Superposition model for 19F isotropic chemical shift in ionic fluorides: from basic metal fluorides to transition metal fluoride glasses

Chemical Physics 249 Ž1999. 89–104 www.elsevier.nlrlocaterchemphys

Superposition model for 19 F isotropic chemical shift in ionic fluorides: from basic metal fluorides to transition metal fluoride glasses B. Bureau ) , G. Silly, J.Y. Buzare, ´ J. Emery Laboratoire de Physique de l’Etat Condense, ´ UPRES - A CNRS no. 6087, UniÕersite´ du Maine, Le Mans, France Received 18 May 1999

Abstract Experimental measurements of 19 F isotropic chemical shifts on a large set of mainly ionic fluoride compounds Žfrom simple metal fluorides to transition metal fluoride glasses. obtained by MAS NMR at 15 kHz spinning rate are simultaneously investigated. First, Ramsey’s theory of the chemical shift with molecular orbitals obtained by Lodwin’s ¨ orthogonalisation method is used to evaluate the isotropic part of the 19 F chemical shift in ionic fluorides for which the crystallographic structure and the atomic radial wavefunctions are known. Then, assuming that the paramagnetic part of the 19 F shielding in a given material is simply the summation of the paramagnetic contributions due to all the cations in the neighbourhood of the considered fluorine, a superposition model of the 19 F isotropic chemical shift is developed. Finally, this empirical approach is applied to complex fluoride compounds of unknown structure and it is demonstrated that it allows to obtain reliable structural informations. More generally, it can be applied to all the ionic fluorides. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction Nuclear magnetic resonance ŽNMR. is one of the most widely used experimental techniques in structural chemistry. In particular, the chemical shift ŽCS. spectra are a fingerprint of the local geometry and the chemical structure of the material under study. From the early days of NMR, the isotropic part of this tensor has been measured in liquids and this measurement has been developed as a tool for molecular structure investigation. In the same time, an important effort has been devoted to relating the isotropic chemical shift and the electronic structure theoretically w1,2x. )

Corresponding author. e-mail: [email protected]

In the solid state, MAS NMR is often used to reduce dipolar broadening and chemical shift anisotropy and measure the isotropic part of the chemical shift as in the liquid state w3x. Recent papers w4–6x clearly demonstrate the widespread utility of fluorine MAS NMR, especially in inorganic systems. In a further step, when we want to use the isotropic chemical shift as a tool for structural investigations, we need some relation between this experimental parameter and the environment of the 19 F nucleus. This can be achieved either by ab-initio or empirical calculations. However, until now, few calculations has been undertaken w7–12x and most of them are restricted to alkali–metal fluorides w7,11,12x. Recently, we applied MAS NMR to transition metal fluoride glasses studied by 19 F NMR w13–15x.

0301-0104r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 9 . 0 0 2 5 3 - 0

90

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

In these works, the great sensitivity of the 19 F isotropic chemical shift to the environment of the fluorine atom was used to investigate the fluorine octahedron network connectivity in various transition metal fluoride glasses ŽTMFG.: PbF2 –ZnF2 –GaF3 ŽPZG., PbF2 –ZnF2 –InF3 ŽPZI., PbF2 –BaF2 –InF3 ŽPBI. glasses and alkali fluoride glasses such as PbF2 –LiF–GaF3 ŽPLG., PbF2 –NaF–GaF3 ŽPNG., PbF2 –KF–GaF3 ŽPKG.. NMR experiments gave evidence for different fluorine atom sites. The 19 F isotropic chemical shifts measured in the glasses were qualitatively interpreted in the light of chemical shifts measured in selected crystalline compounds. From the chemical shift values, it was shown that the fluorines may be classified into three categories: shared and unshared fluorines between two MF6 octahedra and free fluorines which are not implied in these MF6 octahedra. From the relative intensities of the corresponding NMR lines, it was inferred that the octahedra are corner shared and the dimensionality of the network is varied from nearly 1D to 3D according to the glass composition. In the present work, we would like to show that the isotropic 19 F chemical shift values may be established for ionic fluorides in the framework of an empirical model which corroborates our results and may be used as a structural probe. This work is based on the analysis of a large number of 19 F isotropic chemical shifts in inorganic fluorides where the Fy site symmetries and environments are quite different and with increasing complexity: basic fluorides ŽMF, MF2 and MF3 ., fluoroperovskites ŽAMF3 ., fluoroaluminates ŽKAlF4 , RbAlF4 ., barium fluorometallates ŽBaMgF4 , BaZnF4 and Ba 2 ZnF6 ., pyrochlore ŽCsZnGaF6 ., PbF2 –M II F2 –M III F3 ŽM II s Ba, Zn; M III s Ga, In. crystalline and glassy phases and PbF2 –AF–GaF3 ŽA s Li, Na, K. glasses. In some of these compounds, the chemical shifts may be evaluated through theoretical calculations when the radial wavefunctions of the atoms are known. Unfortunately, it is not always the case. When these functions are unknown, an alternative way to correlate isotropic chemical shift values and local environments of Fy has to be developed. The aim of this paper is to work out an empirical model which may be applied when the theoretical approach fails. The paper is organised as follows. A first part is devoted to 19 F isotropic chemical shift theoretical calcula-

tions in inorganic fluorides. They have been derived on the basis of Ramsey’s theory w1,2x using Lowdin ¨ molecular orbitals w16x when the Slater type atomic orbitals were known w17,18x. This approach has produced results which allow to reproduce 19 F chemical shifts in various systems: F2 and HF w19x, C 6 F6 w20x, alkali fluorides w7x, MgF2 w8x, KZnF3 w9x and RbCaF3 w10x. It has also the advantage to be not time consuming compared to ab-initio methods. In a second part, from the theoretical calculations as a starting point, we show that the 19 F isotropic chemical shift may be calculated as a summation of contributions over its surrounding cation neighbours. This superposition model may be worked out without any knowledge about the atomic orbitals. Finally, results obtained by applying this model are compared to experimental measurements in glasses and related compounds and allow us to get structural information about the medium range order.

2. Experimental The measured isotropic chemical shift d iso is expressed from the principal components of the shielding tensor si i Ž i s x, y, z . according to the formulas: d i i s sref y si i where sref is the shielding parameter of C 6 F6 , d iso s 13 Ž d x x q d y y q d z z .. Despite most of them are known from literature w4– 10,13–15,21,22x, all the 19 F isotropic chemical shifts of the fluoride materials which are given further were measured by us at room temperature on the same MSL 300 Bruker spectrometer using MAS at a spinning rate of 15 kHz. The 19 F acquisition and processing parameters are identical to those used in the 19 F experiments in PZG glasses w13x. An estimate of over all accuracy is 3 ppm for crystalline compounds. In TMFG, details concerning line widths, residual dipolar couplings and dispersion of d iso values with the glass composition are discussed in w13–15x.

3. Theoretical calculations of

19

F chemical shifts

In different works where Ramsey’s theory is used, it appears that the approximations are not fully consistent. Furthermore, the different notations and ap-

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

proaches make the comparison between these different works sometimes difficult. So, we undertook the calculations from the original formulas applying the same approximations to a large number of inorganic fluorides. The work by Grosescu and Haeberlen w9x is the one where the different steps are the most clearly explained. We used the same approach. Nevertheless, as some significant discrepancy appears between their results and ours, we summarise the different steps in Appendix A. At the end, the isotropic part of the shielding tensor is expressed as a sum of squares of overlap integrals between atomic orbitals multiplied by some averaged distances on fluorine orbitals and divided by an average excitation energy for the fluorine ions. A computer program written by one of us ŽJ.E.. w23x has been used to evaluate the various multi-centre integrals. The radial part of atomic functions for Liq, Naq, Kq, Rbq, Mg 2q, Ca2q, Zn2q, Al 3q and Fy were calculated by Clementi w17x and for Csq and Ba2q by De Pape w18x. It was found: ² ry1 : 1s s 8.63 a.u., ² ry1 : 2s s 1.41 a.u., ² ry1 : 2p s 1.16 a.u., which results in an isotropic diamagnetic contribution s d ŽC.G.S. units. s 480 = 10y6 s 480 ppm in agreement with w24x. The value ² ry3 : 2p s 6.40 a.u. is in agreement with w8x. Then, the paramagnetic contribution of one neighbour labelled l of the considered 19 F nucleus labelled 0 to the isotropic chemical shift in CGS units is written Žsee Appendix A.: 3088

P

sl s y

D Ž eV .

7Ž

2 Slsp0 d q 7

.

Ž

2 Slpp 0 s q 10

.

Ž

2 Slpp 0 p

2

.

2

pp dp dp y8 Ž Slpp 0 . s Ž S l 0 . p q 7 Ž S l 0 . s q 10 Ž S l 0 . p

y8'3 Ž Sldp0 . s Ž Sldp0 . p Ž ppm . .

Ž 1. 19

Then, the complete expression of the F isotropic chemical shift, with C 6 F6 as a reference, is obtained following literature w8–10x based on the absolute abs scale of Hindermann and Cornwell w25x where sCCl 3F

91

s 188.7 ppm. According to w10,26x C 6 F6 has a shift of 164.2 ppm which corresponds to sCabs s 6 F6 352.9 ppm. At last, we get:

d iso r C 6 F6 Ž ppm . 3088

s y127.1 q

2 7 Ž Slsp0 . d Ý D Ž eV . l

2

2

pp q7 Ž Slpp 0 . s q 10 Ž S l 0 . p 2

pp dp y8 Ž Slpp 0 . s Ž Sl 0 . p q 7 Ž Sl 0 . s 2

q10 Ž Sldp0 . p y 8'3 Ž Sldp0 . s Ž Sldp0 . p ,

Ž 2.

where the summation runs over the different neighbours. The s P or d iso expressions may be written in the form

Ý Sl P

s sy

l

D

Ž ppm. , Ž 3.

Ý Sl d iso r C 6 F6 s y127.1 q

l

Ž ppm. ,

D

with 2

2

2

pp Sl s 3088 7 Ž Slsp0 . d q 7 Ž Slpp 0 . s q 10 Ž S l 0 . p 2

2

pp dp dp y8 Ž Slpp 0 . s Ž S l 0 . p q 7 Ž S l 0 . s q 10 Ž S l 0 . p

y8'3 Ž Sldp0 . s Ž Sldp0 . p . They give evidence for Sl contributions related to one neighbour which depend on its distance to the considered fluorine only. These Sl were calculated for different M–F distances for all the M cations we had the wavefunctions and fitted to expressions of the form: Sl Ž d . s A l exp Ž ya l d . ,

Ž 4.

in agreement with the overlap integrals varying as exponential functions of this distance. A l and a l are given in Table 1. It is noteworthy that a linear

Table 1 Parameters A l and a l of the exponential radial dependence of the ligand contribution Sl s A l expŽya l d . l y3 .

A l Ž=10 al

Liq

Naq

Kq

Rbq

Csq

Mg 2q

Ca2q

Ba2q

Zn2q

Al 3q

64 3.281

207 3.256

455 2.855

568 2.721

773 2.580

159 3.428

364 2.976

798 2.736

162 3.240

114 3.521

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

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relation may be established between a l and the ionic radius r l w27x of the ligand l:

a l s y0.806rl q 4.048 .

Ž 5.

This formula will be applied in the following to cations for which radial wavefunctions are unknown. 3.1. Basic fluorides The 19 F isotropic chemical shift calculation was first applied to some basic fluorides. The measured d iso and the s P parameters calculated according to s P s y127.1 y d iso are listed in Table 2. They are the only relevant quantities which vary from one compound to another. The alkali fluorides which adopt the NaCl structure have one cubic fluorine site which is coordinated to six alkali atoms. Except LiF, d iso increases when descending the first column of the periodic table. Difluorides BaF2 , SrF2 , CaF2 , CdF2 and b-PbF2 adopt the cubic fluorine structure with one cubic fluorine site which is coordinated to four cations.

MgF2 and ZnF2 has the rutile structure with one fluorine atom shared between three MF6 octahedra. a-PbF2 has the orthorhombic PbCl 2 structure with two distinct fluorine sites which are coordinated to four and five lead atoms respectively. It may be outlined that d iso increases when descending the second column of the periodic table with values similar to those observed for the first column despite different crystal structures. Roughly, it may be shown that d iso or s P varies linearly versus the ionic radii of the cations for both the alkali fluorides and the difluorides. The trifluoride compounds GaF3 , AlF3 and InF3 crystallise with R3c symmetry. The networks are built up from corner-shared MF6 octahedra. The F site is axially symmetric and coordinated to two M 3q cations. In this case, d iso decreases from Al to In in the opposite direction of the ionic radius variation. The essential features of the different fluorides useful in the following, such as M–F distances and the number of Fy nearest neighbours, are recalled in Table 2. With these values, the quantities Ý l Sl for the fluorine–cation pair can be calculated when the

Table 2 Metal fluorine distances Ž dŽM–F.., coordination number ŽCN., experimental isotropic chemical shift Žexp. d iso ., and paramagnetic shielding Žexp. s P ., theoretically calculated Žsee Section 3.1. parameters Ž S M , calc. s P , D . and empirical Žsee Section 4. paramagnetic contribution Ž s l . in simple fluorides M–Fn Ž n s 1,2,3.

dŽM–F. ˚. ŽA

Exp. d iso

Exp. sP

Ýl

Calc.

Sl

s P)

Calc. D ŽeV.

sl

CN

LiF NaF KF RbF CsF MgF2 CaF2 SrF2 BaF2 ZnF2 CdF2 b-PbF2 a-PbF2

2.013 2.310 2.672 2.82 3.004 1.991 2.366 2.511 2.685 2.033 2.334 2.573 2.770 2.518 1.797 1.892 2.051

6 6 6 6 6 3 4 4 4 3 4 4 5 4 2 2 2

y37.0 y57.5 35.0 77.0 158 y29.5 58 80.5 153 y36.5 y26.5 129 110 147 y5 y3 y42

y90.0 y69.5 y162.0 y204.0 y285.0 y97.5 y185.0 y207.5 y280.0 y90.5 y100.5 y256.0 y237.0 y274.1 y122.0 y124.0 y85.0

520 672 1328 1585 1997 518 1274 – 2059 670 – – – – 407 – –

y69.5 y89.5 y177.0 y211.5 y266.5 y69.0 y170.0 – y274.5 y89.5 – – – – y54.5 – –

5.8 9.7 8.2 7.8 7.0 5.3 6.9 – 7.4 7.4 – – – – 3.3 – –

y15 y11.5 y27.0 y34.0 y47.5 y32.5 y46.5 y52.0 y70.0 y30.0 y25.0 y64.0 y47.5 y68.5 y61.0 y62.0 y42.5

AlF3 GaF3 InF3

M

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

cation radial wavefunction is known. It is verified that the smaller its distance to the considered Fy and the larger its electronic cloud, the more important the cation contribution Sl . Then, the calculations can be managed in two ways: - On the one hand, we can try to fit the experimental values by assuming a unique value of the D parameter for all the basic fluorides: the best results are obtained by adjusting D s 7.5 eV for all the compounds. This D value is in agreement with the literature w26x where D varies from 11.7 for LiF to 9.1 eV for CsF. The calculated s P values mirror the variation of the experimental ones. Fig. 1 compares the measured and calculated d iso parameters. - On the other hand, D may be adjusted for each compound in order to fit the experimental d iso and s P precisely. In this case, different D values ranging between 3.3 and 9.7 eV, are related to different Fy cationic environments. It appears that small D values are obtained for the basic fluorides where the cations have the smallest ionic radii, i.e., LiF, MgF2 and AlF3 . All the calculated values are given in Table 2. The main result of these calculations is that d iso and s P depend on the Fy-cation bonding essentially.

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3.2. FluoroperoÕskites (AMF3 ; A s K, Rb, Cs, Ba and M s Ca, Li, Mg, Zn) In a second step, the same calculations were done in fluoroperovkites. In Table 3, are collected all the measured compounds for which the atomic radial wavefunctions are known. Except KCaF3 , which adopts the cubic symmetry above 550 K, the studied fluoroperovskites are cubic ŽFm3m space group. at room temperature. The network is built up from MF6 corner-shared octahedra with A in interstitial position. The Fy site is axially symmetric, coordinated to two M 2q ions as nearest neighbours and four alkali ions as next nearest neighbours. In this series, BaLiF3 is an exception: it is an inverted perovskite. So, the Fy site is coordinated to four Ba2q and two Liq ions. The experimentally determined values of d iso are presented in Table 3. The calculations were driven in the same way as for basic fluorides. Only the nearest and next nearest cations were considered. Table 3 gives the essential data needed for the calculations of the contributions of the cations A, ÝA SA and M, ÝM S M , the resulting s P parameters calculated with D s 8.35 eV ŽFig. 2 mirrors the fine agreement between the variations of the experimental and calculated d iso . and the calculated D parameters which give a strict equality between experimental and calculated s P Žor d iso .. They are slightly larger than in the basic fluorides but less dispersed ranging from 7.9 to 9.3 eV. 3.3. Fluoroaluminates AMF4 (A s K, Rb; M s Al) As the radial wavefunctions of Kq, Rbq and Al ions are known, these compounds are also interesting candidates to test our calculations especially in the case of Al 3q for which a very low value of D was deduced from the 19 F chemical shift in AlF3 . RbAlF4 at room temperature has the tetragonal ˚ c s 6.2815 A˚ . structure ŽP4rmmm, a s 5.1227 A, which consists of infinite layers of AlF6 octahedra sharing four Feq atoms in the Ž001. plane; two unshared fluorine atoms denoted Fax lie along the c axis w28x. The structure of KAlF4 is derived from the ideal one by correlated rotations of the AlF6 octahedra around the four-fold axis of the tetragonal phase. At room temperature, the space group is P4rmbm 3q

Fig. 1. Comparison between experimental and theoretically calculated d iso parameters in simple fluorides Ž D s 7.5 eV..

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

94

Table 3 Metal fluorine distances Ž dŽM–F. and dŽA–F.., corresponding coordination numbers ŽM CN and A CN., experimental isotropic chemical shift Žexp. d iso ., and paramagnetic shielding Žexp. s P ., theoretically calculated Žsee Sections 3.2–3.5. parameters Žcalc. s P ) , D ., empirically calculated Žsee Section 4. paramagnetic contribution Žcalc. s P ) ) . in some complex fluorides. ax, eq, s, u and f stand for axial, equatorial, shared, unshared and free fluorines, respectively Calc. D ŽeV.

Calc.

y103 y109 y171 y179 y205 y193

y115 y113 y165 y167 y200 y188

9.3 8.6 8.0 7.8 8.2 8.1

y123 y139 y193 y195 y241 y220

1

y128

y92

6.0

y136

2 2

1

y128

y55

3.6

y127

2.918

4

14

y141

y126

7.5

y178

2

3.509 3.755

2 2

1

y128

y58

3.8

y126

1.999

1

2.636 2.701

1 2

80

y207

y218

8.8

y248

u

1.967

1

2.630 3.015

2 1

65

y192

y201

8.8

y228

s

1.967 2.074

1 1

2.839 3.927

2 2

7

y134

y141

8.8

y160

s

2.104

2

2.824 3.285 3.366

1 1 1

3

y130

y106

6.8

y120

f u s

– 1.968 2.050

0 1 2

2.607 2.941 3.20

4 4 4

167 30 15

y294 y157 y142

y305 y155 y111

8.7 8.3 6.5

y346 y176 y125

A

Exp.

CN

dŽA–F. ˚. ŽA

CN

d iso

2.027 1.987 2.18 2.227 2.262 2.000

2 2 2 2 2 2

2.867 2.810 3.09 3.149 3.199 2.828

4 4 4 4 4 4

y24 y17.5 44 52 78 66

ax

1.76

1

2.845

4

eq

1.81

2

3.411 3.725

ax

1.744

1

eq

1.828

u

KZnF3 KMgF3 KCaF3 RbCaF3 CsCaF3 BaLiF3 KAlF4

RbAlF4

BaZnF4

Ba 2 ZnF6

dŽM–F. ˚. ŽA

M

˚ and c s 6.1592 A˚ w29x. So, there with a s 5.0449 A are two types of fluorine atoms in these fluoroaluminates: Fax fluorines which belong to one AlF6 octahedron called unshared fluorines and Feq fluorines which bridge two octahedra, called shared fluorines. The experimental d iso and the results of the calculations are gathered in Table 3. The calculations were worked out in the same way as for basic fluorides and fluoroperovskites. We present the d iso and s P values calculated with D s 8.35 eV. The agreement between calculated and experimental values is not so fine as in the previous cases but is still satisfactory. The D values which provide a strict

Exp. s

P

Calc. s P)

AMFn

s P))

equality between experimental and calculated d iso and s P are now ranging from 3.6–3.8 eV for shared fluorines as in AlF3 to 6–7.4 eV for unshared fluorines. This result proves the dependence of D on the nucleus environment and the neighbouring cations. The low value of D obtained for shared fluorines may be related to the low contribution of Kq ions to the corresponding s P parameter. 3.4. BaZnF4 BaZnF4 belongs to the orthorhombic space group Cmc2 1 w30x. The Ba2q sites are coordinated by 11

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

Fig. 2. Comparison between experimental and theoretically calculated d iso parameters in fluoroperovskites Ž D s8.35 eV..

Fy ions. The Zn2q ions are surrounded by 6 Fy ions in which four Fy ions bridge other Zn2q ions and 2 Fy ions bond the Ba2q ions. According to recent crystallographic investigations w31x, there are four different fluorine sites: two being shared fluorines, the other two being unshared. The measured d iso , the s P parameters, the bond lengths, the coordination numbers and the results of the calculations are gathered in Table 3. As in the fluoroperovskites, a mean value of D equal to 8.35 eV leads to a satisfactory agreement between experimental and calculated isotropic shielding. The calculated D values which provide a strict equality between experimental and calculated d iso and s P, are in the same range as in perovskites except for the shared fluorines which correspond to a low contribution of the Ba2q ions to the shielding. 3.5. Ba2 ZnF6 This fluoride compound belongs to the space group I422. Its interest lies in the existence of three distinct types of fluorines as encountered in transition metal fluoride glasses w32x: shared, unshared and fluorines which are not embedded in a ZnF6 octahedron called free fluorines. The measured parameters, the crystallographic data and the results of the calculations are summarised in Table 3. As in BaZnF4 and fluoroperovskites, a mean value of D equal to 8.35 eV leads to a satisfactory agreement between experi-

95

mental and calculated isotropic shielding. The calculated D values which provide a strict equality between experimental and calculated d iso and s P, are in the same range as in perovskites except for the fluorines for which the contribution of Ba2q ions to the corresponding shielding is low. From this first part of the work, insofar as we may be confident in the determined D values which depend significantly on the neighbouring cations, it may be concluded that this approach provides an interesting way to calculate the isotropic part of the 19 F shielding tensor in inorganic fluorides. Nevertheless, owing to the lack of Clementi’s atomic orbitals, it is difficult to generalise to lead and gallium fluorides for instance.

4. Superposition model for shift

19

F isotropic chemical

So, in the following, we develop an empirical model supported by the preceding theoretical approach. In this model, the knowledge of the atomic orbitals should not be necessary. As previously outlined in theoretical fluorine chemical shift calculations w9,10x, the major difficulty is that, in the average energy approximation, it is not possible to know D whose value is specific to the shielding and not to other measurable property of the system. To remove this difficulty, we may assume that the isotropic paramagnetic shielding may be written

s Psy Ý l

Sl Ž d .

Dl

Ž ppm. ,

Ž 6.

where Dl is the calculated value for the cation l from the corresponding basic fluoride. This approach may be generalised to cations for which the radial wavefunctions are unknown. Indeed from the above results, the experimental d iso and s P parameters may be expressed as a summation of contributions related to the neighbouring cations as

d iso r C 6 F6 s y127.1 y Ý s l s P s Ý s l . l

l

Ž 7.

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

96

Table 4 Metal fluorine distances Ž dŽM–F. and dŽA–F.., corresponding coordination numbers ŽM CN and A CN., experimental isotropic chemical shift Žexp. d iso ., and paramagnetic shielding Žexp. s P ., empirically calculated Žsee Section 4. paramagnetic contribution Žcalc. s P ) ) . in some complex fluorides. s and u stand for shared and unshared fluorines AMFn

A

Exp.

Exp.

Calc.

CN

dŽA–F. ˚. ŽA

CN

d iso

sP

s P))

2.199 2.200

2 2

3.074 3.111

4 4

y47 y37

y80 y90

y110 y137

u

1.992

1

2.627 2.668

1 2

85

y212

y261

u

1.959

1

2.596 3.006

2 1

77

y204

y244

s

1.961 2.064

1 1

2.809 3.898

2 2

4

y131

y166

s

2.068

2

2.811 3.276 3.350

1 1 1

y5

y122

y125

KCdF3 RbCdF3 BaMgF4

dŽM–F. ˚. ŽA

M

Now, for a given cation l, the s l values are deduced from the experimental isotropic chemical shift measured in the corresponding basic fluoride compounds. They are given in Table 2 for all the studied fluorides. The relation between s l and the metal fluorine distance d in the considered fluoride compound is assumed to take a similar form as deduced from theoretical calculations:

sl s y

Sl Ž d .

Dl

sy

Al

Dl

exp Ž ya l d . ,

Ž 8.

where a l is deduced from the relation Ž5.. At last, the determination of Dl is not necessary. Actually, if s l 0 and d 0 are the cation contribution to the shielding and the cation fluorine distance in the related basic fluoride respectively, we get:

s l s s l 0 exp ya l Ž d y d 0 . .

Ž 9.

At this point, we have to verify this empirical superposition model by applying it to an extended set of fluorides. This model was first applied to AMF3 fluoperovskites. The d iso and s P parameters were calculated according to the following relations:

d iso r C 6 F6 s y127.1 y 2 s M y 4sA , s P s 2 s M q 4sA .

Ž 10 .

Then, we applied the model to fluoroaluminates, BaZnF4 and Ba 2 ZnF6 . The results obtained with the superposition model are compared to the experimental values in Table 3. In Table 4, calculations concerning two cadmium fluoroperovskites and BaMgF4 assumed to be isostructural to BaZnF4 are presented which confirm the applicability of the model. The quite large difference between experimental and calculated s P values for ACdF3 which is obviously out of the range observed for the rest of the family, may be attributed to covalency effects due to large Cd 2q electronic cloud. At last, the model has been applied to CsZnGaF6 which belongs to the pyrochlore family. As in GaF3 , the fluorine atoms are shared between two fluorine octahedra. The Ga3q and Zn2q are statistically distributed among the centres of these octahedra. So there are three different types of environment for the fluorine nuclei: 50% are bonded to one Zn2q ion and one Ga3q ion, 25% to two Zn2q and 25% to two Ga3q ions. The statistical distribution of the two different transition metal ions results in some topological disorder. This is the main point which makes this compound interesting in the framework of the investigation of fluoride glasses. The measured chemical shifts w14x and the results of the calculations are given in Table 5. The Ga–F and Zn–F

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

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Table 5 Metal fluorine distances Ž dŽM–F. and dŽA–F.., corresponding coordination numbers ŽM CN and A CN., experimental isotropic chemical shift Žexp. d iso ., and paramagnetic shielding Žexp. s P ., empirically calculated Žsee Section 4. paramagnetic contribution Žcalc. s P ) ) . in CsZnGaF6 CsZnGaF6 Ga–F–Ga Ga–F–Zn Zn–F–Zn

dŽM–F. ˚. ŽA

M CN

1.89 1.89r2.01 2.01

2 2 2

dŽCs–F. ˚. ŽA

CN

3.225, 3.700

1, 2

mean distances given by EXAFS w33x were used in the calculations. Each fluorine atom has three Csq ions as next nearest neighbours. Once more the agreement between the experimental and calculated s P is quite correct. A linear regression Žcorrelation coefficient: 0.97. taking into account the values in Tables 2–4 provides the following relation:

s P Ž calc. . s 1.15s P Ž exp. . .

Ž 11 .

So, on average, the calculated values are overestimated by about 15%. This is in agreement with a D parameter larger in complex fluorides than in the basic ones as previously observed through the theoretical calculations. This overestimation of the calculated values may be related to a slight covalency effect which is neglected all along this work. To conclude this second part of the paper, we would like to outline that from this superposition model, the 19 F isotropic chemical shifts in a given ionic compound may be inferred quite easily and reliably when its structure Žbond lengths and coordination numbers essentially. is known.

5. The superposition model as a structural probe for complex fluorides The fine agreement between experimental and calculated values reported for a large set of inorganic fluorides supports the extension of this model to compounds whose structure is unknown: crystalline model compounds ŽPb 2 ZnF6 , PbGaF5 , Pb 3 Ga 2 F12 and Pb 9 Ga 2 F24 . and transition metal fluoride glasses ŽPZG, PBI, PZI, PLG, PKG and PKG.. These fluoride compounds have the peculiarity that either their crystallographic structures have never been precisely

Cs

Exp. d iso

Exp. sP

Calc. s P))

16 y2.5 y23.5

y143 y125 y104

y168 y138 y107

determined or they are disordered compounds. In all these lead fluorides, 19 F NMR measurements give evidence for some free, shared andror unshared fluorine ions w13x. The application of the above developed empirical chemical shift calculations to these compounds was achieved in order to get information on bond lengths and coordination numbers. The mismatch between calculated and experimental s P values was neglected in the following. The approach may be exemplified in the case of Pb 2 ZnF6 . 5.1. Pb2 ZnF6 Pb 2 ZnF6 is assumed to have a structure similar to Ba 2 ZnF6 Žspace group I422. w13x. Then, free, shared and unshared fluorine sites are present. The measured 19 F isotropic chemical shifts d iso and the related paramagnetic shieldings s P are presented in Table 6. As in Ba 2 ZnF6 , free, shared and unshared fluorine atoms have 4 neighbouring Pb 2q. The experimentally measured isotropic paramagnetic shielding may be written:

s P s Ý s Zn q Ý s Pb s n Zn s Zn q n Pb s Pb , Zn

Ž 12 .

Pb

n Zn and n Pb are the number of Zn2q and Pb 2q ions respectively, in the neighbouring of the 19 F nucleus under consideration. s Zn and s Pb represent the contribution of one Zn2q and Pb 2q ions to s P, respectively, as defined in Table 2 but modified by the distance effect. In Pb 2 ZnF6 , we have n Zn s 0, 1, 2 for free, unshared and shared fluorines respectively and n Pb s 4. So, assuming that the mean Zn–F distance is equal to that observed in ZnF2 , which is verified in

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

98

Table 6 Experimental isotropic chemical shift Žexp. d iso ., paramagnetic shielding Žexp. s P . and empirically calculated Žsee Sections 5.1 and 5.2. Pb 2q paramagnetic contribution in some lead zincrgallium fluorides. f, u, s stand for free, unshared and shared fluorines respectively Exp. d iso

Exp. sP

ÝPb s Pb

Pb 2 ZnF6

f u s

147 74 29.5

y274 y201 y157

y274 y169 y92

Pb 9 Ga 2 F24

f u

139 59

y266 y186

y266 y124

Pb 3 Ga 2 F12

f u s

141 60 34

y268 y187 y161

y268 y125 y36

PbGaF5

u s

60 40

y187 y167

y125 y42

d Ž Pb–F . s d 0 y

a Pb

Ý s Pb ln

Pb

n Pb s Pb0

,

d Ž Pb–Ff . - d Ž Pb–Fu . - d Ž Pb–Fs . . So, on this example concerning Pb 2 ZnF6 , assuming that a fluorine atom has 4 Pb 2q in its neighbourhood as in Ba 2 ZnF6 , we were able to evaluate the mean distances between the free, unshared and shared fluorines and the surrounding Pb 2q ions. 5.2. PbGaF5 , Pb3 Ga2 F12 , Pb9 Ga2 F2 4 and TMFG

Ba 2 ZnF6 , it is straightforward to deduce the ÝPb s Pb s n Pb s Pb values. They are given for the three types of fluorine atoms in Table 6. The Pb–F distances can be evaluated according to the formula: 1

ues of the Pb–F distances according to the type of fluorine may also be emphasised:

Ž 13 .

˚ and s Pb0 s y64 ppm are the reference d 0 s 2.573 A ˚ y1 is values from b-PbF2 . The value a Pb s 2.880 A ˚ which is the deduced from Eq. Ž5. with r Pb s 1.45 A Pb 2q ionic radius in four-fold coordination. It is found: ˚ d Ž Pb–Fu . s 2.72 A, ˚ d Ž Pb–Ff . s 2.55 A, ˚, d Ž Pb–Fs . s 2.93 A for free, unshared and shared fluorines, respectively which may be compared to the sum of the ionic radii ˚ These distances are slightly smaller than the 2.64 A. corresponding Ba–F distances in the isostructural Ba 2 ZnF6 compound ŽTable 3.. This is in agreement with the Pb 2q ionic radius which is smaller than the Ba2q one. It may be noted that the distance dŽPb– ˚ is nearly equal to the distance Pb–F in Ff . s 2.55 A b-PbF2 . This result enforces the confidence we may have in these calculated distances. The distinct val-

For all these compounds, since the Pb coordination number of the fluorines is not known a priori, the structural information is not extracted from the values of ÝPb s Pb as easily as for Pb 2 ZnF6 . Nevertheless, the aim of this last part of the paper is to demonstrate that it is possible to get some information about coordination and bond length through these values. PbGaF5 , Pb 3 Ga 2 F12 and Pb 9 Ga 2 F24 allow to study the variation of the isotropic chemical shifts of these fluorines when the PbF2rGaF3 ratio is varied. The measured 19 F isotropic chemical shifts s iso and the related paramagnetic shieldings s P are gathered in Table 6. Assuming that the mean Ga–F distance is equal to the corresponding one in GaF3 , the values of ÝPb s Pb which are the Pb 2q contributions to the paramagnetic shieldings may be evaluated. They are given in Table 6. The TMFG should be classified into two families w14,15x. In the first one ŽPZG, PBI, PZI., the transition metal ions are statistically distributed and free, shared and unshared fluorine ratios vary according to the composition. The measured 19 F isotropic chemical shifts d iso averaged over the glass composition, the related paramagnetic shieldings s P and the values of ÝPb s Pb calculated assuming that Ga–F, Zn–F and In–F distances are those measured in the related basic fluorides are given in Table 7. In the case of PBI, the situation is more complex due to the same role played by Pb and Ba. Note that ÝPb s Pb should be written ÝPb s Pb q ÝBa s Ba . This case will be not discussed further. The second family of TMFG ŽPLG, PNG and PKG. contains alkali fluorides and the networks are build up from disconnected domains of AF6 and GaF6 corner shared octahedra w14,15x. In this case, all the fluorines are shared between two

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

99

Table 7 Experimental isotropic chemical shift Žexp. d iso ., paramagnetic shielding Žexp. s P ., and empirically calculated Žsee Section 5.2. Pb 2q or Ba2q paramagnetic contribution in some TMFG. f, u, s stand for free, unshared and shared fluorines respectively Exp. d iso PZG

Exp. sP

ÝPb s Pb

156 69

y283 y196

y283 y164 y134

Zn–F Ga–F

f u u

Zn–F–Zn Ga–F–Zn Ga–F–Ga

s s s

8

y135

y70 y40 y10

PBI

f u s

155 50 y11

y282 y177 y116

y282 y135 y31

PZI

f

154

y281

y281

Zn–F In–F

u u

67

y194

y162 y152

Zn-F–Zn In-F–Zn In-F–In

s s s

0

y127

octahedra. The averaged measured 19 F isotropic chemical shifts d iso , the related paramagnetic shieldings s P and the values of ÝPb s Pb calculated assuming that Ga–F and Li, Na, K–F distances are those measured in the related basic fluorides are given in Table 8. Actually as seen from Eq. Ž13., the calculated values of ÝPb s Pb can mirror not only the Pb–F distances as in Pb 2 ZnF6 , but also the Pb coordination number n Pb which may be different from 4. In order to take these two parameters into account simultane-

y62 y52 y42

ously, the variations of dŽPb–F. were represented as a function of ÝPb s Pb with n Pb as a parameter ranging from 1 Žbottom. to 6 Žtop. ŽFig. 3.. Owing to steric considerations, the value n Pb s 6 is clearly a maximum. On this figure, the calculated values of ÝPb s Pb presented in Tables 6–8 are also reported as vertical lines. Solid, dashed, and dotted lines are related to free, unshared and shared fluorines, respectively. The main point is that no overlap is observed between the values of ÝPb s Pb for the three types of fluorine atoms. This result reinforces the correctness

Table 8 Experimental isotropic chemical shift Žexp. d iso ., paramagnetic shielding Žexp. s P . and empirically calculated Žsee Section 5.2. Pb 2q paramagnetic contribution in some TMFG containing alkali metal fluorides Exp. d iso

Exp. sP

ÝPb s Pb

PLG

Li–F–Li Ga–F–Ga

y19 11

y108 y138

y78 y14

PNG

Na–F–Na Ga–F–Ga

y26 7

y101 y134

y78 y10

PKG

K–F–K Ga—F-Ga

48 6

y175 y133

y121 y9

100

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

Fig. 3. Pb–F distance versus the ÝPb s Pb variable for n Pb s 1 Žbottom. to 6 Žtop.. The vertical solid, dashed and dotted lines represent the experimental values of ÝPb s Pb listed in Table 6Table 7Table 8 for free, unshared and shared fluorines, respectively. The two horizontal ˚ stand for the limiting values of the Pb–F distance. The rectangles represent the domains of compatibility of the lines at 2.3 and 3.7 A experimental results with distance and coordination constraints for the three types of fluorines.

of the fluorine atom classification in the various fluoride compounds w13–15x. It is also in favour of disconnected domains of AF6 and GaF6 corner shared octahedra in the PLG, PNG and PKG glasses, since the corresponding ÝPb s Pb values are all found in the range corresponding to shared fluorines. In the following discussion, we have to keep in mind that according to literature, the range of the ˚ w34x. Pb–F distances is quite large from 2.3 to 3.7 A First, consider the left part of Fig. 3 which is related to free fluorines. The coordination numbers of this kind of fluorine which are compatible with ˚F Pb–F distances are 3 F n Pb F 6, we get 2.4 A ˚ ˚ Ž . d Pb–Ff F 2.7 A with a mean value equal to 2.55 A and an average coordination number equal to 4. These values are compatible with the mean Pb–F distances measured in the various phases of PbF2 . However, the very small dispersion of the dŽPb–Ff . values Žfor an average value of n Pb s 4 as in ˚ F dŽPb–Ff . F 2.56 A˚ . Pb 2 ZnF6 , we found 2.54 A allows us to suggest similar environments for the

free fluorines in all the studied compounds. So, on average, a free fluorine is surrounded by four Pb 2q ˚ ions at a mean distance of 2.55 A. A similar discussion can be done for unshared fluorines. The coordination numbers of this kind of fluorine which are compatible with Pb–F distances ˚ F dŽPb–Fu . F 2.9 A˚ are 2 F n Pb F 6, we get 2.4 A ˚ and an average with a mean value equal to 2.7 A coordination number between 3 and 4. The mean dŽPb–Fu . distance is larger than the mean dŽPb–Ff . one. From our results ŽTables 6 and 7., for a given coordination number, it appears that the distance dŽPb–Fu . is longer when the fluorine atom belongs to a GaF6 octahedron rather than to a ZnF6 one. This may be explained by electrostatic repulsion which is larger between Pb 2q and Ga3q than Pb 2q and Zn2q. For shared fluorines, it seems that all the coordination numbers have to be considered. So, the dis˚F persion of the dŽPb–Fs . may be very large: 2.3A ˚ Ž . Ž . d Pb–Fs F 3.7 A. The mean d Pb–Fs distance is ˚ somewhat larger than the other found equal to 3.0 A

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

two Pb–F lengths. The average coordination number is found between 3 and 4. It may be outlined that the values of ÝPb s Pb about y10 ppm observed in all the studied glasses for fluorines shared between GaF6 octahedra are compatible with coordination numbers smaller than or equal to 4 only. However, for n Pb s 4, ˚ We we get a Pb–F distance as large as 3.7 A. suggest that this unusually large value could be an indication of a small Pb coordination number of the fluorine shared between two GaF6 octahedra. Indeed, for n Pb s 1 and ÝPb s Pb s y10 ppm, we obtain ˚ which is still large compared to dŽPb–Fs . s 3.2 A the values obtained for free and unshared fluorines. In the case of the alkali fluoride glasses ŽPLG, PNG and PKG., this result demonstrates that the distribution of the Pb 2q ions between the disconnected AF6 and GaF6 domains occurs in favour of the former ones and tends to ensure a local electrical neutrality. As previously outlined for unshared fluorines, for a given coordination number, the distance dŽPb–Fu . is longer when the shared fluorine atom belongs to GaF6 octahedra rather than to ZnF6 ones, due to electrostatic repulsion effect.

6. Conclusions A large set of experimental measurements of 19 F isotropic chemical shifts in fluoride compounds obtained by MAS NMR at 15 kHz spinning rate, has been presented. Some of the data were not published previously. First, it was shown that Ramsey’s theory of the chemical shift with molecular orbitals obtained by Lodwin’s orthogonalisation method is well-adapted ¨ to evaluate the isotropic part of the 19 F chemical shift in ionic fluorides. The D parameters which are specific to the shielding in the average energy approximation were calculated in fluorides for which the Slater atomic radial wavefunctions were known. The calculated D values ranging between 3.3 to 9.3 eV are of the right order of magnitude. In a second step, it was shown that it was possible to overcome the failure of a reliable estimation of D in the calculation of the chemical shift. As evidenced from the theoretical approach, the calculations were founded on the assumption that the paramagnetic

101

part of the 19 F shielding in a given material is simply the summation of the paramagnetic contributions due to all the cations in the neighbourhood of the considered fluorine. The paramagnetic contribution to the 19 F shielding of different metal ions were deduced from the experimental measurements of d iso in the corresponding simple fluorides. Then these values were used to calculate the d iso parameters in various fluorides whose crystallographic structures were precisely known. The results were found in fine agreement with the experimental values within a 20% error bar. Finally, this prompted us to apply this empirical approach to lead fluoride compounds with unknown structure in connection with TMFG. Our results confirm the fluorine classification previously proposed in free, unshared and shared fluorines in all the investigated fluorides in which the essential feature is the networks built up from MF6 octahedra. They also confirm the existence of disconnected domains of AF6 and GaF6 corner shared octahedra. The Pb 2q ions are distributed in interstitial positions preferentially in domains containing AF6 octahedra: this tends to insure local charge neutrality in agreement with the composition of the glasses. Information about Pb 2qy Fy distances was obtained: it was clearly demonstrated that the mean Pb 2qy Fy distances depends on the type of fluorines in the following way: dŽPb–Ff . - dŽPb–Fu . - dŽPb–Fs . with mean ˚ respectively. At values equal to 2.55, 2.7 and 3.0 A last, the Pb coordination numbers of the different types of fluorine atoms were also discussed: in average, the three- or four-fold coordinations seem the most probable for the three types of fluorines, in agreement with the mean distances. Nevertheless, we suggested that shared fluorines between GaF6 octahedra might have only a small number of neighbouring Pb 2q ions. The superposition model for 19 F isotropic chemical shift calculation developed and tested in this paper appears to be a valuable tool to relate NMR measurements and local environment of fluorine nuclei in ionic fluorides at least. More generally, this approach can be applied to all the ionic fluorides where the diamagnetic contribution to the 19 F isotropic chemical shift may be assumed constant. It permits to avoid quantum chemical calculations when they would be too time consuming or impossible to

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

102

undertake due to the lack of reliable atomic wavefunctions.

Acknowledgements It is a pleasure to acknowledge A.M. Mercier for her assistance in fluoride synthesis and G. Niesseron, J. Nouet for providing us with CsCdF3 , KCdF3 , BaMgF4 , BaZnF4 , Ba 2 ZnF6 compounds. Appendix A. Theoretical calculation of isotropic chemical shift [9]

19

F

According to Ramsey’s theory w1,2x, the shielding tensor is split into a diamagnetic and a paramagnetic term si j s si dj q si pj , where

si j d s

e2 2 mc 2

¦

c0

Ý rk2di jyr

ki

r3

kj rk

k

;

c0

and

di j p s

e2 2 2

2m c

¦

= cm

¦

= cm

1

Ý Ž E yE . 0 m m Ý k

lk j r k3

;¦ ;5

c0 q c0

Ý lk i c0 k

½¦

c0

;

Ý l k i cm k

Ý k

lk j r k3

;

cm

ye 2 m2 c 2D

Ý k,k

X

¦

c0

,

Ý k

l k i l kX j r k3

c k s Ý A al Ž k . w la , la

where w la is an atomic wavefunction of a-type centred on the l atom. We introduce the charge density wPx matrix whose elements are defined by w1,2x, a )b Plab m s Ý 2 Al Ž k . Am Ž k . .

k

We take into account that the nucleus of interest Ž l s 0. is 19 F. The w 0a wavefunctions are 2s, 2p x , 2p y , 2p z . Following Grosescu and Haeberlen w9x, we retain only the contributions linear in ²1rr : 0 , ²1rr 3 : 0 and ² w la <1rr < w lb : where l stands for a ligand. The expressions of the diamagnetic and paramagnetic parts of the shielding tensor are now written:

sxdx s

e2 2 mc 2

sx px s y

c 0 and cm are the ground Ž E0 . and excited Ž Em . state wavefunctions of the system in the absence of the external magnetic field. ™ r k is the vector from the considered nucleus to the k ™ th electron, Ž r k i is the ith component of this vector.. l k is the angular momentum operator of the k th electron Ž l k i is the ith component of this operator.. e and m are the electron charge and mass respectively. The paramagnetic term involves a sum over the excited states. According to Van Vleck and Frank w35x, Ramsey simplified the above expression w1,2x where E0 y Em - 0 are replaced by an average value D ) 0 which may be seen as a parameter: si pj s

The ground-state wavefunction c 0 is written as a Slater’s determinant of molecular orbitals c k . The doubly occupied molecular orbitals c k are approximated by linear combination of atomic orbitals ŽLCAO–MO.:

;

c0 .

¦

Ý P00a a w 0a a

e 2 "2 m2 c 2D

r2yx2 r3

;

w 0a .

zz ² ry3 : 2 p P00y y q P00z z y P00y y R 00

az qP00y z P00y z q Ý S10 Ž P00y z P0yla y P00y y P0zla .

la ay q Ý S10 Ž P00z y P0zla y P00z z P0yla .

la

1 y "

Ý Ý P0yla P0zlb Im² w lb < l x < w la : l

.

ab

Here, the first discrepancy with Ref. w9x appears: it concerns the last term in the above expression which is written yŽ mr" .Ý l Ý a b P0yla P0zlb <² w lb <1 x < w la :<. The two other diagonal components are obtained by cyclic permutation of the index. Fluorides are known as ionic compounds. However, it is straightforward to show that a purely ionic model cannot account for the 19 F isotropic chemical

B. Bureau et al.r Chemical Physics 249 (1999) 89–104

shift in fluorides. In this case, c j s w1a. Then, P1laa s 2 and Plab m s 0, and the prominent terms are:

sxdx s

e2

Ý mc 2 a

¦

w 0a

r2yx2 r3

;

103

equivalent to the purely ionic approximation, and gives the diamagnetic contribution to the isotropic part of shielding tensor:

w 0a

s d s sxdx s sydy s szdz s

and

sx px s sy py s sz pz s 0. 2e

2

3mc

2

Ý a

¦

w 0a

1 r

;

w 0a

which is independent of the nucleus environment in disagreement with the experimental results. In a second step, molecular orbitals ŽMO. obtained by Lodwin’s orthogonalisation method are ¨ used w16x. They may be expressed as:

c k s cnhs wnh y

1

Ý Slahn w la 2 la

3 q 8

bh a Ý Ý Slab m Sm n w l . . . .

la m b

The overlap integrals will never exceed 0.1. So, this type of MO has been widely used in ionic compounds w8–10x. Nevertheless, this form does not take into account any covalency effect which may be present in metal fluorine bonding. However, as far as only isotropic chemical shift is concerned, covalency may be not considered further. Neglecting all terms containing multiplication of more than two overlap aa integrals, it is easy to calculate P00 s 2 w1 1 ba . 2 x Ž q 4 Ým b Sm0 . Here is the second discrepancy ba . 2 x with Ref. w9x where P00aa s 2w1 q Ým b Ž Sm0 . We agree with the other results which are written Plaa l s ab ma mb . 2w1 q 14 Ý m b Ž Smbal . 2 x, P00 s 2Ý m mŽ Sm0 Sm0 and ab ma mb . Ž Plab 0 s 2 ySl 0 q Ý m m Sm l Sm0 . Overlap integrals are generally expressed by p , s or d atomic functions. Calling l, m and n the director cosines of the fluorine–metal bond in the principal axes of the tensor, we get the following formulas according to McWeeny w36x:

sxdx s

2 e2 3mc 2

3mc 2

Ý

² ry1 :i

is1s,2s,2p

The paramagnetic contribution is written:

The isotropic chemical shift should be written:

sss dqs ps

2 e2

Ý

² ry1 :i

is1s,2s,2p

where only the prominent term is kept which is three orders of magnitude larger than the others. It is

sxpx s y

e 2 "2 2 2

2 Dm c

² ry3 : 2 p 7 Ž 1 y l 2 . Ž Slsp0 . 2d 2

2

2 pp q7 Ž 1 y l 2 . Ž Slpp 0 . s q Ž 7 y l . Ž Sl 0 . p 2

pp 2 dp y8 Ž 1 y l 2 . Ž Slpp 0 . s Ž Sl 0 . p q 7 Ž 1 y l . Ž Sl 0 . s 2

q Ž 7 y l 2 . Ž Sldp0 . p y 8'3 Ž 1 y l 2 . = Ž Sldp0 . s Ž Sldp0 . p . These formula correspond to the shielding due to one ligand with ns, np and nd valence electrons. If one of these electrons is not present, the corresponding overlap integral has to be taken equal to zero. The paramagnetic contribution to the shielding is different from zero and depends on the geometry of the nucleus environment, its isotropic part is deduced from

s p s 13 Ž sxpx q sypy q sz pz . which gives

s psy

e 2 "2 3 D m2 c 2

2 ² ry3 : 2 p 7 Ž Slsp0 . 2d q 7 Ž Slpp 0 .s

2

2

pp pp dp q10 Ž Slpp 0 . p y 8 Ž Sl 0 . s Ž Sl 0 . p q 7 Ž Sl 0 . s 2

q10 Ž Sldp0 . p y 8'3 Ž Sldp0 . s Ž Sldp0 . p . It may be outlined that s p is independent of the angular distribution of the ligands around the considered fluorine.

References w1x N.F. Ramsey, Phys. Rev. 78 Ž1950. 699. w2x N.F. Ramsey, Phys. Rev. 86 Ž1952. 243. w3x E.R. Andrew, A. Bradbury, R.G. Eades, Nature 182 Ž1958. 1659. w4x A.T. Kreinbrink, C.D. Sazavsky, J.W. Pyrz, D.G.A. Nelson, R.S. Honkonen, J. Magn. Reson. 88 Ž1990. 267.

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