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ENERGY LEVELS IN COMPLEXES WITH ALMOST SPHERICAL SYMMETRY
CHAPTER
10
ENERGY LEVELS IN COMPLEXES WITH ALMOST SPHERICAL SYMMETRY T H E R E is no reason for a M.O. in all cases to resemble orbitals adapted to spherical symmetry; this is known from atomic spectroscopy. In the next chapter we shall consider transitions which certainly cannot be described thus. The three ordinary transition groups present a partly filled shell which to some extent is intermediate between the subject of this and the following chapter. On the one hand, the assumption of / = 2 in the electrostatic model or in the expanded radial function model explains almost all the pertinent features of the ligand field bands; on the other hand, the splitting into two (octahedral and tetrahedral complexes) or more subshells produces so large an orbital energy difference that the multiplet term system of the corresponding gaseous ion is not easily recognized.
If we except the recent study by Herzfeld of the energy levels of nitrogen atoms, trapped in solid nitrogen lattice, we have no other case of approximate spherical symmetry in chemistry as conspicun ous as the 4f -transitions in the lanthanides. These absorption bands have been noted since about 1880 for their narrowness (δ between 0-016 and 0-3 kK, since the Franck-Condon principle enforces no vibrational excitation of the nearly parallel potential curves), their low intensity ( e 0 between 0Ό1 and 15, they are all Laporte-forbidden), and their relative small shifts (the nephelauxetic effect) and band splittings (the ligand field effects) as functions of the chemical compound of a given central ion. Many experimental studies have been made of these lanthanide compounds, whose bands can be seen sharply in a spectroscope and are easy to photograph. Thus, Auer von Welsbach and Georges Urbain used them for analytical purposes; H. Becquerel was interested in them as a physicist; H. C. Jones studied the influence of solvents, of temperature etc.; the Ley den laboratory the influence of magnetic fields (the Zeeman effect) ; Gorter et al measured oscillator strengths; and many authors such as 173
174
ENERGY
LEVELS
IN
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Yntema, Prandtl, Stewart, Holleck, etc., continued the investigation to find all the weak bands. To identify the /-values of the energy levels of the corresponding gaseous ions (which have not been studied by atomic spectroscopists, because the spectra,
- D "'F
100
3iif'C
K'G
D -2 ' 2D =2 2
- I, L
-3p
S
:
- G 4 4
S, F
4H
=,P,!h — : — F
:
G
6D
-I
1
-6p
_
k , 4l _ 4
S v
- F -6 g
- H -5 ρ
4
4 ί
4
5 5
— ^
-\D?/ 5
6
.
4K'
-'K hV
-3 G3 L 2
6H
«D'
;jf
4
= 6?ΗΞΞΞ Ι.' Μ3
K
5
5
- S, F
FIG. 2 1 . The relative position of the baricentres of multiplet terms of n z f -systems in units of E according to the assumptions of Elliott, Judd and Runciman (ref. 133). The dotted lines indicate terms with a lower value of S than the maximum value.
which also involve excited states, consist of hundreds of thousands of fines, in contrast to the simple ions with only one or two electrons in partly filled shells) the measurement of the number of sub-levels has been proposed, at very low temperature (suppressing the population of excited sublevels of the ground state) which at most can be 2J + 1 for an even number of f-electrons and J + \ (Kramer's doublets) for an odd number. This is not an infallible method of determining J of the excited state, since vibrational levels may imitate electronic sub-levels. Hellwege used it with great success (1951) for levels of Pr(III) and Eu(III), and Satten (1953) for Nd(III). Many authors have tried to explain the distribution of the sub-levels of each such level by first-order electrostatic perturbations and often obtained good
ENERGY LEVELS IN
175
COMPLEXES
results, if the ligand field parameters were chosen as empirical parameters in the same way as Δ in the octahedral complexes. This is probably a translation of relatively large second-order perturbations contributing to the small ligand field splitting. However, the method of identification of J of each excited level from the distribution of the sub-levels has not been the most fruitful n way of increasing our knowledge of the 4f -energy levels. Rather, it has been useful to compare the overall distribution of spin-allowed and spin-forbidden bands with the Slater-Condon-Shortley theory of energy levels, which for the special case of ^-configurations has been highly simplified by Racah (1949). Ellis (1936) was the first to identify the three bands of Pr(III) at 20-7, 21-4 and 22-6 kK with 3 3 3 2 transitions to P 0 , P 1? P 2 of 4f . In the following years, Bethe and 12 Spedding (1937) studied 4f Tm(III), Spedding (1940) Pr(III), and 3 Satten (1953) 4f Nd(III). Actually, these authors only identified levels with maximum value of S correctly, while the excited levels of the spin-forbidden bands were erroneously assigned, due to the 2 6 assumption of the unrealistic relation F ~ F* ~ 10F . This does not affect the relative distances between the spin-allowed bands, z which are multiples of the linear combination E only (eqn. 36). n Fig. 21 gives the order of multiplet term baricentres of 4f with a 74 2 G 2 plausible ratio i = 0-75F and F = 0-60F . Table 18 gives the values 3 of E in the trivalent lanthanides, increasing smoothly with the atomic number. Gobrecht (1938) identified the /-levels expected in each lanthanide from the lowest multiplet term: f l ? f1 3
2 > 1ff 2
2p
3JJ
3
f>
fl
4J
l
4 > 1ff 0 5J
f 5 ? f9 6JI
6 ? f8f 7p
f7 8
(
1
5
f
)
and determined thus the values of £4f from the total spreading (L + £)£4f of this multiplet. Table 18 gives the somewhat more accurate values, derived by Judd from a partial correction for intermediate coupling effects. Elliott, Judd and Runciman (1957) n have generally discussed how to predict the f -energy levels as k function of F and £4f. However, the exact identification of the definite levels is at present only certain for all the eleven known excited levels of Pr(III), for a certain number in Nd(III) (among those the 2 extremely narrow band at 23-4 kK, having as excited level P£), for the ten first levels of Er(III) (Wybourne (1960) and Kahle (1961)), all the eleven levels of Tm(III) (Jorgensen (1955), Johnsen (1958), and
)
176
ENERGY LEVELS IN
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Gruber and Conway (I960)) again excepting the high-lying S ; and a few others. Fluorescent transitions, which usually go from the lowest excited state with a given S to the ground state (cf. Cr(III) and Mn(IV) ruby lines, Gd(III), the first electron transfer state of 2 + U 0 2 , etc.), may also go to less excited states. This has been of great importance for the study of Sm(III), Eu(III), Tb(III), Dy(III), Pu(III) and Am(III), where the corresponding absorption lines, going from the ground state, are situated in the infra-red. 3
TABLE 18. THE PARAMETER OF INTERELECTRONIC REPULSION £ AND THE LANDE PARAMETER £ NF IN COMPLEXES (ROMAN NUMERALS INDICATING THE OXIDATION NUMBER) AND GASEOUS IONS OF THE LANTHANIDES AND ACTINIDES, IN kK. THE FUNCTION k FOR THE NEPHELAUXETIC EFFECT OF EQN. (137) IS ESTIMATED FROM A COMPARISON OF AQUO AND VARIOUS ANION COMPLEXES
£ 4 2f 4 3f
Pr(III) Nd(III) 5 4 f Sm(III) 4 6 f Sm(II) Eu(IH) 4 7 f Gd(III) 8 4 f Tb(III) 4 9 f DyOU) 4fio Ho(III) 4f i i Er(III) 4f l 2 Tm(III) 4f l 3 Yb(III)
3
0-46 0-48 0-48 0-42
—
0-52
—
0-57
— —
0-63
—
Î4f
0-8 0-86 118 11 1-36
003 002 002
—
001
1-72 1-92 208 2-3 2-6 2-95
£
k
— —
— — —
5f 5 2f
2+
— —
A c3+ Th
2 +x
3
h
3 U(IV) 5f6 U(III) 5f7 Am(III) 5f Cm(III)
0-298 0-36 0-3
—
0-3
Î5!
0-750 1-236 1035 1-6 1-7 3-5
—
k
— — —
004 005
—
002
000
— —
It is the main concept of Seaborg's actiniae hypothesis that the actinide compounds have only 5f-electrons in addition to the closed emanation shell of eighty-six electrons. This hypothesis can be proved with certainty either by extrapolation from atomic spectroscopy or from the absorption spectra of the complexes. The atomic number Z, at which the 5f orbital energy falls below that of 6d is highly dependent on the ionic charge Z 0 — 1 (eqn. 158). Thus the neutral gaseous atoms contain, in most cases, two 7s- and one 6delectron in addition to the 5f-shell. These electrons are removed at 3+ first by ionization, leaving T h with the configuration [Em]5f some 10 kK below that of [Em]6d. It can be safely extrapolated from the 4 + 5+ behaviour of other transition groups that P a , U and all following systems with eighty-seven electrons are most stable as [Em]5f. Further, it can be extrapolated that all actinide ions with the ionic
ENERGY
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COMPLEXES n
charge at least + 3 have the configuration [Em]5f . Thus, while 2 [Em]5f is an excited configuration, extended from 15-96 to 33-68 kK (with one further level ^ 0 at 51-48 kK) above the ground state of 2 + 2 T h , it can be extrapolated that [Em]5f occurs at least 50 kK below 4 + [Em]5f6d of gaseous U .
/
pram
, 1
—
/
1
ι
/
1
1
Ή. L
1 O
3
»r
1
10 1
ρ
3
Th
2
I I°I'i
1
++
gaseous s
°
II ^ 2
12 0 1
1
30
1
II
4 01
2
FIG. 22. Energy levels of 4f - and 5f -systems. Pr(III) aquo ions from refs. 119, 223, and 560. U(IV)2+aquo ions from refs. 122, 268, and 336 (cf. also ref. 95). Gaseous T h from2 refs. 325 and 489. For a general discussion of 5f , see ref. 298.
This behaviour is also found in the chemical compounds. Fried and Hindman (1954) report the same type of transitions in Pa(IV) as described below (eqn. 160) for Ce(III). The actinides with at least two electrons outside the emanation shell have the narrow bands with only small chemical shifts known from the lanthanides. There are observed only from two to five times the ligand field splittings in n n 5f , compared to 4f , and the nephelauxetic effect (see Table 18) is also only about twice as large. The intermediate coupling effects, removing the spin-forbiddenness, are pronounced in the actinide complexes, which have £5f equal to about twice the value of the corresponding £4f. The most conspicuous difference between the Laporte-forbidden
178
ENERGY LEVELS IN n
COMPLEXES
n
bands in 5f and 4f is the intensity, which is some thirty times larger (incorporated in this factor are the band widths, which are on average twice as large). This can be explained by the smaller distance n_1 to the levels with opposite parity (cf. eqn. 116), 5 f 6 d discussed below, and by a larger interaction with the environment.
5
7
( / 2) ( / 2)
2
FIG. 2 3 . The effects of intermediate coupling in f . The determinants were given by Spedding (ref. 550) and applied by Gruen (ref. 188). For large £ nr, the j values 5 and I of the individual f-electrons begin to form pure configurations. The actual cases Pr(III) and U(IV) are shown in Fig. 2 2 .
The absorption spectrum of uranium (IV) aquo ions (see Fig. 22) has a very characteristic distribution of some nine narrow absorption bands in the range 6 to 23 kK, and after that only one narrow band at 40-8 kK. This is exactly as expected from the Slater theory for 2 3 3 X V 3 5f , having the twelve first energy levels ( H, F , G, D9 H, P)
179
ENERGY LEVELS IN COMPLEXES 3
within a range slightly above 4 2 £ + l £ 5 f , while the thirteenth level z follows some 55 E higher. This indicates slightly higher values 2 of E* and ζα in U(IV) (Table 18) than known in Th +. The present writer does not believe that the ligand field effects are so small that each observed sub-level of U(IV) corresponds to exactly one /-level of spherical symmetry; / may very well be intermixed between levels having a distance of, say, 2 kK. On the other hand, the overall 2 distribution of multiplets is closely analogous to the case of 5f in spherical symmetry. Conway (1959) has drawn similar conclusions z for U(IV) in CaF 2, reporting the values E = 300 Κ and £ 5 f = 1870 K.
2
3
4
FIG. 24. The effects of intermediate coupling in p , p , and p in spherical symmetry. As discussed by Kotani (refs. 334 and 559)4 and Griffith 3 2 (ref. 181), the determinants are the same as those for y 5 , y 5 , and y 5 , when these electrons3 have the angular 4dependence characteristic f o r / = 2. 2_ 2_ The position of 5d R e C l 6 and 5d O s C l e in the diagram has previously been discussed by Griffith and the present writer (ref. 295),2 while Moffitt, Goodman, Fred and Weinstock (ref. 409) investigated 5d O s F e 3 and 5d I r F 6.
Unfortunately, most lanthanide and actinide compounds have environments of very low symmetry. The only octahedral complexes 2 x 2 well known are 5f N p F 6 , 5f P u F e and UC1 6 ~. It would be very interesting to interpret the absorption spectra of these complexes in terms of the two orbital energy differences* demanded by group theory between the orbital odd γ2 at lowest energy (see Table 1) and three equivalent, odd γ5 (being ττ-antibonding) and the three 1 equivalent, σ-antibonding odd y 4 (L^-coupling in f produces the levels Γ 7 , Γ8, Γ 7 , Γ 8 , and Γ 6 ). These orbital energy differences are *Axe (I960) demonstrated that these two M.O. energy differences in Pa(IV) in Cs 2ZrCl 6 are 1-50 kK and 2-16 kK, while J 5f is 1-49 kK.
180
ENERGY LEVELS IN
COMPLEXES
only some 2 kK in N p F 6, while ζ 5Γ is some 2-2 kK. P u F e may indicate 3 Ε ~ 0*4 kK. It is remembered that near to the weak field limit (p. 81), the total splitting of a given level of atomic spectroscopy is in most cases considerably smaller than the ligand field parameters, since the M.O. configurations are highly mixed. Goodman and Fred (1959) assume that the energy difference y 2 — y 4 is as large as 21 kK. However, they consider the bands at 23 and 28 kK of N p F e as ligand field transitions. These bands are closely analogous to the electron transfer bands of U F e, which also have vibrational structure, at 27 kK. The relative invariability of all internal 5f-transitions with the environment makes such large values of the ligand field parameters highly improbable.
Λ
FIG. 25. The effect of intermediate coupling and octahedral ligand fields on a single f-electron. The variable is the orbital energy difference between y 4, y 5 and y 2. The position of N p F 6 is indicated, as discussed on this page.
An unsolved problem is the splitting of energy of orbitals with different λ ( = 3 , 2, 1 , 0 ) from the 5f-shell in actinide dioxo ions 2+ + M 0 2 and M 0 2 . If the two oxygen atoms have strongly π- and σ-antibonding effects on the 5f-shell, the orbitals with λ = 1 and 0, respectively, would have higher energy. Actually, the complexes with at least one 5f-electron exhibit extremely narrow, rather
ENERGY LEVELS IN
COMPLEXES
181
intense bands, but it is not certain whether the energy levels involved have mainly retained / from atomic spectroscopy, or if the components with different Ω are strongly separated. Since the 5f-electrons are so much more easily removed in actinide compounds (producing higher and more variable oxidation numbers) than are 4f in lanthanides, it is often felt by chemists that the 5f- and 4f-orbitals have quite different properties, though they both may have / = 3. If we except the more uncertain case of the dioxo ions and concentrate our interest on the usual compounds of + 3 and + 4 , we must conclude that absorption spectra by no means present evidence of a much larger perturbation from the ligands on 5f- than on 4f-electrons. If we apply the argument of inverse proportionality between the parameters of interelectronic repulsion and the average radius of the partly filled shell, r in the actinides is some 60 per cent larger, while the crystallographic ionic radii are nearly the 2 + same. This situation in the actinides also occurs in the case of M n , where f is known to be 0-57 Â and the ionic radius in Mn(II) compounds is 0-80 Â. Actually, the nephelauxetic effect is not conspicuously large in the actinides (Table 18), though it can clearly be observed in the recently published absorption spectra of curium (III), where eqn. (140) is valid for C m F 3, the aquo ion, and Cm(III) in anhydrous LaCl 3. The discovery of narrow bands, belonging to internal F-transitions as in Gd(III), is very valuable, since the interelectronic repulsion parameters are seen to be decreased some 40 per cent, compared to the lanthanides. The intermediate coupling effects (which in a half-filled shell are always second-order perturbations and thus proportional to ζη*) are rather strong. Previously, only the broad bands of radiation-introduced impurities ( H 2 0 2 , Cl 2, etc.) were reported of Cm(III). Asprey and Keenan (1958) measured reflection spectra of A m F 3, A m F 4, 6 C m F 3 and C m F 4. C m F 4 is an f -system, having the sharp peak at 19-9 kK of A m F 3 shifted to 22-1 kK, showing the influence of increasing ionic charge, as previously discussed by the writer. Magnetochemists have often argued for the presence of 6drather than 5f-electrons in the actinides. This was in some cases (e.g. U(IV)) supported by a difference of 40 per cent of the paramagnetic deviation from 1 of the macroscopic, magnetic susceptibility. In these cases, the experimental values are compared to the expresn sions valid, when the ligand field splittings of f -energy levels are n negligible or when the ligand field allows a d -complex to retain a Ν
182
ENERGY LEVELS IN
COMPLEXES
paramagnetic contribution to the magnetic susceptibility, proportional to S(S + 1), divided by the absolute temperature (Curie's law). The magnetic moment μ in Bohr units is in that case g-\/{S{S + 1)} (see eqn. 119) being derived from Curie-Weiss' law for the molar magnetic susceptibility xm Νμ
2
X = CmXm With Xm = 3 ^ _
+ Xcorr
(157)
where Ν is Avogadro's number, # is the Weiss correction (of dimension as temperature) and C m the molar concentration of the paramagnetic substance. The diamagnetic correction xcorr is composed of : (a) a small, negative contribution, common to all materials and approximately proportional to the number of electrons; and (b) a positive contribution, the temperature-independent paramagnetism, characteristic of complexes with low-lying electronic states, with which the ground state has non-diagonal elements of magnetic moment. The expression of this quantity, inversely proportional to the energy of excitation to the low-lying states, is often called Van Vleck's high-frequency formula. Actually, the assumption of negligible ligand field splittings in n 5f -complexes is not valid, because the splittings are to be compared to the Boltzmann energy kT, which is only 0-21 kK at room temperature. When the splittings are some 0-5 to 1 kK, they may produce almost any type of magnetic behaviour, though the effects on 2 the visible absorption bands are not very serious. Thus, the 5f systems generally exhibit too small a paramagnetism (compared to g, as calculated in spherical symmetry) and do not follow eqn. (157) (due to the varying population of sub-levels of the lowest level 3 H 4) , and P u F e is nearly diamagnetic, showing only a feeble temperature-independent paramagnetism. This only proves the existence of M.O. energy differences considerably larger than 0-21 kK, but the differences are not necessarily large compared to those between levels in spherical symmetry. The condition of diamagnetism (or temperature-independent paramagnetism) in systems with an even number of electrons is actually that the ground state is not degenerate (e = 1) and is fairly well removed from other, excited states (producing temperature-independent paramagnetism if the energy distance is some 1 to 50 kK, and strong paramagnetism for lower distances). Ordinarily, the ground states of systems with an
ENERGY
LEVELS
IN
COMPLEXES
183
odd number of electrons, having always at least e = 2, correspond to usual paramagnetism (eqn. 157). However, the gyromagnetic factor g may accidentally assume the value zero for some ground 2 states (2/7^ of N O at low temperature; the component Γ 8 of Γ 5 of 1 octahedral d -systems) leading to only xcorr. Further, the interaction between two ions, each with an odd number of electrons, may be sufficiently large to produce a diamagnetic ground state (eqn. 168). n n _ 1 Laporte-allowed f -> f d transitions are observed in Ce(III), Sm(II), Eu(II), Tb(III), Yb(II), Pa(IV), U(III), Np(III) and Pu(HI). It is evident that the distance between the orbital energies of 4f and 5d, or of 5f and 6d, increases with the oxidation number. The (strongly reducing) divalent lanthanides present these bands at rather low wave numbers (17-9-30-2 kK in Sm(II), 31-2 and 40-4 kK in Eu(II)), while in the trivalent ions they are only known for Ce(III) and Tb(III). This is due to the steady increase of their wave number with Ζ for a given ionic charge, bringing the bands outside the measured range already in Pr(IIl). The reappearance of a band at 45-6 kK of Tb(III) is a half-filled shell effect (eqn. 149), the transition 8 7 being 4f -> 4f*5d. Actually, the hump at f of the ground state energy, X 3 assuming eqn. (146), is 8D, where D = %E is approximately 1 3 £ n (eqn. 36) in f -systems. Consequently, 8 D is about 50 kK in the lanthanides and 30 kK in the actinides (Table 18). The effect on the n _ 1 position of the lowest level of f d is perhaps somewhat smaller, owing to the interelectronic repulsion between the f-shell and the d-electron. The energy differences in kK between the lowest level of n _ 1 n 5f and of 5f 6 d , as derived from gaseous actinide ions and their complexes, is a function of Z 0 (the ionic charge plus one) and the atomic number Ζ for, at most, seven f-electrons: 7(Z -
96) + 18(Z 0 -
1)
(158)
8
Thus, it would be expected that the 5f -system Bk(III) exhibits the first Laporte-allowed band at 31 rather than 61 kK derived from eqn. (158). 3+ The energy levels of gaseous C e are situated at: 2 2
F| F|
0 2-25
2
Df 2 Df
49-74 kK 52-23
The aquo ion is probably nine-co-ordinated,
(159) having
trigonal
184
ENERGY
LEVELS IN
COMPLEXES
microsymmetry, which can be approximated to the linear symmetry, identifying the excited orbitals: e
Ce(IIIK 33-7 39-6 41-7 45-1 47-4
ω
«4 5 2
3
Έ
o
Pa(IV)a 0-
16 710 600 380 270
36-3 39-2 44-8
— (160)
3
£5d of Ce + is seen from eqn. (159) to be 1-00 kK, explaining a reasonable splitting into ω-levels of the λ-levels in eqn. (160). The m/z 800 200
Ι ι +' 4 ι
600
500
400
'
'—ι—ι
1
300 1
1
FIG. 26. The absorption spectra of uranium (IV) aquo ions (cf. Fig. 22), grey uranium (III) aquo ions in 1 M H C 1 0 4 or 2 M HCl, and red uranium (III) chloro complexes in 11 M HCl. The Laporte-allowed 2 3 transitions from 5f to 5f 6d are seen of U(III). The figure is reproduced with permission from Acta Chem. Scand. (ref. 281).
low intensity of the first band must be due to some selection rule of symmetry-forbiddenness. However, the present writer (1956) has previously suggested as an alternative possibility that Ce(III) aquo ions might exist with a lower co-ordination number, producing the
ENERGY
LEVELS IN
185
COMPLEXES
weak band. The intensity of the weak band varies strongly with temperature, with the presence of other ions, and even with the substitution of heavy water as solvent. By comparison of eqn. (159) and (160), it is seen that the energy-difference 4f — 5d is some 19 per 3 + cent smaller in the Ce(III) aquo ion than in the gaseous C e . The decrease is some 28 per cent in complexes of ethylenediaminetetraacetate, citrate, chloride, etc., representing the same analogy to the 2 nephelauxetic effect as the s -» sp transitions discussed below. In the systems with more f-electrons in addition to an excited d-electron, it is not possible at present to assign the broad, Laporte-allowed bands to specific energy levels from among the many possible ones. However, the strongest bands of U(III) aquo ions (see Fig. 26), with maxima at 28-6 and 31-2 kK and a shoulder at 25-5 kK, might suggest parent terms of λ,ω-classification of the 6d-electron (eqn. 160) with a definite level of 5Ρ, e.g. the ground state Ή 4 . Solid-state physicists have been much interested in the absorption spectra and fluorescence properties of thallium (I) ions, embedded in alkali halide crystals. Seitz (1938) proposed that the weaker of two X 3 1 bands was due to the transition S0 — P X and the stronger to S 0 — X 2 PX of the configurations 6s and 6s6p, respectively. Actually, Fromherz had studied for many years similar bands of lead (II) complexes; and Macbeth and Maxwell (1923) found the rather narrow band at 30-5 kK of bismuth (III) in hydrochloric acid. These 2 s -> sp transitions can be observed for the whole series of metals, having oxidation numbers two lower than the group-number in the period system: In(I), Sn(II), Sb(III), T1(I), Pb(II), and Bi(III). If considered as problems of atomic spectroscopy, the energy of the levels with / = / of a given configuration Is are the eigenvalues of a Houston determinant E(/s) + K(/s) - Ε VVO + l)K„,/2 I_ E(ls) - K(/s) - Um — Ε |
VW + mm/2
n
n
f ni U U
°
where in Condon and Shortley's book, the exchange integral K(/s) l is denoted by Gi = G /(2l + 1). The two other levels of a configuration Is with the values of J = I — 1 and / + 1 both belong to the triplet term only and have the energies E(/s) — K(/s) — (/ + 1)ζηι/2 and E(/s) — K(/s) + Ιζηι/2, respectively. 3 3 In the case of sp, the transitions to the two levels P 0 and P 2 are completely spin-forbidden, if no other configurations are taken into
186
ENERGY LEVELS I N COMPLEXES 3
1
account. On the other hand, the two levels P X and P 1 with the original distance 2K(/s) are mixed by the non-diagonal element ζηι/\/2 according to eqn. (161). Table 19 gives the wave numbers of the transitions in the gaseous ions and in the complexes. While the second, spin-allowed transition corresponds to bands with oscillator
TABLE 19. THE WAVE NUMBERS IN k K OF THE TRANSITIONS 2TO THE TRIPLET AND 2 SINGLET STATES OF 5s5p OR 6s6p IN 5S - AND 6S -COMPLEXES
In© Sn(II) Sb(III) Te(IV) T1(I) Pb(II) Bi(III)
4
H 20
43-35 6303 55-20 79-91 66-70 95-95 78-02 111-71 52-39 75-66 64-39 95-34 75-93 114-60
—
—
—
—
— — —
—
— — — — — — — — —
— — — — —
480
41-65
41-4
—
450
— —
37-9
—
—
—
—
— — —
46-7
OH-
enta -
gaseous
—
—
ci-
i-
Br-
33, 36 43 34-7, 38-2 45-0 34-8 43-9 26-8? 33-2? 40-5 510 36-8 510 30-5 450
34 39 32 41
32 41 28 34-5
—
— — — —
— — —
38-1 47-2 33-2 44-9 26-8 38-5
35-5 43 0 27-6 —
(22-2) 29-8
strength Ρ ~ 0-8, the first bands are much weaker (Ρ ^ 0-02) for 2 2 5s - than for 6s -complexes (Ρ ^ 0-15). This agrees well with the higher values of ζ 6 Ρ than of ζ 5 Ρ. In the gaseous atoms and ions, the values are (in kK): 2
2
3
Cd In+ Sn + Hg T1+ Pb + Bi +^ 1-14 2-37 3-79 4-26 8-18 12-39 16-99 ηρ K(rcp, ns) 5-68 9-11 11-25 5-59 8-16 9-66 10-86 [ 2 E ( s p ) - E ( s ) 36-67 53-19 67-56 46-74 64-02 79-86 95-26^
ζ
I
U
}
It is seen from Table 19 that the increase with ionic charge of the gaseous ions of the three parameters of importance: the distance 2 3 E(sp) — E(s ) between the average (not the baricentre) of P X and X P 0 and the ground state; £ np and K(«p, ns) of eqn. (162) do not occur in isoelectronic complexes with different oxidation number.
ENERGY LEVELS IN
COMPLEXES
187
Actually, a certain principle of electroneutrality is acting; in the chloro complexes the parameters are (in kK) : ζ ηρ Κ(/φ,Λ8) 2 E(sp)-E(s )
In(I) Sn(II) Sb(III) T1(I) Pb(II) Bi(III) * — — 1-9 4-2(3-6) 5-7 (4-8) 5-8(4-9) > — — 3-8 2-1(3-6) 2-85(4-8) 2-9(4-9) 39 41 39-4 44-8 43-9 37-8 (163)
The relative size of £ n p and K(«p, m) in complexes can in principle be inferred from the ratio between the oscillator strengths of the spin-forbidden and spin-allowed bands. The values given for 6s6p in eqn. (163) are calculated for ζ 6 Ρ = 2K(6s, 6p) (giving φ of eqn. 2 (23) 21-6° and sin c? = 0-136), while the values in parentheses are calculated from the observed band distances in Table 19 with ζ 6 Ρ = K(6s, 6p). The truth is probably somewhere between these limits. The values for Sb(III) are calculated by the analogous method, assuming ζ 5 Ρ = |K(5s, 5p). Though the values of ζην and the K-integral in eqn. (163), taken separately, are rather uncertain, they give an impression of analogy to neutral Cd and Hg in eqn. (162), as also does the orbital energy difference between s and p. The development of the latter quantity follows closely the nephelauxetic series, decreasing in the order (see Table 19): 4
gaseous ion > H 2 0 > OH~ ~ enta " > Cl" > B r - > I~ (164) The decrease of the three quantities in complexes (eqn. 163 and 164), as compared to gaseous ions (eqn. 162) may be interpreted in two different ways, analogous to the discussion of the nephelauxetic effect. We might ascribe it solely to central-field covalency effects, having the spherical symmetrical part V 0 of the ligand field added to the central field with consequent changes in the orbital energies of ns and «p. This would be a genuine Schläfer-Craig-Magnusson effect (p. 141), which may also act on the orbital energy differences 4f - 5d in Ce(III) and 5f — 6d in U(III) complexes. Notably, the exchange integrals Κ are very sensitive to a relative change of the two radial functions. We may also consider symmetry-restricted covalency. The mere 2 existence of s -complexes is rather surprising, since the s-orbital in any symmetry will be σ-antibonding. In octahedral complexes
188
ENERGY
LEVELS IN
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(eqn. 54), the p-orbital is both σ- and ττ-antibonding. The energy difference might either increase or decrease by this effect, and we may very well ascribe a part of the decrease in eqn. (163) to the formation of covalent bonds. The σ-antibonding character of the ground state is shown by the difficulty in forming amine and cyanide complexes, and by the great tendency to form complexes with the more polarizable halides, C I , Br~ and I~. The anomalously large formation constants of the halide complexes may partly be due to relative instability of the aquo ions. The behaviour with O H is variable; T1(I) does not form complexes at all, or at most, weakly bound ion pairs; Sn(II), Sb(III) and Pb(II) form precipitates, easily soluble in an excess of O H ; and Bi(III) forms precipitates at very low hydroxyl concentration (highly acidic solution), perhaps containing oxygen bridges. However, the value of K(«s, np) in eqn. (163) must not be taken too seriously. Already in the gaseous ion, the different /-levels of the configuration sp might have slightly different radial functions (cf. eqn. 40), and this might develop further in the complexes, making eqn. 161 an inappropriate expression of the energy difference between the mainly triplet and mainly singlet level. Actually, the band widths are some 80 per cent larger in the singlet band of Pb(II) and Bi(III) chloro complexes than in the triplet bands. Williams (1951) suggests that the internuclear equilibrium distance TL is smaller in the excited state, at least the triplet, than in the ground state. The triplet and singlet levels of T1(I) in KI have different fine-structures (Yuster and Delbecq, 1953). This is somewhat surprising, since low microsymmetries (e.g. rhombic) should produce the same number of sublevels of the two levels with J = 1. However, it can be shown that 3 the splitting of P X is only — \ as large as the splitting of Ψν Finally, quite another deviation from the case of atomic spectroscopy may occur: the highest filled M.O. of the ground state may be rather far removed from having the symmetry s. Orgel (1958) studied the stereochemistry of this family and concludes that the lone pair, at least in Pb(II) compounds, has a strong tendency to concentrate at one side of the metal ion, corresponding to strong intermixing with higher values of /. Nevertheless, crystalline PbCl 2, which has this stereochemistry with irregularly varying distances between the ligands, shows the usual spectrum with a triplet and a singlet band. Once more, we cannot estimate the exact form of the M.O. involved from the absorption spectra.
ENERGY
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189
T1(I) ions in solid alkali halides have been much discussed by Wilhams and Johnson (1951, 1952) and In(I), as given in Table 19, is only known in such crystals (Williams, Segall, and Johnson (1957); Dunina, Morgenstern, and Shamosky, (1958); Katz and Nikolsky (1958)). In solution, Merritt, Hershenson and Rogers (1953) studied T1(I), Pb(II) and Bi(III) halide complexes for analytical purposes (and also Hg(II)), and Newman and Hume (1957) made a profound study of Bi(III) chloro and bromo complexes. A further possible case, where the excited state may involve a n n _ 1 configuration from spherical symmetry, is the 3 d -> 3 d 4 s transitions producing Laporte-forbidden, broad absorption bands in strongly reducing complexes of the first transition group. Dainton (1952) correlated the wave numbers of these bands (the hexa-aquo ions of V(II) at 33, Cr(II) at 40, and Fe(II) at 41 kK) with the standard oxidation potentials, suggesting an "inverted" electron transfer process, as discussed in the next chapter. The transitions might be Laporte-allowed with accidentally small dipole moment, or spinforbidden (the K-integrals between 3d and 4s in gaseous atoms are invariably from 1-2 to 1-3 kK). However, it seems probable that the excited orbital has / = 0 to some extent and is the even yx (eqn. 54) to be filled in a (somewhat hypothetical) octahedral complex of Ga(I). The X-ray absorption edges of chemical compounds, e.g. the K-edges, representing the excitation of a ls-electron of a given atom, have been much studied. Stelling (1928-1934) observed a 2 2 variation between 19,955 kK in S ~ and 20,050 kK in S 0 4 - among sulphur compounds and found a regular increase of the wave number with the oxidation number of many elements. Later, Mitchell and Beeman (1952) measured several transition group complexes and found chemical shifts and fine structure of the absorption edge of the metal ion. These transitions were now postulated to go to 4p, 5p, . . . shells, and the Pauling hybridization theory was applied to explain that diamagnetic square-planar nickel (II) complexes have a narrow band at lower energies than the paramagnetic, octahedral Ni(II). This description in terms of an empty or a filled 4p-shell was continued by Kauer (1956), Boke (1957) and Collet (1959). However, it is obvious that the K-absorption spectrum simply indicates the available range of empty, preferably odd y 4 (eqn. 115), orbitals, if M.O. configurations can be used. The energy and wave functions of these empty M.O. depend as well on the ligands as on
190
ENERGY
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the central ion. For instance, aromatic ligands with empty orbitals of relatively low energy show the transitions at lower wave numbers (cf. the Ir-py band to be discussed in the next chapter). Stelling (1932) found one of the two K-edges of covalent chloro-complexes at a lower wave number than that of ionic Cl~. This is not as well explained by CI having an oxidation number below — 1 , as by the available empty orbitals in metal complexes. 2+ 2+ A significant fact is that N i ( N H 3 ) 6 and C u ( N H 3 ) 4 both have 2+ two bands with an energy difference 30 kK, while Z n e n 3 only has one (Cotton and Hansen, 1958). This influence of the partly filled d-shell might be connected with interelectronic repulsion energies, but is probably too large to be explained in that way. It is probably not possible to use ligand field theory on the 4p-shell (Cotton and Ballhausen, 1956) since the splitting is also observed in 2+ 3 the octahedral N i ( N H 3 ) 6 and C o ( N H 3) 6 + . It would be useful to have more experimental facts available, especially on the more loosely bound electrons (2p, 3d, etc.) of complexes of heavy metals. However, the resolution limit is not good, some 20 kK for bands in the K-region, though the relative accuracy of the wave number is very high. Collet (1959) measured the chemical shifts of several 2p absorption bands of rhenium (IV), platinum (II) and platinum (IV) complexes.
10
ENERGY LEVELS IN COMPLEXES WITH ALMOST SPHERICAL SYMMETRY T H E R E is no reason for a M.O. in all cases to resemble orbitals adapted to spherical symmetry; this is known from atomic spectroscopy. In the next chapter we shall consider transitions which certainly cannot be described thus. The three ordinary transition groups present a partly filled shell which to some extent is intermediate between the subject of this and the following chapter. On the one hand, the assumption of / = 2 in the electrostatic model or in the expanded radial function model explains almost all the pertinent features of the ligand field bands; on the other hand, the splitting into two (octahedral and tetrahedral complexes) or more subshells produces so large an orbital energy difference that the multiplet term system of the corresponding gaseous ion is not easily recognized.
If we except the recent study by Herzfeld of the energy levels of nitrogen atoms, trapped in solid nitrogen lattice, we have no other case of approximate spherical symmetry in chemistry as conspicun ous as the 4f -transitions in the lanthanides. These absorption bands have been noted since about 1880 for their narrowness (δ between 0-016 and 0-3 kK, since the Franck-Condon principle enforces no vibrational excitation of the nearly parallel potential curves), their low intensity ( e 0 between 0Ό1 and 15, they are all Laporte-forbidden), and their relative small shifts (the nephelauxetic effect) and band splittings (the ligand field effects) as functions of the chemical compound of a given central ion. Many experimental studies have been made of these lanthanide compounds, whose bands can be seen sharply in a spectroscope and are easy to photograph. Thus, Auer von Welsbach and Georges Urbain used them for analytical purposes; H. Becquerel was interested in them as a physicist; H. C. Jones studied the influence of solvents, of temperature etc.; the Ley den laboratory the influence of magnetic fields (the Zeeman effect) ; Gorter et al measured oscillator strengths; and many authors such as 173
174
ENERGY
LEVELS
IN
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Yntema, Prandtl, Stewart, Holleck, etc., continued the investigation to find all the weak bands. To identify the /-values of the energy levels of the corresponding gaseous ions (which have not been studied by atomic spectroscopists, because the spectra,
- D "'F
100
3iif'C
K'G
D -2 ' 2D =2 2
- I, L
-3p
S
:
- G 4 4
S, F
4H
=,P,!h — : — F
:
G
6D
-I
1
-6p
_
k , 4l _ 4
S v
- F -6 g
- H -5 ρ
4
4 ί
4
5 5
— ^
-\D?/ 5
6
.
4K'
-'K hV
-3 G3 L 2
6H
«D'
;jf
4
= 6?ΗΞΞΞ Ι.' Μ3
K
5
5
- S, F
FIG. 2 1 . The relative position of the baricentres of multiplet terms of n z f -systems in units of E according to the assumptions of Elliott, Judd and Runciman (ref. 133). The dotted lines indicate terms with a lower value of S than the maximum value.
which also involve excited states, consist of hundreds of thousands of fines, in contrast to the simple ions with only one or two electrons in partly filled shells) the measurement of the number of sub-levels has been proposed, at very low temperature (suppressing the population of excited sublevels of the ground state) which at most can be 2J + 1 for an even number of f-electrons and J + \ (Kramer's doublets) for an odd number. This is not an infallible method of determining J of the excited state, since vibrational levels may imitate electronic sub-levels. Hellwege used it with great success (1951) for levels of Pr(III) and Eu(III), and Satten (1953) for Nd(III). Many authors have tried to explain the distribution of the sub-levels of each such level by first-order electrostatic perturbations and often obtained good
ENERGY LEVELS IN
175
COMPLEXES
results, if the ligand field parameters were chosen as empirical parameters in the same way as Δ in the octahedral complexes. This is probably a translation of relatively large second-order perturbations contributing to the small ligand field splitting. However, the method of identification of J of each excited level from the distribution of the sub-levels has not been the most fruitful n way of increasing our knowledge of the 4f -energy levels. Rather, it has been useful to compare the overall distribution of spin-allowed and spin-forbidden bands with the Slater-Condon-Shortley theory of energy levels, which for the special case of ^-configurations has been highly simplified by Racah (1949). Ellis (1936) was the first to identify the three bands of Pr(III) at 20-7, 21-4 and 22-6 kK with 3 3 3 2 transitions to P 0 , P 1? P 2 of 4f . In the following years, Bethe and 12 Spedding (1937) studied 4f Tm(III), Spedding (1940) Pr(III), and 3 Satten (1953) 4f Nd(III). Actually, these authors only identified levels with maximum value of S correctly, while the excited levels of the spin-forbidden bands were erroneously assigned, due to the 2 6 assumption of the unrealistic relation F ~ F* ~ 10F . This does not affect the relative distances between the spin-allowed bands, z which are multiples of the linear combination E only (eqn. 36). n Fig. 21 gives the order of multiplet term baricentres of 4f with a 74 2 G 2 plausible ratio i = 0-75F and F = 0-60F . Table 18 gives the values 3 of E in the trivalent lanthanides, increasing smoothly with the atomic number. Gobrecht (1938) identified the /-levels expected in each lanthanide from the lowest multiplet term: f l ? f1 3
2 > 1ff 2
2p
3JJ
3
f>
fl
4J
l
4 > 1ff 0 5J
f 5 ? f9 6JI
6 ? f8f 7p
f7 8
(
1
5
f
)
and determined thus the values of £4f from the total spreading (L + £)£4f of this multiplet. Table 18 gives the somewhat more accurate values, derived by Judd from a partial correction for intermediate coupling effects. Elliott, Judd and Runciman (1957) n have generally discussed how to predict the f -energy levels as k function of F and £4f. However, the exact identification of the definite levels is at present only certain for all the eleven known excited levels of Pr(III), for a certain number in Nd(III) (among those the 2 extremely narrow band at 23-4 kK, having as excited level P£), for the ten first levels of Er(III) (Wybourne (1960) and Kahle (1961)), all the eleven levels of Tm(III) (Jorgensen (1955), Johnsen (1958), and
)
176
ENERGY LEVELS IN
COMPLEXES X
Gruber and Conway (I960)) again excepting the high-lying S ; and a few others. Fluorescent transitions, which usually go from the lowest excited state with a given S to the ground state (cf. Cr(III) and Mn(IV) ruby lines, Gd(III), the first electron transfer state of 2 + U 0 2 , etc.), may also go to less excited states. This has been of great importance for the study of Sm(III), Eu(III), Tb(III), Dy(III), Pu(III) and Am(III), where the corresponding absorption lines, going from the ground state, are situated in the infra-red. 3
TABLE 18. THE PARAMETER OF INTERELECTRONIC REPULSION £ AND THE LANDE PARAMETER £ NF IN COMPLEXES (ROMAN NUMERALS INDICATING THE OXIDATION NUMBER) AND GASEOUS IONS OF THE LANTHANIDES AND ACTINIDES, IN kK. THE FUNCTION k FOR THE NEPHELAUXETIC EFFECT OF EQN. (137) IS ESTIMATED FROM A COMPARISON OF AQUO AND VARIOUS ANION COMPLEXES
£ 4 2f 4 3f
Pr(III) Nd(III) 5 4 f Sm(III) 4 6 f Sm(II) Eu(IH) 4 7 f Gd(III) 8 4 f Tb(III) 4 9 f DyOU) 4fio Ho(III) 4f i i Er(III) 4f l 2 Tm(III) 4f l 3 Yb(III)
3
0-46 0-48 0-48 0-42
—
0-52
—
0-57
— —
0-63
—
Î4f
0-8 0-86 118 11 1-36
003 002 002
—
001
1-72 1-92 208 2-3 2-6 2-95
£
k
— —
— — —
5f 5 2f
2+
— —
A c3+ Th
2 +x
3
h
3 U(IV) 5f6 U(III) 5f7 Am(III) 5f Cm(III)
0-298 0-36 0-3
—
0-3
Î5!
0-750 1-236 1035 1-6 1-7 3-5
—
k
— — —
004 005
—
002
000
— —
It is the main concept of Seaborg's actiniae hypothesis that the actinide compounds have only 5f-electrons in addition to the closed emanation shell of eighty-six electrons. This hypothesis can be proved with certainty either by extrapolation from atomic spectroscopy or from the absorption spectra of the complexes. The atomic number Z, at which the 5f orbital energy falls below that of 6d is highly dependent on the ionic charge Z 0 — 1 (eqn. 158). Thus the neutral gaseous atoms contain, in most cases, two 7s- and one 6delectron in addition to the 5f-shell. These electrons are removed at 3+ first by ionization, leaving T h with the configuration [Em]5f some 10 kK below that of [Em]6d. It can be safely extrapolated from the 4 + 5+ behaviour of other transition groups that P a , U and all following systems with eighty-seven electrons are most stable as [Em]5f. Further, it can be extrapolated that all actinide ions with the ionic
ENERGY
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COMPLEXES n
charge at least + 3 have the configuration [Em]5f . Thus, while 2 [Em]5f is an excited configuration, extended from 15-96 to 33-68 kK (with one further level ^ 0 at 51-48 kK) above the ground state of 2 + 2 T h , it can be extrapolated that [Em]5f occurs at least 50 kK below 4 + [Em]5f6d of gaseous U .
/
pram
, 1
—
/
1
ι
/
1
1
Ή. L
1 O
3
»r
1
10 1
ρ
3
Th
2
I I°I'i
1
++
gaseous s
°
II ^ 2
12 0 1
1
30
1
II
4 01
2
FIG. 22. Energy levels of 4f - and 5f -systems. Pr(III) aquo ions from refs. 119, 223, and 560. U(IV)2+aquo ions from refs. 122, 268, and 336 (cf. also ref. 95). Gaseous T h from2 refs. 325 and 489. For a general discussion of 5f , see ref. 298.
This behaviour is also found in the chemical compounds. Fried and Hindman (1954) report the same type of transitions in Pa(IV) as described below (eqn. 160) for Ce(III). The actinides with at least two electrons outside the emanation shell have the narrow bands with only small chemical shifts known from the lanthanides. There are observed only from two to five times the ligand field splittings in n n 5f , compared to 4f , and the nephelauxetic effect (see Table 18) is also only about twice as large. The intermediate coupling effects, removing the spin-forbiddenness, are pronounced in the actinide complexes, which have £5f equal to about twice the value of the corresponding £4f. The most conspicuous difference between the Laporte-forbidden
178
ENERGY LEVELS IN n
COMPLEXES
n
bands in 5f and 4f is the intensity, which is some thirty times larger (incorporated in this factor are the band widths, which are on average twice as large). This can be explained by the smaller distance n_1 to the levels with opposite parity (cf. eqn. 116), 5 f 6 d discussed below, and by a larger interaction with the environment.
5
7
( / 2) ( / 2)
2
FIG. 2 3 . The effects of intermediate coupling in f . The determinants were given by Spedding (ref. 550) and applied by Gruen (ref. 188). For large £ nr, the j values 5 and I of the individual f-electrons begin to form pure configurations. The actual cases Pr(III) and U(IV) are shown in Fig. 2 2 .
The absorption spectrum of uranium (IV) aquo ions (see Fig. 22) has a very characteristic distribution of some nine narrow absorption bands in the range 6 to 23 kK, and after that only one narrow band at 40-8 kK. This is exactly as expected from the Slater theory for 2 3 3 X V 3 5f , having the twelve first energy levels ( H, F , G, D9 H, P)
179
ENERGY LEVELS IN COMPLEXES 3
within a range slightly above 4 2 £ + l £ 5 f , while the thirteenth level z follows some 55 E higher. This indicates slightly higher values 2 of E* and ζα in U(IV) (Table 18) than known in Th +. The present writer does not believe that the ligand field effects are so small that each observed sub-level of U(IV) corresponds to exactly one /-level of spherical symmetry; / may very well be intermixed between levels having a distance of, say, 2 kK. On the other hand, the overall 2 distribution of multiplets is closely analogous to the case of 5f in spherical symmetry. Conway (1959) has drawn similar conclusions z for U(IV) in CaF 2, reporting the values E = 300 Κ and £ 5 f = 1870 K.
2
3
4
FIG. 24. The effects of intermediate coupling in p , p , and p in spherical symmetry. As discussed by Kotani (refs. 334 and 559)4 and Griffith 3 2 (ref. 181), the determinants are the same as those for y 5 , y 5 , and y 5 , when these electrons3 have the angular 4dependence characteristic f o r / = 2. 2_ 2_ The position of 5d R e C l 6 and 5d O s C l e in the diagram has previously been discussed by Griffith and the present writer (ref. 295),2 while Moffitt, Goodman, Fred and Weinstock (ref. 409) investigated 5d O s F e 3 and 5d I r F 6.
Unfortunately, most lanthanide and actinide compounds have environments of very low symmetry. The only octahedral complexes 2 x 2 well known are 5f N p F 6 , 5f P u F e and UC1 6 ~. It would be very interesting to interpret the absorption spectra of these complexes in terms of the two orbital energy differences* demanded by group theory between the orbital odd γ2 at lowest energy (see Table 1) and three equivalent, odd γ5 (being ττ-antibonding) and the three 1 equivalent, σ-antibonding odd y 4 (L^-coupling in f produces the levels Γ 7 , Γ8, Γ 7 , Γ 8 , and Γ 6 ). These orbital energy differences are *Axe (I960) demonstrated that these two M.O. energy differences in Pa(IV) in Cs 2ZrCl 6 are 1-50 kK and 2-16 kK, while J 5f is 1-49 kK.
180
ENERGY LEVELS IN
COMPLEXES
only some 2 kK in N p F 6, while ζ 5Γ is some 2-2 kK. P u F e may indicate 3 Ε ~ 0*4 kK. It is remembered that near to the weak field limit (p. 81), the total splitting of a given level of atomic spectroscopy is in most cases considerably smaller than the ligand field parameters, since the M.O. configurations are highly mixed. Goodman and Fred (1959) assume that the energy difference y 2 — y 4 is as large as 21 kK. However, they consider the bands at 23 and 28 kK of N p F e as ligand field transitions. These bands are closely analogous to the electron transfer bands of U F e, which also have vibrational structure, at 27 kK. The relative invariability of all internal 5f-transitions with the environment makes such large values of the ligand field parameters highly improbable.
Λ
FIG. 25. The effect of intermediate coupling and octahedral ligand fields on a single f-electron. The variable is the orbital energy difference between y 4, y 5 and y 2. The position of N p F 6 is indicated, as discussed on this page.
An unsolved problem is the splitting of energy of orbitals with different λ ( = 3 , 2, 1 , 0 ) from the 5f-shell in actinide dioxo ions 2+ + M 0 2 and M 0 2 . If the two oxygen atoms have strongly π- and σ-antibonding effects on the 5f-shell, the orbitals with λ = 1 and 0, respectively, would have higher energy. Actually, the complexes with at least one 5f-electron exhibit extremely narrow, rather
ENERGY LEVELS IN
COMPLEXES
181
intense bands, but it is not certain whether the energy levels involved have mainly retained / from atomic spectroscopy, or if the components with different Ω are strongly separated. Since the 5f-electrons are so much more easily removed in actinide compounds (producing higher and more variable oxidation numbers) than are 4f in lanthanides, it is often felt by chemists that the 5f- and 4f-orbitals have quite different properties, though they both may have / = 3. If we except the more uncertain case of the dioxo ions and concentrate our interest on the usual compounds of + 3 and + 4 , we must conclude that absorption spectra by no means present evidence of a much larger perturbation from the ligands on 5f- than on 4f-electrons. If we apply the argument of inverse proportionality between the parameters of interelectronic repulsion and the average radius of the partly filled shell, r in the actinides is some 60 per cent larger, while the crystallographic ionic radii are nearly the 2 + same. This situation in the actinides also occurs in the case of M n , where f is known to be 0-57 Â and the ionic radius in Mn(II) compounds is 0-80 Â. Actually, the nephelauxetic effect is not conspicuously large in the actinides (Table 18), though it can clearly be observed in the recently published absorption spectra of curium (III), where eqn. (140) is valid for C m F 3, the aquo ion, and Cm(III) in anhydrous LaCl 3. The discovery of narrow bands, belonging to internal F-transitions as in Gd(III), is very valuable, since the interelectronic repulsion parameters are seen to be decreased some 40 per cent, compared to the lanthanides. The intermediate coupling effects (which in a half-filled shell are always second-order perturbations and thus proportional to ζη*) are rather strong. Previously, only the broad bands of radiation-introduced impurities ( H 2 0 2 , Cl 2, etc.) were reported of Cm(III). Asprey and Keenan (1958) measured reflection spectra of A m F 3, A m F 4, 6 C m F 3 and C m F 4. C m F 4 is an f -system, having the sharp peak at 19-9 kK of A m F 3 shifted to 22-1 kK, showing the influence of increasing ionic charge, as previously discussed by the writer. Magnetochemists have often argued for the presence of 6drather than 5f-electrons in the actinides. This was in some cases (e.g. U(IV)) supported by a difference of 40 per cent of the paramagnetic deviation from 1 of the macroscopic, magnetic susceptibility. In these cases, the experimental values are compared to the expresn sions valid, when the ligand field splittings of f -energy levels are n negligible or when the ligand field allows a d -complex to retain a Ν
182
ENERGY LEVELS IN
COMPLEXES
paramagnetic contribution to the magnetic susceptibility, proportional to S(S + 1), divided by the absolute temperature (Curie's law). The magnetic moment μ in Bohr units is in that case g-\/{S{S + 1)} (see eqn. 119) being derived from Curie-Weiss' law for the molar magnetic susceptibility xm Νμ
2
X = CmXm With Xm = 3 ^ _
+ Xcorr
(157)
where Ν is Avogadro's number, # is the Weiss correction (of dimension as temperature) and C m the molar concentration of the paramagnetic substance. The diamagnetic correction xcorr is composed of : (a) a small, negative contribution, common to all materials and approximately proportional to the number of electrons; and (b) a positive contribution, the temperature-independent paramagnetism, characteristic of complexes with low-lying electronic states, with which the ground state has non-diagonal elements of magnetic moment. The expression of this quantity, inversely proportional to the energy of excitation to the low-lying states, is often called Van Vleck's high-frequency formula. Actually, the assumption of negligible ligand field splittings in n 5f -complexes is not valid, because the splittings are to be compared to the Boltzmann energy kT, which is only 0-21 kK at room temperature. When the splittings are some 0-5 to 1 kK, they may produce almost any type of magnetic behaviour, though the effects on 2 the visible absorption bands are not very serious. Thus, the 5f systems generally exhibit too small a paramagnetism (compared to g, as calculated in spherical symmetry) and do not follow eqn. (157) (due to the varying population of sub-levels of the lowest level 3 H 4) , and P u F e is nearly diamagnetic, showing only a feeble temperature-independent paramagnetism. This only proves the existence of M.O. energy differences considerably larger than 0-21 kK, but the differences are not necessarily large compared to those between levels in spherical symmetry. The condition of diamagnetism (or temperature-independent paramagnetism) in systems with an even number of electrons is actually that the ground state is not degenerate (e = 1) and is fairly well removed from other, excited states (producing temperature-independent paramagnetism if the energy distance is some 1 to 50 kK, and strong paramagnetism for lower distances). Ordinarily, the ground states of systems with an
ENERGY
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183
odd number of electrons, having always at least e = 2, correspond to usual paramagnetism (eqn. 157). However, the gyromagnetic factor g may accidentally assume the value zero for some ground 2 states (2/7^ of N O at low temperature; the component Γ 8 of Γ 5 of 1 octahedral d -systems) leading to only xcorr. Further, the interaction between two ions, each with an odd number of electrons, may be sufficiently large to produce a diamagnetic ground state (eqn. 168). n n _ 1 Laporte-allowed f -> f d transitions are observed in Ce(III), Sm(II), Eu(II), Tb(III), Yb(II), Pa(IV), U(III), Np(III) and Pu(HI). It is evident that the distance between the orbital energies of 4f and 5d, or of 5f and 6d, increases with the oxidation number. The (strongly reducing) divalent lanthanides present these bands at rather low wave numbers (17-9-30-2 kK in Sm(II), 31-2 and 40-4 kK in Eu(II)), while in the trivalent ions they are only known for Ce(III) and Tb(III). This is due to the steady increase of their wave number with Ζ for a given ionic charge, bringing the bands outside the measured range already in Pr(IIl). The reappearance of a band at 45-6 kK of Tb(III) is a half-filled shell effect (eqn. 149), the transition 8 7 being 4f -> 4f*5d. Actually, the hump at f of the ground state energy, X 3 assuming eqn. (146), is 8D, where D = %E is approximately 1 3 £ n (eqn. 36) in f -systems. Consequently, 8 D is about 50 kK in the lanthanides and 30 kK in the actinides (Table 18). The effect on the n _ 1 position of the lowest level of f d is perhaps somewhat smaller, owing to the interelectronic repulsion between the f-shell and the d-electron. The energy differences in kK between the lowest level of n _ 1 n 5f and of 5f 6 d , as derived from gaseous actinide ions and their complexes, is a function of Z 0 (the ionic charge plus one) and the atomic number Ζ for, at most, seven f-electrons: 7(Z -
96) + 18(Z 0 -
1)
(158)
8
Thus, it would be expected that the 5f -system Bk(III) exhibits the first Laporte-allowed band at 31 rather than 61 kK derived from eqn. (158). 3+ The energy levels of gaseous C e are situated at: 2 2
F| F|
0 2-25
2
Df 2 Df
49-74 kK 52-23
The aquo ion is probably nine-co-ordinated,
(159) having
trigonal
184
ENERGY
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microsymmetry, which can be approximated to the linear symmetry, identifying the excited orbitals: e
Ce(IIIK 33-7 39-6 41-7 45-1 47-4
ω
«4 5 2
3
Έ
o
Pa(IV)a 0-
16 710 600 380 270
36-3 39-2 44-8
— (160)
3
£5d of Ce + is seen from eqn. (159) to be 1-00 kK, explaining a reasonable splitting into ω-levels of the λ-levels in eqn. (160). The m/z 800 200
Ι ι +' 4 ι
600
500
400
'
'—ι—ι
1
300 1
1
FIG. 26. The absorption spectra of uranium (IV) aquo ions (cf. Fig. 22), grey uranium (III) aquo ions in 1 M H C 1 0 4 or 2 M HCl, and red uranium (III) chloro complexes in 11 M HCl. The Laporte-allowed 2 3 transitions from 5f to 5f 6d are seen of U(III). The figure is reproduced with permission from Acta Chem. Scand. (ref. 281).
low intensity of the first band must be due to some selection rule of symmetry-forbiddenness. However, the present writer (1956) has previously suggested as an alternative possibility that Ce(III) aquo ions might exist with a lower co-ordination number, producing the
ENERGY
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185
COMPLEXES
weak band. The intensity of the weak band varies strongly with temperature, with the presence of other ions, and even with the substitution of heavy water as solvent. By comparison of eqn. (159) and (160), it is seen that the energy-difference 4f — 5d is some 19 per 3 + cent smaller in the Ce(III) aquo ion than in the gaseous C e . The decrease is some 28 per cent in complexes of ethylenediaminetetraacetate, citrate, chloride, etc., representing the same analogy to the 2 nephelauxetic effect as the s -» sp transitions discussed below. In the systems with more f-electrons in addition to an excited d-electron, it is not possible at present to assign the broad, Laporte-allowed bands to specific energy levels from among the many possible ones. However, the strongest bands of U(III) aquo ions (see Fig. 26), with maxima at 28-6 and 31-2 kK and a shoulder at 25-5 kK, might suggest parent terms of λ,ω-classification of the 6d-electron (eqn. 160) with a definite level of 5Ρ, e.g. the ground state Ή 4 . Solid-state physicists have been much interested in the absorption spectra and fluorescence properties of thallium (I) ions, embedded in alkali halide crystals. Seitz (1938) proposed that the weaker of two X 3 1 bands was due to the transition S0 — P X and the stronger to S 0 — X 2 PX of the configurations 6s and 6s6p, respectively. Actually, Fromherz had studied for many years similar bands of lead (II) complexes; and Macbeth and Maxwell (1923) found the rather narrow band at 30-5 kK of bismuth (III) in hydrochloric acid. These 2 s -> sp transitions can be observed for the whole series of metals, having oxidation numbers two lower than the group-number in the period system: In(I), Sn(II), Sb(III), T1(I), Pb(II), and Bi(III). If considered as problems of atomic spectroscopy, the energy of the levels with / = / of a given configuration Is are the eigenvalues of a Houston determinant E(/s) + K(/s) - Ε VVO + l)K„,/2 I_ E(ls) - K(/s) - Um — Ε |
VW + mm/2
n
n
f ni U U
°
where in Condon and Shortley's book, the exchange integral K(/s) l is denoted by Gi = G /(2l + 1). The two other levels of a configuration Is with the values of J = I — 1 and / + 1 both belong to the triplet term only and have the energies E(/s) — K(/s) — (/ + 1)ζηι/2 and E(/s) — K(/s) + Ιζηι/2, respectively. 3 3 In the case of sp, the transitions to the two levels P 0 and P 2 are completely spin-forbidden, if no other configurations are taken into
186
ENERGY LEVELS I N COMPLEXES 3
1
account. On the other hand, the two levels P X and P 1 with the original distance 2K(/s) are mixed by the non-diagonal element ζηι/\/2 according to eqn. (161). Table 19 gives the wave numbers of the transitions in the gaseous ions and in the complexes. While the second, spin-allowed transition corresponds to bands with oscillator
TABLE 19. THE WAVE NUMBERS IN k K OF THE TRANSITIONS 2TO THE TRIPLET AND 2 SINGLET STATES OF 5s5p OR 6s6p IN 5S - AND 6S -COMPLEXES
In© Sn(II) Sb(III) Te(IV) T1(I) Pb(II) Bi(III)
4
H 20
43-35 6303 55-20 79-91 66-70 95-95 78-02 111-71 52-39 75-66 64-39 95-34 75-93 114-60
—
—
—
—
— — —
—
— — — — — — — — —
— — — — —
480
41-65
41-4
—
450
— —
37-9
—
—
—
—
— — —
46-7
OH-
enta -
gaseous
—
—
ci-
i-
Br-
33, 36 43 34-7, 38-2 45-0 34-8 43-9 26-8? 33-2? 40-5 510 36-8 510 30-5 450
34 39 32 41
32 41 28 34-5
—
— — — —
— — —
38-1 47-2 33-2 44-9 26-8 38-5
35-5 43 0 27-6 —
(22-2) 29-8
strength Ρ ~ 0-8, the first bands are much weaker (Ρ ^ 0-02) for 2 2 5s - than for 6s -complexes (Ρ ^ 0-15). This agrees well with the higher values of ζ 6 Ρ than of ζ 5 Ρ. In the gaseous atoms and ions, the values are (in kK): 2
2
3
Cd In+ Sn + Hg T1+ Pb + Bi +^ 1-14 2-37 3-79 4-26 8-18 12-39 16-99 ηρ K(rcp, ns) 5-68 9-11 11-25 5-59 8-16 9-66 10-86 [ 2 E ( s p ) - E ( s ) 36-67 53-19 67-56 46-74 64-02 79-86 95-26^
ζ
I
U
}
It is seen from Table 19 that the increase with ionic charge of the gaseous ions of the three parameters of importance: the distance 2 3 E(sp) — E(s ) between the average (not the baricentre) of P X and X P 0 and the ground state; £ np and K(«p, ns) of eqn. (162) do not occur in isoelectronic complexes with different oxidation number.
ENERGY LEVELS IN
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187
Actually, a certain principle of electroneutrality is acting; in the chloro complexes the parameters are (in kK) : ζ ηρ Κ(/φ,Λ8) 2 E(sp)-E(s )
In(I) Sn(II) Sb(III) T1(I) Pb(II) Bi(III) * — — 1-9 4-2(3-6) 5-7 (4-8) 5-8(4-9) > — — 3-8 2-1(3-6) 2-85(4-8) 2-9(4-9) 39 41 39-4 44-8 43-9 37-8 (163)
The relative size of £ n p and K(«p, m) in complexes can in principle be inferred from the ratio between the oscillator strengths of the spin-forbidden and spin-allowed bands. The values given for 6s6p in eqn. (163) are calculated for ζ 6 Ρ = 2K(6s, 6p) (giving φ of eqn. 2 (23) 21-6° and sin c? = 0-136), while the values in parentheses are calculated from the observed band distances in Table 19 with ζ 6 Ρ = K(6s, 6p). The truth is probably somewhere between these limits. The values for Sb(III) are calculated by the analogous method, assuming ζ 5 Ρ = |K(5s, 5p). Though the values of ζην and the K-integral in eqn. (163), taken separately, are rather uncertain, they give an impression of analogy to neutral Cd and Hg in eqn. (162), as also does the orbital energy difference between s and p. The development of the latter quantity follows closely the nephelauxetic series, decreasing in the order (see Table 19): 4
gaseous ion > H 2 0 > OH~ ~ enta " > Cl" > B r - > I~ (164) The decrease of the three quantities in complexes (eqn. 163 and 164), as compared to gaseous ions (eqn. 162) may be interpreted in two different ways, analogous to the discussion of the nephelauxetic effect. We might ascribe it solely to central-field covalency effects, having the spherical symmetrical part V 0 of the ligand field added to the central field with consequent changes in the orbital energies of ns and «p. This would be a genuine Schläfer-Craig-Magnusson effect (p. 141), which may also act on the orbital energy differences 4f - 5d in Ce(III) and 5f — 6d in U(III) complexes. Notably, the exchange integrals Κ are very sensitive to a relative change of the two radial functions. We may also consider symmetry-restricted covalency. The mere 2 existence of s -complexes is rather surprising, since the s-orbital in any symmetry will be σ-antibonding. In octahedral complexes
188
ENERGY
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(eqn. 54), the p-orbital is both σ- and ττ-antibonding. The energy difference might either increase or decrease by this effect, and we may very well ascribe a part of the decrease in eqn. (163) to the formation of covalent bonds. The σ-antibonding character of the ground state is shown by the difficulty in forming amine and cyanide complexes, and by the great tendency to form complexes with the more polarizable halides, C I , Br~ and I~. The anomalously large formation constants of the halide complexes may partly be due to relative instability of the aquo ions. The behaviour with O H is variable; T1(I) does not form complexes at all, or at most, weakly bound ion pairs; Sn(II), Sb(III) and Pb(II) form precipitates, easily soluble in an excess of O H ; and Bi(III) forms precipitates at very low hydroxyl concentration (highly acidic solution), perhaps containing oxygen bridges. However, the value of K(«s, np) in eqn. (163) must not be taken too seriously. Already in the gaseous ion, the different /-levels of the configuration sp might have slightly different radial functions (cf. eqn. 40), and this might develop further in the complexes, making eqn. 161 an inappropriate expression of the energy difference between the mainly triplet and mainly singlet level. Actually, the band widths are some 80 per cent larger in the singlet band of Pb(II) and Bi(III) chloro complexes than in the triplet bands. Williams (1951) suggests that the internuclear equilibrium distance TL is smaller in the excited state, at least the triplet, than in the ground state. The triplet and singlet levels of T1(I) in KI have different fine-structures (Yuster and Delbecq, 1953). This is somewhat surprising, since low microsymmetries (e.g. rhombic) should produce the same number of sublevels of the two levels with J = 1. However, it can be shown that 3 the splitting of P X is only — \ as large as the splitting of Ψν Finally, quite another deviation from the case of atomic spectroscopy may occur: the highest filled M.O. of the ground state may be rather far removed from having the symmetry s. Orgel (1958) studied the stereochemistry of this family and concludes that the lone pair, at least in Pb(II) compounds, has a strong tendency to concentrate at one side of the metal ion, corresponding to strong intermixing with higher values of /. Nevertheless, crystalline PbCl 2, which has this stereochemistry with irregularly varying distances between the ligands, shows the usual spectrum with a triplet and a singlet band. Once more, we cannot estimate the exact form of the M.O. involved from the absorption spectra.
ENERGY
LEVELS
IN
COMPLEXES
189
T1(I) ions in solid alkali halides have been much discussed by Wilhams and Johnson (1951, 1952) and In(I), as given in Table 19, is only known in such crystals (Williams, Segall, and Johnson (1957); Dunina, Morgenstern, and Shamosky, (1958); Katz and Nikolsky (1958)). In solution, Merritt, Hershenson and Rogers (1953) studied T1(I), Pb(II) and Bi(III) halide complexes for analytical purposes (and also Hg(II)), and Newman and Hume (1957) made a profound study of Bi(III) chloro and bromo complexes. A further possible case, where the excited state may involve a n n _ 1 configuration from spherical symmetry, is the 3 d -> 3 d 4 s transitions producing Laporte-forbidden, broad absorption bands in strongly reducing complexes of the first transition group. Dainton (1952) correlated the wave numbers of these bands (the hexa-aquo ions of V(II) at 33, Cr(II) at 40, and Fe(II) at 41 kK) with the standard oxidation potentials, suggesting an "inverted" electron transfer process, as discussed in the next chapter. The transitions might be Laporte-allowed with accidentally small dipole moment, or spinforbidden (the K-integrals between 3d and 4s in gaseous atoms are invariably from 1-2 to 1-3 kK). However, it seems probable that the excited orbital has / = 0 to some extent and is the even yx (eqn. 54) to be filled in a (somewhat hypothetical) octahedral complex of Ga(I). The X-ray absorption edges of chemical compounds, e.g. the K-edges, representing the excitation of a ls-electron of a given atom, have been much studied. Stelling (1928-1934) observed a 2 2 variation between 19,955 kK in S ~ and 20,050 kK in S 0 4 - among sulphur compounds and found a regular increase of the wave number with the oxidation number of many elements. Later, Mitchell and Beeman (1952) measured several transition group complexes and found chemical shifts and fine structure of the absorption edge of the metal ion. These transitions were now postulated to go to 4p, 5p, . . . shells, and the Pauling hybridization theory was applied to explain that diamagnetic square-planar nickel (II) complexes have a narrow band at lower energies than the paramagnetic, octahedral Ni(II). This description in terms of an empty or a filled 4p-shell was continued by Kauer (1956), Boke (1957) and Collet (1959). However, it is obvious that the K-absorption spectrum simply indicates the available range of empty, preferably odd y 4 (eqn. 115), orbitals, if M.O. configurations can be used. The energy and wave functions of these empty M.O. depend as well on the ligands as on
190
ENERGY
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the central ion. For instance, aromatic ligands with empty orbitals of relatively low energy show the transitions at lower wave numbers (cf. the Ir-py band to be discussed in the next chapter). Stelling (1932) found one of the two K-edges of covalent chloro-complexes at a lower wave number than that of ionic Cl~. This is not as well explained by CI having an oxidation number below — 1 , as by the available empty orbitals in metal complexes. 2+ 2+ A significant fact is that N i ( N H 3 ) 6 and C u ( N H 3 ) 4 both have 2+ two bands with an energy difference 30 kK, while Z n e n 3 only has one (Cotton and Hansen, 1958). This influence of the partly filled d-shell might be connected with interelectronic repulsion energies, but is probably too large to be explained in that way. It is probably not possible to use ligand field theory on the 4p-shell (Cotton and Ballhausen, 1956) since the splitting is also observed in 2+ 3 the octahedral N i ( N H 3 ) 6 and C o ( N H 3) 6 + . It would be useful to have more experimental facts available, especially on the more loosely bound electrons (2p, 3d, etc.) of complexes of heavy metals. However, the resolution limit is not good, some 20 kK for bands in the K-region, though the relative accuracy of the wave number is very high. Collet (1959) measured the chemical shifts of several 2p absorption bands of rhenium (IV), platinum (II) and platinum (IV) complexes.