Section 5-7
Section 5-7 Electron Angular Momentum in Atoms
low-energy end of the list (5-58) and working up in energy. In addition, when filling a set of degenerate levels like the five 3d levels, one half-fills all the levels with electrons of parallel spin before filling any of them. This prescription enables one to guess the electronic configuration of any atom, once its atomic number is known, unless it happens to put us into a region of ambiguity, where different levels have almost the same energy. (Electronic configurations for such atoms are deduced from experimentally determined chemical, spectral, and physical properties.) The configuration for carbon (atomic number 6) would be 1s22s22p2, with the understanding that p electrons occupy different p orbitals and have parallel spins. (Recall that we expect the most stable of all the states arising from the configuration 1s22s22p2 to be the one of highest multiplicity.
The 2p electrons can produce either a singlet or a triplet state just as could the two electrons in the 1s2s configuration of helium. The triplet should be the ground state and this corresponds to parallel spins, which requires different p orbitals by the exclusion principle.)
It is important to realize that the orbital ordering (5-58) used in the aufbau process is not fixed, but depends on the atomic number Z. The ordering in (5-58) cannot be blindly followed in all cases. For instance, the ordering shows that 5s fills before 4d.
It is true that element 38, strontium, has a · · · 4p65s24d0 configuration. But a later element, palladium, number 46, has · · · 4p64d105s0 as its ground state configuration.
The effect of adding more protons and electrons has been to depress the 4d level more than the 5s level.
5-7 Electron Angular Momentum in Atoms
Most of our attention thus far has been with wavefunction symmetry and energy.
However, understanding atomic spectroscopy or interatomic interactions (in reactions or scattering) requires close attention to angular momentum due to electronic orbital motion and “spin.” In this section we will see what possibilities exist for the total electronic angular momenta of atoms and how these various states are distinguished symbolically.
We encountered earlier (Section 4-5) the notion that the total angular momentum for a classical system is the vector sum of the angular momenta of its parts. If the system interacts with a z-directed field, the total angular momentum vector precesses about the z axis, so the z component continues to be conserved and continues to be equal to the sum of z components of the system’s parts. Since quantum hydrogenlike systems obey angular momentum relations analogous to a precessing classical system, it is this z-axis behavior that we focus on as we seek to construct the nature of the total angular momentum from the orbital and spin parts we already understand.
Because it is the total angular momentum that is conserved in a multicomponent classical system, it is the total angular momentum that obeys the quantum rules we have previously described for separate spin and orbital components. If we consider a oneelectron system, the combined spin-orbital angular momentum can be associated with a quantum number symbolized by j (analogous to s and l). Then we can immediately say that the allowed z components of total angular momentum are, in a.u., mj = ±j,
√
±(j − 1), . . . and that the length of the vector is j (j + 1) a.u.
The implication of accepting total angular momentum as the fundamental quantized quantity is that the spin and orbital angular momenta do not individually obey the
Chapter 5 Many-Electron Atoms quantum rules we have so far applied to them—s and l are not “good” quantum numbers.
However, for atoms of low atomic number they are in fact quite good, especially for low-energy states, and we can continue to refer to the s and l quantum numbers in such cases with some confidence. (Classically, this corresponds to cases where there is little transfer of angular momentum between modes.)
5-7.A Combined Spin-Orbital Angular Momentum forOne-Electron Ions
The key to understanding the following discussion is to remember that a quantum number l, s, or j really tells us three things: 1. It equals the maximum value of ml, ms, or mj . If l = 2, the maximum allowed value of ml is 2, and the maximum z component of orbital angular momentum is 2 a.u.
2. It allows us to know the length of the related angular momentum vector, l, s, or j,
√
in a.u. For j , this is given by j (j + 1). If j = 2, the length of the total angular
√
momentum vector j is 6 a.u.
3. It allows us to know the degeneracy, g, of the energy level due to states having this angular momentum. For s, this is 2s + 1. If s = 1/2, gs = 2. The corresponding l degeneracies produce the s, p, d, f degeneracies of 1, 3, 5, 7.
Using the hydrogen atom as our example, let us consider what the total electronic angular momentum is in the ground (1s) state. For an s AO, l = 0, and so there is no orbital angular momentum. This means that the total angular momentum is the same as the spin angular momentum, so j = s = 1/2, mj = ±1/2. The diagram for the vector j, then, looks just like that for s (Fig. 5-3).
Figure 5-6 (a) Maximum z components of orbital and spin angular momenta for a p electron leading to a total z component of 3/2. (b) The four states corresponding to the j = 3/2 vector assuming its possible z intercepts (3/2, 1/2, −1/2, −3/2).
Section 5-7 Electron Angular Momentum in AtomsFigure 5-7 (a) j = 1/2, resulting when l is oriented with its maximum z intercept and s is oriented in its other orientation (other than as in Fig. 5-6). (b) The two states corresponding to the J = 1/2 vector assuming its possible z intercepts (1/2, −1/2).
More interesting is an excited state, say 2p. Now l = 1 and s = 1/2. From l = 1 we can say that the maximum orbital z component of angular momentum is 1 a.u. s = 1/2 tells us that there is an additional maximum spin z component of 1/2. The maximum sum, then, is 3/2 for the z component of j. But, if this is the maximum mj , then j
√
itself must be 3/2 and the length of j must be (3/2)(5/2) = 1.94 a.u. There must be 2j + 1 = 4 allowed orientations of j, with z intercepts at 3/2, 1/2, −1/2, −3/2 in a.u.
(Fig. 5-6).
We are not yet finished with the 2p possibilities. The total angular momentum is the sum of its orbital and spin parts, and we have so far found the way they combine to give the maximum z component. But this is not the only way they can combine. It is possible to have ml = 1 and ms = −1/2. Then the maximum mj = 1/2, so j = 1/2,
√
giving us a vector j of length (1/2)(3/2) a.u. and two orientations (Fig. 5-7).
So far we have identified six states, four with j = 3/2 and two with j = 1/2. This is all we should expect since we have three 2p AOs and two spins, giving a total of six combinations. It seems, though, that we could generate some more states by now letting ml = 0 or −1 and combining these with ms = ±1/2. However, these possibilities are already implicitly accounted for in the multiplicity of states we recognize to be associated with the j = 3/2, 1/2 cases already found. This illustrates the general approach to be taken when combining two vectors: Orient the larger vector to give maximum z projection, and combine this projection with each of the allowed z components of the smaller vector. This gives all of the possible mj (max) values, hence all of the j values. In other words, it gives us all of the allowed vectors, j, each oriented with maximum z component, and it remains only to recognize that these can have certain other orientations corresponding to z intercepts of mj − 1, mj − 2, . . . , −mj .
States can be labeled to reflect all of the angular momentum parts they possess. The main symbol is simply s, p, d, f, g, etc. depending on the l value as usual. A superscript at left gives spin multiplicity (2s + 1) for the states. A subscript at right tells the j quantum number for the states. If an individual member of the group of states having the same j value is to be cited, it is identified by placing its mj value at upper right.
Chapter 5 Many-Electron Atoms Thus, all six of the states discussed above can be referred to as 2p states. The two groups having different total angular momentum are distinguished as 2p3/2 and 2p1/2.
−1/2
One of the four states in the former group is the 2p state.
3/2
m
The general form of the symbol is 2s+1l j . Such symbols are normally called term
j
symbols, and the collection of states they refer to is called a term (except when an individual state is denoted by inclusion of the mj value).
The reason for distinguishing between the 2p3/2 and 2p1/2 terms is that they occur at slightly different energies. This results from the different energies of interaction between the magnetic moments due to spin and orbital motions. For instance, if l and s are coupled so as to give the maximum j , their associated magnetic moments are oriented like two bar magnets side by side with north poles adjacent. This is a higherenergy arrangement than the other extreme, where l and s couple to give minimum j , acting as a pair of parallel bar magnets with the north pole of each next to the south pole of the other. So 2p1/2 should be lower in energy than 2p3/2 for hydrogen.
EXAMPLE 5-5 For a hydrogen atom having n = 3, l = 2, what are the possible j values, and how many states are possible? Indicate the lengths of the j vectors in a.u. What term symbols apply?
SOLUTION If l = 2 (d states), there are five AOs and two possible spins, so we expect a total of ten possible states. The maximum possible values of the z-component of angular moment for orbital and spin respectively are 2 and 1/2. So the maximum value is 5/2 giving a vector length
√
of 35/4 a.u. and six possible z projections, hence six states. The term symbol is 2d5/2. The remaining possible j value is 2 − 1/2 = 3/2, accounting for four more states and giving a vector
√
length of 15/4 a.u. and a term symbol of 2d3/2.
5-7.B Spin-Orbital Angular Momentum for Many-Electron Atoms
Much of what we have seen for one-electron ions continues to hold for many-electron atoms or ions. All the symbolism is the same, except that capital letters replace lowercase: The quantum numbers are L, S, and J , and the main symbol becomes S, P, D, F, G, etc. There is a total orbital angular momentum vector L with quantum number
√
L that equals the maximum value of ML. The length of L is L(L + 1) a.u., and it has 2L + 1 orientations. Vectors S and J behave analogously. When constructing the vectors J, we continue to place the larger of L and S to give maximum z intercept, and add to this the possible z intercepts of the smaller vector. The situation, then, is just as before except that we need to figure out the possible values for ML and MS by combining the allowed values of ml(1), ml(2), . . . and ms(1), ms(2), . . . .8
8This procedure of first combining individual orbital contributions to find L and spin contributions to find S and then combining these to get J is referred to as “L–S coupling,” or “Russell–Saunders coupling.” The other extreme is to first combine l and s for the first electron to give j(1), l and s for the second electron to give j(2), . . .
and then combine these individual-electron j’s to give J. This is more appropriate for atoms having high atomic number (in which electrons move at relativistic speeds in the vicinity of the nucleus), and is referred to as “j –j coupling.” We will not describe j –j coupling in this text. The reader should consult Herzberg [6] for a fuller treatment.
Section 5-7 Electron Angular Momentum in Atoms
For example, if we have found that ML(max) = 2 (which means L = 2) and Ms(max) = 1 (which means S = 1), we have that MJ (max) can be 2 + 1, 2 + 0, and 2 + (−1), or 3, 2, and 1. This means that the possible values of J are 3, 2, 1, giving three different J vectors. Since L = 2 and S = 1, the term symbols for these three J cases are 3D3, 3D2, and 3D1. Notice that the multiplicities of these three terms—7, 5, and 3, respectively, obtained from 2J + 1—total 15 states, which is just what we should expect for the 3D symbol (spin multiplicity of 3, orbital multiplicity of 5). The 15 triplet-D states are found in three closely spaced levels, differing in energy because of different spin-orbital magnetic interactions.
The problem remains, how do we find the ML and Ms values that allow us to construct term symbols? There are two situations to distinguish in this context, and a different approach is taken for each.
1.
Nonequivalent Electrons. The first situation is exemplified by carbon in its 1s22s22p3p configuration. It is not difficult to show that the electrons in the 1s and 2s AOs contribute no net angular momentum and can be ignored: The spins of paired
electrons are opposed, hence cancel, and the s-type AOs have no angular momentum, hence cannot contribute. However, even p, d, etc. sets of AOs cannot contribute if they
are filled because then any orbital momentum having z intercept ml is canceled by one with −ml. The important result is that filled subshells do not contribute to orbital or
spin angular momentum. The remaining 2p and 3p electrons occupy different sets of AOs, hence are called nonequivalent electrons.
Since these electrons are never in the same AO, they are not restricted to have opposite spins at any time—their AO and spin assignments are independent. There are three AO choices (p1, p0, p−1) and two spin choices—six possibilities—for each electron, hence 36 unique possibilities. We should expect, therefore, 36 states to be included in our final set of terms.
We first find the possible L values. ml for each electron is 1, 0, or −1. We orient the larger of the l vectors to give the maximum ml(1) = +1 and orient the second l in all possible ways, giving ml(2) = +1, 0, −1. (Since the vectors have equal length in this case there is no “larger–smaller” choice to make.) The net ML values are +2, +1, 0, and this tells us that the possible L values are 2, 1, 0.
Treating ms values similarly gives Ms = 1, 0, so S = 1, 0.
Thus, we have three L vectors and two S vectors. We now combine every one of the L, S pairs. In each case, we again take the longer in its position of greatest z overlap and combine it with the shorter in all of its orientations. This gives the J values shown in Table 5-2. The appropriate term symbols follow from L, S, and J in each case.
Thus, our term symbols are 3D3, 3D2, 3D1, 1D2, 3P2, 3P1, 3P0,1P1, 3S1 and 1S0 for a total of 7 + 5 + 3 + 5 + 5 + 3 + 1 + 3 + 3 + 1 = 36 states.
In the absence of external fields, these 36 states occur in 10 energy levels, one for each term. These lie at different energies for several reasons. We have already seen, in our discussion of 1s2s helium states, that different spin multiplicities are associated with different symmetries of the spin wavefunction, meaning that the space part of the wavefunctions also differ in symmetry. This has a significant effect on energy, so 1S and 3S, for example, have rather different energies. Different L values amount to different occupancies of AOs, which also has an effect on the spatial wavefunctions, so 3P and 3S have different energies. Finally, we have already seen that different J values correspond to different relative orientations of orbital and spin angular momentum Chapter 5 Many-Electron AtomsTABLE 5-2 L and S Values for Two Nonequivalent Electrons and Resulting J Values and Term Symbols
L
S
J
Term
3
3D3
2
1
3
2
D2 1
3D1
2
0
2
1D2
2
3P2
1
1
3
1
P1 0
3P0
1
0
1
1P1
0
1
1
3S1
0
0
0
1S0
vectors, hence of magnetic moments. For light atoms, this is a relatively small effect, so 3P2, 3P1, and 3P0 have only slightly different energies. The resulting energies for states of carbon in 1s22s22p2, 1s22s22p3p, and 1s22s22p4p configurations are shown in Fig. 5-8. Only the major term-energy differences are distinguishable on the scale of the figure. The line for 3D is really three very closely spaced lines corresponding to 3D3, 3D2, and 3D1 terms.
2.
Zeeman Effect. It was pointed out in Section 4-6 that the orbital energies of a hydrogen atom corresponding to the same n but different ml undergo splitting when a magnetic field is imposed. Now we have seen that spin angular momentum is also present. Therefore, a proper treatment of the Zeeman effect requires that we focus on total angular momentum, not just the orbital component. Since there are 2J + 1 states with different MJ values in a given term, we expect each term to split into 2J + 1 evenly separated energies in the presence of a magnetic field, and this is indeed what is seen to happen (through its effects on lines in the spectrum). For example, a 3P2 term splits into five closely spaced energies, corresponding to MJ = 2, 1, 0, −1, −2, and a1P1 term splits into three energies.
A surprising feature of this phenomenon is that the amount of splitting is not the same for all terms, despite the fact that adjacent members of any term always differ by ±1 unit of angular momentum on the z axis. For instance, the spacing between adjacent members of the 3P2 term mentioned above is 1.50 times greater than that in the 1P1 term. It was recognized that terms wherein S = 0, so that J is entirely due to orbital angular momentum (J = L), undergo “normal splitting”—i.e., equal to what classical physics would predict for the amount of angular momentum and charge involved. On the other hand, terms wherein J is entirely due to spin (L = 0, so J = S) undergo twice the splitting predicted from classical considerations. [This extra factor of two (actually 2.0023) was without theoretical explanation until Dirac’s relativistic treatment of quantum mechanics.]
Terms wherein J contains contributions from both L and S have Zeeman splittings other than one or two times the normal value, depending on the details of the way L and
Section 5-7 Electron Angular Momentum in AtomsFigure 5-8
Energy levels for carbon atom terms resulting from configurations 1s22s22p2, 1s22s22p3p, and 1s22s22p4p.
S are combined. The extent to which a term member’s energy is shifted by a magnetic field of strength B is E = gβeMj B
(5-59)
where βe is the Bohr magneton (Appendix 10) and g is the Land´e g factor, which accounts for the different effects of L and S on magnetic moment that we have been discussing: g = 1 + J (J + 1) + S(S + 1) − L(L + 1)
(5-60)
2J (J + 1)
It is not difficult to see that this formula equals one when S = 0, J = L, and equals two when L = 0, J = S. For the 3P2 term, S = 1, L = 1, J = 2, and g equals 1.5, indicating that, in this state, half of the z-component of angular momentum is due to orbital Chapter 5 Many-Electron Atoms motion, and half is due to spin (which is double-weighted in its effect on magnetic moment).
3.
Equivalent Electrons.
Observe that the energy-level diagram for carbon (Fig. 5-8) shows the 10 expected terms for the excited 2p3p and 2p4p configurations, but not for the ground 2p2 configuration. There are no new terms for the latter case, but some of the terms present for 2p3p or 2p4p are gone, namely 3D, 3S, and 1P. The remaining terms account for 15 states. Evidently 21 states that are possible for a pair of nonequivalent p electrons are not allowed for a pair of equivalent electrons in a p2 configuration. We will see that some of the states that are different for nonequivalent electrons become one and the same for equivalent electrons and must be excluded.
Others are excluded by the Pauli exclusion principle because they would require two electrons to be in the same AO with the same spin.
We now demonstrate the method for discovering the terms that exist for equivalent electrons. This is more difficult than for nonequivalent electrons, even though there are fewer terms. We first list all the orbital-spin combinations (called microstates), strike out those that are redundant or that violate the Pauli exclusion principle, and then infer from the remaining microstates what terms exist.
Taking the p2 case for illustration, we begin with the 36 microstates listed in Table 5-3.
Some of these microstates are equivalent to others.
For instance, 2p1(1)2p1(2)α(1)β(2) is not a different state from 2p1(1)2p1(2)β(1)α(2). These both correspond to a pair of electrons in the same pair of spin orbitals, 2p1 and 2p1. Since electrons are indistinguishable, we cannot expect wavefunctions differing only in the order of electron labels to correspond to different physical states. [The single state that does exist would be accurately represented by 2p1(1)2p1(2)(α(1)β(2) − β(1)α(2)), which is a linear combination of the microstates. But we do not need to go to this level of detail when finding terms. We only need to recognize that there is but one state here and omit one of the microstates as superfluous.] Accordingly, we strike out rows 3, 19, and 35 from Table 5-3, labeling them “R” for redundant.
Another way to recognize this equivalence is to observe that the microstates deemed redundant differ only by an interchange of a pair of electrons. This reveals that the set of four microstates with 2p1(1)2p0(2) is equivalent to the set with 2p0(1)2p1(2).
Therefore, we can strike out rows 13–16. A similar argument removes rows 25–32.
Already we have removed 15 microstates.
Next we look for violations of the Pauli exclusion principle. This leads us to strike out rows 1, 4, 17, 20, 33, and 36, labeling them “P” for Pauli. Our remaining microstates number 15 and are reassembled in Table 5-4, along with values of the quantum numbers for z components of the relevant angular momentum vectors for individual electrons as well as for their sum.
At this stage of the argument, the final column of Table 5-4 (the term symbols) is not yet known. We are about to fill out this column by making use of a simple rule that is based on the diagrammatic device described earlier—placing the larger vector so that it has the maximum z extension and then placing the shorter vector in all its allowed orientations. It is not difficult to see that the maximum resultant z component (MJ ) can be achieved in one and only one way, namely when both vectors give their maximum z projection. This means that the maximum-MJ -member of a given set of states in the same term should be recognized as corresponding to one and only one microstate, because there is only one way to achieve this orientation. So we look for this maximum Section 5-7 Electron Angular Momentum in AtomsTABLE 5-3 Unrestricted List of Space–Spin Combinations for a Pair of Electrons (Same Subshell). R = “Redundant,” P = “Pauli” Electron number
Row number
1
2
1
2
Comment
1
p1
p1
α
α
P
2
p1
p1
α
β
3
p1
p1
β
α
R
4
p1
p1
β
β
P
5
p1
p0
α
α
6
p1
p0
α
β
7
p1
p0
β
α
8
p1
p0
β
β
9
p1
p−1
α
α
10
p1
p−1
α
β
11
p1
p−1
β
α
12
p1
p−1
β
β
13
p0
p1
α
α
R
14
p0
p1
α
β
R
15
p0
p1
β
α
R
16
p0
p1
β
β
R
17
p0
p0
α
α
P
18
p0
p0
α
β
19
p0
p0
β
α
R
20
p0
p0
β
β
P
21
p0
p−1
α
α
22
p0
p−1
α
β
23
p0
p−1
β
α
24
p0
p−1
β
β
25
p−1
p1
α
α
R
26
p−1
p1
α
β
R
27
p−1
p1
β
α
R
28
p−1
p1
β
β
R
29
p−1
p0
α
α
R
30
p−1
p0
α
β
R
31
p−1
p0
β
α
R
32
p−1
p0
β
β
R
33
p−1
p−1
α
α
P
34
p−1
p−1
α
β
35
p−1
p−1
β
α
R
36
p−1
p−1
β
β
P
Chapter 5 Many-Electron AtomsTABLE 5-4 Allowed Space-Spin Combinations and M Quantum Numbers for a Pair of p Electrons (Same Subshell)
Microstate
ml(1) ml(2) ms(1) ms(2)
ML
MS
MJ State term
p
2
1p1αβ
1
1
1/2
−1/2
2
0
2
1D2
p
2
1p0αα
1
0
1/2
1/2
1
1
2
3P2
p
1
1p0αβ
1
0
1/2
−1/2
1
0
1
(1D2 )
p
1
1p0βα
1
0
−1/2
1/2
1
0
1
(3P2 )
p
0
1p0ββ
1
0
−1/2
−1/2
1
−1
0
(3P2 )
p
1
1p−1αα
1
−1
1/2
1/2
0
1
1
(3P1 )
p
0
1p−1αβ
1
−1
1/2
−1/2
0
0
0
(1D2 )
p
0
1p−1βα
1
−1
−1/2
1/2
0
0
0
(3P1 )
p
−1
1p−1ββ
1
−1
−1/2
−1/2
0
−1
−1
(3P2 )
p
0
0p0αβ
0
0
1/2
−1/2
0
0
0
1S0
p
0
0p−1αα
0
−1
1/2
1/2
−1
1
0
(3P0 )
p
−1
0p−1αβ
0
−1
1/2
−1/2
−1
0
−1
(1D2 )
p
−1
0p−1βα
0
−1
−1/2
1/2
−1
0
−1
(3P1 )
p
−2
0p−1ββ
0
−1
−1/2
−1/2
−1
−1
−2
(3P2 )
p
−
−
2
1p−1αβ
−1
−1
1/2
−1/2
−2
0
−2
(1D2 )
MJ and, from its microstate, get the L and S values that go with it. That gives us the information we need to establish the term symbol.
We start, then, by seeking the maximum MJ value in Table 5-4. This is MJ = 2, and it occurs twice (in the first two rows). The first of these goes with ML = 2, MS = 0.
Since these result when L and S are giving their maximum z component, we conclude that L = 2, S = 0. This, then, is a member of the 1D2 term. (It is the 1D2 member of
2
that term, since MJ = 2.) We label this row 1D2 and proceed to select microstates that
2
can account for the other four members of this term. Our choice is controlled by the requirements that (1) the MJ values for the other members must be 1, 0, −1, −2, and (2) we cannot have an |Ms| value larger than zero or an |ML| value larger than 2. (That would be impossible for states resulting from vectors having L = 2 and S = 0.) Our selections are indicated in Table 5-4, with parentheses to indicate that these assignments follow from recognition of the leading member 1D2. (All are symbolized as 1D
2
2.)
There is some arbitrariness in selecting the “inner” members, for which |MJ |
low-energy end of the list (5-58) and working up in energy. In addition, when filling a set of degenerate levels like the five 3d levels, one half-fills all the levels with electrons of parallel spin before filling any of them. This prescription enables one to guess the electronic configuration of any atom, once its atomic number is known, unless it happens to put us into a region of ambiguity, where different levels have almost the same energy. (Electronic configurations for such atoms are deduced from experimentally determined chemical, spectral, and physical properties.) The configuration for carbon (atomic number 6) would be 1s22s22p2, with the understanding that p electrons occupy different p orbitals and have parallel spins. (Recall that we expect the most stable of all the states arising from the configuration 1s22s22p2 to be the one of highest multiplicity.
The 2p electrons can produce either a singlet or a triplet state just as could the two electrons in the 1s2s configuration of helium. The triplet should be the ground state and this corresponds to parallel spins, which requires different p orbitals by the exclusion principle.)
It is important to realize that the orbital ordering (5-58) used in the aufbau process is not fixed, but depends on the atomic number Z. The ordering in (5-58) cannot be blindly followed in all cases. For instance, the ordering shows that 5s fills before 4d.
It is true that element 38, strontium, has a · · · 4p65s24d0 configuration. But a later element, palladium, number 46, has · · · 4p64d105s0 as its ground state configuration.
The effect of adding more protons and electrons has been to depress the 4d level more than the 5s level.
5-7 Electron Angular Momentum in Atoms
Most of our attention thus far has been with wavefunction symmetry and energy.
However, understanding atomic spectroscopy or interatomic interactions (in reactions or scattering) requires close attention to angular momentum due to electronic orbital motion and “spin.” In this section we will see what possibilities exist for the total electronic angular momenta of atoms and how these various states are distinguished symbolically.
We encountered earlier (Section 4-5) the notion that the total angular momentum for a classical system is the vector sum of the angular momenta of its parts. If the system interacts with a z-directed field, the total angular momentum vector precesses about the z axis, so the z component continues to be conserved and continues to be equal to the sum of z components of the system’s parts. Since quantum hydrogenlike systems obey angular momentum relations analogous to a precessing classical system, it is this z-axis behavior that we focus on as we seek to construct the nature of the total angular momentum from the orbital and spin parts we already understand.
Because it is the total angular momentum that is conserved in a multicomponent classical system, it is the total angular momentum that obeys the quantum rules we have previously described for separate spin and orbital components. If we consider a oneelectron system, the combined spin-orbital angular momentum can be associated with a quantum number symbolized by j (analogous to s and l). Then we can immediately say that the allowed z components of total angular momentum are, in a.u., mj = ±j,
√
±(j − 1), . . . and that the length of the vector is j (j + 1) a.u.
The implication of accepting total angular momentum as the fundamental quantized quantity is that the spin and orbital angular momenta do not individually obey the
Chapter 5 Many-Electron Atoms quantum rules we have so far applied to them—s and l are not “good” quantum numbers.
However, for atoms of low atomic number they are in fact quite good, especially for low-energy states, and we can continue to refer to the s and l quantum numbers in such cases with some confidence. (Classically, this corresponds to cases where there is little transfer of angular momentum between modes.)
5-7.A Combined Spin-Orbital Angular Momentum forOne-Electron Ions
The key to understanding the following discussion is to remember that a quantum number l, s, or j really tells us three things: 1. It equals the maximum value of ml, ms, or mj . If l = 2, the maximum allowed value of ml is 2, and the maximum z component of orbital angular momentum is 2 a.u.
2. It allows us to know the length of the related angular momentum vector, l, s, or j,
√
in a.u. For j , this is given by j (j + 1). If j = 2, the length of the total angular
√
momentum vector j is 6 a.u.
3. It allows us to know the degeneracy, g, of the energy level due to states having this angular momentum. For s, this is 2s + 1. If s = 1/2, gs = 2. The corresponding l degeneracies produce the s, p, d, f degeneracies of 1, 3, 5, 7.
Using the hydrogen atom as our example, let us consider what the total electronic angular momentum is in the ground (1s) state. For an s AO, l = 0, and so there is no orbital angular momentum. This means that the total angular momentum is the same as the spin angular momentum, so j = s = 1/2, mj = ±1/2. The diagram for the vector j, then, looks just like that for s (Fig. 5-3).
Figure 5-6 (a) Maximum z components of orbital and spin angular momenta for a p electron leading to a total z component of 3/2. (b) The four states corresponding to the j = 3/2 vector assuming its possible z intercepts (3/2, 1/2, −1/2, −3/2).
Section 5-7 Electron Angular Momentum in AtomsFigure 5-7 (a) j = 1/2, resulting when l is oriented with its maximum z intercept and s is oriented in its other orientation (other than as in Fig. 5-6). (b) The two states corresponding to the J = 1/2 vector assuming its possible z intercepts (1/2, −1/2).
More interesting is an excited state, say 2p. Now l = 1 and s = 1/2. From l = 1 we can say that the maximum orbital z component of angular momentum is 1 a.u. s = 1/2 tells us that there is an additional maximum spin z component of 1/2. The maximum sum, then, is 3/2 for the z component of j. But, if this is the maximum mj , then j
√
itself must be 3/2 and the length of j must be (3/2)(5/2) = 1.94 a.u. There must be 2j + 1 = 4 allowed orientations of j, with z intercepts at 3/2, 1/2, −1/2, −3/2 in a.u.
(Fig. 5-6).
We are not yet finished with the 2p possibilities. The total angular momentum is the sum of its orbital and spin parts, and we have so far found the way they combine to give the maximum z component. But this is not the only way they can combine. It is possible to have ml = 1 and ms = −1/2. Then the maximum mj = 1/2, so j = 1/2,
√
giving us a vector j of length (1/2)(3/2) a.u. and two orientations (Fig. 5-7).
So far we have identified six states, four with j = 3/2 and two with j = 1/2. This is all we should expect since we have three 2p AOs and two spins, giving a total of six combinations. It seems, though, that we could generate some more states by now letting ml = 0 or −1 and combining these with ms = ±1/2. However, these possibilities are already implicitly accounted for in the multiplicity of states we recognize to be associated with the j = 3/2, 1/2 cases already found. This illustrates the general approach to be taken when combining two vectors: Orient the larger vector to give maximum z projection, and combine this projection with each of the allowed z components of the smaller vector. This gives all of the possible mj (max) values, hence all of the j values. In other words, it gives us all of the allowed vectors, j, each oriented with maximum z component, and it remains only to recognize that these can have certain other orientations corresponding to z intercepts of mj − 1, mj − 2, . . . , −mj .
States can be labeled to reflect all of the angular momentum parts they possess. The main symbol is simply s, p, d, f, g, etc. depending on the l value as usual. A superscript at left gives spin multiplicity (2s + 1) for the states. A subscript at right tells the j quantum number for the states. If an individual member of the group of states having the same j value is to be cited, it is identified by placing its mj value at upper right.
Chapter 5 Many-Electron Atoms Thus, all six of the states discussed above can be referred to as 2p states. The two groups having different total angular momentum are distinguished as 2p3/2 and 2p1/2.
−1/2
One of the four states in the former group is the 2p state.
3/2
m
The general form of the symbol is 2s+1l j . Such symbols are normally called term
j
symbols, and the collection of states they refer to is called a term (except when an individual state is denoted by inclusion of the mj value).
The reason for distinguishing between the 2p3/2 and 2p1/2 terms is that they occur at slightly different energies. This results from the different energies of interaction between the magnetic moments due to spin and orbital motions. For instance, if l and s are coupled so as to give the maximum j , their associated magnetic moments are oriented like two bar magnets side by side with north poles adjacent. This is a higherenergy arrangement than the other extreme, where l and s couple to give minimum j , acting as a pair of parallel bar magnets with the north pole of each next to the south pole of the other. So 2p1/2 should be lower in energy than 2p3/2 for hydrogen.
EXAMPLE 5-5 For a hydrogen atom having n = 3, l = 2, what are the possible j values, and how many states are possible? Indicate the lengths of the j vectors in a.u. What term symbols apply?
SOLUTION If l = 2 (d states), there are five AOs and two possible spins, so we expect a total of ten possible states. The maximum possible values of the z-component of angular moment for orbital and spin respectively are 2 and 1/2. So the maximum value is 5/2 giving a vector length
√
of 35/4 a.u. and six possible z projections, hence six states. The term symbol is 2d5/2. The remaining possible j value is 2 − 1/2 = 3/2, accounting for four more states and giving a vector
√
length of 15/4 a.u. and a term symbol of 2d3/2.
5-7.B Spin-Orbital Angular Momentum for Many-Electron Atoms
Much of what we have seen for one-electron ions continues to hold for many-electron atoms or ions. All the symbolism is the same, except that capital letters replace lowercase: The quantum numbers are L, S, and J , and the main symbol becomes S, P, D, F, G, etc. There is a total orbital angular momentum vector L with quantum number
√
L that equals the maximum value of ML. The length of L is L(L + 1) a.u., and it has 2L + 1 orientations. Vectors S and J behave analogously. When constructing the vectors J, we continue to place the larger of L and S to give maximum z intercept, and add to this the possible z intercepts of the smaller vector. The situation, then, is just as before except that we need to figure out the possible values for ML and MS by combining the allowed values of ml(1), ml(2), . . . and ms(1), ms(2), . . . .8
8This procedure of first combining individual orbital contributions to find L and spin contributions to find S and then combining these to get J is referred to as “L–S coupling,” or “Russell–Saunders coupling.” The other extreme is to first combine l and s for the first electron to give j(1), l and s for the second electron to give j(2), . . .
and then combine these individual-electron j’s to give J. This is more appropriate for atoms having high atomic number (in which electrons move at relativistic speeds in the vicinity of the nucleus), and is referred to as “j –j coupling.” We will not describe j –j coupling in this text. The reader should consult Herzberg [6] for a fuller treatment.
Section 5-7 Electron Angular Momentum in Atoms
For example, if we have found that ML(max) = 2 (which means L = 2) and Ms(max) = 1 (which means S = 1), we have that MJ (max) can be 2 + 1, 2 + 0, and 2 + (−1), or 3, 2, and 1. This means that the possible values of J are 3, 2, 1, giving three different J vectors. Since L = 2 and S = 1, the term symbols for these three J cases are 3D3, 3D2, and 3D1. Notice that the multiplicities of these three terms—7, 5, and 3, respectively, obtained from 2J + 1—total 15 states, which is just what we should expect for the 3D symbol (spin multiplicity of 3, orbital multiplicity of 5). The 15 triplet-D states are found in three closely spaced levels, differing in energy because of different spin-orbital magnetic interactions.
The problem remains, how do we find the ML and Ms values that allow us to construct term symbols? There are two situations to distinguish in this context, and a different approach is taken for each.
1.
Nonequivalent Electrons. The first situation is exemplified by carbon in its 1s22s22p3p configuration. It is not difficult to show that the electrons in the 1s and 2s AOs contribute no net angular momentum and can be ignored: The spins of paired
electrons are opposed, hence cancel, and the s-type AOs have no angular momentum, hence cannot contribute. However, even p, d, etc. sets of AOs cannot contribute if they
are filled because then any orbital momentum having z intercept ml is canceled by one with −ml. The important result is that filled subshells do not contribute to orbital or
spin angular momentum. The remaining 2p and 3p electrons occupy different sets of AOs, hence are called nonequivalent electrons.
Since these electrons are never in the same AO, they are not restricted to have opposite spins at any time—their AO and spin assignments are independent. There are three AO choices (p1, p0, p−1) and two spin choices—six possibilities—for each electron, hence 36 unique possibilities. We should expect, therefore, 36 states to be included in our final set of terms.
We first find the possible L values. ml for each electron is 1, 0, or −1. We orient the larger of the l vectors to give the maximum ml(1) = +1 and orient the second l in all possible ways, giving ml(2) = +1, 0, −1. (Since the vectors have equal length in this case there is no “larger–smaller” choice to make.) The net ML values are +2, +1, 0, and this tells us that the possible L values are 2, 1, 0.
Treating ms values similarly gives Ms = 1, 0, so S = 1, 0.
Thus, we have three L vectors and two S vectors. We now combine every one of the L, S pairs. In each case, we again take the longer in its position of greatest z overlap and combine it with the shorter in all of its orientations. This gives the J values shown in Table 5-2. The appropriate term symbols follow from L, S, and J in each case.
Thus, our term symbols are 3D3, 3D2, 3D1, 1D2, 3P2, 3P1, 3P0,1P1, 3S1 and 1S0 for a total of 7 + 5 + 3 + 5 + 5 + 3 + 1 + 3 + 3 + 1 = 36 states.
In the absence of external fields, these 36 states occur in 10 energy levels, one for each term. These lie at different energies for several reasons. We have already seen, in our discussion of 1s2s helium states, that different spin multiplicities are associated with different symmetries of the spin wavefunction, meaning that the space part of the wavefunctions also differ in symmetry. This has a significant effect on energy, so 1S and 3S, for example, have rather different energies. Different L values amount to different occupancies of AOs, which also has an effect on the spatial wavefunctions, so 3P and 3S have different energies. Finally, we have already seen that different J values correspond to different relative orientations of orbital and spin angular momentum Chapter 5 Many-Electron AtomsTABLE 5-2 L and S Values for Two Nonequivalent Electrons and Resulting J Values and Term Symbols
L
S
J
Term
3
3D3
2
1
3
2
D2 1
3D1
2
0
2
1D2
2
3P2
1
1
3
1
P1 0
3P0
1
0
1
1P1
0
1
1
3S1
0
0
0
1S0
vectors, hence of magnetic moments. For light atoms, this is a relatively small effect, so 3P2, 3P1, and 3P0 have only slightly different energies. The resulting energies for states of carbon in 1s22s22p2, 1s22s22p3p, and 1s22s22p4p configurations are shown in Fig. 5-8. Only the major term-energy differences are distinguishable on the scale of the figure. The line for 3D is really three very closely spaced lines corresponding to 3D3, 3D2, and 3D1 terms.
2.
Zeeman Effect. It was pointed out in Section 4-6 that the orbital energies of a hydrogen atom corresponding to the same n but different ml undergo splitting when a magnetic field is imposed. Now we have seen that spin angular momentum is also present. Therefore, a proper treatment of the Zeeman effect requires that we focus on total angular momentum, not just the orbital component. Since there are 2J + 1 states with different MJ values in a given term, we expect each term to split into 2J + 1 evenly separated energies in the presence of a magnetic field, and this is indeed what is seen to happen (through its effects on lines in the spectrum). For example, a 3P2 term splits into five closely spaced energies, corresponding to MJ = 2, 1, 0, −1, −2, and a1P1 term splits into three energies.
A surprising feature of this phenomenon is that the amount of splitting is not the same for all terms, despite the fact that adjacent members of any term always differ by ±1 unit of angular momentum on the z axis. For instance, the spacing between adjacent members of the 3P2 term mentioned above is 1.50 times greater than that in the 1P1 term. It was recognized that terms wherein S = 0, so that J is entirely due to orbital angular momentum (J = L), undergo “normal splitting”—i.e., equal to what classical physics would predict for the amount of angular momentum and charge involved. On the other hand, terms wherein J is entirely due to spin (L = 0, so J = S) undergo twice the splitting predicted from classical considerations. [This extra factor of two (actually 2.0023) was without theoretical explanation until Dirac’s relativistic treatment of quantum mechanics.]
Terms wherein J contains contributions from both L and S have Zeeman splittings other than one or two times the normal value, depending on the details of the way L and
Section 5-7 Electron Angular Momentum in AtomsFigure 5-8
Energy levels for carbon atom terms resulting from configurations 1s22s22p2, 1s22s22p3p, and 1s22s22p4p.
S are combined. The extent to which a term member’s energy is shifted by a magnetic field of strength B is E = gβeMj B
(5-59)
where βe is the Bohr magneton (Appendix 10) and g is the Land´e g factor, which accounts for the different effects of L and S on magnetic moment that we have been discussing: g = 1 + J (J + 1) + S(S + 1) − L(L + 1)
(5-60)
2J (J + 1)
It is not difficult to see that this formula equals one when S = 0, J = L, and equals two when L = 0, J = S. For the 3P2 term, S = 1, L = 1, J = 2, and g equals 1.5, indicating that, in this state, half of the z-component of angular momentum is due to orbital Chapter 5 Many-Electron Atoms motion, and half is due to spin (which is double-weighted in its effect on magnetic moment).
3.
Equivalent Electrons.
Observe that the energy-level diagram for carbon (Fig. 5-8) shows the 10 expected terms for the excited 2p3p and 2p4p configurations, but not for the ground 2p2 configuration. There are no new terms for the latter case, but some of the terms present for 2p3p or 2p4p are gone, namely 3D, 3S, and 1P. The remaining terms account for 15 states. Evidently 21 states that are possible for a pair of nonequivalent p electrons are not allowed for a pair of equivalent electrons in a p2 configuration. We will see that some of the states that are different for nonequivalent electrons become one and the same for equivalent electrons and must be excluded.
Others are excluded by the Pauli exclusion principle because they would require two electrons to be in the same AO with the same spin.
We now demonstrate the method for discovering the terms that exist for equivalent electrons. This is more difficult than for nonequivalent electrons, even though there are fewer terms. We first list all the orbital-spin combinations (called microstates), strike out those that are redundant or that violate the Pauli exclusion principle, and then infer from the remaining microstates what terms exist.
Taking the p2 case for illustration, we begin with the 36 microstates listed in Table 5-3.
Some of these microstates are equivalent to others.
For instance, 2p1(1)2p1(2)α(1)β(2) is not a different state from 2p1(1)2p1(2)β(1)α(2). These both correspond to a pair of electrons in the same pair of spin orbitals, 2p1 and 2p1. Since electrons are indistinguishable, we cannot expect wavefunctions differing only in the order of electron labels to correspond to different physical states. [The single state that does exist would be accurately represented by 2p1(1)2p1(2)(α(1)β(2) − β(1)α(2)), which is a linear combination of the microstates. But we do not need to go to this level of detail when finding terms. We only need to recognize that there is but one state here and omit one of the microstates as superfluous.] Accordingly, we strike out rows 3, 19, and 35 from Table 5-3, labeling them “R” for redundant.
Another way to recognize this equivalence is to observe that the microstates deemed redundant differ only by an interchange of a pair of electrons. This reveals that the set of four microstates with 2p1(1)2p0(2) is equivalent to the set with 2p0(1)2p1(2).
Therefore, we can strike out rows 13–16. A similar argument removes rows 25–32.
Already we have removed 15 microstates.
Next we look for violations of the Pauli exclusion principle. This leads us to strike out rows 1, 4, 17, 20, 33, and 36, labeling them “P” for Pauli. Our remaining microstates number 15 and are reassembled in Table 5-4, along with values of the quantum numbers for z components of the relevant angular momentum vectors for individual electrons as well as for their sum.
At this stage of the argument, the final column of Table 5-4 (the term symbols) is not yet known. We are about to fill out this column by making use of a simple rule that is based on the diagrammatic device described earlier—placing the larger vector so that it has the maximum z extension and then placing the shorter vector in all its allowed orientations. It is not difficult to see that the maximum resultant z component (MJ ) can be achieved in one and only one way, namely when both vectors give their maximum z projection. This means that the maximum-MJ -member of a given set of states in the same term should be recognized as corresponding to one and only one microstate, because there is only one way to achieve this orientation. So we look for this maximum Section 5-7 Electron Angular Momentum in AtomsTABLE 5-3 Unrestricted List of Space–Spin Combinations for a Pair of Electrons (Same Subshell). R = “Redundant,” P = “Pauli” Electron number
Row number
1
2
1
2
Comment
1
p1
p1
α
α
P
2
p1
p1
α
β
3
p1
p1
β
α
R
4
p1
p1
β
β
P
5
p1
p0
α
α
6
p1
p0
α
β
7
p1
p0
β
α
8
p1
p0
β
β
9
p1
p−1
α
α
10
p1
p−1
α
β
11
p1
p−1
β
α
12
p1
p−1
β
β
13
p0
p1
α
α
R
14
p0
p1
α
β
R
15
p0
p1
β
α
R
16
p0
p1
β
β
R
17
p0
p0
α
α
P
18
p0
p0
α
β
19
p0
p0
β
α
R
20
p0
p0
β
β
P
21
p0
p−1
α
α
22
p0
p−1
α
β
23
p0
p−1
β
α
24
p0
p−1
β
β
25
p−1
p1
α
α
R
26
p−1
p1
α
β
R
27
p−1
p1
β
α
R
28
p−1
p1
β
β
R
29
p−1
p0
α
α
R
30
p−1
p0
α
β
R
31
p−1
p0
β
α
R
32
p−1
p0
β
β
R
33
p−1
p−1
α
α
P
34
p−1
p−1
α
β
35
p−1
p−1
β
α
R
36
p−1
p−1
β
β
P
Chapter 5 Many-Electron AtomsTABLE 5-4 Allowed Space-Spin Combinations and M Quantum Numbers for a Pair of p Electrons (Same Subshell)
Microstate
ml(1) ml(2) ms(1) ms(2)
ML
MS
MJ State term
p
2
1p1αβ
1
1
1/2
−1/2
2
0
2
1D2
p
2
1p0αα
1
0
1/2
1/2
1
1
2
3P2
p
1
1p0αβ
1
0
1/2
−1/2
1
0
1
(1D2 )
p
1
1p0βα
1
0
−1/2
1/2
1
0
1
(3P2 )
p
0
1p0ββ
1
0
−1/2
−1/2
1
−1
0
(3P2 )
p
1
1p−1αα
1
−1
1/2
1/2
0
1
1
(3P1 )
p
0
1p−1αβ
1
−1
1/2
−1/2
0
0
0
(1D2 )
p
0
1p−1βα
1
−1
−1/2
1/2
0
0
0
(3P1 )
p
−1
1p−1ββ
1
−1
−1/2
−1/2
0
−1
−1
(3P2 )
p
0
0p0αβ
0
0
1/2
−1/2
0
0
0
1S0
p
0
0p−1αα
0
−1
1/2
1/2
−1
1
0
(3P0 )
p
−1
0p−1αβ
0
−1
1/2
−1/2
−1
0
−1
(1D2 )
p
−1
0p−1βα
0
−1
−1/2
1/2
−1
0
−1
(3P1 )
p
−2
0p−1ββ
0
−1
−1/2
−1/2
−1
−1
−2
(3P2 )
p
−
−
2
1p−1αβ
−1
−1
1/2
−1/2
−2
0
−2
(1D2 )
MJ and, from its microstate, get the L and S values that go with it. That gives us the information we need to establish the term symbol.
We start, then, by seeking the maximum MJ value in Table 5-4. This is MJ = 2, and it occurs twice (in the first two rows). The first of these goes with ML = 2, MS = 0.
Since these result when L and S are giving their maximum z component, we conclude that L = 2, S = 0. This, then, is a member of the 1D2 term. (It is the 1D2 member of
2
that term, since MJ = 2.) We label this row 1D2 and proceed to select microstates that
2
can account for the other four members of this term. Our choice is controlled by the requirements that (1) the MJ values for the other members must be 1, 0, −1, −2, and (2) we cannot have an |Ms| value larger than zero or an |ML| value larger than 2. (That would be impossible for states resulting from vectors having L = 2 and S = 0.) Our selections are indicated in Table 5-4, with parentheses to indicate that these assignments follow from recognition of the leading member 1D2. (All are symbolized as 1D
2
2.)
There is some arbitrariness in selecting the “inner” members, for which |MJ |