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Crystal field strength: Angular overlap analysis of ligand field effects in some rare-earth compounds
Volume 129, number 4
5 September 1986
CHEMICAL PHYSICS LETTERS
CRYSTAL FIELD STRENGTH: ANGULAR OVERLAP ANALYSIS OF LIGAND FIELD EFFECTS IN SOME RARE-EARTH COMPOUNDS M. FAUCHER, D. GARCIA Laboratoire des El&ments de Transition dam les Solides, Equipe de.Recherche 1 place A. Brian4 92195 Meudon-Bellevue, France
associbe au CNRS 60210,
and C.K. JIBRGENSEN D$artement
de Chimie &fin&ale, Analytique et Appliquke, Universiti de Get&e,
CH-1211 Geneva 4, Switzerland
Received 7 April 1986; in final form 11 June 1986
An expression for the crystal field strength is defined in terms of one-electron energies and crystal field parameters. The relative contribution of kinetic energy and Coulombic interaction in ligand field effects is analyzed. The angular overlap model is applied to 11 rareearth or rareearthdoped compounds.
1. Introduction
2. Crystal field strength
IIIref. [ 1] , a covaloelectrostatic model was utilized to derive crystal field parameters Bt for a series of rare-earth compounds. The use of reliability and scale factors (one for each k value) enabled us to obtain good agreement between experimental and calculated values. Our aims in this work are the following: (a) To define a unique crystal field invariant in terms of one-electron energies and Bi parameters, and consequently unique scale and reliability factors. (b) In ref. [2] , it was qualitatively shown that kinetic energy effects are responsible for a large part of ligand field effects. The scale and reliability factors hereabove defined are now utilized for a more quantitative analysis. (c) To analyze the experimental results by means of the angular overlap model [3] in order to deduce u and n antibonding energies that can be transferred from compound to compound.
It is questionable whether a unique parameter, likely to be deduced from experimental values, can describe the crystal field strength. At first sight the best choice for an I electron is
(FE, )/W+ 1)
(1)
provided there exists a zeroenergy reference, i.e. one or several non-bonding orbitals (I, m). This is generally not the case and one has to refer to barycentered energies. A second possibility consists of the sum of the squares of barycentered energies, weighted by the number of levels:
E2=1
c
21+ 1 -I<~
E2
m’
It is possible to express eq. (2) in terms of Bi parameters. Firstly we state that E2 is just equal to (3)
0 009-2614/86/$ 03.50 Q Elsevier Science Publishers B.V. (North-Holland physics publishing Divison)
387
CHEMICAL PHYSICSLETTERS
Volume 129, number 4
Expression (3) may be evaluated utilizing the Hmm, coefficients of the interaction matrix since in an nth degree secular equation in E, the coefficient of End2 is equal to the sum of the cross products of the roots:
mm,
=
&-l)“-“e.MLz(2k t 1) k
L
the agreement between a set of observed data and a set of calculated values (the reliability factor) is [7] : 2, [Fm(exp.) - sF,(calc.)] 2 ‘I2
r=
(
E, KJexp.)]
1
’
’
In crystallographic experiments, the data F, are structure amplitudes but they may be any other experimentally accessible quantities; in the present case for instance, one-electron energies
With Hmmlgiven by
H
5 September 1986
(L , M) stands for a ligand orbital. The evaluation of eq, (4)? utilizing eq. (5) is straigbtfon&rd-and the final expression for E2 is:
where the scale factor s is the quantity which minimizes eq. (9). Its value is: ~m~m(exp.)~m(c~c.) S=
zm[E,(calc.)J 2
’
(10)
LlMlLlM2 ii?2 = -!-
21+ 1
c
k+O,q
eL1MreL2M2(-l)ml+ms(2ktl)
r = f 1 - ~2~2(c~c.)~~2(exp.)] ‘j2,
Transposed in terms of Bq parameters [4], this becomes: l/2 8= (21t l)k.z2&(; ‘0 $W] 7 (7) [ which is the same invariant as the one derived by Kibler [S] , except for a (21 t l)lfz factor. Kibler demonstrates the relation between the “crystal field strength” and the crystal field parameters, in a different way, utilizing symmetry-adapted parameters. Eq. (7) is also close to the rotational invariant defmed by Chang et al. [6] :
i c .-i-l@12
S=
3 k+u 2kt i
lf2.
3. !&le and rehbiiity factor; experimental and cslculated v&es A widely utilized expression for the measure of 388
Eq. (9) can also be written as:
63)
(11)
From expressions (10) and (1 l), it is obvious that a perfect agreement between calculated and observed values would correspond to s = 1 and t = 0. Table 1 lists the values of the crystal field strength according to eqs. (2) or (7) and of the global scale and reliability factors for the calculated one-electron energies of the 11 compounds investigated in ref. [l] (LiYF4:Nd3+,KY3F,,:Eu3+,YOCl:Eu3+, BaFCl: Sm2+, SrFCl : Sm2+, NdzOS , Ndz02S, NdAlO3, LaA103: Eu3+, L&13: Nd3+ and Y2O3: Ed+) with and wi~out ~o~ombic ~teraction, For compounds containing trivalent lan~~des’with oxygen or fluoride as nearest neighbours (aluminates excepted), the average scale factors, with and without Coulombic interaction, are equal to 1.35 and 0.91 respec.. tively. Again for the same compounds, all the r values decrease when Co~ombic iteration is added, roughly by a factor of 2. For the two Sm2+ fluoride compounds, the reliability factor is also divided by 4 when Coulombic interaction is added, but the scale factor of SrFCl: Sm2+ is too small indicating that kinetic energy effects are somewhat overestimated. On the other hand, for I&13, the scale and reliability factors are good, showing kinetic energy effects only, probably because the charges are smeared out.
Volume 129, number 4
CHEMICAL PHYSICS LETTERS
5 September 1986
Table 1 Crystal field strength, global scale and reliability factors and Coulombic energy contribution rare earth or rareearthdoped compounds E(exp) (cm-t)
s (KE)
0 (KE + CI)
1OOr (KE)
1OOr (KE + CI)
AEIE
Ndz%
422
1.46
0.82
31.1
20.0
52.4
w@,S
252
1.31
0.93
25.5
14.5
31.7
83
0.52
0.52
40.0
19.5
8.9
RaFCl : Sm2*
73
0.88
0.95
51.7
27.9
9.0
320
1 A4
0.88
49.5
26.3
58.7
KYaFto:E$+
249
0.88
0.71
56.7
25.9
45.3
LiYF4:Nd3*
363
1.20
0.99
14.1
8.4
18.7
C-Y, 0, : l&J+
522
1.78
1.12
27.0
11.4
41.6
LaCl,:Ndj+
143
0.86
0.38
10.5
83.4
52.0
NdAlO,
387
2.34
1.05
17.3
56.1
58.7
LaA103: Eu3+
365
5.41
2.72
4.5
37.1
49.8
YOCl : Eu3+
The NdAlO, case is not well understood since the scale factor becomes perfect with the introduction of Coulombic interaction whereas the reliability factor worsens. The LaAlO, : Eu3+ case is even more puzzling as it has the highest scale factor of the series (5.41), but we suspect that this is due to a shrinkage of the coordination polyhedron around europium. KE*Ci
EXP.
.-
,.-’
This hypothesis is supported by the fact that the structure of EtAlO (orthorhombic perovskite) is different to that of LaAlO, (rhombohedral perovskite). Figs. l-3 represent calculated and experimental oneelectron energies for LaCl3 : Nd3+, LiYF4 : Nd3+ and C-Y203 : Eu3” (C!2,,site). Calling the calculated kinetic energy contribution to a oneelectron energy level Ec2, and the sum of the
KE KEGI
EXP
KE
‘...
‘-._
_*
-.
,.:
**._
:
,
_.
*. ..-..y~
::-:..
_ ____,. ____.. ..‘..::::-..::...:;~:::-._
--.
energies for eleven
Compound
SrFCl : Sm2+
B.
to oneelectron
.._.,_
_...
.____._ _
Ei
-._..-.
. . . .._____.* -.m._,_
... .
E”
‘_.._ . ._
A2
--..
,.-.. ,.**
-. -.
*..’
.._*
_.: -.
;:
-,
:.: .,
I: 1oocm-’ r
-.
..
--.* *.._
‘... ‘-._ Y i
100cm-1
--
. ..... -.
*. ._.a-
__.-
,.--
.-
;;
_
.._....... . . .
_ . . . . . . .m..
_
_._-.____
*..-
i
Fig. 1. Oneelectron energies for LaCl, : Nd*, with the contribution from the kinetic energy (on the right), with the kinetic energy and Coulombic interactjon (middle of the f&ure) compared with experimentally determined energies (on the left).
-mme-....-*..--...
--_.
_-__
_..___._.......... -_.._ e-m.. . . . . ..____.
-*--
.__ _ ______ _ _ _ . . _.z
. .. . -
__..I
_e.. /.__.-_m-
E
62
E
4
Fig. 2. As fu. 1 for LiYF4:Nd3+. 389
CHEMICAL PHYSICS LETTERS
Volume 129, number 4
KE*Ci
EXP A, --_
---___
__
61
82
___----
-
--__ ---
____----_
Al __--B2-
--.
--.
--_
___--
--w
-_m __
_-
- ---=_r; __*_*_*_
AZ--------___-_>
energy to ligand field effects, we are justified in analyzing the experimental results by means of the angular overlap model [3,8] . The experimentally available quantities are the Bk parameters utilized to interpret the coupled states of the rare earths. The angular overlap model is restricted to (I and R effects due to the p shells of the rare-earth ligands. There are too many unknown antibonding energies to take into account the effect of s shells. Therefore we have 2n unknown quantities per compound, n being the number of independent ligands. The equations to be solved are of the form:
KE
-_ -.
----_
------____
___---
61
-----._
_ _ _ - /-
_ _--
5 September 1986
*’ _’ /*
-_-_--____
*-’
B;=T(;,,L.
v
-’
Fig. 3. As fii. 1 for Yz0,:Eu3+.
1- 5
EC1 EC21 g
E&)
Bi = c
(seat + + e,&*(L),
Bi =I?
(9 eaL + % enL)Ct*(L),
L
Coulombic energy and of the kinetic energy contributions Ecl, then a mean value for the strength of the Coulombic energy contribution can be defined as:
AEIE= 100
+ H emL)C~*@),
i 200 cm’ A
.
(12)
The A&‘/Evalues for the eleven investigated compounds are reported in the last column of table 1. They range between 9 and 59%, the mean value being 39%.
4. Angular overlap analysis In view of the important contribution of kinetic
(13)
for each ligand type. The C’t are 2n/2k + 1 times the corresponding spherical harmonic. When the system ls redundant, normal equations are written. On the other hand, for Nd203, the number of unknowns is larger than the number of equations so that a constraint is established (i.e. the ratio e,/eJ is assumed to be the same for two sets of nearest neighbours. For europium doped Y203, a slightly “idealized” structure cancelling the imaginary ligand field parameters is designed. The small distortions from C2v symmetry were ignored, resulting in the distances and angles listed in table 2. In table 3 we report the rareearth ligand distances R, the squared overlaps S2 (calculated with the ana-
Table 2 Coordination polyhedron of Y3+in Y,O, Real structure R (A)
e (deg)
1
2.3371
57.247
2
2.2677
69.03
3
390
Idealized structure
2.2435
136.06
v (de& 87.413 -92.586
R (A)
e
(deg)
2.332
57.2
-2.313 -182.313
2.26
69.0
94.9 -85.10
2.242
P @et4 90 -90 0 -180
136.09
90 -90
Volume 129, number 4
CHEMICAL PHYSICS LETTERS
5 September 1986
Table 3 Seventy-two internuclear distances between a lanthanide and the nearest neighbour nuclei. There are approximately 18 equivalent sites in 9 compounds given in the order of ref. [ 21. The valueof e, and e&z,, are derived from the oneelectron energy differences compiled there, and the squares of the overlap integrals from ref. (11 Contact
Compound
Nd-9 Cl
Nd&s
Nd-3 0 Nd-O Nd-3 0
1-x Cl 3
Nd203
ea (cm-r)
ei&c7
R (A)
104s2 PO
104s2 pn
(Sp#po)2
K = e,/S& (ev)
229
0.22
2.952
1.393
0.709
0.51
204
705 450 259
0.48 [0.48] ?
2.301 2.400 2.657
4.490
3.427 1.685
2.513 1.741 0.675
0.56 0.51 0.40
195 163 190
Nd4 0 Nd-3 S
Nd,O,S
507 71
0.44 0.56
2.362 2.964
3.790 1.390
2.000 0.843
0.53 0.61
165 63
Sm4 F Sm-5Cl
Sm.$r 1,FCl
157 90
0.28 0.26
2.494 3.08
2.837 1.090
0.747 0.329
0.26 0.30
102
Sm4 F Sm-5Cl
SmxBal,FCl
136 71
0.29 ?
2.649 3.21
1.649
0.864
0.363 0.246
0.22 0.28
102 102
Err-4 0 Eu-5 Cl
EuxYI _xoQ
549 90
0.39
2.278 3.017
4.290
0.816
1.747 0.253
0.41 0.31
159
?
Eu4 F Eu-4 F
455 236
0.40 0.43
2.196 2.331
4.661 2.977
1.687 0.899
0.36 0.30
121 98
Nd-4 F Nd4 F
505
500
0.16 0.29
2.246 2.293
4.368 3.769
2.065 1.709
0.47 0.45
143 164
881 391
0.41 0.29
2.25 2.33
4.651 3.662
1.968 1.387
0.42 0.38
235 132
Err-4 0 Eu-2 0
E%Y2.$3
lytical formulas given in ref. [ 11, the antibonding energies e, and e, and the K = e,,/S& values in eV. We should obtain constant K values for a given contact type since all the distance dependence is contained in the squared overlap. Instead of this, the K values are scattered and the general trend is a decrease for increasing distances. The most outstanding example occurs for Y2O3 : EU3+ for which the individual “strength” of K for the four nearest neighbours at 2.25 A is nearly twice that for the two slightly farther oxygens at 2.33 A. This may account for ah the contributions which have been ignored: interaction with the ligand s shell (the overlaps with the 4f orbitals are only slightly lower than for the p0 shell), with farther neighbours, the Coulombic interaction etc. ... . One interesting fact is that the mean value for K is fairly constant and equal to 180 eV for Nd-0 and 184 eV for Eu-0 contacts. It equals 153 eV for Nd-F but decreases drastically for Eu-F in the single investigated compound: KY3Flo : Eu3+. Expressions (13)
.
69
137
together with the K valuet~in table 3, can be used to obtain initial values of one-electron energies or crystal field parameters for other compounds.
References [l] D. Garcia and M. Faucher, J. Chem. Phys. 82 (1985) 5554.
[ 2 ] C.K. Jdrgensen, M. Faucher and D. Garcia, Chem. Phys. Letters 128 (1986) 250. [3] C.K. Jdrgensen, R. Pappalardo and H.H. Schmidtke, J. Chem. Phys. 39 (1963) 1422. [4 ] M. Faucher and D. Garcia, J. LessCommon Metals 93 (1983) 31. (51 M. Kibler, Phys. Letters A 98 (1983) 343. [6] N.C. Chang, J.B. Gruber, R.P. Leavitt and C.A. Morrison, J. Cbem. Phys. 76 (1982) 3877. [ 71 International tables for Xlay crystallography, Vol. 4 (Kynoch, Birmingham, 1974) p. 288. [8] C.K. Jtirgensen, J. Phys. (Paris) 26 (1965) 825.
393
5 September 1986
CHEMICAL PHYSICS LETTERS
CRYSTAL FIELD STRENGTH: ANGULAR OVERLAP ANALYSIS OF LIGAND FIELD EFFECTS IN SOME RARE-EARTH COMPOUNDS M. FAUCHER, D. GARCIA Laboratoire des El&ments de Transition dam les Solides, Equipe de.Recherche 1 place A. Brian4 92195 Meudon-Bellevue, France
associbe au CNRS 60210,
and C.K. JIBRGENSEN D$artement
de Chimie &fin&ale, Analytique et Appliquke, Universiti de Get&e,
CH-1211 Geneva 4, Switzerland
Received 7 April 1986; in final form 11 June 1986
An expression for the crystal field strength is defined in terms of one-electron energies and crystal field parameters. The relative contribution of kinetic energy and Coulombic interaction in ligand field effects is analyzed. The angular overlap model is applied to 11 rareearth or rareearthdoped compounds.
1. Introduction
2. Crystal field strength
IIIref. [ 1] , a covaloelectrostatic model was utilized to derive crystal field parameters Bt for a series of rare-earth compounds. The use of reliability and scale factors (one for each k value) enabled us to obtain good agreement between experimental and calculated values. Our aims in this work are the following: (a) To define a unique crystal field invariant in terms of one-electron energies and Bi parameters, and consequently unique scale and reliability factors. (b) In ref. [2] , it was qualitatively shown that kinetic energy effects are responsible for a large part of ligand field effects. The scale and reliability factors hereabove defined are now utilized for a more quantitative analysis. (c) To analyze the experimental results by means of the angular overlap model [3] in order to deduce u and n antibonding energies that can be transferred from compound to compound.
It is questionable whether a unique parameter, likely to be deduced from experimental values, can describe the crystal field strength. At first sight the best choice for an I electron is
(FE, )/W+ 1)
(1)
provided there exists a zeroenergy reference, i.e. one or several non-bonding orbitals (I, m). This is generally not the case and one has to refer to barycentered energies. A second possibility consists of the sum of the squares of barycentered energies, weighted by the number of levels:
E2=1
c
21+ 1 -I<~
E2
m’
It is possible to express eq. (2) in terms of Bi parameters. Firstly we state that E2 is just equal to (3)
0 009-2614/86/$ 03.50 Q Elsevier Science Publishers B.V. (North-Holland physics publishing Divison)
387
CHEMICAL PHYSICSLETTERS
Volume 129, number 4
Expression (3) may be evaluated utilizing the Hmm, coefficients of the interaction matrix since in an nth degree secular equation in E, the coefficient of End2 is equal to the sum of the cross products of the roots:
mm,
=
&-l)“-“e.MLz(2k t 1) k
L
the agreement between a set of observed data and a set of calculated values (the reliability factor) is [7] : 2, [Fm(exp.) - sF,(calc.)] 2 ‘I2
r=
(
E, KJexp.)]
1
’
’
In crystallographic experiments, the data F, are structure amplitudes but they may be any other experimentally accessible quantities; in the present case for instance, one-electron energies
With Hmmlgiven by
H
5 September 1986
(L , M) stands for a ligand orbital. The evaluation of eq, (4)? utilizing eq. (5) is straigbtfon&rd-and the final expression for E2 is:
where the scale factor s is the quantity which minimizes eq. (9). Its value is: ~m~m(exp.)~m(c~c.) S=
zm[E,(calc.)J 2
’
(10)
LlMlLlM2 ii?2 = -!-
21+ 1
c
k+O,q
eL1MreL2M2(-l)ml+ms(2ktl)
r = f 1 - ~2~2(c~c.)~~2(exp.)] ‘j2,
Transposed in terms of Bq parameters [4], this becomes: l/2 8= (21t l)k.z2&(; ‘0 $W] 7 (7) [ which is the same invariant as the one derived by Kibler [S] , except for a (21 t l)lfz factor. Kibler demonstrates the relation between the “crystal field strength” and the crystal field parameters, in a different way, utilizing symmetry-adapted parameters. Eq. (7) is also close to the rotational invariant defmed by Chang et al. [6] :
i c .-i-l@12
S=
3 k+u 2kt i
lf2.
3. !&le and rehbiiity factor; experimental and cslculated v&es A widely utilized expression for the measure of 388
Eq. (9) can also be written as:
63)
(11)
From expressions (10) and (1 l), it is obvious that a perfect agreement between calculated and observed values would correspond to s = 1 and t = 0. Table 1 lists the values of the crystal field strength according to eqs. (2) or (7) and of the global scale and reliability factors for the calculated one-electron energies of the 11 compounds investigated in ref. [l] (LiYF4:Nd3+,KY3F,,:Eu3+,YOCl:Eu3+, BaFCl: Sm2+, SrFCl : Sm2+, NdzOS , Ndz02S, NdAlO3, LaA103: Eu3+, L&13: Nd3+ and Y2O3: Ed+) with and wi~out ~o~ombic ~teraction, For compounds containing trivalent lan~~des’with oxygen or fluoride as nearest neighbours (aluminates excepted), the average scale factors, with and without Coulombic interaction, are equal to 1.35 and 0.91 respec.. tively. Again for the same compounds, all the r values decrease when Co~ombic iteration is added, roughly by a factor of 2. For the two Sm2+ fluoride compounds, the reliability factor is also divided by 4 when Coulombic interaction is added, but the scale factor of SrFCl: Sm2+ is too small indicating that kinetic energy effects are somewhat overestimated. On the other hand, for I&13, the scale and reliability factors are good, showing kinetic energy effects only, probably because the charges are smeared out.
Volume 129, number 4
CHEMICAL PHYSICS LETTERS
5 September 1986
Table 1 Crystal field strength, global scale and reliability factors and Coulombic energy contribution rare earth or rareearthdoped compounds E(exp) (cm-t)
s (KE)
0 (KE + CI)
1OOr (KE)
1OOr (KE + CI)
AEIE
Ndz%
422
1.46
0.82
31.1
20.0
52.4
w@,S
252
1.31
0.93
25.5
14.5
31.7
83
0.52
0.52
40.0
19.5
8.9
RaFCl : Sm2*
73
0.88
0.95
51.7
27.9
9.0
320
1 A4
0.88
49.5
26.3
58.7
KYaFto:E$+
249
0.88
0.71
56.7
25.9
45.3
LiYF4:Nd3*
363
1.20
0.99
14.1
8.4
18.7
C-Y, 0, : l&J+
522
1.78
1.12
27.0
11.4
41.6
LaCl,:Ndj+
143
0.86
0.38
10.5
83.4
52.0
NdAlO,
387
2.34
1.05
17.3
56.1
58.7
LaA103: Eu3+
365
5.41
2.72
4.5
37.1
49.8
YOCl : Eu3+
The NdAlO, case is not well understood since the scale factor becomes perfect with the introduction of Coulombic interaction whereas the reliability factor worsens. The LaAlO, : Eu3+ case is even more puzzling as it has the highest scale factor of the series (5.41), but we suspect that this is due to a shrinkage of the coordination polyhedron around europium. KE*Ci
EXP.
.-
,.-’
This hypothesis is supported by the fact that the structure of EtAlO (orthorhombic perovskite) is different to that of LaAlO, (rhombohedral perovskite). Figs. l-3 represent calculated and experimental oneelectron energies for LaCl3 : Nd3+, LiYF4 : Nd3+ and C-Y203 : Eu3” (C!2,,site). Calling the calculated kinetic energy contribution to a oneelectron energy level Ec2, and the sum of the
KE KEGI
EXP
KE
‘...
‘-._
_*
-.
,.:
**._
:
,
_.
*. ..-..y~
::-:..
_ ____,. ____.. ..‘..::::-..::...:;~:::-._
--.
energies for eleven
Compound
SrFCl : Sm2+
B.
to oneelectron
.._.,_
_...
.____._ _
Ei
-._..-.
. . . .._____.* -.m._,_
... .
E”
‘_.._ . ._
A2
--..
,.-.. ,.**
-. -.
*..’
.._*
_.: -.
;:
-,
:.: .,
I: 1oocm-’ r
-.
..
--.* *.._
‘... ‘-._ Y i
100cm-1
--
. ..... -.
*. ._.a-
__.-
,.--
.-
;;
_
.._....... . . .
_ . . . . . . .m..
_
_._-.____
*..-
i
Fig. 1. Oneelectron energies for LaCl, : Nd*, with the contribution from the kinetic energy (on the right), with the kinetic energy and Coulombic interactjon (middle of the f&ure) compared with experimentally determined energies (on the left).
-mme-....-*..--...
--_.
_-__
_..___._.......... -_.._ e-m.. . . . . ..____.
-*--
.__ _ ______ _ _ _ . . _.z
. .. . -
__..I
_e.. /.__.-_m-
E
62
E
4
Fig. 2. As fu. 1 for LiYF4:Nd3+. 389
CHEMICAL PHYSICS LETTERS
Volume 129, number 4
KE*Ci
EXP A, --_
---___
__
61
82
___----
-
--__ ---
____----_
Al __--B2-
--.
--.
--_
___--
--w
-_m __
_-
- ---=_r; __*_*_*_
AZ--------___-_>
energy to ligand field effects, we are justified in analyzing the experimental results by means of the angular overlap model [3,8] . The experimentally available quantities are the Bk parameters utilized to interpret the coupled states of the rare earths. The angular overlap model is restricted to (I and R effects due to the p shells of the rare-earth ligands. There are too many unknown antibonding energies to take into account the effect of s shells. Therefore we have 2n unknown quantities per compound, n being the number of independent ligands. The equations to be solved are of the form:
KE
-_ -.
----_
------____
___---
61
-----._
_ _ _ - /-
_ _--
5 September 1986
*’ _’ /*
-_-_--____
*-’
B;=T(;,,L.
v
-’
Fig. 3. As fii. 1 for Yz0,:Eu3+.
1- 5
EC1 EC21 g
E&)
Bi = c
(seat + + e,&*(L),
Bi =I?
(9 eaL + % enL)Ct*(L),
L
Coulombic energy and of the kinetic energy contributions Ecl, then a mean value for the strength of the Coulombic energy contribution can be defined as:
AEIE= 100
+ H emL)C~*@),
i 200 cm’ A
.
(12)
The A&‘/Evalues for the eleven investigated compounds are reported in the last column of table 1. They range between 9 and 59%, the mean value being 39%.
4. Angular overlap analysis In view of the important contribution of kinetic
(13)
for each ligand type. The C’t are 2n/2k + 1 times the corresponding spherical harmonic. When the system ls redundant, normal equations are written. On the other hand, for Nd203, the number of unknowns is larger than the number of equations so that a constraint is established (i.e. the ratio e,/eJ is assumed to be the same for two sets of nearest neighbours. For europium doped Y203, a slightly “idealized” structure cancelling the imaginary ligand field parameters is designed. The small distortions from C2v symmetry were ignored, resulting in the distances and angles listed in table 2. In table 3 we report the rareearth ligand distances R, the squared overlaps S2 (calculated with the ana-
Table 2 Coordination polyhedron of Y3+in Y,O, Real structure R (A)
e (deg)
1
2.3371
57.247
2
2.2677
69.03
3
390
Idealized structure
2.2435
136.06
v (de& 87.413 -92.586
R (A)
e
(deg)
2.332
57.2
-2.313 -182.313
2.26
69.0
94.9 -85.10
2.242
P @et4 90 -90 0 -180
136.09
90 -90
Volume 129, number 4
CHEMICAL PHYSICS LETTERS
5 September 1986
Table 3 Seventy-two internuclear distances between a lanthanide and the nearest neighbour nuclei. There are approximately 18 equivalent sites in 9 compounds given in the order of ref. [ 21. The valueof e, and e&z,, are derived from the oneelectron energy differences compiled there, and the squares of the overlap integrals from ref. (11 Contact
Compound
Nd-9 Cl
Nd&s
Nd-3 0 Nd-O Nd-3 0
1-x Cl 3
Nd203
ea (cm-r)
ei&c7
R (A)
104s2 PO
104s2 pn
(Sp#po)2
K = e,/S& (ev)
229
0.22
2.952
1.393
0.709
0.51
204
705 450 259
0.48 [0.48] ?
2.301 2.400 2.657
4.490
3.427 1.685
2.513 1.741 0.675
0.56 0.51 0.40
195 163 190
Nd4 0 Nd-3 S
Nd,O,S
507 71
0.44 0.56
2.362 2.964
3.790 1.390
2.000 0.843
0.53 0.61
165 63
Sm4 F Sm-5Cl
Sm.$r 1,FCl
157 90
0.28 0.26
2.494 3.08
2.837 1.090
0.747 0.329
0.26 0.30
102
Sm4 F Sm-5Cl
SmxBal,FCl
136 71
0.29 ?
2.649 3.21
1.649
0.864
0.363 0.246
0.22 0.28
102 102
Err-4 0 Eu-5 Cl
EuxYI _xoQ
549 90
0.39
2.278 3.017
4.290
0.816
1.747 0.253
0.41 0.31
159
?
Eu4 F Eu-4 F
455 236
0.40 0.43
2.196 2.331
4.661 2.977
1.687 0.899
0.36 0.30
121 98
Nd-4 F Nd4 F
505
500
0.16 0.29
2.246 2.293
4.368 3.769
2.065 1.709
0.47 0.45
143 164
881 391
0.41 0.29
2.25 2.33
4.651 3.662
1.968 1.387
0.42 0.38
235 132
Err-4 0 Eu-2 0
E%Y2.$3
lytical formulas given in ref. [ 11, the antibonding energies e, and e, and the K = e,,/S& values in eV. We should obtain constant K values for a given contact type since all the distance dependence is contained in the squared overlap. Instead of this, the K values are scattered and the general trend is a decrease for increasing distances. The most outstanding example occurs for Y2O3 : EU3+ for which the individual “strength” of K for the four nearest neighbours at 2.25 A is nearly twice that for the two slightly farther oxygens at 2.33 A. This may account for ah the contributions which have been ignored: interaction with the ligand s shell (the overlaps with the 4f orbitals are only slightly lower than for the p0 shell), with farther neighbours, the Coulombic interaction etc. ... . One interesting fact is that the mean value for K is fairly constant and equal to 180 eV for Nd-0 and 184 eV for Eu-0 contacts. It equals 153 eV for Nd-F but decreases drastically for Eu-F in the single investigated compound: KY3Flo : Eu3+. Expressions (13)
.
69
137
together with the K valuet~in table 3, can be used to obtain initial values of one-electron energies or crystal field parameters for other compounds.
References [l] D. Garcia and M. Faucher, J. Chem. Phys. 82 (1985) 5554.
[ 2 ] C.K. Jdrgensen, M. Faucher and D. Garcia, Chem. Phys. Letters 128 (1986) 250. [3] C.K. Jdrgensen, R. Pappalardo and H.H. Schmidtke, J. Chem. Phys. 39 (1963) 1422. [4 ] M. Faucher and D. Garcia, J. LessCommon Metals 93 (1983) 31. (51 M. Kibler, Phys. Letters A 98 (1983) 343. [6] N.C. Chang, J.B. Gruber, R.P. Leavitt and C.A. Morrison, J. Cbem. Phys. 76 (1982) 3877. [ 71 International tables for Xlay crystallography, Vol. 4 (Kynoch, Birmingham, 1974) p. 288. [8] C.K. Jtirgensen, J. Phys. (Paris) 26 (1965) 825.
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