Quantum number, azimuthal electron-spin

EQUIVALENT ELECTRONS. For an atom, electrons in the same orbital (whereby they have the same principal quantum number and ihe same azimuthal quantum number). For a molecule, electrons having the same quantum numbers, apart from spin, and the same symmetry s or u. 

When the Schrodinger equation is solved, it yields many solutions— many possible wave functions. The wave functions themselves are fairly comphcated mathematical functions, and we do not examine them in detail in this book. Instead, we will introduce graphical representations (or plots) of the orbitals that correspond to the wave functions. Each orbital is specified by three interrelated quantum numbers n, the principal quantum number I, the angular momentum quantum number (sometimes called the azimuthal quantum number) and mi, the magnetic quantum number. These quantum numbers all have integer values, as had been hinted at by both the Rydberg equation and Bohr s model. A fourth quantum number, nis, the spin quantum number, specifies the orientation of the spin of the electron. We examine each of these quantum numbers individually. 

The shell theory has had great success in accounting for many nuclear properties (3). The principal quantum number n for nucleons is usually taken to be n, + 1, where nr, the radial quantum number, is the number of nodes in the radial wave function. (For electrons n is taken to be nr + / +1 / is the azimuthal quantum number.) Strong spin-orbit coupling is assumed,... 

Figure 10.5 Energy levels of atomic orbitals, n is the principal quantum number, and the 5, p, d notation indicates the azimuthal quantum number (/). For / = 1 and above the orbital is split into multiple suborbitals (indicated by the number of lines), corresponding to the values of the magnetic quantum number m Each of these lines can hold two electrons (corresponding to spin up and spin down ), giving rise to the rules for filling up the orbitals.

According to quantum mechanics, electrons in atoms occupy the allowed energy levels of atomic orbitals that are described by four quantum numbers the principal, the azimuthal, the magnetic, and the spin quantum numbers. The orbitals are usually expressed by the principal quantum numbers 1, 2, 3, —increasing from the lowest level, and the azimuthal quantum numbers conventionally eiqiressed by s (sharp), p (principal), d (diffuse), f (fundamental), — in order. For instance, the atom of oxygen with 8 electrons is described by (Is) (2s) (2p), where the superscript indicates the munber of electrons occupying the orbitals, as shown in Fig. 2-1. 

The energy states of atoms are expressed in terms of four quantum numbers j it, the principal quantum number /, the azimuthal quantum number m, the magnetic quantum number and mt or s, the spin quantum number. According to Pauli s exclusion principle, no two electrons can have the same values for all the four quantum numbers. 

The only quantum number that flows naturally from the Bohr approach is the principal quantum number, n the azimuthal quantum number Z (a modified k), the spin quantum number ms and the magnetic quantum number mm are all ad hoc, improvised to meet an experimental reality. Why should electrons move in elliptical orbits that depend on the principal quantum number n Why should electrons spin, with only two values for this spin Why should the orbital plane of the electron take up with respect to an external magnetic field only certain orientations, which depend on the azimuthal quantum number All four quantum numbers should follow naturally from a satisfying theory of the behaviour of electrons in atoms. 

The Sommerfeld model for Ne is shown in figure 2.8. The He atom presented a special problem as the quantum numbers restrict the two electrons to the same circular orbit, on a collision course. One way to overcome this dilemma was by assuming an azimuthal quantum number k = for each electron, confining them to coplanar elliptic orbits with a common focal point. To avoid interference they need to stay precisely out of phase. This postulate, which antedates the discovery of electron spin was never seen as an acceptable solution to the problem which eventually led to the demise of the Sommerfeld model. 

Here E denotes the energy of the bound electron after deducting the rest energy, and Eq is the rest energy mc is the radial quantum number (Sommerfeld) is identical with Bohr s azimuthal quantum number /c, and corresponds therefore to the H + 1 of wave mechanics. Since, however, as we have just seen, two terms with dilierent I but the same j always coincide when we take the spin into account, discrimination between the terms by means of the quantum number is identical with discrimination by means of j we therefore have = i + I- The principal quantum number is then found as the sum n=- n>r + The constant a is given by... 

Another concept which is frequently employed is this. Two electrons are said to be equivalent when they possess the same n and the same 1. Two equivalent electrons must therefore, in accordance with Pauli s principle, differ from each other in the direction of I, or in the spin direction and only certain definite values, not all values, of /X and a are possible for them. It is otherwise with two electrons which are not equivalent, but differ either in the principal or the azimuthal quantum number (or in both) in this case all values of /x and o are possible. 

The terminology and symbolism used to specify the various quanmm numbers are not too informative. The numbers are known as the principal (n), azimuthal (1), magnetic (mi), spin (s) and magnetic spin (m ), quantum numbers. The first three are integers, such that, for one set of eigenfunctions, is a positive number, I is always less than n and w has a total of (2/+ 1) allowed values, clustered about zero. For n = 2 and / = 1 it follows that m/ has the three possible values +1, 0 and -1. The quantum numbers s and m have half-integer values. All electrons have 5=5 and Wj = 5. 

In electron configurations with the same main and azimuthal quantum numbers, the highest total spin configuration is the most stable. 

The lanthanides are f-elements and the lanthanide ions have all their valence electrons that fill 4f orbitals. The f orbitals are depicted in Figure 1. There are seven f orbitals each can accept two electrons with an opposite spin. Electrons in the 4f subshell have their principal quantum number = 4, their azimuthal quantum number = 3, can have their... 

The formal names of these numbers are principal, azimuthal, magnetic, and electron spin. We use the name and number of the principal quantum number, but not the other three. All, however, are described to the extent necessary to specify the distribution of electrons in an atom. 

Atomic orbital (AO) (Sections 1.10, 1.11, and 1.15) A volume of space about the nucleus of an atom where there is a high probability of finding an electron. An atomic orbital can be described mathematically by its wave function. Atomic orbitals have characteristic quantum numbers the principal quantum number, n, is related to the energy of the electron in an atomic orbital and can have the values 1, 2, 3,. The azimuthal quantum number I, determines the angular momentum of the electron that results from its motion around the nucleus, and can have the values 0, 1, 2,. .., (n — 1). The magnetic quantum number, m, determines the orientation in space of the angular momentum and can have values from 3-/ to /. The spin quantum number, s, specifies the intrinsic angular... 

We have two atoms with m and m electrons respectively their azimuthal quantum numbers are / and their character on reflection w and to, and their partitions z and z. We put r and r for — z and m — z (the r s then mean something like the core momentum). Our task consists in that we want to know the terms of the molecule for fixed nuclei (electronic terms) which result from these atoms by adiabatically uniting the nuclei, i.e. their representation char2u teristics under permutation of the electronic centres of mass and reflection-rotation about the Z-axis. The spin-free eigenfunctions of the atoms are and wherein the index C distinguishes between the ( ) — (Jli) eigenfunctions that belong to the partition z. 

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