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Handout 1: a More Detailed Look at Electronic Structure

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Handout 1: a More Detailed Look at Electronic Structure Handout 1: A More Detailed Look at Electronic Structure. (this handout was originally part of a graduate course on f-element chemistry but the basic principles all still apply to 324) The electron configuration of an atom gives only a very crude description of the electronic energy levels of that atom since each electron configuration (4f2, 4f15d1 etc.) gives rise to several states which differ in energy and have different values of orbital and spin angular momentum (L and S, respectively). In order to get a more detailed picture of the electronic state of an atom, we must consider perturbations to the energy of a single electron state when more than one electron is present in a given orbital. For the lighter elements (through the lanthanides) the biggest perturbation is due to electron-electron repulsion (e2/r) between electrons in the same orbital. For the transition metals, crystal field effects are the next most important perturbation leading to splitting of the Russell-Saunders states (see below) which depends on the symmetry of the ligand crystal (electrical) field. Spin-orbit coupling is generally insignificant for transition metals. In the lanthanide series, spin-orbit coupling is much more important than crystal field effects and this has important consequences in optical spectroscopy and magnetic behaviour. The way in which the energy levels of a d2 and a f2 ion are affected by electron-electron repulsion, spin-orbit coupling and crystal field interactions are shown qualitatively in Figure 1 and 2. We turn now to the problem of determining the electronic structure of a f2 ion. 1. Electron-electron repulsion (e2/r): The LS coupling scheme is a set of rules which allow us to predict fairly accurately the energy states (also known as multiplets) which arise as a result of taking electron repulsion into account. This scheme is useful when electron- electron repulsion is the dominant perturbation to the single electron energy levels. Thus it is useful for both the transition metals and the lanthanides but breaks down for the actinides where the spin-orbit term becomes comparable in size. Just as a single electron has a set of quantum numbers (n, l, ml, ms) which describe its state, we can define new quantum numbers (L, ML, MS) which describe the atom as a whole *. We define L, the total orbital angular momentum of the atom, as L = l1 + l2 + .... Thus ML can take on values from +L to -L (from {l1+l2+...} to {-l1-l2-...}). Similarly we define S, the total spin angular momentum, as S = s1 + s2 + ... and MS can take on values from S to -S (ie. from {s1+s2+...} to {-s1-s2-...}). At this degree of approximation, all states with the same value of L and S will have the same energy. We denote this group of states (known as a multiplet or term) by giving it a special symbol called a term symbol. Since we referred to an electron with a given value of l by letter (l=0, s; 1, p; 2, d; etc.) we do the same for states with a given value of L (L=0, S; 1, P; 2, D; 3, F; 4, G; etc.). In addition we specify the unique value of S for a given multiplet by attaching the spin multiplicity 2S+1 (the total number of spin states, ie different values of MS possible for this value of L). For example a state with L=3 (ML = 3,2,1,0,-1,-2,-3) 3 and S=1 (MS = 1,0,-1) is a F multiplet. The total degeneracy of a multiplet is given by (2L+1)(2S+1) so for 3F the total degeneracy of the multiplet is 21. In the following example LS coupling is used to determine the terms arising from a f2 electron configuration. LS coupling for an f2 ion: For each f electron: l = 3 (ml = 3,2,1,0,-1,-2,-3) and L = l1 + l2 so that L = 6 (ML = 6,5,4,3,2,1,0,-1,-2,-3,-4,-5,-6). Also s = 1/2 (ms = +1/2, -1/2) and S = s1 + s2 = 1 (MS = 1,0,-1). We then construct a table of all possible combinations of MS and ML for the two electrons (known as microstates) taking into account the fact that certain combinations are not allowed by the Pauli exclusion principle (remember: no two electrons can have the same values for all four quantum numbers). In order to simplify the table, we will specify the values of ML by number and MS by + or - + + (for +1/2 and -1/2). Thus (3 ,2 ) refers to a microstate where electron #1 has ML = 3 and MS = +1/2 while electron #2 has ML = 2 and MS = +1/2. * N.B. Closed shells do not contribute since by definition ML = MS = 0. MS +1 0 -1 M L 6 (3+,3-) 5 (3+,2+) (3+,2-)(3-,2+) (3-,2-) 4 (3+,1+) (3+,1-)(2+,2-) (3-,1-) (3-,1+) 3 (3+,0+)(2+,1+) (3+,0-)(2+,1-) (3-,0-)(2-,1-) (1+,2-)(0+,3-) 2 (3+,-1+)(2+,0+) (3+,-1-)(2+,0-) (3-,-1-)(2-,0-) (1+,1-)(0+,2-) (-1+,3-) 1 (3+,-2+)(2+,-1+) (3+,-2-)(2+,-1+) (3-,-2-)(2-,-1-) (1+,0+) (1+,0-)(0+,1-) (1-,0-) (-1+,2-)(-2+,3-) 0 (3+,-3+)(2+,-2+) (3+,-3-)(2+,-2-) (3-,-3-)(2-,-2-) (1+,-1+) (1+,-1-)(0+,0-) (1-,-1-) (-1+,1-)(-2+,2-) (-3+,3-) -1 (-3+,2+)(-2+,1+) (-3+,2-)(-2+,1-) (-3-,2-)(-2-,1-) (-1+,0+) (-1+,0-)(0+,-1-) (-1-,0-) (1+,-2-)(2+,-3-) -2 (-3+,1+)(-2+,0+) (-3+,1-)(-2+,0-) (-3-,1-)(-2-,0-) (-1+,-1-)(0+,-2-) (1+,-3-) -3 (-3+,0+)(-2+,-1+) (-3+,0-)(-2+,-1-) (-3-,0-)(-2-,-1-) (-1+,-2-)(0+,-3-) -4 (-3+,-1+) (-3+,-1-)(-2+,-2-) (-3-,-1-) (-3-,-1+) -5 (-3+,-2+) (-3+,-2-)(-3-,-2+) (-3-,-2-) -6 (-3-,-3+) From the table we observe that the highest value of ML is 6. This microstate must be generated by a multiplet with L=6, that is an I term. Now note that for this value of ML there is only one possible value of MS so that this must be a singlet level: 1I. Having identified this term, we then remove all microstates which arise from it in order to identify the remaining energy levels (ie. we delete any one 1 microstate for each possible value of ML and MS arising from a I state). After this has been done, the remaining microstates are as shown in the table below: MS +1 0 -1 M L 6 5 (3+,2+) (3+,2-) (3-,2-) 4 (3+,1+) (3+,1-)(2+,2-) (3-,1-) 3 (3+,0+)(2+,1+) (3+,0-)(2+,1-) (3-,0-)(2-,1-) (1+,2-) 2 (3+,-1+)(2+,0+) (3+,-1-)(2+,0-) (3-,-1-)(2-,0-) (1+,1-)(0+,2-) 1 (3+,-2+)(2+,-1+) (3+,-2-)(2+,-1+) (3-,-2-)(2-,-1-) (1+,0+) (1+,0-)(0+,1-) (1-,0-) (-1+,2-) 0 (3+,-3+)(2+,-2+) (3+,-3-)(2+,-2-) (3-,-3-)(2-,-2-) (1+,-1+) (1+,-1-)(0+,0-) (1-,-1-) (-1+,1-)(-2+,2-) -1 (-3+,2+)(-2+,1+) (-3+,2-)(-2+,1-) (-3-,2-)(-2-,1-) (-1+,0+) (-1+,0-)(0+,-1-) (-1-,0-) (1+,-2-) -2 (-3+,1+)(-2+,0+) (-3+,1-)(-2+,0-) (-3-,1-)(-2-,0-) (-1+,-1-)(0+,-2-) -3 (-3+,0+)(-2+,-1+) (-3+,0-)(-2+,-1-) (-3-,0-)(-2-,-1-) (-1+,-2-) -4 (-3+,-1+) (-3+,-1-)(-2+,-2-) (-3-,-1-) -5 (-3+,-2+) (-3+,-2-) (-3-,-2-) -6 Now note that the highest remaining ML value is 5 and that MS can take on all three possible values - these microstates must arise from a 3H term. Proceeding as before we would delete these microstates from the table and then look for the next highest value of ML. We would continue this process until all the microstates in the table had been deleted. For the f2 ion, the complete set of levels arising from electron-electron repulsion are 1I, 3H, 1G, 3F, 1D, 3P and 1S (Figure 2). 2. Spin-Orbit Coupling Spin and orbital angular momenta can interact (couple) with one another and states which have different values for the combined angular momentum will differ in energy. Ligand electrical fields can restrict the motion of electrons about the nucleus thus quenching the orbital angular momentum and greatly reducing the magnitude of spin-orbit coupling. This is the case for the transition metals. In the lanthanide series however, the 4f orbitals are screened from the ligand electrical field by the filled 5s and 5p shells and spin-orbit coupling is not quenched. We define a new quantum number J = L+S which represents the total angular momentum. Since this is a vector addition, J can take on values from L+S to L-S. So for the 3H multiplet of an f2 ion, J = 4,5 or 6. The energy of states with different values of J will differ, so we append the value of J to the lower right of the term symbol to indicate a group of states with the same energy (in the absence of crystal field splitting or an external magnetic field). The ordering of the energy levels can be determined from theory but the calculations are extremely complex. Fortunately, a set of rules are available which predict the ground state for the various lanthanide ions.
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