Hund’s third rule

Hund s third rule 7 = L - 5 for a less-than-half-filled shell J = L + S) for a more-than-half-filled shell and 7 = 5 for a half-filled shell (L = 0). 

Since the shell is less than half-filled, the state with the lowest /-value is the ground state (Hund s third rule), so this is / = 5/2. 

These are the five energy states for the carbon atom referred to at the beginning of this section. The state of lowest energy (spin-orbit coupling included) can be predicted from Hund s third rule ... 

Hund s third rule predicts which J value corresponds to the lowest energy state. The d orbitals are less than half-filled, so the minimum J value for F, 7 = 2, is the ground state. Overall, it is F2. 

From Hund s third rule Box 20.6), the value of J for the ground state is givenhy L—S) for a sub-shell that is less than half-filled, and by (F + 5) for a sub-shell that is more than half-filled. 

Hund s third rule is a relativistic correction to the first two rules, introducing a splitting of the terms given by the previous rules. The energy operator (Hamiltonian) commutes with the square of the total angular momentum J = L - - S, and therefore, the energy levels depend rather on the total momentum Jp = J J + This means that they depend on the mumal orientation of L and S (this is a relativistic effect due to the spin-orbit coupling in the Hamiltonian). The vectors L and S add in quantum mechanics in a specific way (see... 

Closely related to the Pauli exclusion principle is the third rule, Hund s rule, which states that when electrons occupy orbitals of equal energy (e.g., the five 3d orbitals), one electron enters each orbital until all the orbitals contain one electron. In this configuration, all electrons will have parallel spin (same direction). Second electrons then add to each orbital so that their spins are opposite to the first electrons in the orbital. Atoms with all outer orbitals half-filled are very stable. 

Notice in Table 1-1 that carbon s third and fourth valence electrons are not paired they occupy separate orbitals. Although the Pauli exclusion principle says that two electrons can occupy the same orbital, the electrons repel each other, and pairing requires additional energy. Hund s rule states that when there are two or more orbitals of the same energy, electrons will go into different orbitals rather than pair up in the same orbital. The first 2p electron (boron) goes into one 2p orbital, the second 2p electron (carbon) goes into a different orbital, and the third 2p electron (nitrogen) occupies the last 2p orbital. The fourth, fifth, and sixth 2p electrons must pair up with the first three electrons. 

In the third row K uses a doubly occupied 4s band of bonding orbitals, and Ca uses two s-p hybrid orbitals. In Sc the 3d-electron is comparable in energy to the 4s, and the d orbitals form a five-fold degenerate band. The free atoms of the first transition series all have high spin d-electrons in accordance with Hund s rule. We expect similar behavior in the solid state, so that only 2 1/2 electrons will fill the bonding d levels without double occupancy. This is the case for Sc, Ti and V. 

The orbital capacities and order of filling of atomic orbitals are governed, respectively, by the Pauli exclusion principle and Hund s rules of maximum angular momentum. In its simplest form, the exclusion principle states that no two electrons in the same atom can have four identical quantum numbers. Hence, if an orbital is specified by n, l, and m, it can accommodate a maximum of two electrons, one with s = +lh and one with s = h. A third electron would have... 

From these first two rules, we can assign electrons to atomic orbitals for atoms that contain one, two, three, four, or five electrons. The single electron of a hydrogen atom occupies a I5 atomic orbital, the second electron of a helium atom fills the Is atomic orbital, the third electron of a lithium atom occupies a 2s atomic orbital, the fourth electron of a beryllium atom fills the 2s atomic orbital, and the fifth electron of a boron atom occupies one of the 2p atomic orbitals. (The subscripts x, y, and z distinguish the three 2p atomic orbitals.) Because the three p orbitals are degenerate, the electron can be put into any one of them. Before we can continue to larger atoms—those containing six or more electrons—we need Hund s rule ... 

The spin-orbit coupling constant X is positive if the 4f subshell is less than half filled and negative if it is more than half filled. A new quantum number/, associated with the total angular momentum J = L + S, has to be introduced with values ranging from (L + S) to (L — S). As a consequence, each term is further split into a number of spectroscopic levels ( s+i)p g cjj (2/ + 1) multiplicity. Again, the sum of these multiplicities must be equal to the multiplicity of the term. For instance, the ground term of Eu is split into Fo, Fi, p2, F3, F4, F5, and Fg with multiplicities 1 + 3 + 5 + 7 + 9 + 11+13 = 49. The ground level can be found with third Hund s rule ... 

Write down the detailed electronic configurations of the elements of the third period as given by Hund s rules. 

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