Chapter 8 Multielectron Atoms Spin and Term Symbols 8.1 Electron Spin Electron spin is a form of angular momentum different from orbital angular momentum. It represents the top-like spinning behavior of electrons. The spin quantum number m s was introduced to account for the experimental observation of a doublet in the atomic spectrum of sodium. The Schrödinger equation (without spin) predicted only a single line. Recall the eigenequations for orbital angular momentum: ˆL Y m l (θ, φ) = h l(l + 1)Y m l (θ, φ) l = 0, 1,,... ˆL z Y m l (θ, φ) = m hy m l (θ, φ) l m l 1
Similarly, we can write eigenequations for spin angular momentum: ) ( α Ŝ β }{{} α or β ( α = h s (s + 1) β ) s = 1 Ŝ z ( α β ) ( α = m s h β ) s m s s m s = ± 1 Note: For α, m s = 1 ; and for β, m s = 1. The eigenfunctions α(σ) and β(σ) are never given a real form, but can be formally associated with α = Y 1/ 1/ and β = Y 1/ 1/ (i.e. use formal association for orthogonality relations and relation with spin operators...) We now have a fourth quantum number m s, the spin quantum number, and two spin functions α(σ) and β(σ) (where σ is a spin variable ). This gives us one more degree of freedom that of spin. Note that α and β are eigenfunctions of both Ŝ and Ŝz, which are Hermitian operators, and therefore they are orthogonal: α (σ)β(σ)dσ = β (σ)α(σ)dσ = 0 We assume they are also normalized: α (σ)α(σ)dσ = β (σ)β(σ)dσ = 1 Spin angular momentum has no classical analog, so do not try to attach a classical meaning to it.
8. Pauli Exclusion Principle Now that we have introduced spin, a many (N) electron wave function can be written: Ψ( r 1 σ 1, r σ,... r N σ N ) = Ψ(1,,...N) Here, r 1 = (x 1, y 1, z 1 ) represents the spatial varibles of electron 1, and σ 1 is the spin variable of electron 1. So when one writes Ψ (1,,...N)Ψ(1,,...N)dτ, it means:... Ψ (1,,...N)Ψ(1,,...N)dx 1 dy 1 dz 1 dσ 1,...dx N dy N dz N dσ N Note: This is a 4N-dimensional integral! (Remember (function) dτ means integration over the entire space of all the varibles.) The Pauli Principle states that all electronic wavefunctions must be antisymmetric under the interchange of any two electrons. This is an implication of two facts: 1. Electrons are indistinguishable: e.g. Ψ(1, ) = Ψ(, 1) Ψ(1, ) = Ψ(, 1). Two electrons with the same spin can t be on top of one another: Ψ(1, 1) = 0 3
8.3 Term Symbols Term symbols are used to describe atomic spectra and predict the ground state electron configuration. Knowing only the number of electrons does not give us a detailed enough description. e.g. Carbon: 1s s p leaves us with unanswered questions: which p orbitals have electrons? what is the spin state? Using atomic term symbols gives us a detailed specification of the electron configuration. Term symbols have the form: S+1 L J L: total orbital angular momentum S: total spin angular momentum S + 1: spin multiplicity J: total angular momentum (Russel-Saunders coupling of L and S) where J = L + S,..., L S L = 0 1 3 S P D F M L = i m li = L z M s = i m si = S z Note: For a filled shell (i.e. ns, np 6, nd 10,...) the term symbol is always: 1 S 0 4
8.3.1 An Example What are the possible term symbols for an np 3 electron configuration? We don t have to worry about fully occupied subshells, because their contribution to M L and M S is always zero. (See pg. 95 for the explanation.) Thus we have enough information to solve this problem, only knowing the unfilled subshell. L =, 1, 0 since M L can be -, -1, 0, 1, or. For L = S = 1 since there is one unpaired electron (the one in the m l = 0 orbital). Thus J = 5, 3. So we have D 5 and D 3. For L = 1 S = 1 since there is one unpaired electron (the one in the m l = 1or1 orbital). Thus J = 3, 1. So we also have P 3 and P 1. Finally, for L = 0 S = 3 since all three electrons are unpaired (there is one in each of the m l orbitals (-1, 0, and 1)). Thus J = 3. So there is one more term symbol, 4 S 3. 8.4 Finding the Number of Microstates The number of microstates is given by: G! N!(G N)! where G is the number of spin orbitals and N is the number of electrons. Note: If N = G, the number of states is 1. The number of states per term symbol can also be found: (L + 1)(S + 1) 5
8.5 Hund s Rules Hund s Rules tell us how to determine the ground state of a configuration based on its term symbols. Hund s Rules 1. The state with the larger value of S (higher multiplicity) is more stable.. For states with the same S, the one with the larger L is more stable. 3. For states with the same S and L: if subshell is less than half filled, the smaller J is more stable if subshell is more than half filled, the larger J is more stable 8.5.1 An Example Apply Hund s rules to the term symbols found in 8.3.1. 4 S 3 is the most stable, by # 1. 8.6 More Examples 8.6.1 Zr Zr has a ground state configuration of [Kr]4d 5s. What is its ground state term symbol? We only need to concern ourselves with the unfilled subshell (4d ). According to Hund s Rules, we are looking for the highest multiplicity, thus as many spins as possible should be parallel. In this case, all of the spins can be parallel, giving S = 1. There are many states like this though, so we move on to the next rule, which tells us to find the highest L. A state like: 1 0 6 1
has the maximum value of L (under the constraint that the spins are all parallel), which is 3. Since S = 1 and L = 3, J can be 4,3, or. By # 3, the ground state has the lowest J, since this subshell is less than half full. Thus the ground state term symbol for Zr is 3 F. 8.6. Fe Fe has a ground state configuration of [Ar]3d 6 4s. What is its ground state term symbol? There are six electrons we need to worry about, but only 5 orbitals, thus there will be at least one spin pair, so the maximum S =. The maximum L is from a state like: 1 0 1 Since S = and L = + 1 + 0 + 1 + + =, J can be 4, 3,, 1, or 0. Because this subshell is more than half filled, the ground state has the highest J, which is 4. Therefore the ground state of Fe is 5 D 4. 8.7 Atomic Spectra Term symbols allows us to predict atomic spectra by following a few simple selection rules. Selection Rules for Atomic Spectra L = ±1 S = 0 J = 0, ±1 except J = 0 J = 0 7
8.7.1 An Example List all of the allowed transitions between the sodium states S 1 D 1, F 7, F 5., P 3, P 1, D 3, S 1 S 1 P 3 P 3 P 1 P 1 D 3 P 3 P 1 D 3 D 1 D 3 D 1 F 5 A few examples of transitions that are not allowed (and why) are: S 1 D 3 D 3 F 7 because L = because J = 8