate sublevels. The number of electrons present in a given sublevel is limited to 2,6,10, and 14 for s, p, d, and/, sublevels, respectively. It follows that, since each orbital can be occupied by a maximum of 2 electrons, Aere is only 1 orbital in the 5 sublevel, 3 orbitals in the p sublevel, 5 in the d and 7 in the /. The value of the principal quantum number and the letter designating the azimuthal quantum number are written in sequence to designate both the shell and subshell. For example. Ad represents the d subshell of the fourth shell. 

The magnetic quantum number is also known as the orientational quantum number. It designates the orientation of orbitals in space relative to each other and distinguishes orbitals within a subshell from each other. It is called the magnetic quantum number because the presence of a magnetic field can result in the appearance of additional lines among those emitted by electronically excited atoms (i.e., in atomic emission spectra). The possible values of m in a subshell with azimuthal quantum number Z are given by m, = +1, +(Z - 1). 0. -(Z - 1), -Z. As examples, for Z = 0, the only possible value of m, is 0, and for I = 3,m, may have values of 3, 2,1,0, -1, -2, -3. 

---

Quantitative analysis, infrared, 250 Quantitative presentation of data, 14 Quantum mechanics, 259, 260 and the hydrogen atom, 259 Quantum number, 260 and hydrogen atom, 260 and orbitals, 261 principal, 260... 

The expectation values of various powers of the radial variable r for a hydrogenlike atom with quantum numbers n and I are given by equation (6.69)... 

There are several ways of indicating the arrangement of the electrons in an atom. The most common way is the electron configuration. The electron configuration requires the use of the n and / quantum numbers along with the number of electrons. The principle quantum number, n, is represented by an integer (1,2,3. ..), and a letter represents the l quantum number (0 = s, 1 = p, 2 = d, and 3 = f). Any s-subshell can hold a maximum of two electrons, any p-subshell can hold up to six electrons, any d-subshell can hold a maximum of 10 electrons, and any f-subshell can hold up to 14 electrons. 

The relationship between orbital size and quantum number for the hydrogen atom. 

Here, the relevant angular momentum vectors and quantum numbers are L (7), the total orbital angular momentum of the atom, obtained as the vector coupling of those corresponding to the core and to the outer electron(s), S (S), the total spin, and J (7), the total angular momentum for a given atomic level [8] ... 

Fig. 2-4.—Bohr orbits for the hydrogen atom, total quantum number 2, 3, and 4. These orbits are represented as having the values of angular momentum given by quantum mechanics.

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. 

The data for a paramagnetic many-electron atom with quantum numbers S, L, and / are... 

Many calculations for atoms have led to the development of a number of recipes for deciding the best values of and n. A further important issue is the size of the basis set. A minimal basis set of STOs for an atom would include one function for each SCF occupied orbital with different n and / quantum numbers in equation (6.56) for the chlorine atom, therefore, the minimal basis set would include s, 2s, 2p, 3s and 3p functions, each with an optimised Slater orbital exponent . A higher order of approximation would be to double the number of STOs (the double zeta basis set), with orbital exponents optimised ultimately the Hartree-Fock limit is reached, as it has been for all atoms from He to Xe [13]. 

One of the most commonly employed procedures has been to simply extrapolate the molecular coupling from the available atomic parameters using the so-called atoms-in-molecules approach (72). Here (r,) is assumed to be constant for electrons with the same n and / quantum numbers. The values of are then assumed to be equal to the spin-orbit coupling constants n/, which are derived from atomic spectral data. This approach has been employed by Wadt (73) in all-electron studies, and by other groups (32,74) in effective potential calculations involving the rare-gas dimers and dimer ions. Ermler and co-workers used this approach coupled with AREP calculations to determine spectroscopic properties for various states of Au2 (42), Hg2, and HgTl (75). 

A necessary condition to be used is the Pauli exclusion principle, which states that no two electrons in the same atom can have the same set of four quantum numbers. It should also be recognized that lower n values represent states of lower energy. For hydrogen, the four quantum numbers to describe the single electron can be written as n = 1, l = 0, mt = 0, ms = +1/2. For convenience, the positive values of mt and ms are used before the negative values. For the two electrons in a helium atom, the quantum numbers are as follows ... 

Scattering theory concerns a collision of two bodies, that may change the state of one or both of the bodies. In our application one body (the projectile) is an electron, whose internal state is specified by its spin-projection quantum number v. The other body (the target) is an atom or an atomic ion, whose internal bound state is specified by the principal quantum number n and quantum numbers j, m and / for the total angular momentum, its projection and the parity respectively. We... 

Unfortunately, 5 is used in two ways to designate the atomic spin quantum number and to designate a state having L = 0. Chemists are not always wise in choosing their symbols ... 

TABLE n-3 Examples of Atomic States (Free-ion Terms) and Quantum Numbers... 

The subscript A on a perimeter orbital defined in Equation (2.2) can be viewed as a quantum number related to the z component of orbital angular momentum associated with an electron in the orbital The selection rule for one-electron transitions between perimeter orbitals is similar to that familiar from atoms the quantum number k is allowed to increase or decrease by I. Thus promotions from to, and to, are allowed,... 

---