Nephelauxetic Effects in the Electronic Spectra of Pr3+ - The Journal

Article pubs.acs.org/JPCA

Nephelauxetic Effects in the Electronic Spectra of Pr3+ Peter A. Tanner* and Yau Yuen Yeung Department of Science and Environmental Studies, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong S.A.R., People’s Republic of China S Supporting Information *

ABSTRACT: The well-known nephelauxetic series of ligands describes the change in interelectronic repulsion of the central metal ion, which is reduced on going from the vapor to crystalline state. This study examines the trends and quantifies the mechanism of this series for the lanthanide ion Pr3+, with the 4f2 electronic configuration. A new and concise measurement by a single parameter, σee, is introduced to quantify the overall strength of interelectronic repulsion, as the alternative to the Slater parameters, Fk (k = 2, 4, 6). Energy parameters have been derived from the literature electronic spectra of Pr3+, in the free ion and in various crystalline hosts, with new calculations in some cases. It is found that at least the first 12 of the 13 multiplet terms of Pr3+ must be well-determined to obtain reliable parameter values. The shifts of various energy levels for changes in the Slater parameters are not uniform in direction. For the various Pr3+ solid-state systems, the change in σee is only up to ∼5%, with the magnitude of σee in the order F− > Cl− > O2− ≈ Br− > C, and decreasing with lower coordination number of the ligand. The decreases of the Slater parameters from the free ion values are reasonably estimated by considering the dielectric constant of the medium. In particular, the magnitude of σee (and of the spin−orbit coupling constant) is proportional to the polarizability of the ligand for F−, Cl−, O2−, and Br−. The data point scatter for oxide systems is accounted for by considering the individual ligand bond distances. A fair estimation of nephelauxetic effects can be made from some luminescence transition energies, by contrast with Eu3+ systems where crystal field effects also play a major role. In conclusion, the nephelauxetic effect of Pr3+ is due to the polarization of the ligand by one 4f electron, and the interaction of the other electron with the induced multipolar moments, of which the dipole moment dominates.

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INTRODUCTION The contraction of free ion parameter values, notably concerning interelectronic repulsion and spin−orbit coupling, from those in the gas phase to those in crystalline materials has been termed the nephelauxetic effect. Jørgensen1 studied the spectra of transition metal complexes and arranged a nephelauxetic series for ligands: F− < O2 − < Cl− < N3 − < Br − < I− < S2 − < Se 2 −

in order of decreasing electronegativity of the corresponding element. The d-orbitals of transition metals are more widely extended than the core-like 4f orbitals of lanthanide ions, and it has been of interest whether similar behavior could be found for Ln3+ systems. Moreover, the mechanism of the nephelauxetic effect, the so-called cloud expansion effect, is still controversial, and it still receives attention for d- and f-electron systems.2,3 A decrease in interelectron repulsion on going from the free ion to the crystal would be expected to lead to a contraction, not an expansion, of the electron cloud.4 Free ion and crystal field calculations are performed for lanthanide ions by using a semiempirical Hamiltonian comprising atomic, crystal field, and some additional terms representing various interactions.5 The free ion energy levels of Pr3+, as experimentally determined by Sugar,6 are shown diagrammatically in Figure 1. For the calculation of these © 2013 American Chemical Society

Figure 1. 4f2 free ion J-multiplet energy levels of Pr3+.6 The electronic ground state is 3H4. The calculated value is given for 1S0.

energy levels, the Hamiltonian and the parameters have been described in detail previously, and the reader is referred elsewhere. 5,7,8 The notation employed herein for the Received: August 28, 2013 Revised: September 23, 2013 Published: September 24, 2013 10726

dx.doi.org/10.1021/jp408625s | J. Phys. Chem. A 2013, 117, 10726−10735

The Journal of Physical Chemistry A

Article

parameters is Fk (k = 2,4,6) for the Slater parameters (representing interelectron repulsion interactions); ζ4f for the spin−orbit coupling constant (representing the interaction between electron spin and orbital angular momenta); α, β, γ for the two-body configuration interaction parameters; and Bkq for the crystal field parameters. Some additional parameters in the atomic Hamiltonian fine-tune the term energies. The Mk (k = 0, 2, 4) parameters describe the spin−spin and spin−other orbit interactions between electrons, whereas the electrostatically correlated spin−orbit interaction, Pk (k = 2, 4, 6), allows for the effect of additional configurations upon the spin−orbit interaction. Although these additional parameters have been employed in our calculations, they are not of significant importance in this work. Wong and Richman9 noticed that the barycenters of multiplets in the spectra of Pr3+ in LaCl3 were shifted downward by ∼1000 cm−1 for LaBr3:Pr3+. The comparison of the interelectronic repulsion showed a 0.5% decrease for the Slater parameters in the bromide system. These authors indicated two explanations for these changes. First, an Ionic Model envisages mixing between 4f2 and a higher configuration, with the 4f orbital extending slightly outside the ionic radius. Wensky and Moulton10 argued that the host-dependent changes in Fk and ζ4f arise from screening terms provided by these higher configurations. The orbitals, such as 6s and 6p, of higher configurations extend much more into space than 4f and are more susceptible to distortion by the surrounding ligands. The extent of compression of the 4f orbital then depends upon the metal ligand distance, or the change of ionic radius from one crystal to another. Because the interelectronic repulsion varies as r−1, whereas the spin−orbit coupling varies as r−3, the effect of the compression will be greatest for the former. This model therefore predicts more significant changes in the Fk parameters as compared to ζ4f, from one crystal to another. The second model, cited by Wong and Richman, the Covalent Model, takes into account the fact that 4f electrons are shielded by various mechanisms, with the direct mixing of 4f electrons with ligand electrons being of note. Many authors have associated nephelauxetic effects with covalency,11−13 but Newman14 has argued that the reduction of Slater parameters is not related to covalency resulting from ligand charge interpenetration of 4f orbitals, because otherwise the crystal field parameters calculated using Hartree−Fock 4f orbitals would greatly underestimate the experimental values. Newman calculated the covalency contributions to be an order of magnitude too small to account for Slater parameter shifts from the free ion to the crystal. It was discovered from the pressure-dependence of the optical spectra of Pr3+ that both Fk and ζ4f decrease with increasing pressure.15 The decreases of Slater parameters follow the order ΔF2/F2 > ΔF4/F4 > ΔF6/F6, and the relative decrease Δζ4f/ζ4f is smaller than that of F2, but similar to those of F4 and F6. Holzapfel’s group15,16 employed a combination of the Central Field Model and the Symmetry-Restricted Covalency Model to attempt to explain the results for ROCl:Pr3+ (R = Pr, La, Gd) and LaCl3:Pr3+ under pressure. The Central Field Model envisages that Fk and ζ4f decrease from the gas to the condensed phase due to an isotropic expansion of the 4f orbitals. This expansion occurs because of the screening of the lanthanide nuclear charge by the spherically symmetric interpenetration by ligand orbitals. Newman has pointed out that a spherically symmetric medium outside and centered on the Pr3+ ion would not contribute a potential gradient.4 The

Symmetry-Restricted Covalency Model interprets the 4f orbital expansion as nonisotropic, caused by a symmetry-dependent admixture of ligand orbitals. It was concluded that neither model can explain the changes of Fk and ζ4f under pressure, and that the dielectric screening model fits the experimental data better, but that a microscopic basis of this model is lacking. Newman has proposed that the mechanism of Morrison et al.17 involving the dielectric screening of the interelectronic repulsion by the crystal medium can account for the reduction of Slater parameters, and that it effectively includes the Ionic Model. This model envisages the free ion being placed in a sphere of radius slightly greater than the ionic radius, with dielectric constant of unity. In a crystal, the surrounding solid has a greater dielectric constant, and this changes the interaction potential between electrons. Correction terms to the Slater parameters for d-electron systems were given by Morrison et al.17 and for f-electron systems by Newman.14 Assuming the validity of this dielectric screening model, Angelov18 utilized the Slater parameter shifts from the free ion values together with the dielectric permittivity of crystals doped with, or containing Pr3+, to calculate the radial integrals ⟨rk⟩ (k = 2, 4, 6) for Pr3+ in certain solid-state materials. The values for ⟨r4⟩ and ⟨r6⟩ each varied by more than a factor of 2 for the different crystals. Angelov noted that the nephelauxetic ratios for different crystals, βk = Fk(crystal)/Fk(free ion), were k-dependent, but indicated stronger nephelauxetic effects for Pr3+ than for Er3+. We note that it is probable that the errors in fitted Slater parameters do not justify the calculation of accurate values for radial integrals. A revisit to the mechanism of the nephelauxetic effect in delectron systems has recently been made by Tchougréeff and Dronskowski,19 who pointed out the lack of coincidence of the nephelauxetic and spectrochemical series, and suggested an interpretation based upon the polarization of the ligands. Our previous study investigated the validity of the use of optical transition energies to elucidate nephelauxetic effects, with the case study of Eu3+ in crystals.8 It was discovered that crystal field effects, and not electron−electron repulsion effects, are mainly responsible for dramatic changes in the 5D0−7F0 energy. However, we recognized that a less-complicated study could be based upon Pr3+, where certain energy parameters (Tk) are not present in the empirical Hamiltonian (because a correction is then not needed for fitted Fk parameters), and where the energy level scheme is much simpler and experimentally more completely determined, so that fitted Fk parameters are more accurate. The 4f2 configuration only encompasses 13 multiplet terms (Figure 1) and a maximum of 91 energy levels, by comparison with 4f6 Eu3+ (up to 3003 energy levels). Hence, in the present study of Pr3+ systems, we have investigated the effects of Slater parameter variations due to missing energy level empirical data, the effects of parameter changes upon energy level positions, various correlations concerning interelectronic repulsion parameters and ligand type or coordination number from literature data sets, and the validity of utilizing certain electronic transition energies as indicators of nephelauxetic effects. The mechanism of the nephelauxetic effect for 4fN ions is also clarified by simple calculations using the variables of dielectric constant, polarizability, ligand metal−bond distance, and coordination number. Moreover, to clarify the notion of interelectron repulsion, a novel concise parameter has been introduced and defined. 10727



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Article

THEORETICAL SECTION Overall Strength of Electrostatic Repulsion. The Slater parameters Fk are related to the Racah parameters Ei by the relations:7 F 2 = (75/14)[E1 + 143E2 + 11E3]

(1)

F 4 = (99/7)[E1 − 130E2 + 4E3]

(2)

6

1

2

3

F = (5577/350)[E + 35E − 7E ]

σee2 =

(11)

The normalization constants Ni(n) for other 4f configurations with n electrons are related to those of 4f2 by the following relation (for i = 1, 2, or 3): Ni(n)/Ni(2) ≡

(3)

∑

Hee =

E ei =

i = 0,1,2,3

N (n) ≡

∑

∑ (2Sψ + 1)(2Lψ

+ 1)⟨ψ |ei|ψ ′ ψ ′|ei ′|ψ ⟩ (5)

Alternatively:

Tr[eî eî ′] = δii ′

(6)

For the 4f2 electronic configuration, the normalization constants are given as follows:

ΔF k =

(7) i

E ̂ = Ni·E

and

1 N

∑ (E ψ

(8)

− EAVE)2 (9)

ψ

ΔF k =

where Eψ is the eigenvalue for the Hamiltonian Hee as given in eq 4 and N is the total number of energy levels. Because the quantity EAVE is simply E0e0, if it is removed (i.e., to make the Hamiltonian to be traceless Tr[Hee] = 0), then we can get a very simple expression for σee:

∑

∑

αbNbe 2 R b2k + 4

(k + 1)|⟨r k⟩|2

Δζ4f = constant ×

(17)

∑ b

αbNb R b6

(18)

Crystal Field Strength. Traditionally, it is convenient to concisely quantify the overall crystal field strength in terms of the Bkq crystal field parameters by defining the scalar crystal field strength parameter, Nv:23

Ni2(Ei)2

i = 1,2,3

(16)

where Nb is the number of ligands with polarizability αb at a distance Rb from the metal ion. A corresponding equation has been given for the decrease of the magnitude of the spin−orbit coupling constant from the free ion to the value in a crystal, Δζ4f:22

1 Tr[(Hee − E 0e0)2 ] N 1 = ∑ (Eî )2 N i = 1,2,3 1 N

(15)

r k 2 e2 (ε − 1) [(k /(k + 1)) + ε] R2k + 1 4πε0r0

b

σee2 =

=

(14)

where the correction ΔFk is in cm−1; the radial integrals ⟨rk⟩ are in (atomic units)k; the effective radius, taken as the Ln3+ ionic radius, R, is in atomic units; e is the electronic charge; and r0 is the Bohr radius. From the Clausius−Mossotti relation, the variation of a quantity with dielectric constant is expected to give rise to another relationship with molecular polarizability. The effect of ligand polarization upon the decrease of the Slater parameters may be roughly estimated by the formula:21,22

i

The average standard deviation (also known as the second moment) σee of all energy levels due to electrostatic repulsion may be used as a good measure of its strength, using the formula: σee2 =

12! × 2 n! (14 − n)!

F k = F k(FI) − ΔF k

1890 N02 = 91; N12 = ; N22 = 1 081 080; N32 = 13 860 13

Note that: e eî = i Ni

(13)

Application of these formulas shows, for example, that the value of σee lies in the range of 8000−9000 cm−1 for Pr3+ systems, as compared to, for example, 12 000−13 000 cm−1 for Tm3+ systems. Correction to Slater Parameters from Morrison’s Electrostatic Shielding Model. A formula derived by Morrison et al.17 for d-electron systems accounted for the modification of Slater parameters for an ion in a crystal (Fk), as compared to the free ion, F k (FI). The formula was subsequently given by Newman for Ln3+ f-electron systems:14

ψ ,ψ ′

= Ni2δii ′

14! n! (14 − n)!

N (n)/N (2) ≡

(4)

where the second Ei (with cap) refers to the orthonormalized operators. The trace operation on the Racah operators is: Tr[eiei ′] =

(12)

or

i E ̂ eî

i = 0,1,2,3

10! (n − 2) ! (12 − n)!

For i = 0, an additional factor of (n + 1)/(n − 1) is needed due to the definition of the e0 operator = n(n − 1)/2. The total number of levels for each 4f configuration is given by:

The nephelauxetic effect involves a change in interelectronic repulsion on going from the vapor to the condensed phase, so that to quantify such an effect it is desirable to have one parameter that represents the overall electrostatic repulsion. This may be achieved by using orthonormalized Racah operators. The energy for electrostatic repulsion, in Racah’s notation,20 is given by: i

⎤ 1 ⎡ 1890 1 2 (E ) + 1 081 080(E2)2 + 13 860(E3)2 ⎥ ⎢⎣ ⎦ 91 13

(10)

For 4f2, we then have the following specific formula: 10728


The Journal of Physical Chemistry A ⎡ ⎤1/2 4π 2⎥ ⎢ (Bkq ) Nv = ∑ ⎢⎣ ⎥⎦ 2k + 1 k ,q

Article

vibrational fingerprint then provides an aid for the location of the respective pure electronic transition. The reader is referred to the figure of the 3H4 → 3P0 vibrational sideband of LaF3:Pr3+ (Figure 3 in ref 28) to illustrate that vibronic sidebands can be extensive even for some transitions of Pr3+ situated at a low symmetry site. The spurious effects upon parameter values for incomplete spectral data sets of Pr3+ in crystals are now examined. The most accurate and complete data set available for Pr3+ in the solid state is that for this ion diluted into the LaCl3 crystal, from the careful studies of Sarup and Crozier29 and Rana and Kaseta.30 A new analysis of this data, including the effects of spin−spin interactions, has recently been published.5 The data set has been reanalyzed herein with the omission of certain multiplets to determine the consequent effects and changes upon parameter values. The detailed description is provided in the Supporting Information, and the main conclusions are now listed. In summary of our calculations, we do not further consider literature data where levels from both the 1I6 and the 1S0 multiplets are absent, even though lower levels have been assigned, because the fitted Slater parameters would be misleading. Only three data sets are available for Pr3+ systems which include assignments for 1S0. We therefore consider the values of Slater and spin−orbit coupling parameters for other systems without 1S0 assignment, but cautiously remember the fact that the fitted Fk parameters may be 200−500 cm−1 too high. Effects of Parameter Changes upon Energy Levels. Because the analysis of the LaCl3:Pr3+ system is the most accurate and comprehensive one available, the effects of parameter changes upon the energy levels were investigated with a view to finding the most sensitive levels. Each energy parameter Fk (k = 2,4,6), ζ4f, α, β, γ, B20, B40, B60, and B66, was individually increased by 10% from its best-fit parameter value for LaCl3:Pr3+, and the calculation was repeated to observe significant changes in energy level positions. It was observed that changes in the crystal field parameters only modified energy level values slightly, by a few cm−1, and that no levels crossed. The shifts of 1D2 levels for 10% increases in F2, F4, or F6 are about 1380 cm−1 upward, 180 cm−1 downward, and 230 cm−1 upward, respectively, whereas 3P0 shifts up by 2137 cm−1, up by 378 cm−1, and down by 559 cm−1, respectively. The corresponding shifts of other levels are described in the Supporting Information. The shifts of different multiplets differ considerably, and also differ for each Slater parameter. In conclusion, there is no level that provides a clear indication of simultaneous decreases in Fk and ζ4f, although 3 P0 shifts down appreciably for a 10% overall decrease in these parameters. The shift of 3P0 is evident from the relation:

(19)

and a similar definition for the rank-k crystal field strength (or so-called quadratic rotational invariant):24 ⎡ 1 sk = ⎢ ⎢⎣ 2k + 1

⎤1/2

2⎥

∑ |Bkq| ⎥ q

⎦

(20)

where the summation is over the magnitude squared for every q component of the crystal field parameter. Similar to our present definition of σee in eq 9, there is a better equivalent parameter for Nv, which is defined as: 2

σCF = [∑ sk2 l||C(k)||l ]1/2 q

(21)

where the square of the reduced matrix element for the spherical harmonic operator C(k) inside the summation sign takes the values of 28/15, 14/11, and 700/429 for k = 2, 4, and 6, respectively. This is also called the normalized crystal field strength parameter because it carries a clear physical meaning for representing the spread (or standard deviation) of crystal field splittings in the case of a one-electron configuration.

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RESULTS AND DISCUSSION Free Ion Energy Levels and Parameters. The energy levels of the J-multiplets of the 4f2 configuration of Pr3+ in the vapor state have been reported by Sugar6 and by Crosswhite et al.25 Because of the parity selection rule, the 4f2−4f2 transitions are electric dipole forbidden in the gas phase so that the energy levels were obtained from 4f5d → 4f2 emission spectra. Remarkably, the two studies showed agreement to within 0.3 cm−1. The data set of Crosswhite et al. is more complete because it includes the 1S0 energy at 50 090.3 cm−1, but the assignment of this level has been questioned,16 and it is not included in the Atomic Spectra Database of NIST.26 This level has also been omitted in the more recent analysis of the Pr3+ free ion data by Wyart et al.27 The parameter fits to the free ion data sets of Crosswhite et al. and Sugar are described in the Supporting Information, and these provided further evidence for discarding the assignment of 1S0 by Crosswhite et al. The major outcomes of these fits are the values of Slater and spin−orbit parameters for the Pr3+ free ion (in cm−1): F2 = 72 222 ± 47; F4 = 52 940 ± 122; F6 = 35 097 ± 138; ζ = 764.4 ± 0.1. Effects of Missing Energy Levels upon Energy Parameters. There are experimental difficulties in collecting complete and accurate electronic energy data sets for Pr3+ in crystals. Although there are only 13 2S+1LJ multiplets in the 4f2 configuration of Pr3+, only a few studies have succeeded in assigning levels in each of these multiplets. This is partly because the highest multiplet, 1S0, is usually inaccessible because it lies within the host band gap or within the 4f5d configuration of Pr3+ in many crystals. Assignments for 1I6 also prove to be difficult unless 1S0 → 1I6 emission is observed, because the 3H4 → 1I6 structure in the ultraviolet absorption spectrum is usually weak and broad. In many cases, assignments are complicated by the appearance of vibronic transitions, and the distinction of such structure from weak electronic transitions is difficult. Naturally, these transitions dominate the intensity for Pr3+ situated at centrosymmetric sites, but the

E(3P) = F0 + 45F2 + 33F4 − 1287F6 = F 0 + 0.2F 2 + 0.0303F 4 − 0.174825F 6

(22)

2

so that, as from our calculations, F is expected to have the major influence upon 3P0 and its effect complements that of ζ4f. It therefore appears plausible that the 3P0−3H4 energy might serve as an indicator of nephelauxetic effects. We also consider 1 D2−3H4 and 1S0−3H4 from literature experimental data as alternative candidates. The Pr3+ site in LaCl3 has C3h symmetry. Our previous investigations of nephelauxetic effects in Eu3+ systems indicated the importance of J-mixing of the 7F0 ground state in low 10729



Article

Table 1. Fk, ζ4f Parameters, 1S0, 3P0 → 3H4 Ground State, and Lowest 1D2 → 3H4 Transition Energies, and Second Moment σee (All in cm−1) for Pr3+ in the Free Ion and Solid-State Systemsa system

F2

F4

F6

ζ4f

Pr free ion LaF3 LiYF4 KCaF3 K2YF5 KY3F10 BiLiY4 K2LaCl5 LaCl3 Cs2NaPrCl6 ThCl4 ThBr4 K2LaBr5 LaBr3 CsCdBr3 Cs2NaYBr6 SrLaAlO4 Na5Pr(MoO4)4 LaGa3O6 La2Be2O5 Y2SiO5 Gd2SiO5 Pr(H2PO2)3 Y3Ga5O12 Pr3Ga5O12 Gd3Ga5O12 YPO4 GaPrGe2O7 Pr(H2O)2[glut]3·4H2O LiYO2 PrVO4 Pr2O3 Y3Al5O12 Y4Al2O9 LaAlO3 YAlO3 KY(WO4)2 KGd(WO4)2 Pr2Mg3(NO3)12·24H2O K2Pr(NO3)5·(H2O)2 Pr(AP)6I3 PrP5O14 La2O3 La(BO2)3 SrLaGa3O7 La(C2H5SO4)3·9H2O PbMoO4 Cs2NaPr(NO2)6 Gd2Ti2O7 La2Ti2O7 YCl3·6H2O Sr5(PO4)3F PrOCl LaOCl GdOCl La2O2CN2 YOCl La2O2S (Me3SiC5H4)3Pr N(n-Bu)4+(MeCp3)PrCl−

72 222 68 993 68 734

52 940 50 563 50 785

35 097 32 805 33 237

764.4 747.7 748.4

3+

68 282 68 439 67 776 67 947 68 354 66 817 68 431 67 550

50 391 50 226 49 846 50 576 50 310 48 400 50 775 49 076

33 158 32 973 32 787 33 468 33 799 31 332 33 673 32 311

739.5 748 744 742 739 748.4 746.7 741.1

66 570 67 871 68 039

44 503 49 296 49 047

30 328 32 299 32 212

756 737.6 742.0

68 601 67 466 67 148 67 359 67 824 68 228 68 111 66 902

50 508 49 923 50 390 49 476 49 661 48 725 49 700 50 610

33 266 32 505 32 505 32 282 32 473 32 006 32 441 33 927

742.9 745 745 746 745.8 739 726.5 708.4

66 454 67 007 67 270 69 304

49 287 49 014 47 566 56 430

32 476 32 120 31 460 37 374

741.5 722.6 754.3 755

69 644

54 173

34 167

763.4

66 566 67 914

49 529 49 339

32 575 31 937

736 743.3

69 165

53 840

37 824

727.9

67 288 67 291 67 107 67 379

49 747 50 141 49 896 51 814

32 926 32 967 32 881 33 943

741 742 741 746

66 141

49 023

32 188

745.1

S0−3H4

1

46 962

46 773 46 451

P0−3H4

3

20 925 20 860 20 907 20 919 20 728 20 853 20 530 20 475 20 626 20 517 20 428 20 432 20 372 20 393 20 486 20 427

Nexpt

σee

σCF

ref

16 872 16 740

74 46

8915 8520 8495

16 619 16 654 16 744 16 570 16 631 16 666 16 710 16 652 16 634 16 571 16 540

0 835 1053 1004 1065

86 61 36 52 42 70 35 37

8437 8451 8371 8405 8440 8238 8461 8329

27 35 71

8118 8369 8381

677 351 779 836 701 536 312 865 612 754 797 1173

76 68 56 62 50 50 36 64

8474 8341 8320 8319 8373 8394 8405 8293

784 1550 1535 1543 749 690 810

27 51 49 42

8217 8271 8265 8689

1030 1689

* * 31, 32,* 33 34 35 36 37,* * 38 39 40, 41 42,* 9,* 43 44 45 46 47 48 49 49 50 51 51 51 52 53 54 55 56 57 58, 59 60 61, 62 63, 64 65 65 66 67 68, 69 70 57 71 72 73 74 75 76 76 77 78 15, 79 15, 80 15 81, 79 76

16 669

20 685 20 501 20 499 20 609 20 716 20 597 20 577 20 591 20 481

16 617 16 493 16 507 16 610 16 656 16 404 16 468 16 430 16 461

20 649 20 463 20 249 20 254 20 534 20 668 20 594 20 417 20 344 20 382 20 863 20 762 20 949 20 798 20 274 20 698 20 648 20 687 20 500

16 690 16 356

20 582 20 462 20 657 20 806 20 245 20 283 20 385 20 165 20 046 20 196 20 008 10730

D2−3H4

1

16 342 16 413 16 479 16 694 16 380 16 314 16 346 16 880 16 898 16 726 16 370 16 676 16 536 16 709 16 583 16 822

16 694 16 841 16 301

67 64 11 18

8677

42 54

8235 8377

25

8603

20

1219 970 945 376 1087

1016 813

866

16 272

26 32 30 31

8316 8326 8300 8375

738 774 705 876

16 567 16 289

19 42

8178

1211

82 83,*



Article

Table 1. continued system Cp3Pr·CNC6H11 Cp3Pr(NCCH3)2 Cp3Pr·n-BuAc Cp3Pr·MeTHF

F2

F4

F6

ζ4f

65 646

48 151

32 172

736.6

66 207

49 184

32 543

742

S0−3H4

1

P0−3H4

3

20 008 20 067 20 121 20 109

D2−3H4

Nexpt

σee

σCF

ref

16 429 16 458 16 495 16 462

46 38

8102

1672

42

8187

1431

84−86,* 87 88 86, 89

1

Nexpt is the reported number of levels in the energy level fit. A star indicates that the fit was performed herein. Glut, [O2C(CH2)3CO2]; AP, antipyrene; Me, methyl; Bu, butyl; Ac, acetate; Cp, η5-cyclopentadienyl; THF, tetrahydrofuran. The transition energies refer to the zero phonon lines of the transition from the lowest upper level to the lowest ground-state level. Starred references refer to revised parameter values from calculations in this work. a

Figure 2. Boxplots of second moment due to electrostatic electron repulsion versus (a) ligand type and (b) coordination number of oxygen. The number of data, N, is indicated. (c,d) Ligand type for 9- and 8-coordination.

symmetry systems, thereby pushing it downward in energy.8 Further calculations were therefore performed for LaF3:Pr3+, where the Pr3+ ion has C2 site symmetry to ascertain the importance of crystal field effects for Pr3+ systems. The 3H4 ground state of Pr3+ can J-mix with 3H5 through rank 2, 4, and 6 crystal field parameters. However, the extent of J-mixing is predicted to be small due to the ∼2000 cm−1 gap between the ground state and 3H5. The calculations confirmed these predictions. A 10% increase in any one of the nine C2v crystal field parameters for LaF3:Pr3+ produced calculated energies for the 3 P 0− 3 H4 (1) and 1 D2(1)−3H4(1) transitions in the ranges of 20 942−20 948 and 16 896−16 904 cm−1, respectively. The energy shifts are negligible as compared to those generated by changes in Slater parameters. The wave function of the electronic ground state 3 H4(1) in LaF3:Pr3+ has the composition of 97.3% 3H4 character, with only about 2.0% 1G4 and 0.04% 3H5 character mixed in. Similarly, the 3P0 wave function is very pure, comprising 98.5% 3P0, 0.2% 3P2, and 0.1% 1S0. The wave function of 1D2(1) in LaF3:Pr3+ is the least pure of these three, comprising 89.5% 1D2, 7.6% 3P2, and 2.4% 3F2.

Changes in Parameter Values with Ligand Type and Coordination Number. Literature data for fittings of Pr3+ energy levels in crystals have been critically examined, and studies without the inclusion of 1I6 levels in the energy level fittings have been excluded in Table 1. Some studies of Pr3+ systems, such as for Pr(H2O)2[O2C(CH2)3CO2]3·4H2O,54 or Pr3+ doped into YAG,58,59 LiYO2,55 or La(C2H5SO4)3·9H2O,73 appear to have anomalously low fitted values of ζ4f. Also, literature parameter data for SrLaAlO445 include very low values for F4, F6, whereas that for La(C2H5SO4)3·9H2O73 includes very high values for these two parameters. We have not deleted these systems from Table 1 so that we do not bias our conclusions, but they are viewed with caution. In many cases, the reported energy level data for Pr3+ systems have progressed through modifications due to reassignments over the years, and we have taken the most recent or the most reliable report available for each case. Many of the assignment problems relate to the difficulty in distinguishing vibronic and pure electronic structures, even for low symmetry systems. For example, there have been numerous studies of Pr3+ at the C2 site in LaF3 (for example, refs 90−93). However, the energy level fittings of Carnall et al.,92 as well as subsequent ones, have 10731



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Table 2. Calculated Shifts of Slater Parameters (in cm−1) from the Free Ion Values (Eq 16) as a Function of Ionic Radius of Pr3+a

not taken into account the symmetry irreps of energy levels. Therefore, we have studied the reported optical polarization data, the inelastic neutron scattering data,94 and the comparison with the assignments for PrF395 in refining our energy level data set for this system. The optical spectroscopic data for Pr3+ doped into LiYF4 have similarly undergone revisions throughout the years. Subsequent energy level data reported for the 1I6 levels by two groups showed complete disagreement.96,97 We have partially resolved this by recognizing that the mechanism for the spectral data reported by Xie et al.97 is not from two-color two-photon spectroscopy but rather from excited-state absorption from 1G4. The data set for PrI398 have been excluded because the assignments are invalid and because the energy levels cannot be fitted with sensible parameter values. At 10 K, emission does not occur from the seven Pr3+ energy levels, above and including 3P0, because they are too closely spaced. Furthermore, the irreps for odd-J multiplets are incorrect, and no account of forced electric dipole selection rules was made in formulating the given assignments. We have repeated some of the energy level calculations for the systems in Table 1, as indicated by stars, and our parameter values are listed in those cases. For the moment, we focus upon the Fk and σee values for the systems in Table 1. Figure 2a displays the relationship between the ligand type and σee values in a boxplot. The free ion value of 8915 cm−1 is reduced to ca. 8500 cm−1 in fluoride hosts, and then progressively lower to the value near 8150 cm−1 for η5cyclopentadienyl compounds where the effective charge on each carbon atom is 0.2. The LnOCl:Pr3+ systems (Ln = La, Gd, Pr: not shown in the figure), with the coordination LnO4Cl5, exhibit σee values in the region of 8313 ± 13 cm−1. The reduction from the free ion value to fluoride systems is thus less than 5%, and the entire variation over solid-state systems is less than 5% of the free ion value. There is much scatter for the values of σee for each ligand type, as exemplified by oxygen in Figure 2a, where considerably more data are available. This scatter arises from several reasons, including the different Pr−O distances and coordination numbers of Pr3+ in the crystals, and is subsequently investigated. The effect of coordination number is shown for oxygen systems in Figure 2b. The larger Pr3+ coordination numbers (such as 12, for Pr2Mg3(NO3)12·24H2O and LaAlO3) generally present greater σee values. Figure 2c,d shows that the range of σee values for different ligands, but with the same coordination number, is only several hundred cm−1. The involvement of the medium dielectric constant is anticipated to be of importance. Although the model of Morrison et al. is very simple, its application can provide Slater parameter shifts, ΔFk (eq 16), of the correct order of magnitude, in addition to illustrating certain other points. The only experimental data required are the dielectric constant of the material and the radius of a fictitious cavity surrounding the rare earth ion, taken as the ionic radius. Table 2 collects calculated values of ΔFk as a function of the ionic radii of Pr3+ for different coordination numbers, with the same dielectric constant of the crystal. The shift from the free ion value of each parameter increases with decrease in coordination number, just as has been demonstrated from the fitted parameter values for oxygen systems in Figure 2b, and the comparison of Figure 2c,d. Taking the appropriate dielectric constants and ionic radii, the Fk values for some Pr3+ systems in Table 3 are also calculated using eq 16. The comparison is made in the table

R(VI) R(VIII) R(IX)

ΔF2

ΔF4

ΔF6

10 067 5196 4185

5451 1657 1123

13 790 2470 1407

⟨r2⟩ = 1.09 (a.u.)2; ⟨r4⟩ = 2.82 (a.u.)4; ⟨r6⟩ = 15.73 (a.u.)6; ε = 13.8; R(VI) = 1.871 a.u.; R(VIII) = 2.135 a.u.; R(IX) = 2.23 a.u. a

Table 3. Calculated Slater Parameters in cm−1 (Eq 16) for Some Pr3+ Systemsa system LaF3

LaCl3

LaBr3

Y3Al5O12

calcd exptl % diff. calcd exptl % diff. calcd exptl % diff. calcd exptl % diff.

F2

F4

F6

σee

68 037 68 993 −1.39 68 349 68 439 −0.13 68 277 68 431 −0.23 67 141 67 007 0.20

51 817 50 563 2.48 51 907 50 226 3.35 51 886 50 775 2.19 51 322 49 014 4.71

33 690 32 805 2.70 33 804 32 973 2.52 33 778 33 673 0.31 32 686 32 120 1.76

8446.2 8519.6 −0.86 8481.2 8450.7 0.36 8473.1 8460.7 0.15 8342.7 8270.5 0.87

a Free ion Fk values as in Table 1; ionic radii and ⟨rk⟩ values as in Table 2; ε(LaF3) = 13.8; ε(LaCl3) = 8.53; ε(LaBr3(calc)) = 9.373; ε(Y3Al5O12) = 11.7. Exptl denotes the fitted values from Table 1.

with the values of Fk fitted from energy level parametrizations. The lower values for Y3Al5O12:Pr3+ are reproduced. The relation between σee and crystal field strength Nv is weak (R2adj = 0.13, N = 34), and also σee vs σCF (R2adj = 0.263, N = 34). The fit improves with decreasing rank of crystal field strength: σee and S2 (R2adj = 0.34, N = 35), σee and S4 (R2adj = 0.112, N = 35); σee and S6 (R2adj = 0.003, N = 35). However, in conclusion, the major changes in interelectronic repulsion are not due to crystal field effects. Although the data are scattered (Figure 3a), there is a significant linear relation between the ligand polarizability and σee or the Slater parameter F2 (N = 37), as expected from eq 17. These results are analogous to the linear relationship discovered by Newman for the d-electron system Co2+ between ligand polarizability and Racah parameter B.4 Newman concluded from this relation that the nephelauxetic effect is due to ligand polarizability. Figure 3a therefore confirms Newman’s suggestion and demonstration for d-electron systems4 that the 4f interelectronic repulsion is linearly related to ligand polarizability. The data for the F2 parameter for oxide systems in Table 1 spread over ∼3000 cm−1. This may be accounted for by considering the individual Pr−O bond distances, as in eq 17. Figure 3b shows that a reasonable linear relation is found for the plot of ΔF2 and ∑b(Nb/R8b), for oxide systems where crystallographic data are available to us. Equation 17 is in fact an approximation because only dipolar interactions are considered while higher order multipolar interactions are neglected. Furthermore, the oxide data set in Figure 3b comprises some Pr3+ compounds, in addition to doped crystals. For the latter, the host metal−oxide bond distances have been taken so that the bond length changes upon introduction of the guest ion were not taken into account. 10732



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Figure 4. Plot of spin−orbit coupling constant versus ligand polarizability from all data.

transition energies are fairly representative of interelectronic repulsion effects. The plot of 3P0−3H4 ground-state energy versus the Slater parameter F2 has a higher adjusted coefficient of determination, R2adj, Figure 5c. This contrasts with the scenario for Eu3+, where the 5D0−7F0 energy is not a reliable indicator of interelectron repulsion because the ground-state 7 F0 energy is severely depressed in low symmetry systems. As reasoned above, crystal field effects are relatively unimportant in tuning the energies of 3P0 or 3H4, so that for the Sr5(PO4)3F:Pr3+ system (Table 1), the 3P0 state is at 20 806 cm−1 above the ground state, and this is not an extreme outlier in Figure 5a. The corresponding Sr5(PO4)3F:Eu3+ system exhibits anomalous 5D0−7F0 transition behavior due to crystal field J-mixing.

Figure 3. Plot of (a) second moment, σee, versus ligand polarizability from all data; (b) ΔF2 as a function of ligand distance for oxide systems, as eq 17.

Analogous plots for F4 and F6 for oxide systems provided values of R2adj ≈ 0.5, but the data sets comprise some spurious parameter values, with some magnitudes greater than those for the free ion. The plot of the spin−orbit coupling constant ζ4f versus σee is displayed in Supporting Information Figure S1. The variation of ζ4f is mainly from 736 to ∼750 cm−1 (i.e., only about 2%). The free ion datum is included at 764 cm−1, together with three other high data points at 763 cm−1 in Pr2Mg3(NO3)12.24H2O (PrO12), 755 cm−1 in LaAlO3 (LaO12) and Y2Al4O9 (major site, YO7 assumed) (Table 1). If the outliers marked 1−4 in the figure are removed, there is a linear relation between σee and the spin−orbit coupling constant, as indicated in the figure caption. Figure 4 shows that ζ4f does also vary linearly with ligand polarizability, according to eq 18. Estimation of Nephelauxetic Effects from Optical Spectra of Pr3+. From the parameter fitting calculations we have carried out, the variations of 1S0, 3P0, and lowest 1D2 state transition energies to the electronic ground state for Pr3+ systems in crystals are mainly due to changes in Slater and spin−orbit coupling parameters. We now investigate the experimental evidence whether these transition energies can be employed to reflect nephelauxetic effects. For this purpose, the concise, single parameter σee was introduced above to describe the overall strength of electrostatic interactions. There are only three accurate data available for the 1S0−3H4 ground-state transition energy (Table 1), although emission from 1S0 has been observed in many fluoride systems such as LiCaAlF6, YF3, and CaF2. Hence, we do not consider this transition energy further. The plots of the 3P0 (Figure 5a) and 1 D2 (Figure 5b) transition energies to the electronic ground state against σee serve to show that generally these two

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CONCLUSIONS Interelectronic repulsion for the two 4f electrons of Pr3+ is reduced on going from the vapor to crystalline state. A new parameter has been introduced to represent the total extent of interelectronic repulsion, and it exhibits a linear variation with the extent of screening of charge in the crystal. As was previously suggested by Morrison,22 this nephelauxetic effect is due to the polarization of the ligand by one 4f electron, and the interaction of the other electron with the induced multipolar moments, of which the dipole moment dominates. The presence of a polarizable ligand perturbs the central potential of Pr3+. In fact, the same interaction has been employed to account for the host-dependent shift of different configurations of a lanthanide ion.22 These conclusions have been made from the very few reasonably complete electronic energy level data sets of Pr3+ in crystals. The dependence of the electron repulsion parameters upon the completeness of the energy level data set has been highlighted and critically examined. By contrast with the electronic spectra of Eu3+, the measurement of the emission and/or absorption spectra of Pr3+ systems in the visible region can provide an approximate guide to the ligand nephelauxetic series. A Reviewer has asked the question concerning the nephelauxetic effect: Is it cloud expansion or contraction? The nephelauxetic effect has generally been described as a cloud expansion effect, but Newman considered that a decrease in interelectronic repulsion would lead to a contraction. The 10733



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Figure 5. Plots of 3P0−3H4 transition energy against σee (a) and F2 (c); 1D2−3H4 transition energy against σee (b).

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success of the dielectric screening model shows that the effect is simply due to the screening of electronic charge by the crystal medium. The fact that reasonable values of shifts of Slater parameters, ΔFk, can be calculated from changes in dielectric constant, but assuming constant radial expectation values (Tables 2, 3), implies that the changes in the “cloud” are minor. An uncertainty in eq 16 is the meaning of R, taken as the ionic radius. As noted in the Introduction, Angelov and coworkers have utilized eq 16 to calculate values of radial integrals from experimental values of ΔFk. However, many literature data sets do not comprise a sufficient number of energy levels to provide reliable values of Slater parameters. In conclusion, we consider that the mechanism of the nephelauxetic effect is embodied in eq 17, and it is intricately related to ligand polarizability and the distortion of the electron cloud. Hence, regarding the Reviewer’s question, we simply do not have the relevant evidence to answer it fully, and its resolution may come from ab initio calculations of the Slater parameters and the radial expectation values, which properly incorporate the effect of ligand polarizability as suggested in this Article.

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ASSOCIATED CONTENT

S Supporting Information *

Free ion energy levels and parameters. Effects of missing energy levels upon energy parameters. Effects of parameter changes upon energy levels. Figure S1: Plot of second moment, σee, versus spin−orbit coupling constant ζ4f for the Pr3+ data set. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from HKIEd is gratefully acknowledged. 10734



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Nephelauxetic Effects in the Electronic Spectra of Pr3+ - The Journal

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