Download 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.
determining the electronic structure of a f2 ion.
We turn now to the problem of
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 electronelectron 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)
and S=1 (MS = 1,0,-1) is a 3F 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
(3+,3-)
6
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-)
(-1+,-1-)(0+,-2-)
(1+,-3-)
(-3-,1-)(-2-,0-)
-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-)
(-3-,-3+)
-6
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
microstate for each possible value of ML and MS arising from a 1I state). After this
has been done, the remaining microstates are as shown in the table below:
MS
+1
0
-1
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-)
M
L
6
(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. These are known as
Hund's rules:
1.
S is equal to its maximum value (consistent with the Pauli exclusion
principle).
2.
L is equal to its maximum value (consistent with rule 1).
3.
For a less than half-filled shell, J = Jmin = L-S
For a more than half-filled shell, J = Jmax = L+S
Thus for an f2 ion, electron repulsion gives the terms 1I, 3H, 1G, 3F, 1D, 3P and 1S.
Applying rule #1 gives us a choice of the three triplet states, 3H, 3F and 3P.
Applying rule #2 tells us that the 3H term will give rise to the lowest energy level.
From spin-orbit coupling, we get the J levels 3H4, 3H5 and 3H6. Then applying rule
#3 for a less than half-filled shell tells us that the ground state is the 3H4 level.
It should also be pointed out that J can take on values from MJ = J to -J and
the total degeneracy of each J level is 2J+1. These energy levels are referred to as
the free ion energy states because they do not reflect interactions with the ligand
environment.
3.
Crystal Field Splitting
When a lanthanide ion is placed in a crystal field, the J levels do split apart
in energy although the total splitting is generally much less than spin-orbit
coupling and often smaller than kT at room temperature (ca. 200 cm-1). Just how
the levels split depends upon the value of L and the point symmetry of the crystal
field at the metal ion. For example, a D state in an octahedral environment would
split into two energy levels analogous to the T2g and Eg sets for a single d electron.
Low symmetry fields (which are the case in most compounds) completely
lift the degeneracy of an even electron ion but, for reasons we will not go into here,
each level remains two-fold degenerate in an odd electron ion. Such levels are
referred to as Kramer's doublets and the ion is known as a Kramer's ion. As an
example, the splitting of an Yb3+ ion (4f13) in a low symmetry crystal field is
shown in Figure 3.
Figure 1: Splitting of Electronic Levels for d2 ions
1S
1A
1A
1
1E
1
3A
1E
1G
d2
{
1T
1D
3F
Free
Ion
e2/r
(RS)
{
}
t 22
2
1A
1
3T
{
}
t 2e
1
1T
1T
3P
2
}
e2
1T
3T
1
3T
1E
1T
1
2
1
2
2
3A
2
3T
2
3T
1
1A
1
1E
1T
2
3T
1
Weak
field
Strong Infinitely
field
strong
field
Crystal Field
Figure 2: Splitting of Electronic Levels for f2 ions
1S
3P
0
2
1
0
1I
6
1D
2
1G
4
f2
3F
3H
4
3
2
6
5
4
e2/r
(RS)
J
Spin-orbit
coupling
External
Field
Figure 3: Splitting of Yb3+ in a Low Symmetry Ligand Field
Energy (cm-1)
(2)
+80
+62
(2)
(2)
-67
-75
(2)
2F
7/2
(8)
Free Ion
kT at room temperature is 207 cm-1 so
all levels are populated at normal temperatures