Ligand hyperfine structure in the ESR spectra of the ionsMoOF2-5 and CrOF2-5Verbeek, J.L.

DOI:10.6100/IR88291

Published: 01/01/1968

Document VersionPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differencesbetween the submitted version and the official published version of record. People interested in the research are advised to contact theauthor for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ?

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 21. Aug. 2018

LIGAND HYPERFINE STRUCTURE IN THE

ESR SPECTRA OF THE IONS MoOF~­

AND Cr0F2· 5

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR

IN DE TECHNISCHE WETENSCHAPPEN AAN DE

TECHNISCHE HOGESCHOOL TE EINDHOVEN, OP

GEZAG VAN DE RECTOR MAGNIFICUS,

DR. K. POSTHUMUS, HOOGLERAAR IN DE AFDELING

DER SCHEIKUNDIGE TECHNOLOGIE, VOOR EEN

COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP

DINSDAG 5 MAART 1968, DES NAMIDDAGS

OM 4 UUR

DOOR

JOHANNESLEGNARDUS VERBEEK

GEBOREN TE 'S-GRA VENHAGE

DRUKKERIJ J . H. PASMANS . 'S-GRAVENHAGE

Promotor: Prof.Dr. G.C.A. Schuit

CONTENTS

General Introduetion

Chapter l

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Summary

Samenvatting

Heferences

Epiloog

Introduetion The Theory of Electron Spin Resonance

Ligand Hyperfine Structure in MOX~--ions

Analysis of ESR Spectra of Polycrystalline Samples

The ESR Spectra of CrOF ~- and MoOF ~­

Discussion of Available Spectroscopic­and ESR Data

Final Remarks and Conclusion

Curriculum Vitae

Appendix

5

6 8

13

25

37

53

66

69

70

71

74

75

77

5

GENERAL INTRODUCTION

The progress of coötdination chemistry got a new impetus when in the post-World War period, high! y sensitive electronic devices be­came avoilobie on a large scale.

Most important in this respect is the development of NMR and ESR, the latter being the main tooi for the investigations to be describ­ed in the following Chapters.

The great value of ESR in complex chemistry is its ability to recognize symmetry patterns, with unpaired electrens as an inter­mediary. That includes also information on the electronic structure in the ground state of a paramagnetic molecule, and on the hyperfine interactions, i.e. interachons of electrens with each ether or with the nuclei of constituent atoms in the molecule.

For several reasons, the complexes of the type MOX ;-, with M a transition metal ion and X a halogen, might draw some attention. One reasen is that their structure deviates only slightly from an octahedral configuration, a structure that has been extensively investigated during the last two decades.One may then consider those molecules as a first step in the generalization of the problem of chemica! bonding. The validity of methods to describe really octahedral molecules may be checked, and conclusions can be drawn in that respect. ESR is one of the major experimentcri aids in such kinds of research.

The well-resolved ligand-hyperfine structure, as observed in the Fluorine complexes with a structure as mentioned above, might be expected to bear subtie information on those hyperfine interactions. lt will be shown, however, that a classica! model describes the details of these spectra with a surprisingly high degree of accuracy.

A secend reasen for the choice of the present subject is found in catalysis. Intermediate compounds, formed in the course of a cata­lytic reaction, aften exhibit symmetries lower than cubic. Information on structure and stability of such intermediates, is of crucial importance for the understanding of the fundamentals of the catalytic process. A number of catalysts containing transition metal ions strongly bonded to oxygen atoms, with a remarkably short metal-oxygen distance, indic­ate the fundamental importance of model-compounds of the type we are going to describe.

6

CHAPTER I

Introduetion

In this chapter the shape of the molecules that are the subject of the work to be described in this thesis, will be considered first. The chemistry of the elements V,Cr,Mo,W,Nb,Ta and, probably, Re, has provided us with a number of ions and molecules in which the central ion consists of a pair MO with a remarkably short M-0-distance; well known examples of this kind of pairs are found in, for instance, V2 0 5 , Cr0 2 Cl 2 and VOS0 4 .5H 20ll.

Although only few data exist regarding the exact geometry, there is a fair amount of evidence for the structure of what we will occasion­ally call metalyl-compounds.

The structure of the blue VOS0 4 .5H 20 is known 2 l to consist of a VO-group, with V-0-distance of 1.67 Ä, perpendicular to a square of four water-c,xygens, now such that the V-0-distance is 2.3 Ä, thus forming a square pyramid with V in the centre of the base; this pyramid is completed to form a distorled octahedron by placing a sixth 0, from a so;--ion, in the remaining axial position. This structure is shown in fig.l.l and obeys the transformation properties of the point-group C4 v. The ions CrOX~- and MoOX~-. where X is F or Cl, are supposed to exhibit a similar structure; the nature of the axial ligand however, is rather uncertain. As will be pointed out in Chapter 2, we are not interested in its exact nature, so we will not discuss this point to any depth.

x l x k55:7 X I

I I I I I I I

* fig. I. 1: Shape of molecule s MOX~-.

7

Ballhausen and Gray3 l discussed the bonding scheme on the basis of an approximative Molecular Orbital calculation. They arrived at an energy diagram that is shown in fig.I.2. We shall use this picture as a guide for the electronic structure of the ions we are studying.

fig. I. 2 : Molecular Orbital scheme according to Ballhausen & Gray.

A detailed account of the assumptions we use 1 throughout Chopters l-4 1 concerning the chemica! bonding in the compounds just mentioned will be gi ven in Chapter 5.

Gray and Hare4 l applied the scheme on MoOcl;- and were led to acceptable results 1 which we will discuss in Chapter 5.

According to this level scheme 1 the unpaired electron accupies the b2 level. Static susceptibility- 1 ESR- and spectroscopie data can be explained by this scheme. It is also in accordance with the pre­dictions of the Crystal Field Theory.

Theoretica! aspec ts of the interpretation of ESR-spectra are found in m any text-book S 1 such as those of Slichter5 l 1 P ake 6 l 1 Carrington and McLachlan 7 l 1 while special mention should be made of the book by Abragam 8 l1 which covers the whole field of magnetic resonance in great detail.

8

We shall not attempt to reproduce the theory in its full length but shall confine oursel ves to a short survey of these aspects that are of direct importance for the subject we are dealing with.

The theory of electron spin resonance.

a) Qualitatieve aspects.

Electron Spin Resonance (ESR), or Electron Paramagnetic Reson­ance (EPR), is the spectroscopy of the transitions between Zeeman­levels, using centimeter wavelengths (X-band, 3 cm.) or millimeter wavelengths (Q-band, 0.8 cm.).

A dynamica} description in full detail of the resonance-phenom­enon may be found in reierences 5-8. We will present here the more phenomenological description.

A spin-degenerate doublet will be split by a magnetic field; this is the well-known Zeeman-splitting. Consiclering the spin as a mag­netic dipole with dipole moment p. , its. energy in a magnetic field s will be

( l.l)

We may write p. 8 = yS, where S is the spin-angular menturn in units 'Îl ( = h/27T), and y = wL /H ff ti ; alternatively we write armo e ec ve

lls = g,BS

so that y1i. = gf3, ,8 being the Bohr-magneton, and g the gyromagnetic ratio, the ratio between magnetic- and mechanica! moment.

From l.l fellows that a parallel position of magnetic moment and field is the most favourable one; the anomaly of the electron' s magnetic moment requires then the spin to be antiparallel to the field.

For a free electron (S = Y2) we have the situation of fig.I.3, where the direction of the externcri field is taken as the z-direction. An electron passes from the lower to the upper level by "flipping" its spin. These two spin states, with m = -Y2 and +'12 respectively, s are not connected by the interaction H S , since this operator has z z diagonal matrix elements only between the two states.

On the contrary, the operators S and S have matrix elements x y connecting the two spin states, and no diagorral elements. A micro-wave field in the xy-plane, that is perpendicular to the externcri field, produces a magnetic field with components H cos c.;t, and so inter-x,y

9

~·--------------~·----~•~ H 0 fiq.l.3: Splitting of a spin doublet in a magnetic field.

actions H cos c.;t are introduced; this time-dependent perturbation causes thexelectron to invert its spin.

The orbital motion of the electron has so far been neglected. It is important however, since the presence of orbital angular momenturn causes the g-values to depart from the free-electron value 2.0023. When an ion is placed in the surroundings of ligands, the motion of its electrens is modified. In sectien b) it will be indicated how the action of a crystal field tends to cancel the orbital motion, or, to "quench" it. Experimentally this effect is revealed by the magnetic susceptibility of many first-row transition metal compounds, which turns out to be almost exactly the "spin-only"-val1Je9 >.

An interaction counteracting this quenching, is the coupling between the electran's spin- and orbital moment. The classica! Hamil­tonian for this interaction is

~

H so

where E = electric field, p = momenturn and S = spin angular momentum. Assuming E to be spherically symmetrie, we write for it

whence

E(r) =_!_E(r) r

Ex p =_lli!l r x p =!_E(r) L r r

10

and finally

~

H so À(r) L.S. ( 1.2)

E(r) following Coulomb's law, causes À(r) to obey an inverse­cube law with respect to r. The effect of the LS-coupling can be shown to be a mixing of higher orbitals into the ground state orbital, lxy> in our case, by the x- and y-components of L.

The expectation value <L > of L will then no longer be zero, and the value of g departs from 2.0023.

T he I xy > ground state of our tetragenall y deformed octahedron is modified te be

The matrix elements of the Zeeman-operator ,BH. (L + 2S) are then

<xy+ I + xy >

,BH ( 1- ---=4-'--'-À-2t

E 2 2 x -y

,BH 0-_1_) +i,BH (1-~) x E Y E

x:a- ya

,BH (1-~)-i,BH (l-l) x E Y E

X2t y:<>

_,BH (1- 4À :a- E 2 2

x -y

where the energies are relative to E . xy Solving the secular determinant (see Chapter 5) for three parti-

cular direcUons of H, the following vcrlues of g are found: H = (O,O,Hz) g = ,BH(2- 8À/E 2 2); z x -y

H = (Hx,O,O) gx = ,BH(2- 2À/Exz); (1.4)

H = (O,Hy,O) gy = ,BH(2- 2À/Eyz).

Since E = E we may write g if =: g and g1 = g = g • XZ 'tZ 11 Z X y

The result now obtained is usually presented via the introduetion of a Spin-Hamiltonian

ll

Pryce 10 l derived the general form of a Spin Hamiltonian

H=E +2,L3S.H-I IA . . (\. S.+j3HJ){\.S.+/3H 1) (1.5) o nJ!oij 1J J 1

where A .J = <0 IL.In><n ILJ lü>/(E -E ) 1 1 n o

acting wi thin the manifold of non-modified crystal field functions, i.e. functions that have not been acted upon by the LS operator.

Interaction between the elec tron spins in a particular ion, and interactions between electron spins and nuclear s pins, also contribute to the height of the electronic energy levels. Abragam and Pryce 11 l derived in an analogous way these fine- a nd hyperfine terms. Rejecting quadrupale terms and terms representing the interaction between nu­clear spins and the magnetic field, the following terms must be added to the Spin Hamiltonian 1.5:

2:: {(-\.2A .. - pl 1J) S.S1. +2/3(8 .. - À. A1J) H .S . -. . 1) t 1) 1 J 1,)

( 1.6)

where u11. = _ .!_E . kl I [ <0 IL . In > <n ILJL k + LkL . 10 >/ (E -E ) ] 2 1 n ,to 1 J n o

and P, À., p, Ç, k arE> constants. In our case the S.S.-terms,representing the interactions between

1 J the various s pins in the ion, ha ve no me aning, s ince only one electron is present in the valenee orbitals .

Us uall y the Spin Hamiltonian is writte n in a parametrie represent­ation that lends it s elf well to comparison with actual s pectra.

For axial symmetry, and one electron present, the Spin Hamil­tonian is then

H = j3 [g //HS +g i(H S +HS )]+A,,SI +AI(SI +SI) 1/ ZZ ~X X yy 1/ZZ ~ X X yy

( l. 7)

The parame ters gff ' g.L, A and Al.. are those defined by e qua tions 1.5 and 1. 6. 11

12

When an LCAO-description is used, the coefficients of the vari­ous molecular orbitals, as well as the spin orbit interaction with the ligand nuclei, enter into the formulae, complicating them a good deal, without essentially al tering them 12- 15 l.

Direct calculation of g- and A-tensors within the scheme outlined above, is risky. There is a delicate balance between the various inter­achons within the molecule, and the atomie wave functions we have at our disposal, are inadequate to deal with such subtie effects.

A notorious example is found in the predietien of vcrnishing iso­tropie hyperfine interactions in the ESR-spectrum of S-state ions, like Mn 2 +, which turned out to be wrong. The Manganeus ion exhibits a well-developped hyperfine structure 6 l. T he electrens occupying d-type orbitals, with noclal planes through the central-ion nucleus, have zero density there. The dipolar coupling between electronic- and nuclear spins averages to zero due to the spherical symmetry of the S-state wave function.

To circumvent this fundamentally wrong result, a mechanism was proposed in which the electrens were partially promoted into a higher s-type orbital, in this way obtaining a non-vanishing density at the nucleus 16 l. A Ie ss artificial treatment includes electron correlation. Watsen and Freeman 17 l stressed this particular point of view. Their escape out of the difficulty is to pass from a restricted- to an un­restricted Hartree-Feek treatment, which they call EPHF, Exchange

Polarized Hartree Fock. The restrietion to be abandonned is the re­quirement that the orbital parts of two conesponding electrens should be the same for spin-up- and spin-down-state. Then, at the nucleus, it is no longer true that f (0) ~ = f (0) f; the net-density at the nucleus explains the presence of an isotropie hyperfine interaction. The ex­change interaction, causing like spins to avoid-, and unlike spins to attract each ether, gives rise to a eertcrin amount of polarization of the core of electrons, and so breaks down the simple picture that lead to the wrong predie ti on.

The computational effort and experience required by such a cal­culation are very large and far beyend the scope of this present work.

13

Chapter 2

LIG AND HYPERFINE STRUCTURE IN MOX~--I ONS

The purpose of this chopter is to account for the complicated structure, observed in the ESR spectra of molecules with the general formula MOX~-, where M = Cr,Mo and X = F in the cases to be dis­cussed presently. The arguments we use will be mainly qualitative.

A general treatment of ligand hyperfine structure ( 1-hfs) was given by Tinkham 18a,b); Watanabe also pays some attention to this. phenomenon in his hook "Operator Methods in Ligand Field Theory"19 l. Their way of handling the problem consists in the introduetion of an additional term into the Spin Hamiltonian

HI-hfs = ~IN.AN.s

where N = number of nuclei with I f 0;

IN= magnetic moment of nucleus N;

S = spin magnetic moment.

Instead of following this line we shall attempt here to describe the phenomenon encountered with the aid of a more pictorial model. In order to do this, we make some assumptions that may be somewhat crude, but seem to be justified, at least to a first ·approximation, by the good agreement between theoretica! expectations and experimentcri evidence.

The main assumptions are l) The 1-hfs is entirely due to the dipole-dipole-interaction between

ligand-nuclei and electron spin; 2) In the ground state of the ions the unpaired electron accupies the

d orbital of the central ion; this assumption is based on the re­s~fts of Ballhausen and Gray 3 l for the Vanadyl ion, and of Gray and Hare 4 l for CrOCl~- and MoOCl~-. In Chopter 5 we will pay closer attention to this aspect.

Consider the dipole-dipole interaction energy in its usual form 8 l

(2.1)

14

with

I,S = the interacting dipole moments;

riJ = unit vector along the line connecting the mid-points of I and S;

R = the distance between the mid-points of I and S.

As Abragam 8 l shows, this is the partial result of the solution of the Schrödinger equation using the Hamiltonian

l may be put equal to zero because of the quenching o f the orbital moment by the crystal field (in first approximation). The last term in the above Hamiltonian represents the Fermi contact term which, due to the presence of Dirac's Delta function, equals zero when the electron is not in the immediate vicinity of the magnetic nucleus.

In spite of the assumed cl-type ground orbital, we wi ll retain this interac tion in the discussions , having in .mind the arguments given at the end of the previous Chapter. We are not going to try to make an estimate of the magnitude of the contact interaction separately.

The R-3 dependenee of the dipole term in combination with the d - ground orbital, suggests a further assumption: xy

the influence of the axial halogen ligand is neglected in the con-struction of the ESR s pectrum, its distance from the interacting electron being much larger than that of the four ligands in the xy­plane.

Befere going into more detail, we will consider the electron to be a m agnetic point dipole in the middle of the square of four halogen ligands.

We are now in a situation to predict the splittings in the energy level s cheme of the electre ns by an externally applied magnetic field, with spec ial reference to some particular orientations of this field. It should be noted first that equation 2.1 may alternatively be written as

l) When Hd i s alongtheaxisof the dipole, S, it equal s + 2S/ R 3 (fig. 2.l). 2) When Hd i s on a line perpendicular to S, it equals -S/ R 3 (fig. 2.2). 3) When Hd ïs perpendicular to S, the angle a between S and rij equals

54°44'; taking Sin the z-direction we obtain thi s resul t by requiring the z-compon e nt of Hd to be zero:

15

from H = R- 3 [-S +3S cos 2 a) = 0

d,z z

fellows

whence

I Hel !4--- A---~ ' ,. + .. , __ ,. .. +~~ f3h ~---A----~ Hel

fig.2.1

s .. s_

R3 fig.2.2

As will be pointed out in Chapter 3, the significant information is obtained from the molecules with orientation of their z-axis either parallel with, or perpendicular to Hext•

In order to derive the directions of the hyperfine fields at the ligand nuclei, it is convenient to start with a point dipale at the metal ion site and subsequently to correct the result for the spatial distri­bution of the electron in the actual wave function.

External field parallel with the z-axis.

The external field Hext is applied in the direction that coincides with the C 4 v principal symmetry axis of the molecule. For this reason we expect the four ligands 1,2,3 and 4 to be equivalent, corresponding to case 2 of the foregoing section. This leads to the dipale fields denoted by Hd in fig.2.3:

16

z

l.sh

y

fig.2.3: Local fields at ligand sites for Hext parallel with "'- axis.

Figure 2.3 has Hd>Hext' a magnitude that may be expected, as will be pointed out in the discussion in Chapter 5, and that is found exper­imentally from the results to be presented in Chapter 4. He is drawn toa, probably, exaggerated scale.

The electran's spin and magnetic moment being of opposite sign, the spin state with m8 = -Y2 is the one with lewest energy in a mag­netic field, i.e. 11 is parallel with H t (cf. Chapter la). Adding or e ex subtracting the energy of interaction with the ligand nuclei, a maximum lowering of the energy of the electron is obtained when all four ligand nuclei are parallel with the resultant magnetic field. This situation I 4 is chqracterized by M1 = .:z: m1 = 2. Hence there will be five energy

1:1

levels with the level having m s = - Y2 as the centre. An analogous rea~oning applies to the 11 spin-flipped" level with m = + Y2 . s

For each of the resulting situations the energy of the complex is given by

4 4

E = -p. .H t- L: lli · . H t- L: /J.Ii'(H +Hd) e ex i:l 1 ex i:l c

The above conditions lead to the following level scheme (fig.2.4): Using the selection rules t.M1 = 0; t.m s = l, this scheme imposes the five transitions as indicated in the figure. The relative intensities of the lines w ill be 1 : 4 : 6: 4 : 1, the ratio of the statistica! weights of the levels.

17

+2 +1

0 -1 -2

fig.2.4 : Energy levels and allowed transitions for H ext parallel with a- axis.

Hext <Hd; .6E = Yr-11 ( IHd I + !He IJ.

Extemal field parallel with the x-axis.

Contrary to the former case there are some complications ansmg when we consider the events on bringing H t from the x-axis to the

ex y-axis, staying in the xy-plane. Again for reasons to be explained in the next Chapter, we confine ourselves to two special cases, narnel y H 1// x-axis (or y-axis) and H 1 making an angle of 45° with the

ex ex x-axis.

a) Hext // x-axis.

The x-axis lying in one of the symmetry planes of C 4 v, we ex­peet the four ligands no longer to be equivalent. Actually, ligands l and 3 are equivalent, and likewise ligands 2 and 4. This is illustrated in fig.2.5.

1

y

fig.2.5: Local fields at inequiva!e nt sites I and 2 for Hext parallel with the x-axis.

18

It has been indicated previously that Hd 2 = - I-'2Hdl' Then we expect a larger splitting of the spin-energy levels due to ligands l and 3 and a smaller splitting, superimposed on the former due to ligands 2 and 4. For I = 1--2 this means a threefold splitting plus a second threefold splitting, in the way as given by fig. 2.6 . Maintaining the

L

_U 0 -0 0 0 ~

- 0 ~' li.J.

~~· ~ t l t l 4 2 I ~ 1-,;._~

·~~===

- . - 0

0 + 0 0 0

• • • 0 . -

fig. 2.6: Energy levels and allowed transitlans with their relative intensities. Hext parallel with x- or y-axis.

requirem ent of non-changing nuclear spins when "flipping" the elec­tron spin, we have t.M 1 = 0; t.M 2 = 0; t.ms = l, leading to the nine transitions of fig . 2.6.

The relative intensities of the transitions are again in the ratio of the statistica! weights of the conesponding levels, so as 1: 2: 1: 2:4:2:1:2:1 (fig.2.7).

1/ 16

1/ 8

l/ 16

1/ 8

1/4

1/8

1/1 6

1/8

1/ 16

fig. 2.7: Relative weight of the levels arising from the ninefold splitting when Hext is parallel with x- or y-axis.

19

In the spectrum there are now to be expected three equidistant lines, each line hoving two symmetrically located satellites. Interchanging M1 and M2 gives the situation for Hexl 'y-axis.

b) H ext under 45° with the x-axis.

A different situation arises when Hext' in the xy-plane, makes an angle of 45° with the x-axis. This situation is shown in fig.2.8.

x y

fig . 2 . 8: Local dipole;fields for Hext in the xy-plane, making

an angle of 45° with x- and y-axis.

Short solid arrows: Hext;

Long solid arrows: dipalefield

Broken arrows: resultant field.

The dipole fields at the four ligand sites are equal in magnitude though. having different directions, that are, however, determined by the sym­metry of the molecule. Like in the cas e with He xt#'z-axis , there wil! be a fivefold splitting of the spin-energy levels, again leading to five absorption lines with relative intensities l: 4: 6:4: l.

The electron in a spatial distribution.

a) So far we considered the electron to be a point dipole at the

origin of the coördinate s ystem. This is evidently a somewhat unreal­istic representation. The electron moves in a wave function of the d -type. For the ease of the argument we will first completely neglect

xy the in-plane pi-bonding, neither will we take into account the small effects on the wave function due to the L S-coupling.

20

The general farm of the d wave function exhibits four lobes . xy poin ting between the x- and y-axes, and hoving their centre of gravity just in the xy-plane. If we know the exact wave tunetion descrihing the motion of the electron, we are able to calculate the average dis­tanee of the electron from the nucleus, or, what is more useful, the average value of r- 3 , <r- 3 > :

Provided the geometry of the molecule has been established in detail, we should be able to calculate exactly the energy of interaction be­tween electren and ligand nucleus; our present problem would then be completely solved.

Many complications arise, however, when we tried to treat the problem along these lines, the most obvious one being that we have no exact wave functions available. The best SCF functions are prim­arily correlated with the best energy, and it is a well-known fact that the wave functions, according to perturbotien theory, are one order less accurate than the energy associated with them. Furthermore, an electron in a molecule is no longer a free electron (free in the sense of belonging to one particular atom), as it is subject to an impressive number of interactions.

Any calculation of molecular wave fundions may use atomie ( = ionic) wave fundions as a storting point (see Chopter 5) but, as was pointed out in Chopter l, only the most sophisticated calculations provide us with more or less reasonable functions *.

An attempt to evaluate the magnetic fields at a ligand site, using semi-empirica! wave functions, stands little chance of succeeding. The best one can hope is that an estimate of the wave functions from, for instance, the observed g-values of the molecule, will provide not too bad an estimate of the hyperfine interactions. But such estimates are very difficult to make and seldom unequivocal.

More complications arise from the unpleasant fact that only very few strudural data exist on the geometry of the molecules we are dealing with 2 oa,b,cl, So even intramolecular distonces are to be estim­ated, thus rendering a calculation speculative.

*

Reviewing these considerations it will now be clear why the

The most advanced calculations of this type were carried out by Shulman

and Sugano 21 ) and by Watsen and Freeman 22 l. Neither of these calcul­ations provides completely satisfactory numerical values of the hyperfine interactions, although qualitatively they seem to be correct.

21

emphasis in this thesis lies on the experimentcri data on hyperfine­interactions. The model, proposed to understand the meaning of the ESR spectra, should not be assigned any rigorous quantitative value. The estimates derived from the arguments in section b) hereafter will at most be order-of-magnitude estimates. It will also be found that deviations may occur when we try to fit the parameters to the spectrum.

b) In order to arrive at some qualitative insight in the ratio of the

hyperfine fields Hd,Hdl'Hd 2 and Hd 45 , we will have to consider the electron as a charge distributed over the region, covered by t)le wave­tunetion descrihing this electron. Passing from the picture of a point dipole to that of a smeared-out dipole, we reconsider the formula

from which follows, as we saw already, that H l. p. for a = 54 °44'. In fig.2.9 one can see how the interaction energy between dipoles p. 1 and p. 2 changes sign on the passage of the surface of the cone when dipole ll 2 travels from position 1 via 2 to position 3.

~t __ _ til

fig. 2.9: Interaction between {ll and {l2 is zero in point (2);

the interactlens have eppesite signs w hen fl 2 is in

point (I) or point (3).

In other words, the projection along the p. 1-directionof the mag­netic field at llt, due to ll 2 , changes sign. So, when the density of the distributed electron lies partly within such a cone, erected at a ligand site, the contribution to the magnetic field at that site, arising from the density within the cone, has its sign opposite to that of the density outside the cone. Thence, if a considerable part of the wave function

22

is found within the cone, a large deviation from point-dipale behaviour will occur. In fact the situation is not quite that serieus since the region of highest density is, for bonding wave functions, somewhere between the bonded ions, and in extreme cases only, very near the ligand. So we expect no large deviations, as has been qualitatively indicated in the figures.

i) Electron polarized along the z-axis.

In this case no significant part of the d -wave function is like­xy

ly to be covered by the cone (fig.2.10).

fig. 2.10: Covering of dxv -wave function by a cone with an

opening of I 09° 28' and axis perpendicular to xy-plane.

ii) Electrem polarized along the x-axis.

In this situation we encountered two inequivalent sets of sites: (1,3) and (2,4). The xy-plane and its intersectien with the cone is shown in projection on the xy-plane, in fig.2.lla,b. It is immediately clear that the deviation from point-dipale behaviour will be larger for sites 2 and 4 than for sites 1 and 3.

iii) Electron polarized in the 45°-direction.

T he electron' s magnetic moment may be decomposed into its x­and y-components 2-Y.y 1JS. At site 1 the x-component produces a

e

23

a fig. 2.11.

a. Coverlog of dxy -wave funclion by a cone with opening of

109°28' and axis along x-axis, through ligand 1.

b. idem, through ligand 2.

fiefd 2-Y.Hd1 in the positive x-direction, the y-component a field 2-Y.Hd 2 in the negative y-direction (fig.2.12).

(The ad dition pd to the subscripts in fig. 2.12 is to indicate the direct­ions of the fields when the electron behaves as a point-dipole.)

It was shown in ii) that, most probably, Hd 2 will be reduced

Hrosult.

fig. 2.12: Loca1 fields at ligand site 1 for Hext in xy-p1ane

making an angle of 4 5 ° with the x-axis.

Hd d = dipo1e field from electronic point dipole p. in origin;

Hd 45 = dipo1e field from an electrooie dipo1e distributed over a d -wave function. xy

24

more than Hd 1 when the electron is spread-out over an actual wave junction of dxy-type. This means that the true Hd 45 lies closer to the x-axis than Hd 45 does. Vectorial addition of the external field Hext brings the resuftant field still closer to the x-axis, Generally speaking, the angle between the two will be of the order of 15°-25°, a value small enough to cause no serious difficulties in the analysis of a pow­der-ESR-spectrum, as will be considered in the next chapter. To con­clude this section, a rough and provisional estimate of the magnetic fields at the ligand sites, due to the electron, is proposed:

IHdll ~ 1.7 Hd

1Hd 2 1 ~ 0.7 Hd

1Hd45 I~ 1.3 Hd

In the course of the analyses of the recorded spectra, departure from these values wil! be of not too great concern to us, regarding the naieve and qualitative nature of the arguments used. N evertheless, they will be very useful in serving as a guide in trying to fit the para­meters to the actual spectrum.

25

Chapter 3

ANALYSfS OF ESR SPECTRA OF POLYCHRYSTALLINE SAMPLES

The form of the Spin-Hamiltanion 1. 7 shows the convenience of the use of single-crystals to obtain the g- and A-values for the special directions separately. It is, however, difficult to obtain magnetically diluted crystals of the compounds we are interested in; th is circum­stance necessitates the use of randomly oriented samples, in this case quickly frozen dilute solutions. We have then to consider the shape of the spectra due to such a random orientation of the paramagnetic een tres.

In the following review of the theory we wil! omit the hyperfine interachons for reasans of simplicity and brevity. lts inclusion in the problem i s straightforward and fellows analogous lines as what is going to be clone for the g-values.

When we are dealing with samples of cubic (site) symmetry, there is no problem at all since the orientation of the symmetry axes with respect to the external magnetic field is irrelevant because of the relation gx = g = gz.

Crystals with axial- or lower symmetry require us to know the precise orientation of the crystal axes in the field as the principal g-values cease to be equal. This anisotropy makes that for any value of H t such that (hv/ ,{:lg )<H t<(hv/,{:lg. ),a resonance line is ex max ex m1n observed. The equivalence of the x- and y-axis in cases of axial symmetry, makes the polar angle <Pirrelevant. So, crienting a crystal with its cluster-C4 -axis along H , we find g , in axial symmetrical

V Z Z

cases mos tly called g 11 ; whenever the axis is orien ted perpendicular to H , we cbserve g = g = g.L. When the crystal is rotated about,

e xt x y_ say, the y-axis , the observei:i g·value varies between g// and g.L, obey-ing an angular dependenee to be explained presently.

Because in a polycrystalline sample all orientations are assumed to he randomly distributed, the ESR spectrum will show all these poss­ibie g-values simultaneously. Now the question arises as to the shape of the spectrum. Sands 23 l and Kneubühl24 l solved this problem for axial and rhombic symmetry respectively.

For axial symmetry that is a rather simple proble m, and for the sake of completeness, an outline of the salution will be given now. Our first task is to write g as a tunetion of the angular coordinates [) and P._._

26

In Chopter l the Spin-Hamiltanion was presented to be

H = /3 [ g H s + g4 H s + g H s ] XXX YY zzz (3.la)

This may be rewritten in a dyadic notatien as

H = /3H. g. S (3.lb)

with g = i.g .i+]' .g ·J· +k.g .k

XX yy ZZ (3.lc)

Camparing this to the formula for the "spin flip" of a free electron

A

H = 2/3H. S

we may write the previous formula as

(3.2a)

from which fellows

Heff = H. g/2 (3.2b)

Transferming now the coordinate system in the sense that the new z-axis coincides with the direction of Heff' we find

where Heft now is the length of the vector Heff. For S = ~. the solutions are ± /3Heff' and thus

Inserting (3. 2b) into (3.4) we find

(3.3)

(3.4)

(3.5)

Expressing H ,H and H in the polar angles 8 and c/J, this becomes x y z

27

If we write g for the square root in (3.6), the familiar result hv = g,BH, like tor a free spin, is found again; this g, however, in­cludes Zeeman- and LS-effects and so no longer necessarily equals 2.0023. In the axial case we were considering, formula (3.6) reduces to

(3.7a)

or

(3. 7b)

Th is formula is of cru ei al importance for the anal ysis of powder spec­tra we are aiming at.

Apart from a possible 8-dependence of the transition probability, the intensity of the absorption by the polycrystalline sample is pro­porticnol to the number of crystals hoving their axes making an angle between e and e + d8 with H t' ex

Plotting the coordinates e and cj; of each crystal in the sample on a unit-sphere, we obtain a uniform dis tribution of points because of the assumed random orientations. The number of crystals hoving their c 4v-axis with e as above, will now be proportienol to 27Tsin8d8, as is illustrated by fig.3.l.

fig. 3.1: Area on unit-sphere representing the number of molecules with z-axis making an angle between e and e + d8 with a fixed direction in space.

28

Since the tot al number of crystals is proportion al to 4rr 1 the area of the unit-sphere 1 the fraction dN will be

dN = (N/2)sin8d8 = (N/2)d( 1- cos8) (3.8)

where N = total number of crystals in the sample. (The choice of ( 1- cos8) instead of simp1y -cos8 is for later con­venience).

F ollow ing Sands 23 l we denote the fixed microwave frequency in the experiment by v and so H = hv /2. Some straightforward algebra

0 0 0

produces the relation between dN and dH 1 the field interval within which resonance occurs at the frequency v0 :

P lotting dN/ dH vs. H 1 we find the absarptien spectrum for a pol y­crystalline sample with zero linewidth (fig.3.2).

11 ..1. fig. 3 . 2: dN/dH vs. H for an axial molecule; zero-line

width assumed.

Two features are still lacking in the problem. One 1 already touched upon 1 is the 8-dependence of the transition probability. The ether is the finite linewidth of the absarptien lines due to relaxation phenomena and to the mutual interactions between the paramagnetic centra in the sample.

The farmer point is dealt with by Bleaney's formula 25 a 1 b 1 cl

(3.10)

which must be reexpressed in H for combination with (3~9).

29

The latter point is extremely important, since the overall shape of a spectrum depends very strongly on the linewidth, especially when a number of partially overlapping lines are present.

In the absence of strong interachons between a paramagnetic ion and its surroundings, the lineshape will be Lorentzian:

(3.11)

Here f(H) is a measure of the intensity of the absorption as a function of the magnetic fields strength1 H is the centre of the symmetrie

0

absorption line and b is the half-width at half-power. This line width has not yet been included in fig.3. 2; the absorption line for each value of H is represented there by a "stick" of zero-width. To incorporate (3.11) in (3.9) has the effect of providing each stick with a finite width, such that each stick is broadened into a Lorentzian line. Care should be taken that the area under each Lorentz curve represents the same transition probability as the stick it is deduced from. 1bers and Swalen 26 l carried out the complete analytica! treatment of the problem and arrived at the complicated formulae

(3.12a)

I(H) ,.,_, [(A,8 1 + Bb,82}/2bP)]L2+ [(-A,82 + Bb,8 1}/bP)] T 2

(8/Hlln[ H#/(H.!_-(Hl-H3/I.)] -D[(Hl-H2}/(H1H11)] #

where

p = [(H2- Hl- b2)2 + 4H2b2] ~;

[3 1 = [Y2P + Y2-(H 2 -Hl- b2 ) ]Y.;

t-H-H L 1 = ln [(t 2 + b2}/(8~ + <j:~}] - 1_

t :H-H#

(3.12b)

30

t-H-H ,T 1 = [tan- 1(b/t) + tan- 1(<P/81)] - l.

t=H-H#

e1 = y 1 t + &1 - (t2 - 2Ht + H2 - HD~~,

82 = y2t + 82 + (-t 2 + 2Ht +Hl- H2)Y.

'Yl (bj32 - Hj31)/P)

81 [ -Hbj32 + (H 2 - HD/31]/P

/32 =- [ YaP - Ya(H 2 - Hl- b2)]Y.

L2 = [ln[(t2 + b2)/(8~ +<P~ )] ] t=H-Hl. t:H-H#

T 2 = [tan- 1(b/ t) - tan- 1(<P:/ 82)] t:H-Hl. t =H- H#

<Pl = y2t + 02

ct2 = 'Yl t + 81

'Y2 = (b/3 1 + H,82)/P

82 = (-Hb/31 - (H 2 - Hi_)/32)/P

The ESR spectrometer delivering first derivative curves of the absorption, we are mainly interested in the calculations of that line. This may be clone in two ways: 1) I(H) is calculated for a large number of H-values and the result

is differentiated numerically; 2) the formulae are differentiated and subsequently the calculation is

carried out.

We choose the second way to save computer time. Mr. W. Konijnendijk differentiated the formulae and wrote an ALGOL program for the EL-X8 computer of the Rekencentrum of the Technological University Eind­hoven. The punched-tape output was fed off-line into a CALCOMP plotter and plotted pointwise. The program is given in the Appendix. To show the shape of the spectra, and the influence of linewidth, some were calculated and are presented in fig.3.3a-f.

31

en lil Ü

x

A_) )

~vv

-4 3200 3250 3300

veldslerkle

a

veldslerkle

b

y V

3350 3-tOO

C\1

I 51

x

32

C\1

151 0 1-----~-+----t-+--------i------1 x

veldslerkle c

veldslerkle

d

33

2~-------.--------.--------,------~

3250 veldslerkle

8

l'il 0 x

- 8 3200

~

~

3250 veldslerkle

3300 3350 3400

e

~

~ -3300 3350 3400

2-fig . 3. 3a- f: Sim uloled ESR- s p ectr a of Mo0C l 5 for

li n e wid th varying between I and I 00 Oe .

34

The relative ease of handling analytically the problem of powder spectra for axial molecules, is lost when we move on to the rhombic case.

That problem was treated by Kneubühl 24 l without accounting for Lorentzian lineshape. His results are, with g3>g 2 >g 1:

I(H} "-' [ (H 1 H 2H 3 H- 2}/[(HÎ-H 2 )~ (H~- H~)~]J K(k) for H ~ H2 (3.13a)

I(H) "-' [ (H 1 H2 H3 H- 2 )/ [ (HÎ- H~)~ (H 2 - H~)~ ]] K(l/k) H . ~ H 2

(3.13b)

K(k) and K( 1/k} are complete elliptic integrals of the first- and secend kind respectively.

A rhombic "stick" diagram, analogous to the one in fig.3.2, is shown in fig.3.4a.

lntroducing the linewidth in the manner indicated before, the spectrum is modified to the one in fig.3.4b, the first-derivative of which is shown in fig.3.4c.

V

fig. 3.4: ESA- spectrum of a randomly oriented s ample of rhombic molecules. a: upper figure, breken line: absorption spectrum.

Stick diagram with zero-linewidth. b: upper figure, solid line: absorption spectrum.

Line width included. c: lower figure: first derivative of b.

dS(H) as in fig. 3.6.

In all the c ases we mentioned so far, hyperfine structure will complicate the pictures in s uch a way that 21 + 1 curves are going to be superimposed (I is nuclear spin quanturn number). An example is shown in fig.3.5.

35

fig. 3. 5 : Stick diagram of the ESR spectrum of a randoml y oriented sample of axial molecules with hyperfine structure. g// >gj_; A// >AJ_; I = 3/2. (From ref.27).

As long as axial molecules are considered, the only variabie of orientation is the angle e so that a plot of B vs. H t provides all the

ex desired inform ation.

As soon as </> also influences the value of g, which happens in rhombic molecules, we are forced to make a 3-dimensional plot, viz. e and </> vs. Hext' The general shape of the curved plones that represent the relation between e and </>, and the value of H t where resonance

ex occurs, is given in fig.3.6.

dH

'f

H

c

a1.•1--· 1

fig. 3.6: Plot of the resonance field H vs. a(= 1-cos el and </>; M 1 and M2 are defined in Chapter 2.

dS(H) is the projection of the intersectien o f the "field­s andwich" .dH with the curved surface, and a meesure of t.,he intensity o f absorption. 6g = 0.

36

In order to deri ve the shape of the absorption line from this kind of picture, a "field sandwich" of thickness dH is moved upward along the vertic al a x is, storting at H = 0. As soon as it intersects the curved plane, absorption of microwave energy starts. The intensity of the absorption is proportional to the number of 11 sticks" that corresponds to the intersected area. Since the number of absorbing molecules in a rhombic case will be proportional to sin eded<j; = d( 1- cose)d<t(cf. 3.8), it is advantageous to plot ( 1- cose) instead of e itself, as then the intersected area is a measure of the intensity of the absorption. The chöice of (l- cose) is arbitrary; this tunetion is 0 at the origin, the same situation as w hen e was used as the coordinate. Th is facilitates comparisons.

An analytica! treatment of rhombic ESR spectra including line­width and both central-ion- and ligand-hyperfine-structure, is extremely complicated. Simulation of this kind of spectra will most conveniently be performed by a method that calculates g and the intensity for a large number of values of e and <P. foliowed by numerical differentiation of the obtained curve 28 l.

However, the very many details in some spectra, require a high accuracy of the counting- and differentiating procedures; as the field range to be covered is, in our cases, also rather large (0.5 - lkOe) one is forced to calculate some 10 4 different g-values. This causes the program to consume a fair amount of computer-time; experiences with simple rhombic cases, without ligand hyperfine struc ture, suggest an estimate of %-1 hour per spectrum. A best fit of the parameters im­poses a number of attempts, and so it is to be expected that a satis­factory computational check of the interpretation of our spectra will take at least 3 hours per spectrum, provided a lucky initia! guess was made. Not accounted for in this estimate is the linewidth, which enters as an extra adjustable parameter. This again may lead to an increase in the time.

In view of these arguments, we refrained from simulation of the 1-hfs spectra.

Returning to fig.3.6, it will be clear that each hyperfine line can be represented by a separate plane. When analyzi ng the 1-hfs in CrOF;­and MoOF;-, in the next Chapter, We shall use this three-g-value de­

scription in order to interpret the 1-hfs s ignals as "ps eudorhombic" signals.

37

Chapter 4

THE ESR SPECTRA OF CrOF~- AND MoOF~-

Experimental

The ions MOX~-, the geometry of which has already been de­scribed in Chapter l, have been known for quite a long time. Weinland and Fiederer29 l report the preparatien of (C 5H5NH)Cr0Cl 4 , which corr:­pound is the starting material for the fluorine complex.

Angell, J ames and Wardlaw 30l prepared MoOCl~-5 with various cations. A summary of the prescriptions is given ir{ the following sections.

A salution of Cr0 3 in a mixture of concentrated HCl and glacial Acetic Acid is saturated with HCl-gas at 0°C. An equivalent amount of Pyridine, pre-treated with HCl-gas, is added to this salution, and the brownish-red precipitate is filtered off and dried over concentrated sulphuric acid; this is to maintain a HCl atmosphere in the desiccator, as long as the compound is moist. Not until it is absolutely dry, the HCl is removed with NaOH. Then the powder is stored in a sealed bottie and kept in darkness. The compound is under these circum­stances, stabie for at least a couple of weeks.

(C 5H 5NH)MoOC1 4 :

Ammonium Molybdate, (NH 4 ) 6Mo 7 0 24 .4H 20, in concentraled HCl, is reduced by Zn or Hg. An equivalent amount of Pyridine, pre-treated with HCl-gas, is added to the deep-blue solution. After cooling to 0°C, a stream of HCl-gas is passed through until saturation. The pre­cipitate, consisting of bright-green needles, is filtered off and washed with a small volume of concentrated HCl. The product may be re­crystallized from warm concentrated HCl. Well-s haped crystals appear when the salution is allowed to cool down slowly. After tiltration the crystals are treated in the same way as the Cr-comple x. The coumpound is stabie for at least a year, probably even much Jonger.

Af ter sol ving the Chlorine complexes in 38% HF -solution, an instantaneous exchange of Cl by F takes place. The Cr-solution turns

38

faintly yellow, and the Mo-solution light blue. The spectra in the visible and UV region will be discussed in Chopter 5.

No attempts were made to isolate the CrOF~--containing com­pounds; the Molybdenum complex, however, was isolated from a salution of [ (MoO(OH) 3 ] 2 in concentrated H ydrofluoric acid. Aft er ad dition of an equivalent amount of NH 4 HF 2 the liquid was slowly evaporated, and a light blue crystalline solid separated out.

The compounds containing Cr53 or Mo98 had to be prepared from less suitable storting materials. Cr53 was available as the oxide Cr~ 303 , containing more than 95% Cr53 ; Mo98 was available as the metal, containing more than 98% Mo98 • In order to convert those mater­ials into the desired complex compounds, they were treated in the following manner.

The Molybdenum metal was heated in a stream of Oxygen at a tempercriure of 600°C. The Mo0 3 formed by this action, was dissolved in a small volume of concentrated hydrochloric acid. In .this .solution the Mo was reduced as described in the beginning of this Chapter.

The Cr(III)-Oxide was fused with N a2 C0 3 under ample exposure to air. The Chromate then formed, was dissolved in glacial Acetic acid and treated in the previously described way.

ESR measurements on HF-containing samples require extreme care as to avoiding severe damage to the microwave cavity. The usual sample-tubes of quartz glass being inadequate in this case, a suitable material had to be looked for. Teflon turned out to be most conven:ient to this purpose. For the X-band measurements it was sufheient to use tubes like the one shown in fig.4.l.

II:::A -~~ j--sl-----~·1

fig. 4.1: Teflon sample tube for ESR measurements.

Rod B is screwed into the sample campartment A, after A has been completely filled with the sample solution. The liquid i s then quickly frezen by dipping it into liquid Nitrogen. After some time it is trans­ferred to the caoled cavity.

Spectra of the solutions at room temperature can be obtained by sucking the liquid up into so-called 11 spaghetti tubing" of 0. 3mm

39

inner diameter. The double-folded piece of tubing is then placed in cernpartment A. This manner of werking enables one to avoid the use of a special liquid-sample-cell.

For Q-band measurements the tube is essentially the same. The dimensions however, differ appreciably. Particularly the thickness of the walls must be as small as possible, in order to avoid an ex­cessive filling factor of the cavity, which would result in the imposs­ibility of tuning it.

In the experiments to be described, a wal! of O.lmm was used. The sample must be in "spaghetti tubing" also at low temperatures. It is clear that the higher Q-band sensitivity is lost by the smaller sample volume; at both frequenceis the lower limit of the number of detectable spins is a bout 10 11 •

ESR measurements at X-band were carried out with a Varion V-4500 Spectrometer. The frequency was determined by a Hewlett­Packard Electronic Counter Model 5245L, coupled with the microwave bridge thr.ough Hewlett-Packard Frequency Converters, Models 25908 and 52538. Q-band measurements were carried out using the Varian V-4561 Microwave Bridge, and special pole tips on the Varian 9 inch magnet. The X-band cavity was cooled by leading the vapour of boiling Nitrogen through the Varian Variabie Temperature Accessory; the tem­perature was regulated by the Varian V-4561 EPR Temperature Con­troller.

Cooling of the Q-band cavity by conduction, as prescribed by the manufacturer, turned out to be unsatisfactory. The lowest temperature we could obtain was -ll0°C. Considerably lower temperatures were reached by blowing the vapour of boiling N itrogen on top of the cavity, which is located in the narrow neck of a Dewar vessel. Heat exchange with the surrounding air was reduced by placing a piece of Styropore in the neck of the Dewar.

Since no device for the meesurement of magnetic fields was available to us, we had to preeeed in the following way, in order to determine the field strength corresponding to various points in the spectrum. The sweep range was determined by the use of a solution of MnC1 2 in water; A was taken as 96 Oe. The sweep was assumed linear over the whole range.

Using Diphenyl-Picryl-Hydrazyl (DPPH), one point of the recorder chart was marked, and start- and end-values of the sweep range were calculated with the aid of this value.

In the calculations the following values were assigned to the relevant constants : l Oe= 0.25.77- 1.103 Am- 1•

Planck's constant h = 6.6256.10- 34 Js; Bohr's magneton f3 = 1.16530.10- 2 9 Vsm;

40

Spectra of dilute solutions.

A useful aid in unraveling the complex spectra of glassy samples is given by the spectrum of the unfrozen solution at room temperature. (sometimes a more elevated temperature may be required to reduce the viscosity of the solvent sufficiently).

In the spectra to be presented in this Chapter, it is usually poss­ibie to locate g by means of the c-hfs, that, fortunately, is rather large and thus spread over a w ide field range. Si nee a re lation exists be­t ween g 11 , g.l and the isotropie <g> as measured in a liquid solution, we are able to calculate the location of g.l from the values of <g> and g //" We w ill now consider the required re1ation between the various g-values.

In Chapter I the Spin-Hamiltonian for an axial molecule was given. Lifting out of this formula ( l. 7) the Zeeman part, we retain

Writing out the inner product H.S, we find by substitution

Hz =(3 [(gl/-g.l)H S +g.LH.S] eeman 11 z z

Suppose the z-axis of a molecule to be oriented as in fig.4. 2.

:r

fig. 4.2: Definition of axes and coördinates .

Then, for H 2 S 2 we may take HqSq;Hq = Hcos6l and H .S = HS 2 •

Inserting this into (4.2) gives us (4.3):

(4.2)

Hz = (3H [(6g cos 2 6l + g.l)S + ~ 6g sin6l cos6l (S+e- 1cP+S e+14>)] eeman z -

(4.3)

41

When the molecule is tumbling rapidly, and does so rapidly enough for the correlation time to be s mall compared to the microwave frequency, the molecule exhibits an average orientation with respect to the magnetic field H. This means that we may take the average of the Hamiltonian. Averaging of (4.3) results in

<H> z . = ,8H(6g/3 + g, )S = ,BH(g 'IJ/3 + 2g, / 3)S (4.4) eeman ...1... z 11 ..L.. z

Hence the isotropie g-value is given by

(4.5)

Reasoning along the same lines, an analogous formula is found for the hyperfine parameters A11 , A1.and <A> :

The ESR Spectra.

Ligand hyperfine structure in ESR spectra has not aften been considered , in contrast to the Nitrogen hyperfine structure, which is found in many biochemically important compounds.

Tinkham lBa ,b) interpreled the ESR spectra of impurities in ZnF 2 and observed Fluorine hyperfine structure. Kon and Sharpless 14 l inter­preled minor details in the spectrum of CrOCl~- as due to the elec­tran's inte raction with the nuclear moment of Cl. The rather low mag­netic moment of the Cl-nucleus as compared with the other halogens (see Table 4.1) suggests a better developped 1-hfs character with the ligands F and Br (or I, but very little is known about the Iodine complexes; e.g. see ref.37).

T ABLE 4.1 (from ref. 32)

isotape abundance nu cl. magne tic I moment •

F 19 100.00% 2.6273 l/2

Cl3S 75.40% 0.82091 3/ 2

Cl 37 24.60% 0.68330 3/ 2

Br 79. 50.57% 2.0991 3/ 2

BrBl 49.43% 2.2626 3/ 2 I 127 100.00% 2.7937 5/ 2

• B OH R mag n eto n s.

42

Kon and Sharpless 3 ll and van Kemenade, Verbeek and Cornaz33•34 •35 l i.nvestigated the ESR spectra of, respectively, the Bromine- and Fluo­rine-complexes of the axial type we are studying here. The lower I makes the Fluorine spectra less complicated. The nature of the Bra­mine spectra is ambiguous, due to a certain amount of doubt as to the Bramine complexes being of the same monomeric character as the Chlorine- and F luorine complexes 36 l. We will return to this important question in the next Chapter.

We will focus our attention on the ions Cror;- and Moor;-. For the interpretation of the c-hfs, data in Table 4. 2 are important.

T ABLE 4.2

Isotape Nat.Abund. Nucl.Spin Nucl.Magn. * moment

Cr53 9.54% 3/2 -0.47354

Mogs 15.78% 5/2 -0.9099

Mo97 9.60% 5/2 -0.9290

Mo 98 - - -

* nuclear magnetons

The ESR spectra of MoOF~- in concentrated HF at room temperature , are shown in fig.4.3a,b for X- and Q-band respectively, and for natura} Mo only, since Mo98 possesses no nuclear spin (Table 4.2).

Assuming the two outermost signals to be the lowest- and highest hyperfine transitions, we find <A>x = 68.5 Oe, and <A > 0 = 68.3 Oe, and we adopt the average value <A> = 68.4 Oe. The X-band spectrum shows <g>x = 1.907, while the Q-band spectrum shows <g> 0 = 1.905. We adopt <g> = 1.906.

The assumption made here concerning the outermost signals, is a reasonable one w hen we take into account their apparent! y equal widths. McConnel1 38 l and Rogers and Pake 39 l pointed out how each hyperfine line may acquire its own particular line width, due to relax­ation effects dependent on m1 ( = projection of I along the axis of refer­ence). From their theory follows, that there is a particular hyperfine line with the minimum width, for two sufficiently differing magnitudes of the extern al magnetic field. If, by coincidence, our outermost s ig­nals at X-band exhibit equal linewidths , then equal linewidths are improbable for the Q-band spectrum, provided this mechanism acts here. Since at both X- and Q-band equal widths are observed , we ex-

43

peet the effect not to be present and conclude that there are no c-hfs lines outside the region shown in fig.4.3.

fig. 4.3 : ESR-spectrum of a salution of MoOF;- in 38% HF; room temperature.

a. X-band; g = 1.907; A = 68.5 Oe. b. Q-band; g = 1.906; A = 68.3 O e .

N ext we pay attention to the spectra of CrOF ~- under the same conditions as above, which are given in fig.4.4a,b,c. The complex containing notural Cr will show only a very weak c-hfs (fig.4.4a). Therefore Cr53 (>95% Cr53 ) was used (fig.4.4b,c). The Q-band spectrum provides us with the values of <g> 0 = 1.964 and <A> 0 = 23.1 Oe.

The X-band spectrum pres ents a peculiar picture. However, assuming the indicated points to be the relevant ones, a good agree­ment is foundwiththepreviousvalues: <g>x=l.964and<A>x=23.3 Oe. We see no obvious reasen for the abnormal X-band spectrum. The non­linearity of the bas e line of both X- and 0- band spectrum, suggests the presence of some ether species, having nearl y the sa me g-val ue, but net showing a resolved hyperfine structure. Dilute s olutions of Mn Cl 2 in water, that almest exclusively contain Mn(H 2 0)~ +, same­times show the same kind of base line deviations.

For the interpretation of the glass-spectra we now dispose of the data in Table 4.3:

TABLE 4.3

MoOF;- CrO F ;-

<g> 1.906 1.964

<A> 68.4 Oe 23.2 Oe

44

a

b

c

2-fiq. 4.4: a. ESA-spectrum of CrOF 5 in 38% H F ; Q-band.

q = 1.964. 53 2-b. ESA-spectrum of Cr OF 5 in 38% HF; X-band.

q = 1.964; A = 23 . 3 Oe. 53 2-c. ESA-spectrum of Cr OF 5 in 38% HF; Q-band.

q = 1.964; A = 23.1 Oe.

Spectra tak e n at room temperature.

45

The lower magnetic moment of Cr corresponds, in first approximation, to a smaller <A>, which is confirmed by the en tri es in this table. The low-temperature spectra of MoOF~- and CrOF~- are shown in fig.4.5a,b.

a

2-lt..toOFs l i:~ ~·t. HF

• 123 K

3050 H~

b

2-(CrOF 5l in 3&•t. HF

123•K

3050 H

3!i50

fig. 4.5

The interpretation of such spectra depends in its initial stage mainly on a procedure of trial and error, guided by the provisional estimates of the various parameters involved, as indicated in Chapter 2. It was also shown there how the profiles of the components. of the spectrum acquire a rhombic character.

The first step in the trial and error procedure is the assignment of the c-hfs. This being established , patterns of 5 and 9 lines have to be looked f~H; the general appearance of those patterns is shown in fig.4.6a,b. A five-line-pattern has to be present around g11 and its c-hfs com­ponents. Both five- and nine-line-patterns are to be found around the

46

gl. -components. This search implies the greater part of the trial and ,error; having found, for instance, the five-line-pattern around gl., the ether splittings should have magnitudes of an order as expressed by the provisienol estimates.

a b

fig. 4.6: a. Schematic five-line absarptien spectrum.

fig. 4.7:

b. Schematic nine-line absarptien spectrum.

Relati ve intensities as indicated; zero-linewidth.

(+1.-1).(-t •• l)

M,.M,•(O.O)

A •4

A•2

I I : -H

(1.-l~)o(-1.0) ' : : <I.•II•(>I.O): I •• A4 :

' ,, I : : 1 11 :: !

2-ESR-spectrum of a quickly frasen salution of CrOF 5 in 38% HF; T = 123°K. Upper spectrum: graphically estimated spectrum. Lower spectrum: experimental result. M 1 and M 2 as previousl y defined.

47

A naive reasoning might lead us to the assumption that the un­paired electron, residing in a d -orbital, will have an average distance

p xy of about l-2 A from the ligands in the xy-plane. This situation gives rise to di po lar fields H d of the order of 5-1.104 Oe. The anal ysis of, for instanee the ESR spectrum of CrOF ;- leads to the interpre totion given in fig.4. 7, i.e. the parameters have the right order of magnitude. This result is perhops not necessarily unique, but as far as the resol­ution of the spectrum allows us to say so, it is the only analysis that fits satisfactorily to the experimen tal data. The first-deri va ti ve spec­trum, obtained with the estimated parameters by drawing graphically the components in the ratio of their intensities as prescribed by figs. 2.4 and 2.7, shows a good resemblances to the actual spectrum.

The parameters are collected in fig.4.8, which represents three of the four faces of the cube in fig.3.6. The field-parameters H 1 2 3 4 are the field-increments required to pass from one absorption' Óf 'a certain kind to the next higher absorption. They may be related to the internol fields Hdl' Hd 2 ' Hd 45 and Hd by the formula

in which m = mass of the proton and p

fig. 4.8

m = mass of the electron. e

11, "·

.L

f3=o <f;=o

I/

", 'll I

_,

' f3=n/ 2 cp=O

"· ....

b

f3 =7T/ 2 <f; =n/ 4

.l

"· ~

• ,

I I

" " _, _,

I • 11

_,

I _,

I

f3=n/2 cf;=n/2

48

All the relevant values are compile.d in Table 4.4.

TABLE 4.4

16 kOe

30 kOe

17 kOe

19 kOe

Internol magnetic fieldsin CrOF~-.

Of course, the interpretation given thus far may be subject to some doubt as to the various assignments made in the foregoing anal­ysis. Two particular points require further investigation, viz. the c-hfs and the assigned g-values.

The crude information provided by polycrystalline samples leaves one method of some power available to make a closer inspection, just as. in the case of the solution spectra, i.e. Q-band spectra. With both natura!- and enriched samples available, we wil! be able to recognize the c-hfs either by its enhancement (for Cr 53 l, or by its disappearance (for Mo98 ).

Returning to the case of the Cr-complex, an attempt can now be made to "predict" the Q-band spectrum from the X-band data. It was pointed out in Chopter 2 how the distance between g 11 and g..L is larger there by a factor of about 3.8, the magnitudes of the internol fields remaining unchanged in first approximation. T hen, the positions of the various absorptions relative to either g 11 or g..L, are unaltered, and so the spectrum becomes elongated along the field axis. Some c-hfs components for the Cr-complex are shown in fig.4.10.

Hoving performed the 11 translation" of fig.4. 7, we arrive at fig. 4.9, where the 11 translation11 is compared with the actual spectrum; only notural Cr is presenled there in order to allow of straightforward com­paris on with fig.4.6. In fig.4.10 the X-band spectra of CrOF~- and Cr 530F~- are compared.

Generally speaking, the Mo-complex shows the same feature s as the Cr-complex. The interprelotion follows the same lines and for that reason we merely present the results in figs.4.ll-4.12. Table 4.5 contoins the same data for Mo as Table 4.4 does for Cr.

X-band

49

TABLE 4.5

Hd 16 kOe

Hdl 43 kOe

Hd2 19 kOe

Hd4S 14 kOe

Internul magnetic fieldsin MoOF~-.

100 Oe

~', I '-:-<!'

Q-band I ,-" 1...,._, J,:. ·'· -~-. 1'1~-- --,cqolA·~

·-- ./ I I I I I 1<11 Jl•.•l-1'1.•> I

r ··--· ~ ---- ~---·: --- u~·-· i··· :

--------

100 Oe

I I I I I I

I I I I I I I I I I I I

V

50

I I I

: _ /10xenlarged

I~ I

9.54%Cr53

I I

')95%Cr53

! ' ---

fig. 4.10: Comparison of X-band spectra of CrOF~- (upper 53 2-spectrum) and Cr OF 5 (lower spectrum).

Same important c-hfs lines are a1so indicated:

A:J.c-hfs of (-1,1) + (1,-1); B:l.c-hfs of (0,0) C: #c-hfs of (0,0) D:#' c-hfs of (0,1) + ( 1,0);

E: #'c-hfs of (1,1);l.c-hfs of (0,-1) + (- 1,0);

F :J..c-hfs of (-1,1) + (1,- 1);

AJ..= 17 Oe; A j' = 46 Oe.

It m ay be noted incidentally how the Q-band spectra show remark­ably less detail than the X-band spectra; this situation is worse for the Me-complex than it is for the Cr-complex. We wil! return to this point in Chapter 6.

On a firs t glance, the observer might be s truck by the more symmetrie s hape of the spectrum of the Cr-complex, compared with that of the Me-complex. From that observation the condusion could be drawn, that the C r-complex would have a smaller value of !J.g = = lg#'- gl.l than the Me-complex. According to our analysis this turns

51

out to be true indeed. This circumstantia1 fact again supports the interpretation that has been built upon the rather diffuse information provided by polycrystalline samples.

The final Table of this Chapter, Table 4.6, summarizes the data on g- and A-values as the y were arri ved at in the course of the anal­ysis in this Chapter.

TABLE 4.6

CrOF~- MoOF~-

g// 1.953 1.881

g.l. 1.969 1.918

All 46 Oe 106 Oe

ÄJ.. 17 Oe 56 Oe

fig. 4.11: Ana1ysis of some important c-hfs-lines of Mo. 0 X-band spectrum of MoOF ~- in 38% HF; T = 123 K.

A: 1/ c-hfs of (0,0); B: 1/ c-hfs of ( 1, 1); C: .1. c-hfs ol (-1,1) + (1,-1); 0: .1. c-hfs of (0,1) + ( 1,0); E: .1. c-hfs of (0,0).

F: J..c-hfs of (0,0); G: .l.c-hfs of (1,-1) + (-1,1); H: .l.c-hfs of (0,-1) + (-1,0); J : #' c-hfs of ( 0, 1) + (1, 0); AJ..= 56 Oe; A#' = 106 Oe.

52

(-1 . 1).{1 .

X-band

100 Oe

"'"' I ' ~ J . ............... _

-t .- ·----1"-·..._, ---1."11).11,0) :· · -·-. -----,Co.<>) A•lo

A•" I I '~"--'..._ --·-·--. I . ( . I ' - -..1:!: 1) A•l ..,

;- --f----1----~l...J. ----·1 I I

Q-band

fig. 4.1 2 : X- and Q-ba nd spectra of MoOF~- in 38% HF; T = 123°1<.

C ompared with theore tic a! a bsorptio n spectra (ze ro linewidth).

53

Chapter 5

DISCUSSION OF .AVAILABLE SPECTROSCOPie- AND ESR DATA

C 4 v site-symmetry of the Vanadyl-, Chromyl-, Molybdenyl- and Tungstenyl-halogeno-complexes has nat been proved for all cases. There is, however, a fair amount of evidence as to the justification of that assurnption. For the crystalline state the work of Wendling 20 l should be mentioned; on the basis of X-ray data and IR-spectroscopy, the structure of some of the alkali metal-metalyl-halogeno-complexes has been established as corresponding to C 4 v site symmetry indeed.

T he ESR s peetra of frezen sol utions 31 l appear to have axial symm etry without exception in the series mentioned above. Measure­ments of DeArmond e.a. 15 ) on single crystals of (NH 4 ) 2(InC1 5H20) doped with V4 + or Mo 5 + also point into this direction. Unfortunately, the Molybdenyl- and Vanadyl-groups bath seem to occupy two different orientations in the cell, due to the smaller size of V and Mo. This phenomenon weakens their conclusions somewhat; yet it remains re­asonable to ossurne C 4 v symmetry. Also the interpretation of the op ti cal spectra strongly resembles the one given for VOSO 4 .SH 2 0 by Ballhausen and Gray3 l, which supports the reasonableness of the assumption.

The theoretica! calculation of g- and A-values, and of internal magnetic fields, was discussed in Chopter 1 to be very difficult, owing to the complexity of the problem arising when we try to evaluate "good" molecular wave functions from first principles. Nevertheless, molecular electronic quantities are anyhow necessary, whatever kind of speetral or magnetic property is to be described. Using such quan­tities, trends in properties may sametimes be derived and subsequently tested by experimentcri data. For instance, some well-established example may serve as a check of the correctness of an assumption. When we find (gj_- g#)fluoride>(gj_- g//)chloride' as is the case for the Mo- and Cr-complexes we just discussed, it is useful to see whether this result is in accordance with the trends expected from optical and other data. For all those sorts of application, knowledge of the vis­ible- and near-UV spectra of the compounds is indispensible. Fortun­ately, there is an extensive litterature on that subject, s ome of the papers dealing with the chemica! behaviour, ether ones with their electronic structure. The latter refer to the pioneering calculation of the Vanadyl-aquo-complex. (We will further refer to this paper as B&G).

54

Befere going into the discussion of our results and some data available in the litterature, we will give a short description of the tiNo kinds of approximations that are, for practical reasons, to be considered when one wishes to calculate electronic properties of com­plex molecules.

a) Crystal Field Theory.

T his is the first successful theory for the de scription of the bonding in complex molecules. The basic idea was put forward by Becquerel 65 }, formulated mathematically by Bet he 6 6 l and applied to magnetic properties by Van V leek 9 l.

T he ligands, surrounding the central ion in an often symmetrie arrangement, raise an electric field of conesponding symmetry. This electric field partially lifts the degeneracy of the electronic energy levels of the free ion. For instance, the five 3d wave functions are split by a field of octahedral symmetry, into a low-lying triplet and a high-lying doublet, respectively of t 2 and e symmetry. The mere g g fact of the splitting is imposed by the symmetry, and so is generally valid, but its sign and magnitude should be given by the application of Schrödinger's equation, using a suitable form for the electric po­tential arising from the ligands.

Expressing the wave functions of the unperturbed ion* as spher­ical harmonies multiplied by a radial function, we have

1l1 = R(r) Y[(8, cp) where Y[W, cp) = PT(cos8)eim4>,

Then the suitable form of the potential V is an expansion in spherical harmonies too.

The energy calculation then leads to integrals over such spher­ical harmonies, since we ask for E = < 1l' I V Iw >. 'Th is is a complicated though straightforward thing to do, and tabulotions can be found in the litterature 59 •67 •68 l.

Localizing the electron in the valence-shell of the central ion, the model is limited in its predictions of electron jumps to transitions within the d-shell; the important phenomenon of charge-transfer from ligand to metal, or vice versa, cannot be explained. This makes clear why the crystal field model was so successful in the explanation of magnetic susceptibilities, where only the ground state is important, and much less so for speetral data. We will indi c ate presently the the inadequacy of the model to deal with g-values as they are en­countered in ESR.

free ion, not subject to ligand-interactions.

55

b) Molecular Orbital Theory.

This theory is based on ideas of Mulliken 69 l and Van V leek 70 l and imposes no localization of the electrons. The formalism used is known as LCAO-method, Linear Combination of Atomie Orbitals. The basic theerem is the .one expressing the general possibility of expanding a tunetion as a linear combination of terms of a complete orthogonal set. A natura! choice may be the orthogenol set provided by atomie wave functions. For practical purposes only a limited num­ber of terms of the set is used, inevitably leading to an inaccuracy of the results, depending on the nature of the incomplete set. Actual calculations use, in the case of transition metals , for instanee nd,(n +l)s and (n+l)p functions, with n=3,4,5, and for the ligands ns and np. The latter are not generally orthogenol to the farmer.

In an approximative way one could now praeeed as fellows. 1) A guess is made as to the distribution of the electrens over the

available atomie wave functions; the wave functions are chosen to fit as well as possible to the assumed charge distribution. Usually Pauling's electroneutrality principle47 l is assumed to be valid, leading to a choice of functions representing a charge be­tween 0 and +1 on the central ion. A good calculation should, how­ever, not make such an a priori assumption.

2) This guess de termines the coefficients of the various atomie orbitals in the molecular orbitals (m.o.'s). A m.o. is a linear combination of the chosen atomie orbitals. There i s one m.o. for each electron, including spin. The square of the coefficient is proportional to the number of electrens in the atomie orbital.

3) Minimization of the energy, calculated by the application of the Schrödinger equation, by determination of the coefficients in an appropriate way, leads to a set of simultaneaous equations; a non­trivia! salution of these equations requires the determinant of the coefficients in those equations to vanish. This de terminant is the so-called s ecular determinant 53 •5 4 l. lts general farm is

. . . = 0

....••.... H -ES nn nn

with Hij <4;i lH l4;i > where H is the Hamiltonian, and

S ij <4;i lq;i > the ove rlap.

56

4) Eigenvcrlues E and Eigenveetors c are determined, and from the Eigenveetors a new charge distribution is calculated, using the Mulliken Populotien Analysis58 l, This contributes to a eertcrin wave fundion a charge proportienol to the square of the coefficient c of that atomie function in the m.o., plus half the "overlap charge'', the latter being proportienol to e.c., where c. and c. beleng to the

l J 1 J two overlapping wave functions </J1 and <Pr Consicier for instanee a simple normali;ed m.o. ll' = c 14J 1 + c 2 4J2 • T·he charge density is then given by ll'll' dT for any space element. Assuming the m.o.'s to be real functions, we have

The total charge in the m.o. is then this density integrated over the entire space

To fundion </; 1 is now attributed a charge cf + c 1c 2 S12 , and to tunetion </; 2 a charge c~ + c 1c 2S12 •

Generally, the new charge distribution differs from the initia! one. Then again 1) A new set of m.o.'s is constructed for this new charge distrib­

ution etcetera, until output-charge = input charge, i.e. self con­sistency with respect to charge di stribution.

Clearly this procedure is rather time-consummg. Especially the computation of the energy integrals is troublesome, since multicentte integrals arise.

Wolfsberg and Helmholtz4 S) were the first to propose a semi­empirica! method, avoiding the difficult energy integrals and the need for the repHition of the calculation of a new overlap matrix.

In a slightly modified version, this scheme was used by Ball­hausen and Gray3 l in the earlier mentioned Vanady 1 calculation •

We will now briefly outline their approximation to the problem. It is a rather rigid method, as it uses only a very limited basis- set of atomie wave functions; it possesses no flexibility towards shift of charge, either to or from the central ion. Atomie wave functions have been calculated by Richardson e.a. 48 for the fi rst transition series, and by Bosch e.a. 49 > for the second, as analytica! approxim­ations of numerical Hartree-Feek functions. Halogen functions are given by Watson 50 l and Clementi 5 ll, In our particular case the basis s et cons i s ts of metal-3d,4s,4p and ligand-3s,3p for the halogen, and

57

ligand-2s,2p for Oxygen. Oiatomic overlaps are computed in the con­ventional way. To save time and effort it is advantageous, but not ne­cessary, to sort out the functions with respect to the irreducible representations of the point-symmetry group the molecule belongs to40 •41 l. For the central ion single functions can be selected, but for the ligands it is necessary to construct the appropriate linear com­binations of ligand functions een te red on the various ligand-sites4 0 •4 2 • 4 3 l.

The next step is then to assembie the group overlaps from the previously determined diatomic overlaps.

The calculation of the energy-integrals being rather complicated and time consuming, Wolfsberg and Helmholtz approximated the dia­gonol matrix elements Hii, of type <</J1 lH I<P1>, with </;1 an atomie functîon or a ligand combination, by taking the negative of the Valenee State lonization Energy (VSIE) which can be calculated from the speetral data in Moore's compilation 5 2 l. The VSIE is calculated for various configurations and various charges of the metal ion, and then the VSIE is expressed as a function of its charge, for each configur­ation, by the quadratic approximation

VSIE = - (Aq2 + Bq + C),

where A,B and C are constants, and q the charge. A guess is now made as to the expected number of electrans in the d,s and p orbitals in the central ion, a number that may take any pos­itive value, restricted by the requirement that the net-charge on the me tal ion is non-negative. Off- diagonal terms H 1j are estimated by the re lation

where G . . is the group-overlap between functions i and i. and f an 1]

adjustable parameter mostly assigned the value 2; sametimes a lower value is chosen for pi-bonding.

The quantities we have now at our disposal, are ploeed into the secular determinant53 •54 l. In this way we arrive at

i.e . a blocked determinant in the case that we us ed symmetry-adapted func tion s:

58

0 0

0

From the Eigenvalues, the Eigenveetors are determined, and from the latter the charge dis tribution over the metal-orbitals using the Mulliken Population Analysis 5 8 l; this new distribution is compared t o the pre­vious charge distribution. When the difference exceeds a certain limit, a new determinant, with unchanged overlaps but adjusted energy inte­grals, is constructed and subsequently solved. This procedure is re­peated until the output-charge dis tribution differs from the input-charge distribution by less than the chosen limit. The computer program we used, is basedon an unpublished report by Helge Johansen* 55 l; some minor changes and amplifications were added in the process of re­writing for the EL-X8 computer.

Some objections can be raised against the actual B&G calcul­ation. In the first place, using a tetrahedral hybrid of H2 0 ,B&G com­pletely negleGt in-plane pi-bonding (see Table 5.1); that may be ad­missab1e in their special case, but is certainly not generally correct.

Ass uming a Cr-Cl-distance of 4.3466 Bohr radii, we find S(d77, p77) = 0.0854 and S(p77,p77) = 0.2153 (Cr alw a ys comes firs t between brackets), so this pi-interaction is, in our cases, far fro m neglige­able. In the secend place, the use of hybrid orbitals, if permitted at all, raises an objection. Prior to discussion of this point, it is ne­cess ary for a goed understanding, to reproduce the wave functions used by B&G w ith their Vanadyl calculation.

The ligands in the plane are numbered rotating in the positive direction 1 to 4, storting on the positive x-axis; the Oxyge n is num­ber 5 and the remaining axial ligand number 6. T hey u se then the selection shown in Table 5. 1:

We are indebted to Professor Ballhausen for putting at our disposai a copy of this report.

59

TABLE 5.1

Representation Metal orbHals Ligand orbitals of group C 4 v

al 3dz2 +4s 0"5

4s - 3d 2 z ~(o-1 + 0"2 + 0"3 + 0"4)

4pz 0"6

e 3d 3d xz' yz 7Ts(2px' 2py)

4px' 4py 2-Y.( ) (]"1 - (]"3 I 2-Y.ca- - a- J 2 4

b1 3d 2 2 x - y ~(o -a-1 2 +O" -0)

3 4

b2 3d -xy

3d 2 + 4s and its orthogonal counterpart 4s - 3d 2 are hybrids con-z z structed in s uch a way that VSIE (cjJ)/S(cjJ) =minimum; c/Jis the angle in the normalized form of the hybrid

sin(<f). (s) ± cos(cjJ). (d)

A reasonable alternative choice for the hybrid is the one with maximum overlap; since VSIE(cjJ) varies rather smoothly between pure-s and pure-d functions, and S( c/J) varies rather sharply near maximum overlap, the condition put forward by B&G is governed mainly by the overlap. In this latter way, the hybrids we re constructed f~r CrOCI ~-, assuming a Cr-0 distance of 3.2127 Bohr radii, and found to be 0.7996 Is> + 0.6007 ld>. Richardson's wave functions were used fora charge + l on the Cr. T he orthogenol hybrid is now 0.60071 s > - 0. 7996 ld>.

Along the same lines we find a s-p-hybrid on the Oxygen, chosen by requiring maximum overlap with d 2 of Cr; this hybrid turns out to be 0.8160 Is> +0.57811 p> =: a-5 • Maxim~m overlap with pof Cr determines o-1 2 3 4 as 0.6440 Is> +0.7650 lp>.

' 'Group overlaps con now be found following the usual formalism 42 l. B&G used those quoted in Table 5.2.

Some overlaps are now seen to be neglected, an important one being that between 4s + 3dz 2 and the a 1-combination; it is nearly as large as the overlap between 4s - 3dz 2 and the same o--combination, which might have been seen be forehand, consiclering the rather large amount of s-character in the hybrid. The neglected overlap is 0.4920, the non-neglected one being 0.5880. Overlap for in-plane bonding (not appearing in Table 5.2) is of the order of 0.2. Fenské 6 l paid attention

G(e77)

G(b 1)

G(Ia 1)

G(IIa 1 )

G(IIIa 1)

G(eo-)

60

TABLE 5.2

= S(3d77 I 2p77)

= 3'f. S(3do-,o-)

= S(sdo-,o-5 )

= 2-Y.(2S(4s,o-) + S(3do-,o-))

= S(4po-,o-)

= 2Y. S(4po-,a)

I to the same 'feature in the calculation of Bedon e.a. 5 7l on the ion T ·F3-

l 6 • In view of these facts one is inclined to con si der B&G' s as­

sumptions rather arbitrary; it becomes questionable whether the good fit of the calculated data to the experimentsis based on unsuspected physics.

The calculation of CrOCl~- might as well have been c arried out in a more straightforward manner. U s ing as a basis set Cr 3d,4 s and 4p, 0 2s,2p and Cl 3s,3p, one could go through the complete B&G formalism, without introducing any a priori fixation of wave functions, as had been done by the construction of hybrid orbitals. T his can be accounted for by taking into considerstion that, if maximum overlap really reflects minimum energy, the a priori hybrids of the B&G method should result, once self-consistency of the charge dis tribution has been obtained.

Before storting the calculations, our program was tested on both Mn04 and CrF~-; the results were in splendid agreement with these reported by Balihousen and Gray43 l.

However, applying the form alism to CrOCl~-, an extremely un­relaistic m.o. scheme was obtained , s howinga [). = 10 Dq of 1700 cm- 1

and the second crystal field transition far into the ultraviolet. It is not our purpose to discuss the validity of semi-empirica!

methods of molecular calculations, but it should be realized that any semi-empirica! calculation is loaded with a eertcrin amount of pre­suppositions which, at least partially, determine the final result of the calculation. It has also been pointe d out56 l that the B&G method strongly exaggerates pi-bonding. A crude trial-calculation using the B&G hybridization scheme, s howed the s ame order o f energy levels as for the Vanadyl ion, except for the lewest anti-bonding a 1 , which was found strongly shifted downward, until just below the non- bonding b2 level. The sign of lO Dq was again correct and its magnitude 16000 cm- 1, too large by approximate ly 3000 cm - 1•

61

1 The antibonding b ~' lying extremely high in the previous calcu1-ation, even above the highest a;, was found to be shifted now down­warcis to its expected region, i.e. between e* and Ila*1• However, this . . ~ b 1 still lies fairly high above e , and there seems to be no reasonable

• • 'TT

possibility to .have b 1 and e change their relati ve positions. The Vanadyl level scheme has already been reproduced in fig.l.2

and we will adhere to this order of levels also in the CrOX~- and MoOX~-. This order is in accordance with the simple crystal field picture for a not too streng tetragonal distortion, as far as intra-d­shell transitions are concerned; the crystal field picture is gi ven in fig.S.l.

E

fig. 5.1. a 2

c) Discussion of q-values.

It is a well-known fact that crystal field theory is inadequate as an interpreting tool in ESR probletns. The g-values depend, often very subtly, on the contribution of several excited states 12- 15 •31 •42 l, some of these giving positive-, ethers giving negative contributions to the g-shift; the crystal field model prediets negative shifts exclusively.

The simplest account for covalency is the introduetion of a reduction factor in the g-shift, usually assumed to be the square of the coefficient of the appropriate metal wave function in the m.o. 3 •27l, This is certainly insufHeient for many cases. For MoOF ~- versus MoOCl ~- this can be illustrated by a crude order-of-magnitude estim­ate. We find there /J.g = lg11 - g.ll for MoOF~- to be approximately +0.037, whereas for MoOCl~- we find -0.025. The speetral data of both complexes are very similor, and for each of them we use

"E "E 2 104 1 and t: = 600 cm- 1 59 l* u b2-e =u b2-bl = · cm- :.

* Ç = Spin-orbit coupling parameter.

62

The central-ion coefficients are called c(~) and c\~i, the sub­scripts re terring to the ex ei ted levels, the superscripts to tne ligand, i.e. Cl or F. We have then, according to crystal field theory 60 l

8Ç(cá~l)2 2Ç(c(F))2 + E = 0.037

i)EB2-Bl EB2-E

and

SÇ(ck~l) )2 +

2Ç(ckC1)2 = -0.025

[)EB 2-Bl i)EB2-E

Rewriting this we find

0.019 4Ç(c~l)2 ç +

c(Fl)2 ( E (i)E HcJr'l 2

B2-Bl i) EB 2-E

and idem for Cl. The level of e-symmetry is associated with the, presumably

strong, pi-bonding between central ion and Oxygen, and is, most pro­bably, much the same in all cases; we s hall choose er:'= 0.9 in both instances. Then

(CB(Clll)2 I (c(EC1))2 ( t:l "E ) 0 016 !::. 0 82-E + ·

Inserting the assumed vcrlues and using the Vanadyl Eigenveetors for the Fluorine complex, we have

whence

(c~l)2 1 (c~ l)2

(ck~ll) 2 I (c(~ ll)2

(600/2.104 ) - 0.024 = 0. 13 (60012.104 ) + 0.016

0.36 c( Cl) I c (Cl)

Bl E

63

and so

ck~I) I c~CI) "' (0.946/0.902)/0.36 = 2.9

We see now that, assuming cE to be approximately constant, eBI must increase by a factor 3, passing from the fluorine- to the chlorine-com­plex. This, however, is rather improbable, regarding the similarity of the spectra. Th en the alternative could be to have cE decrease con­siderably. Now, eBI belengs to the in-plane d 2 2 wave function of x -y the central ion, and is governed by the o--bonding between ligands l-4 and the metal. In this regard it seems rather unreasonable to expect the Metal-Oxygen-pi-bonding to be weakened at the expense of the in-plane-pi-bonding, the more so because the chemistry of the com­pounds shows us the very stabie character of the Metal-Oxygen bond, in contrast to the easy interchange of halogens.

Clearly the crystal field model has to be abandonned in favor of a more sophisticated approach. This could consist in carrying through the full perturbation treatment with the m.o.'si 2 • 13 l, taking into account in-plane-pi-bonding and spin orbit interaction of the elec­trens with the ligand nuclei44 l,

For thecompoundsweare dealing with here, Kon and Sharpless31 l introduced a werking hypothesis of a very special kind. They drew attention to the low-intensity band, found in the optica! spectrum of all metalyl compounds in the region between 20000 and 30000 cm- 1,

and hoving a molar extinction coefficient of crbout 600; this value is too large for a crystal field transition, but much too small for an allowed charge transfer transition. This band undergoes a shift to lower wavenumbers, associated with an increase of g11 , when Br takes the place of Cl.A s hift to higher wavenumbers is associated with a decrease of g 11 , and occurs w hen Mo is replaced by W.

To explain this effect, they assign this low-intensity band to a transition b 1-b;• thus meeting at the same time the criticism on the band assingments by Gray and Hare4 l, as put forward by Allen e.a. 45 l, For the present transition to be the suggested torbidden charge transfer transition, the b 1 level s hould be less stabie than the e( 7T) level, a not unlikely s ituation, they argue, consiclering the easy replacement of the halogen ligands by other halogens, or different kinds of ligands hov­ing a comparable electron affinity. Furthermore, this b 1 ---b; transition plays an important role with respect to g11 , producing a positive shift of g 11 , according to their formula 1 4 • 3 1 l

where

64

+ (2á(3 1Çm + af31Ç 1)(2a1(3 1 + a(311)

~Eb1-b2 *

I

a I a = coefficients of metal orbital and ligand combination 1

respectively 1 in bl symmetry; /3 1 1(31

1 = idem, but for b2 symmetry;

Ç m = spin-orbit coupling parameter of metal;

Ç 1 = idem of ligand.

F oglio 61 l defends a different standpoint. F or MoOCl ~- he proves to be able to account for the g-values by consiclering the mixing of the function of a non-magnetic electron (i.e. an electron in a tilled orbital) into the half-filled antibonding b; orbital.

This author did not try to make a comparison with the compounds arising by the replacement of Cl by Br or F, or of Mo by Cr or W. This contines the good agreement of his calculation w ith experimentcri data, to MoOCl~-.

In conneetion to the proposition put forward by Kon and Sharpless , care has to be taken to avoid false arguments. Bas ing themselves on the speetral shifts, they may have overlooked the feature of di- or poly-merization. In this conneetion the work of Sacconi and Cini62 l and of Hare, Bernal and Gray 63 l on Mo(V) in HCl, should be mentioned. These outhors study Mo(V) in HCl in the concentratien range ·between 2 and 12 Mole/ l. Sacconi and Cini then re port a sharp decline of the magnetic susceptibility x when the concentratien of HCl drops from 7 to 4 Mole/ l. Hare e.a. showed a marked difference be tween the rate s of decline of x and the intensity of the ESR signal in the same range. In conjunction with the spectroscopie data of Haight 64 l it is concluded that in this range of molarities a paramagnetic dimer is formed, eventually disappearing at a concentratien of 2 Mole/ l. Since this process of coupling of monomeric units is associated with a change in the 28 2 _. 28 1 transition, the coupling is assumed to take place in the xy-plane, and, being s trongly pH-dependent, to be effe ctu­ated by water oxygens. Moreover, no ESR line broadening is observed, and so the coupling is probably streng.

Taking a Bramine ligand as being more loosely bonded than a Chlorine lig and ,_ the stahiliz ing coupling mig:ht be expected to occur

65

at a higher molarity (of HBr) for the Bramine complex. This has actl,l­ally been found by Allen and Neumann 36 l who, from equilibrium con­stants, derived with the aid of speetral data, suggest double halogen bridges between the Mo(V)-comp1ex units, to be the common phenom­enon already at molcrities as high as 8 Mole/1.

On these grounds it is doubtful whether the ESR spectrum given by Kon and Sharpless 31 l belongs to monomeric species.

On the other hand, the assumption of monomeric species being present in 38% HF-solutions, which assumption was an essenhal one for the interpretation of the 1-hfs, seems to be justified now.

T his part of the discu ss ion imposes still another limitation to the possibilities of checking our model for the 1-hfs in molecules of the kind we have been discussing in this thesis. Comparison of the 1-hfs parameters, going from F, via Cl to Br as ligands, would allow of camparing the actual trend to the one predicted by our model. How­ever, since Cl causes no observable 1-hfs, and the spectrum of the Br-complex is most probably not of the desired monomeric species, this check is ruled out.

66

Chapter 6

FINAL REMARKS AND CONCLUSION

In Chopter 4 a line-broadening was observed in the Q-band ESR spectra. There are two major causes on line-broadening. The firs t one is 'the in teraction between a paramagnetic unit and its neares t neigh­bours, which occurs for instanee in crystals of the, magneti cally un­diluted, paramagnetic compound. Since our experiments dealt with dilute solutions (l% or les s ), this mechanism plays no role of any importance; moreover, the same kind of broadening should have been observed in the X-band spectrum.

The secend cause lies in relaxation phenomena, three types of which are recogni zed; a detailed description is found in van Reyen1 s thesis27 l. According to his results, a relaxation mechanism of the Raman-type (a two-phonon process) is to be expected over the entire temperature range of interest for our purposes.

As soon as the relaxation rate exceeds 108 s- 1, a broadening starts, leading to complete disappearance of the signal at a rate of about 10 1 1 s- 1• Van Reyen 1 s formulae further indicate a À. 2-depend­ence of the Raman-rate. Since >dor Mo is roughly twice that fo r Cr, the Me-complex will be s lightly more sensitive to broadening effects than the Cr-complex. For Cr the 10 8 limit lies at about 150°K , so just in the temperature region where our measurements had to be carried out. The chances thàt at Q-band the temperature was actually some 10 or 20 degrees higher, are certainly not negligeable. We might ossurne then that an extremely delicate situation can arise, where the broaden­ing in the Cr-complex can hardly be observed yet, while the Me-com­plex has just proceeded far enough to be in the region of clearly observable broadening. A rather small broadening may be sufheient to obscure many tiny details on the flanks of the larger signals , thus over-emphasizing this broadening.

T he most attractive explanation would have been the one that takes into account the considerabl y higher applied magnetic field; s uch a field causes larger splittings and, at a first glance , a higher relax­ation rate would be expected. The direct relaxation process, however, that is proportional to H4 , is of no importance at all at temperatures above l0°K.

We are aware of the rather speculalive character of the reason­ing we used, but we see no ether explanation of the whimsical broaden­ing observed in the spectra.

67

In fig. 6.1, a graph is gi ven of the probabili ty of finding an electron at a certain distance from the metal-ion. The 3d-functions of Cr 1 + and Mo 1 + are plotted as far as their radial parts R(r) are con­cern ed. The electron density is represented as r2 R 2 (r), and is plotted with Bohr radii along the abcis. (One Bohr radius is 0.529148 Ji.).

fig . 6.1 .

0.5 1.0 1.5 2.0 2.5 }.0 4.5 •••

68

Using the metal-halogen distonces Cr-Cl = 2,3 A and Mo-Cl = 2.4 A, 'it is easy to see that the electron spends the major part of the time at a distance of about 1.5-1.6 A from the ligand nucleus. (Another way to put this is to say that the major fraction of the electronic charge has the indicated distance from the ligand nucleus). This result in­dicates that the fields, generated by the electron, at a ligand site, are of the right order of magnitude to be in accordance with the internol fields that were derived in the analysis of the ligand hyperfine struct­ure.

Conclusion

Looking back at the arguments, used by the various authors, it will have become clear that the present situation regarding the bond­ing in tetragonal complexes of the type MOX ~-, is far from satisfact­ory.

When we began our attempts to interpret the ligand hyperfine structure in the ESR signals, we entertained hopes of being able to derive data concerning the chemical bonding in the complexes we discussed in the foregoing Chapters. The result, ho wever, did not come up to our expectations, since the simple model we introduced, described the ligand hyperfine structure in an even semi-quantitative manner. With such a classica! model already representing the actual situation saUsfactoril y, it will be clear that onl y very refined quanturn mechanica! methods con describe the fundamental aspects of the chemical bonding adequately.

Those methods should then complement our given interpretation in such a way as to have it include a reasonably quantitative de­scription. It will be necessary to go through a complete self-con­sis tent calculation, preferabl y of the unrestric ted type, in order to get insight in the apparently very complex situation encountered in the electronic structure of the rather simple molecules we were dealing with.

Excited states, being of great importance for the g-values, will have to be treated in a comparable rigoreus manner. This leads to additienol difficulties, which probably are still beyond the reach of present computational methods.

69

SUMMARY

The work described in this thesis consists of two closely rel­ated parts, the first being an attempt to interpret the l~gand hyperfine structure in ions of the general type MOX~-, with M a transition metal ion, and X a halogen; this is clone by means of a simple but pictorial model. The secend part contoins a brief review of reeent opinions on the nature of the electronic structure of these complexes.

Aft er a résumé of relevant parts of the theory of electron spin resonance, a model of ligand-electron-interaction is proposed, using a simple dipole-dipole-interaction mechanism.

As a consequence of the experimentcri limitations, eenstraining us to the use of polycrystalline samples, the peculiarities of the ESR spectra of such samples are discussed in some detail.

The theoretica! framewerk now being complete, the experimentcri data are presented, and an interpretation is given. A check of this interpre tation lies in the camparisen of X- and Q-band spectra, and in the use of isotapes of the metals. Within the restrictions imposed by the lack of experiments with single crystals, the given interpretation is satisfactory.

In an attempt to decide between diverging opinions in the litter­ature, semi-empirica! calculations of the electronic structure of the present ions were eerried out. The results of such calculations turned out to be extremely sensitive towards eertcrin presuppositions , and appeared inadequate to make positive statements as to the correctness of any of the opinions put forward by the various authors. Same pro­positions from the litterature are presented in some detail. The con­dusion is drawn that a reliable ab-initia calculation, for instanee of the unrestricted Hartree-Feek type, may be the only way out of the difficul ties.

70

SAMENVATTING

Het werk dat in deze dissertatie wordt beschreven, omvat twee nauw verweven delen. Het eerste bevat een poging om de liganden­hyperfijnstructuur, zoals deze wordt waargenomen bij de ESR spectra van ionen met de algemene formule MOX~-, met M een overgangsme­taal en X een halogeen, te verklaren. Daartoe wordt een eenvoudig maar beeldend model ontwikkeld.

Het tweede deel bevat een overzicht van gangbare meningen omtrent de electronische structuur van de onderhavige complexe ionen.

Na een overzicht over de benodigde theorie van de ESR spectro­scopie wordt het model van de ligancien-electron-wissel werking inge­voerd, waarbij gebruik wordt gemaakt van een eenvoudig dipool-dipool­wisselwerkingsmechanisme. Als een gevolg van de omstandigheden waaronder de experimenten moesten worden uitgevoerd, waarbij slechts van monsters met een statistische verdeling der orientaties van de moleculen, gebruik kon worden gemaakt, worden de eigenaardigheden van de ESR spectra van zulke monsters aan een nauwkeurige beschou­wing onderworpen.

Vervolgens worden de experimenten beschreven en verklaard binnen het raam van de theorie zoals die in het voorafgaande werd beschreven.

Een nadere controle op de interpretatie werd gevonden in het vergelijken van de ESR spectra, verkregen met frequenties in de X­en Q-band, en in het gebruik van isotopen der metalen.

Binnen de beperkingen, opgelegd door het ontberen van metingen aan één-kristallen, is de gegeven interpretatie bevredigend.

In een poging te beslissen tussen uiteenlopende opvattingen in de litteratuur, werd getracht semi-empirische berekeningen uit te voe­ren van de electronische structuur van de beschouwde ionen. De uit­komsten van zulke berekeningen bleken buitengewoon gevoelig te zijn voor gemaakte vóór-aannamen, en. bleken ontoereikend om uitspraken te kunnen doen ten gunste van enige opvatting. Enkele voorstellen uit de litteratuur worden nader aangehaald. Uiteindelijk wordt de con­clusie getrokken, dat de enige weg om het probleem eenduidig op te lossen, l e idt naar een ab-initia berekening, bijvoorbeeld van het un­restricted Hartree-Feek type.

71

REFERENCES

l. A.F .WELLS, "Structura1 Inorganic Chemistry", Ciarendon Press, Oxford, 1962.

2. H.B.PALMA-VITTORELLI, M.U . PALMA, D.PALUMBO and F.SGARLATA, Nuovo Cimento 3 718 (1962).

3. C.J.BALLHAUSEN and H.B.GRAY, Inorg.C hem. 1 111 (1962). 4. H.S.GRAY and C.R.HARE, lnorg.Chem. 1363 (1962). 5. C.P .SLICHTER, "Princip1es of Magnetic Resonance", Harper and

Row, New York, 1964. 6. G .E .PAK E, "Paramagnetic Resonance" , Benjamin, New York, 1962. 7. A.CARRINGTON and A.D.McLACHLAN, "Magnetic Resonance11 ,

Harper and Row, New York, 1967. 8. A.ABRAGAM, 11The Princip1es of Nuclear Magnetism11 , Oxford

University Press, 1961. 9. J.H.VAN VLECK, 11Electric and Magnetic Susceptibilities",

Oxford University Press, 1932. 10. M.H. L .PRYCE, Proc.Phys.Soc. A63 25 (1950). 11. A.ABRAGAM and M. H.L.PRYèE, Proc.Roy.Soc. A205 135 (1951). 12. A . H.MAKI and B.R.McGARVEY, J.Chem.Phys. 29 31 (1958). 13. D.KIVELSON and R.NEIMAN, J.Chem.Phys. 35 149 (1961) . 14. H.KON and N.E.SHAR P LESS, J.Chem.Phys. 42. 906 (1965). 15. K.DeARMOND, B . B.GARRETT and H .S. GUTOWSKY, J. Chem.

Phys. 42 1019 (1965). 16. A.ABRAGAM, Phys .Rev. 79 534 (1950). 17. A.J.FREE.MA.N and R.B.FRAENKEL, 11Hyperfine Interactions",

Academie Press, New York, 1967, Chapter 2 and references therein.

18a. M. TINK HAM, Proc.Roy.Soc • . A236 535 (1956). 18b. M. TINK HAM, Proc.Roy.Soc. A236 549 (1956). 19. H.WATANABE, 11 Üperator Methods in Ligand Field Theory",

Prentice-Hall, Englewood Cliffs, 1966. 20a. E.WENDLING, Bull.Soc.Chim. 1967 5. 20b. E.WENDLING, Bull.Soc.Chim. 1967 8. 20c. E.W E NDL!NG, Bull.Soc.Chim. 1967 16. 21. R. G.S HULMAN and S .SUGAN O , Phys.Rev. 130 506,512,517 (1963). 22. R.E.WATSON and A.J. FR EEMAN, Phys. Rev. 134 A1526 (1964). 23. R . H. S ANDS, Phys .Rev. 99 1222 (1955) . 24. F.K. KN E UBÜH L , J .Chem. P hys. 33 1074 (1960). 25a. B . BLEANEY, Proc.Phys.Soc . .A63 407 (1950). 25b. B.BL E ANEY, Phil.Mag. 42 441 (1951). 25c. B .BLE ANE Y , Proc.Phys .Soc. A75 621 (1960).

72

26. J.A.IBERS and J.D.SWALEN, Phys.Rev. 127 1914 (1962) . . 27. L.L. VAN RE YEN, Thesis Technological University Eindhoven,

1963. 28. G.SCHOFFA and G.BÜRK, Phys.Stat.Sol. 8 557 (1965). 29. R . F.WEINLAND and M.FIEDERER, Berichte 39 4042 (1906). 30. F .G.ANGELL, R.G.J AMES and W. WARDLAW, J .Chem.Soc. 1929

2578. 31. H.KON and N.E. SHARPLESS, J.Phys.Chem. 70 105 (1966). 32. N.M.R.-Tab1e, 5th edition, Varion Associates, Palo Alto/Cf., 1965. 33. J.T.C. VAN KEMENADE, J.L.VERBEEK and P.F.CORNAZ, Rec.

Trav.Chim. 85 629 ( 1966). 34. J .L. VERHEEK and P.F .CO RNA Z, Rec. Trav.Chim. 86 209 (1967). 35. J.L.VERBEEK and A.T.VINK, Rec.Trav.Chim. 86 913 (1967). 36. J.F.ALLEN and l-!.M.NEUMANN, Inorg.Chem. 3 1612 (1964). 37. I.N .MAROV, YU .N .DUBROV, B .K .BEL Y AEV A and A.N.ERMAKOV,

Doklady Akad.Nauk SSSR 171 385 (1966) (in Russian), or Dok­lady Phys.Chem. 171 745 (1967) (translation).

38. l-I.M.McCONNELL, J.Chem.Phys. 25 709 (1956). 39. R.N.ROGERS and G.E.PAKE, J.Chém.Phys. 33 1107 {1960). 40. F .A.COTT ON, "Chemica! Applications of Group Theory", Wiley

Interscience, New York, 1965. 41. M.l-!AMERMES!-1, "Group Theory", Addison Wesley Publishing

Company Inc., Reading/ Mass., 1962. 42. C.J.BALL!-!AUSEN, "Introduction to Ligand Field Theory",

McGraw Hill Book Company Inc., New York, 1962. 43. C.J.BALLHAUSEN and H.S. GRAY, "Molecular Orbital Theory",

Benjamin, New York, 1964. 44. R.LACROIX and G.EMCH, Helv.Phys.Acta 35 592 (1962). 45. E.A.ALLEN, 8 . J .B RISDON, D.A.EDWARDS, G. W .A.FOWLES

and R.G.WILLIAMS, J .Chem.Soc. 1963 4649. 46. M.WOLFSBERG and L.HELMH OL T Z , J.Chem.Phys. 20 837 (1952). 47. L.PAULING, "The Nature of theChemical Bond", Cornell Univers­

ity Press, Ithaca/N.Y., 1960. 48a. J. W.RICH ARDSON, W.G.NIEUWPOORT, R.R.POWELL and

W.F.EDGELL, J.Chem.Phys. 36 1057 (1962). 48b. J.W.RICHARDSON, R.R.POWELL and W.C.NIEUWPOORT,

J .Chem.Phys. 38 796 ( 1963). 49. H.BASCH, A.VISTE and H .B .GRAY, Theor.Chim.Acta 4367 (1966). 50. R.E. WATSON, Phys.Rev. 118 1036 (1960) and Phys.Rev. 119

1934 (1960). 51. E.CLEMENTI, J.Chem.Phys. 40 1944 (1964). 52. C.E.MOORE, U.S.National Bureau of Standards Circular 467,

U.S.Government Printing Office, Washington O.C., 1949, 1952.

73

53. A.MESSIAH, "Quantum Mechanics", North Holland Publishing Company, Amsterdam, 1965.

54. H.EYRING, J. WALTER and G.E.KIMBALL, "Quantum Chemistry", lOth printing, John Wiley and Sons, New York, 1961.

55. H . JoHAN SEN, "Algol Programs for Molecular Calculations", Unpublished Report, University of Copenhagen.

56. R.FENSKE, lnorg.Chem. 4 33 (1965). 57. H.D.BEDON, S.M.HORNER and S.Y . TYREE, Inorg.Chem. 3 647

(1964). 58. R . S . MULLIKEN, J.Chem.Phys. 23 1833 (1955). 59. J .S.GRIF FITH, "The Theory of Transition Metal Ions", Cam­

bridge University Press, 1961. 60. A.CARRINGTON and H.C.LONGVET HIGGINS, Quart.Rev. 14

427 (1960). 61. M.E.FOGLIO, Proc.Phys.Soc. 91 620 (1967). 62. L.SACCONI and R.CINI, J.A.C.S. 76 4239 (1954). 63. C.R.HARE, I.BERNAL and H.B.GRAY, Inorg.Chem. 1831 (1962). 64. G.P . HAIGHT, J.Inorg.Nucl.Chem. 24 663 (1962). 65 . J .BECQUEREL, Z.Phys. 58 205 (1929). 66. H.BETHE, Ann.Physik 3 135 (1929). 67 . W.LOW, "Paramagnetic Resonance", Solid State Physics Supple-

ment 2, Academie Press, New York, 1960. 68. M.T.HUTCHINGS, Solid State Physics 16 227 (1964). 69. R.S . MULLIKEN, Phys.Rev. 40 5 (1932). 70. J.H.VAN VLECK and A.SHERMAN, Rev.Mod.Phys. 7 167 (1935).

74

EPILOOG

Aan het tot stand komen van dit proefschrift hebben verscheidene mensen hun onmisbare medewerking verleend, waarvoor ik hier mijn dank wil betuigen.

In de eerste plaats komt Dr.P.F.Cornaz, Ecole Po.Iytechnique de l'Université de Lausanne, Zwitserland, die gedurende de cursus 1965-1966 als gast in onze groep werkte. Zonder de langdurige dis­cussies met hem, en zonder zijn grote kennis van magnetische reso­nantie, zou het werk althans niet in de huidige vorm aan U voorgelegd zijn.

Het chemisch-preparatieve- en -analytische werk komt voor het grootste gedeelte uit handen van Mej. M.Kuijer, daarin aanvankelijk bijgestaan door Mevr. F.A.M.G.Metz-van Elderen. In het bijzond~r

voor de bereiding van de isotoop-bevattende verbindingen uit minder geschikte uitgangsmaterialen ben ik haar veel erken telijk~eid ver­schuldigd. Haar kritische instelling en enthousiame hebben er toe bij­gedragen dat diverse moeilijkheden in korte tijd konden worden opgelost.

De Heer A. T. Vink, Kandidaat Physisch Ingenieur, heeft gedu­rende een stage op het Laboratorium voor Anorganische Chemie en Katalyse, op buitengewoon voortvarende wij ze de Q-band metingen verricht en op substantiele wijze bijgedragen aan de ontrafeling daarvan.

Ir. Th.P .M.Beelen herschreef en testte het Algol-programma voor de semi-empirische berekeningen, en de Heer W.Konijnendijk, Kandi­daat Scheikundig Ingenieur, schreef het simulatieprogramma voor ESR­spectra en bracht enige wijzigingen aan in het daartoe gebruikelijke mathematisch formalisme.

Het Eindhovens Hogeschool Fonds dank ik voor de financieële steun waardoor ik in staat was een vruchtbare discuss ie te hebben met Dr.H.Kon en Dr.N.E.Sharpless, beiden verbonden aan de National Institutes of Health te Bethesda/Md., tijdens een kort verblijf aldaar.

Een aantal der gebruikte figuren verscheen eerder in het Recueil des Travaux Chimiques des Pays Bas, Joumal of the Netherlands Chemica} Society.

Tenslotte dank ik alle ongenoemde collega's en overige perso­neelleden wier bijdragen aan discussies, experimenten en apparatuur een prettige s am enwerking en goede voortgang mogelijk maakten.

Deze epiloog, met alles wat daaraan vooraf ging, heeft veel ge­vergd van het geduld van mijn echtgenote. Voor haar begint na deze slotzin ongetwijfe ld weer een volgend hoofdstuk.

75

CURRICULUM VITAE

Op verzoek van de Senaat der Technische Hogeschool volgen hier enkele persoonlijke gegevens.

Na fragmentarisch Lager Onderwijs gedurende de oorlogsjaren be­zocht ik het Gemeentelijk Lyceum, thans Grotius Lyceum, in mijn geboortestad den Haag. Na het behalen van het Diploma Gymnasium-,8 in 1952 werd ik ingeschreven aan de Rijks Universiteit te Leiden. Na het behalen van het Kandidaatsexamen F werd ik aangesteld als assistent bij de Anorganische Chemie. In 1959 volgde het Doctoraal examen met hoofdvak Physische Chemie, bijvak Anorganische Chemie en Organische Chemie als "derde richting". In aansluiting daarop aanvaardde ik een benoeming tot leraar aan het Gemeentelijk Lyceum te Eindhoven. In 1963 aanvaardde ik een benoeming tot Wetenschappelijk Medewer­ker bij de sectie Anorganische Chemie en Katalyse aan de Technische Hogeschool Eindhoven. In 1964 bezocht ik de Ferienkurs über Theoretische Chemie te Konstanz o.l.v. Prof.H. Hartmann en in 1966 de Summerschool on Theoretica! Chemistry te Oxford o.l.v. Prof.C.A.Coulson. Door een toelage van de NATO en het Eindhovens Hogeschoolfonds kon ik in 1967 deel nemen aan het NATO Advanced Institute aan de McGill University te Montreal o.l.v. Prof.A.J.Freeman en Prof.R.Stevenson. Het thans beschreven onderzoek werd uitgevoerd in de jaren 1966-1967.

APPENDIX

bes:in ~. Afd. Anorganische Chemie, T.H. E1Ddhoven. SDIJLATIEPROORAJoiiA B. S1mulat1eprograaaa voor een epr spektrum van een polykr1atall1Jne stof met axiale eyaaetrie dua met tvee s-arden; real h, hl, hh, hbegin, b, a, da, dx, p, dp, e, de, t, dt, k, dk, 111, dm, r, dr, a, ds, u, du, v, dv, v, dv, z, dr., -- 1, dl, t, dt, c, y, g, J, ap, ar, ss, at, vr, tt, int, ha, bc, he, x, dd, rt, at, arta, aba, tnta, abaa, gm,

nu, 1, ti, ia, ia1a, hhkv, ckv, vvzz, uuvv; in}eger n, pa, JJ, kk, ke, ee, b&Dd, pp, qq; real array hhl[0:10], hll[0:10], ah[0:10], &1[0:10], bb[0:10], gev[0:10, 0:10, 0:10], ah1[0:10], &11[0:10],

bh1, bl1[0:10]; integer !:!!:!!.l rn[0:10], 11[0:10], ga[0:10]J

START: ha :• read; hbegin :• read; pe :• read; kk :• reed; b&Dd :• reed; nu :• read.J PRmrrEXT({: Ao4o611~,~); NLCR; NLCR; tor JJ :• 0 jtep 1 untU kk do begin hhl[JJ :• reed; hll[JJJ :• reed; ah[JJ] :• reed; al[JJ] :• reed; bb[JJ] :• read; rn[JJ] :• read; U[JJ] :• read;

ga[JJ] :• read; ahi[JJ] :• read; al1[JJ] :• reed; bh1[JJ] :• read; bl1[JJ] :• read; PRINTTEXT(~ JJ • ~); FIXT(2,0,JJ); . NLCR; FIXT(5, 2, hhl[JJ]); SPACE(5); FIXT(5, 2, hll[JJ]); SPACE(5); FIXT(4, 2, ah[JJ]); SPACE(5); FIXT(4, 2, al[Jj]); SPACE(5); FIXT(~, ~. bb[JJ]); SPACE(5); FIXT(2, 0, rn[Ji]); SPACE(5); FIXT(2,0,11[JJ]); SPACE(5); FIXT(2,0,ga[JJ]); NLCR; FIXT(4,2,ah1[JJ]); SPACE(5); FIXT(4,2, al1[JJ ); SPACE(5); FIXT(4,2,bhi[JJ]); SPACE(5); FIXT(4,2,bl1[JJ]); NLCR ~

~· ~ tor' JJ :• 0 step 1 until kk do tor pp :• 0 step 1 untU lt[JJ] !!2 tor qq :• 0 step 1 unt11 ga[JJ] do begin gev[JJ,pp,qq] :• read; FI~2,2,gev[JJ,pp,qql};'SPACE(5) end; NLCR; PRINTTEXT( 1: veldsterkte afgeleide abaorpt1elrurve abaorptielrurve g-vaard.e ~); NLCR; !2!: n : • 0 step 1 until pa do begin h :• hbegin + n x ha; int :• 0; he :• h x h; aba :• 0; inta :• 0; abaa :• o;

tor JJ : • 0 atp 1 until kk do begin b :• bb JJ}; bc :• b xb;

tor ke :• 0 step 1 until rn[JJ] do begin ror pp :• 0 step 1 until iiTJJ] do

begin ror qq : • 0 atef 1 until gi[ JJ] do begin hh :• hhl JJ]--:;:--rite- 0,5 xrn[JJll x ah[JJ] + (pp- 0.5 x 11[JJ]) x ah1[JJ] +

(~- 0.5 x ga[JJ]) x bh1[JJ]; hl :• hll[JJ] + (ke - 0.5 x rn[JJ]) x al[JJ] + (pp- 0.5 x 11[JJ]) x al1[JJ] +

(qq- 0.5 x ga[JJ]) x bl1[JJ]; 1t hh < hl then

begin c :• he + bc; ckv :• ex c; hhkv :::--iih x hh; y :• he- hhkv- bc; a :• (he- bc)/(ckv) + 1/(hl x hl); da :•- 2 x h/ckv; dx :• 2 x (1 - 4 x hc/c)/ckv; p :• aqrt(y X y + 4 X he x bc); dp :• 2 x h X (y + 2 x bc)/Pl tt :• p x p; e :• - sqrt(0.5 x p + 0.5 x y); de :• (0.25 x dp + 0.5 x h)/e; r :• aqrt(0.5 x p - 0.5 x y); d! :-- (0.5 x h- 0.25 x dp)/r; k :• (b x r- h x e)/p; dk :• (dt x b- e-h x de)/p- (b x r- h x e) x dp/tt;

m :• (b x e + h x f)jp; dm :• (b x de + f + h x ~)/p- (b x e + h x r) x dpftt; x :• he - hhkv; r :• (- h x b x r + x x e)fp; dr :• (- b x r- b x h x ~ + 2 x e x h + x x de)/p + {b x b x t- x x e) x dp/tt; e :• (- h x b xe-x x r)jp; de :· (- b x e-b x h x de - 2 x r x h- x x ~)/p + (b x b x e + x x r) x dp/tt; g := h- hh; j :• h- hl; vr :• sqrt(hl x hl- hhkv); u :• k x g + r; du :• k + g x dit + dr; v :• k x j + r- vr; dv :• k + j x dk + dr; v :• m x g + a; dv :• m + g x dm+ ds; ~ :ft m x j + s; dz := m + j x dm+ de; dd :• hh/hl; rr :• sqrt(1- dd x dd); vvzz :• v x v + z x z; uuvv :-u x u+ v x v; 1 :• ln(((g x g + bc) x vvzzj(uuvv x (J xj + bc)))); dl :• 2 x gf(g x g + bc) - 2 x J/(J x J + bc) - {2 x u x du + 2 x v x dv)juuvv + (2 x v x dv + 2 x z x d~)fvvzz; t :• arctan(v/u) - arctan(z/v) + arctan(b/g) - arctan(b/J); if 0 <U Á V< 0 then t :• t + ~.14159265; 1f 0 <U Á 0 <V then t :• t- ~.14159265; 1f hh <hA h <hl then t :• t- ~.14159265T" - - --dt :• b x {1j(j x J""""+'"bc) -1/(g Xg + bc)) + (u x dv- v x du)fuuvv + (z x dv- v x dz)jvvzz; 1 :• gev[JJ,pp,~) x (((a x~+ da x r + da x b x de-ex b x dx) x l + (a x r + da x b x e) x dl .• (a x de+ ex da-da x b x d.f + dx x b x f) x 2 x t + (a x e -de. x b x r) x 2 x dt)/(2 x b x p) - ((a x r + de. x b x e) x l +(a x e- da x b x r) x 2 x t) x dp/(2 x b x tt)- dx x {arctan{dd/!!)- 1.570796~)/hh + da x vr/(hhkv x hl)); 1nta :• 1nta + 1; ia :• gev[jj,pp,qq] x ( a x r x. l/{2 x b x p) + da x e x l/{2 x p) + a x ex t/{b x p) - da x r x tfp

+ (da/hh) x (arctan{dd/ff) - 1.57079) + vr/(hhkv x hl x e)); absa :• absa + ia

el se begin c :• he + be; hhkv :• hh x hh;CkV :• c x e; y :• he- hhkv- be;

a :• (he- bc)/ekv + 1/(hl x hl); da :•- 2 x h/ckv; dx :• 2 x (1 -4 x he/c)/ekv; p :• ~rt(y x y + 4 x he x bc); dp :• 2 x h x (y + 2 x bc)jp; tt :- p x p; e :•- sqrt(0.5 x p + 0.5 x y); de :• {0.25 x dp + 0.5 x h)je; t :• sqrt(0.5 x p- 0.5 x y); ~ :• - {0.5 x h - 0.25 x dp)jr; k :- (b x r - h x e)jp; dk :• (~x b - e - h x de)/p- (b x r- h x e) x dp/tt; m :• {b x e + h x r)jp; dm :• (b x de + r + h x d.f)/p- (+ b x e + h x r) x dpftt; x :• he- hhkv; r :• (- h x b x f + x x e)jp; dr :•- (b x r + b x h x ~- 2 x e x h- x x de)/p- (- h x b x f + x x e) x dp/tt; s :•- ( h x b x e + x x f)jp; ds :•- ( b x e + b x h x de + 2 x f x h + x x df)/p + (h x b x e + x x r) x dp/tt;

-.J co

g :• h- hh; J :• h- hl; wr :• sqrt(hhkv- hl x hl); u :• m x g + s; du :• g x dm+ m + ds; v :• m x J + 11 + vr; dv :• J x dm + m + ds; v :• k x g + r; dv :• g x dk + k + dr; z :• k x J + r; dz :• J x dk + k + dr; vvzz :• v x v + z x z; uuvv :• u x u + v x v; 1 :• ln(((g x g + bc) x vvzz)/(uuvv x (j x J + bc))); dl :• 2 x g/(g x g + bc) - 2 x J/(J x J + bc) - (2 x u x du + 2 x v x dv)/uuvv + (2 x v x dv + 2 x z x dz)/vvzz; t :• arctan(z/v)- arctan(v/u) + arctan(b/g)- arctan(b/J); if u< 0 then t :• t + 3.14159265; 1f h < hh A hl< h then t :• t + 3.14159265; dt :• b XW(J x J + bc) -1/(g x g~ bc}) -(u x dv -"""YX du)/uuvv- (z x dv- v x dz)/vvzz; ii :•- gev[jj,pp,qq] x (((a x de + da x e + b x f x dx- b x da x df) x 1 + ( a x e- da x b x f) x dl + (-a x df - f x da-da x b x de + dx X b x e) x 2 x t + (-a x f- da x b x e) x 2 x dt)/(2 x b x p) -((a x e- da x b x f) x 1 + (-a x f- da x b x e) x 2 x t) x dp/( 2 x b x tt) - (dx/hh) x ln(hl/(hh- vr))- (da x vr)/(hhkv x hl)); int :a int + 11; ia1a :• - gev[jJ,pp,qq] x ( a x e x 1/(2 x b x p) - da x f x 1/(2 x p) -a x f x t/(b xp)

- da x ex t/p + (da/hh) x ln(hl/(hh- vr)) - vr/(hhkv x hl x c)); ab~ : • abs + ia1a

em em; iiit: • int + inta; abs : • abs + absa; gm : • 0. 714489 x nu/h;

FIXT(5,1, h); SPACE(5); FLOT(6,3,int); SPACE(5); FLOT(6,3,abs); SPACE(5); FIXT(2,6, gm); NLCR; 1f bam • 1 then begin FLOP( 6, 3, int); PUNLCR em

em; ee :" re ad; if ee • 1 then begin 1f bam • 1 then begin RUNDVr; RUNCUI'; RUNovr end; NEW PAGE; goto START end; PRINTI'EXT(~ einde ~)

STELLINGEN

De bewering van Symons dat de door hem verkregen smelt Cr 5 + bevat in de vorm van tetraedrische ionen CrO ~-, dient in twijfel te worden getrokken.

1) N.Bailey en M.C.R.Symons, J.Chem.Soc. 1957 203. 2) L.L.van Reijen, Proefschrift Eindhoven 1963. 3) 1 .L.Verbeek, 9th Europeon Congress on Molecular

Spectroscopy, Kopenhagen 1965.

II

De F 19 NMR-spectra van Buslaev e.a .. bevestigen de C 4 v structuur van het ion TiQF ~-.

Yu.A.Buslaev, D.S.Dyer en R.O.Ragsdale, Inorg.Chem. 6 2208 ( 1967).

III

De conclusies van Asimov e.a. met betrekking tot de coordinatie en valentietoestand van overgangsmetaalionen in Borosilicaat-glazen, gaan aanzienlijk verder dan hun experimentele resultaten toelaten.

1) T.G.Asimov, Yu.N.Viktorova, S.A.Zelentsova en V.V.Zelentsov, Russ.J.Phys.Chem. 41 212 (1967).

IV

Er is geen reden om aan te nemen dat La 3 + in zeolieten van type X en Y, verhit op 400°C, een voorkeur zouden hebben voor plaatsen met de hoogst mogelijke zuurstof coordinatie.

I) T.I.Barry en L.A.Lay, J.Phys.Chem.Sol. 27 1821 (1966). 2) H.Bruins Slot en J .L .Verbeek, te publiceren. 3) H.Bruins Slot, Intern Rapport Lab. voor Anorganische

Chemie en Katalyse, T.H.Eindhoven.

V

De toepassing van reflectiespectroscopie bij het onderzoek van fasen­diagrammen kan leidep tot c;onclusies die afwijken van die, verkregen met behulp van meer conventionele methoden.

I) W .P.Doyle en F .Forbes, 1 .Inorg_.Nucl.Chem , 27 1271 (1965).

2) A.H.W.M.der Kinderen, Verslag van afstudeerwerk T .H .Eindhoven, 1966.

VI

De spectra in het gebied 10000-30000 cm- 1 van Co 2 + in ZnO en in Al 20 3 zijn niet met elkaar in overeenstemming wanneer in beide ge­vallen uitsluitend een tetraedrische coordinatie door zuurstof wordt aangenomen.

VII

De interpretatie van de ESR spectra van geodenteerde moleculen van Kopernitraat, door Kasai e.a. wordt niet overtuigend gesteund door hun spectrum-simulatie, en is ook anderszins niet eenduidig.

P.H . Kasai, E . B.Whipple en W.W.Weltner, J.Chem.Phys, 44 25Bl (1966),

VIII

De rol van de coordinatiechemie bij het vaststellen van de betekenis van sporenelementen in levende organismen, alsmede bij de bestrijding van vergiftigingsverschijnselen en van deficienties, kan van grote be­tekenis zijn.

J .Schubert, Scientific American 1966.

IX

Het afwijzen van kunstmatige geboortenregeling door als alternatief te stellen een verhoogde voedselproductie, houdt een ernstige onder­schatting in van het ontwikkelingspeil, nodig om zulk een productie­verhoging te realiseren.

Paus Paulus VI.

x

De huidige organisatievorm van de medische voorzieningen in Neder­land voldoet niet aan de eisen die een geordende samenleving daaraan moet stellen.

XI

Het vervagen van de verschillen tussen de programma's in de Natuur­wetenschappen van Technische Hogescholen en Universi.teiten leidt tot de wenselijkheid het bestaande verschil in naamgeving op te heffen.