An interpretation of the U.V. and E.S.R. spectra of some chlorocomplexes of tervalent titanium

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AN INTERPRETATION OF THE U.V. AND E.S.R.

SPECTRA OF SOME CHLOROCOMPLEXES

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AN INTERPRETATION OF THE U.V. AND E.S.R. SPECTRA OF

SOME CHLOROCOMPLEXES OF TERVALENT TITANIUM.

P R O E F S C H R I F T

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECH-NISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H.R. VAN NAUTA LEMKE, HOOGLERAAR IN DE AFDELING DER ELEKTRO-TECHNIEK, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN TE VERDEDI-GEN OP WOENSDAG 7 FEBRUARI 1973 TE 14.00UUR.

DOOR

/ 9

^P

-^o^^

HERMAN KAREL OSTENDORF

doctorandus in de scheikunde geboren te Semarang

1973

Drui< P.IVI. van Hooren, Heerlen.

BIBLIOTHEEK T U Delft P 1888 3088

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Dit proefschrift is goedgekeurd door de promotor:

Prof.dr.ir. L.L. van Reijen

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Aan Gé

Aan C.R.M.

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Dit proefschrift is tot stand gekomen dankzij de hulp en medewerking van velen. De hoofddirectie van DSM te Heerlen en de leiding van het Centraal Laboratorium ben ik zeer erkentelijk voor de gelegenheid die mij werd geboden dit werk voor te bereiden en het in deze vorm te kunnen publiceren. De heren H.J.M. Bartelink, J.P.C, van Heel, K.G.H. Ramaekers en H.J.M. Slangen ben ik veel dank verschuldigd voor het met zorg en aandacht uitvoeren van de experimenten die aan dit werk ten grondslag liggen. Ik ben ook zeer dankbaar voor de vele discussies met chefs en collega's; steeds weer waren zij bereid kritisch te luisteren en mee te denken. In het bijzonder moet ik hier dr. E. Konijnenberg en de andere medewerkers van de werk-groep complexchemie noemen.

Ir. J.J.M. Potters heeft de overlapintegralen van appendix C uitgerekend, waarvoor ik hem zeer erkentelijk ben. Een grote steun heb ik ook aan mijn vrouw gehad die het Engels op vele plaatsen verbeterd heeft. De typografische verzorging was bij de heren H.E. Horst en J.C.M. Steuns in goede handen; de illustraties werden ver-zorgd door de heer G. Schuier en het typewerk werd op bewonderenswaardige wijze uitgevoerd door mevr. G.A.M. Nelissen-Stevens en mevr. A.J.M. Steins-Rikken. De drukkerij P.M. van Hooren tenslotte heeft op voortvarende manier de f o t o off-set verzorgd.

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In homogeneous catalysis complexes of a metal ion w i t h more than one type o f ligand usually play an important role. If the stoichiometry of such a complex is known there are often t w o or more possibilities for the configuration; e.g. a complex ML K w i t h t w o ligands of type L and four ligands of type K in an octahedral structure may occur in either of the t w o stereoisomeric configu-rations: the ligands L in c/s-position or in frans-position. In this thesis this problem is investigated by studying the configurations of t w o series of complexes of

tervalent titanium in detail. The t w o series of complexes are [TiCI (CH CN) J ^ ' " and [ T i C I (CH^OH)^ j ' ^ w i t h x = 2, 3, 4 and y = 1, 2, 3, 4. It will be shown** that the configuration of the ligands around the titanium atom can be deduced from the U.V. and E.S.R. spectra of these compounds.

The tervalent titanium ion has one electron outside closed shells in a 3 d-orbital. In spherically symmetric surroundings the ground state than is ^ D . When the ion is p u t in an environment of lower symmetry the orbital degeneracy of the ^ D level is destroyed; if the symmetry is low enough (e.g. orthorhombic) all orbital degeneracy is lost and there are five different d-orbitals w i t h different energies. This is often the case when complexes are made of the Ti^"*" ion w i t h more than one type of ligand in the first co-ordination sphere. The energy differences between the different orbitals will be governed by the symmetry of the configuration, so reversely the symmetry of the configuration can be deduced if the energy differences are k n o w n .

The energy differences between the different d-orbitals have been studied spectroscopically in the visible and near U.V. region and also by Electron Spin Resonance spectroscopy. In the U.V. spectrum of a complex of tervalent titanium d-d-transitions are f o u n d in the range 10-25 k K * ) . These transitions correspond to the transition f r o m the ground state to the higher sublevels of the ^D term. The components o f the g-tensor of the E.S.R. spectrum are sensitive t o the energy differences between the lower sublevels, so the relative positions of the complete set o f sublevels of the ^ D term can be deduced from the combination of U.V. spectroscopy and E.S.R. measurements. Therefore a study of the U.V. spectra and the E.S.R. data must also give an idea of the configuration of the ligands around the titanium a t o m .

The spectroscopical properties of complexes of titanium and other transition metals have been interpreted in terms of the crystal field theory since

1929' . Later on Van V l e c k ^ ' has studied the magnetic properties of the transition metal ions and has explained many of their properties in terms of the crystal field theory. In the early 1950's many E.S.R. spectra of salts of transition metals have been measured and their g-values and fine structure were explained w i t h this t h e o r y ^ ' ' ' ' . The titanium ion in a trigonal axial environment has been

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studied in detail by Gladney and Swalen^'. In this and similar w o r k on titanium complexes the complex studied had a rather high symmetry: cubic or axial with only small deformations of lower symmetry. The titanium complexes to be studied in this thesis will have relatively low symmetry (mostly C ) though some of the properties of an octahedral environment can still be recognized.

Another approach to the interpretation of the spectroscopical and magnetic properties of transition metal complexes, used with considerable success, is the molecular orbital method. Within terms of this approximation the energy levels of the complex are computed by applying the variation principle to wave-functions which are linear combinations of atomic orbitals. Ab i n i t i o Hartree-Fock calculations have been made of some complexes w i t h rather high symmetry'' •'''* • ' ' These calculations are very complicated and only possible with a large computer. Moreover, the configuration of the complex must be known exactly from the beginning. Semi-empirical calculations in the formalism of the M.O. theory have been tried too, using the extended Huckel theory or the CNDO m e t h o d (complete neglect of differential o v e r l a p ) ' " ' ' ' • ' ^ ' ' ^ ' . The results obtained with these semi-empirical methods depend strongly on the starting values for the diagonal matrix elements, generally estimated from spectroscopic data of the free transition metal ion. Also the semi-empirical methods require lengthy numerical calculations. In this thesis a M.O. approach is developed which gives the results in analytical expressions. The parameters involved are further defined in such a way, that in principle, they can be transferred from one complex t o another containing the same ligands. This analytical approach is thought useful, because an open and direct check is possible of the effect on the final results of the various assumptions involved.

The interpretation of the data usually proceeds in two steps. In the first step the relative positions of the sublevels of the ^ D term of the titanium ion are determined from the g-values of the E.S.R. spectrum and the d-d-transitions in the U.V. spectrum. This step does not depend on the method to describe the splitting of the ^ D term. In a second step parameters are sought w h i c h govern this splitting. For the second step the crystal field approximation or the M.O. approxi-mation can be used. In this work both approaches are used to determine the configuration of the chlorocomplexes of t i t a n i u m - I l l with acetonitrile and methanol and the results and parameters are compared and discussed.

The physical picture used in the crystal field theory is very simple. The ligands are treated as point charges generating an electric field that has the symmetry properties of the configuration of the ligands in the complex studied. This field is treated as a perturbation on the m o t i o n s of the d-eiectron of the transition metal ion. The wavefunctions and energies in the unperturbed situation are those of the ion in the gas phase and are k n o w n from the atomic spectroscopy.

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In chapter 2 of this thesis the crystal field problem will be treated for some specific configurations to be used in the later chapters. We will treat the effective charge and the effective distance between the titanium atom and the ligand as parameters to be determined by the experimental data. Moreover, we will assume that these effective charges and effective distances remain the same in complexes of different compositions of the same ligands. So, for instance, we will take the effective charge and the effective distance of the Cl-ligand to be the same in the complex [TiCI^ ( C H 3 C N ) J ' ' ' C r as in the complex [TiCI^ (CH^CN) J .

In the M.O. approximation the bonding between the metal atom and the ligands is treated in about the same way as the chemical bonding between unlike atoms in a simple covalent bond. The results of the ab initio calculations show that an orbital formed from a linear combination of the d-orbitals of the transition metal and appropriate ligand orbitals is quite a good approximation to the best orbital obtained by a full self consistent calculation. Following this idea Schaffer has described a perturbation method for the treatment of the weak covalent bonding as formed in the coordination compounds' * ' . This paper describes the angular overlap model already used by J0rgensen and his co-workers' ' ' ^ ° ' ^ ' , in the light of group theory.

In chapter 3 of this thesis this method will be explained and developed to give useful analytical expressions for the energies of the orbitals w i t h mainly d-character of some complexes t o be used in the later chapters of this work. In this approximation only the 3 d-orbitals of the metal and the appropriate p-orbitals of the ligands are used as basis. The 4 s- and 4 p-orbitals of the metal are neglected throughout. By using a minimal basis set of functions the number of parameters to be determined by the experimental data is kept as low as possible, and a situation is avoided where there are many more unknown quantities than experimental data. The overlap integrals are treated as parameters to be determined by the experimental data. The unperturbed energy of the titanium atom is treated as a free parameter too. In the complexes t w o types of bonding are of interest: a-bonds and ff-bonds. With each type of bonding an overlap integral is associated which has been assumed to be only dependent on the bond formed and not on the composition of the complex; e.g. the o-overlap parameter and the 7r-overlap parameter of the Ti-CI bond in the complex cation of

[TiCI (CH CN) ] C\' have been assumed to have the same values as the overlap parameters in the complexes [TiCI (CH^CN) ] and [TiCI^ (CH OH) ]'*'C\'.

In chapter 4 the order of the energy levels in complexes of different composition is discussed. T o get some idea of the order of the energy levels in the complexes the relative magnitudes of the parameters introduced in the chapters 2 and 3 are estimated from some general principles (among these the spectrochemical series). Also we give expressions for the components of the g-tensor in chapter 4 to be used in the chapters 5 and 6 of this work. In the last section of chapter 4 an analysis is made of the energy parameters used and in particular suggestions are given for the values of the energies to be used in the diagonal elements of the M.O. method.

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In chapter 5 the formalism introduced in the chapters 2-4 is applied to the spectroscopical data of the chlorocomplexes of titanium III w i t h acetonitrile. The experimental E.S.R. data of the complexes [TiCI (CH CN) ]"*" and

[TiCI (CH CN) ] were published in a paper by the author 22); ^he spectral data of these complexes and the E.S.R. data of the complex [TiCI (CH CN) 1" were taken from the literature^^' ^"^ ^ ^ ' ^^K

In chapter 6 the formalism of the chapters 2-4 is applied to the series of chlorocomplexes of titanium-Ill w i t h methanol. The results of measurements of the U.V. spectra and the components of the g-tensors were published by Brubaker and co-workers^'''.

Finally in chapter 7 the results obtained in the chapters 5 and 6 are discussed critically and compared with each other. There also some examples are given of the use of the parameters obtained to interprete E.S.R. and U.V. spectra of complexes not yet fully understood.

In the appendices A and B some mathematical problems and their solutions as used in this work are explained in detail. In appendix C numerical data are given for the overlap integral of a Ti 3 d-orbital w i t h ligand p-orbitals as a function of the distance.

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CHAPTER 2. THE CRYSTAL FIELD THEORY

2.1. introduction

In the crystal field approach the ligands are treated as point charges which generate an electric f i e l d ; this electric field has the symmetry properties of the configuration as assumed for the complex. This electric field acts as a perturbation on the movements of the d-electrons of the transition metal which is the central ion of the complex. The wavefunctions of the d-level of the central ion - as found w i t h the theory of atomic structure - are the basis functions for a perturbation calculation.

The potential of the electric field can be expanded into a series of spherical harmonics centered on the metal atom :

V(r,Ö,^)= 2 2 r S . a Z . J Ö ' ^ ' <2-1)

k=o a ^"- ^"•

with

- 4 7 r e - b i ^ 2 k a < ® i ' * i )

i

r, 0 and lO are the coordinates of the electron and R , 0 . and fl> are the coordinates

I I I

of the i-th ligand which has an effective charge of - b e , when - e is the charge of an electron. Z, are linear combinations of the spherical harmonics:

4 m 4 ^ 2 [ Y e . ^ " ^ e - m ] '2-3)

4m = iV2[Ye.^- Y^.^]

The spherical harmonics Yn are defined in the usual way^'^ ; Yn is the complex conjugate of Yp

Now, if a configuration of the ligands is assumed, 0 . and 't>. are k n o w n ; b and R. are treated as parameters to be determined by the experimental data. They can be interpreted as the effective charge of the ligand and the distance between the metal atom and the center of gravity of the negative charge on the ligand. With these values for 0 . and <I>| the quantities 7. can be expressed in b and R..

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In the expression 2-1 there are many terms that give no contribution to the matrix elements of this operator w i t h the d-functions:

| M z ^ > = R n(r)Y, (0,^) = R „ ü ( r ) Z , , . ( Ö , ^ ) | M x ^ - y ^ > = R ^ g ( r ) — [ Y ^ ^ ( 9 , . ^ ) + Y ^ _ ^ ( ö , ^ ) ] = R^j(r)Z^^ (0,^)

V2

I M x y > = R n e ' ' ' ' : 7 f'^2 2 < ^ ' ' ^ ' " ' ^ 2 - 2 < ^ ' ^ ' l =^J'K2^^-^^ <2-4) i M y z > = R ^ g ( r ) y - [ Y ^ i ( 9 , v 5 ) + Y ^ ^ ( ö , ^ ) ] = R^^(r)Z^ ^ (0,^) | M z x > = R „ e ( r ) y - [ Y ^ ^ ( 9 , ^ ) - Y ^ ^ ( 9 , ^ ) ] = R^^(r)Z^ ^ (0,^) All matrix elements have the f o r m :

271 TT

ƒ R n V ^ ' ^ ' ^ n K ^ ^ ' d r ƒ ƒ Z e ^ ( 0 , , . ) - ^ Z g , ^ , s i n O d 0 d ^ u 0 0 ••

Now the expansion of \/(r,6,^) is in spherical harmonics, so the angular integral of 2-5 is that of the product of three spherical harmonics. However, it is well known that the integral of a product of three spherical harmonics Yp , Yp, , and Yf,,, ,, is zero, unless m = m ' -t- m " and S, C' and C" are such t h a t they can

v; m

form a triangle. In the d-functions 2 = 2, so only the terms with k = 0, 2 and 4 in the expansion 2-1 are needed. Of those terms w i t h k = 0, 2 and 4 some will be zero because of the symmetry of the configuration.

oo

The integral J R * p ( r ) r ' ^ R n(r)r^dr is abbreviated r in the following sections 0

The products Z . Z for all possible combinations of m and n are

2m 2n "^ ,

given by G r i f f i t h in terms of Z , Z and Z ' . With the help of this table

^ ' 4p ' 2q 00 "^

and the orthogonality relations of the Z-functions the angular integrals of the type

27r TT

ƒ ƒ Z , ^ Z ^ „ Z ^ „ s i n 0 d 0 d ^

0 0 _ _

can be evaluated readily. With the abbreviations r and r"* all matrix elements w i l l be a sum of terms in the quantities:

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become sums of terms in the other two quantities. T o simplify the formulae the parameters X^ and X ' are introduced:

b..e^r^

and X '

R^ R'

X X

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These t w o quantities will be treated as the basic parameters in the ligand field treatment in stead of the quantities b and R . If X^ and X ' are known b and R

X X X X

can easily be calculated.

2.2. Application to octahedral stereoisomers with different symmetries

In the remaining sections of this chapter three configurations will be treated in detail. These configurations have been chosen such that all the needed configurations for the chlorocomplex of titanium w i t h methanol and acetonitrile can be derived by simple substitutions.

2.2.1. The complex MGHL^

The complex M G H L w i t h G and H situated in trans-positions can be depicted as in fig. 2 - 1 .

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G H L3 L4 R = RQ R L 0G = 0 180° 90° 90° 90° 90° 1 ' G = -180-1/3 180+ V' -V5

The symmetry of the ligands is that of the point group C Grouptheory

gives that the d-orbitals |Mz^ > , |Mx^ • y^ > , IMxy > , |Mzx > and |Myz > transform according t o the irreducible representations respectively A , A , A , B and B .

1 2

The coordinates of the ligands are:

effective charge: - b ^ e - b ^ e -b|ê -b, e -bê -bjê

In the C symmetry the crystal field operator takes the f o r m :

V(r,0,^)=V47:[r„„Z„„(0,VP) + r^^Z^„(0,^)r^+r;^Z^^(0,^)r^ +

+ I\oZ4„(Ö.^)r' +i;%Z:,(0,^)r^ + r : , Z ^ , ( 0 , ^ ) r ^ + (2-6) + higher terms which can be neglected].

With the definition of the parameters X^ and X"" given in the preceding section we can compute the coefficients in equation 2-6 w i t h equation 2-2.

b_e^ b^e^ 4 b, e^

r =-^— + " ^

00 R^ R^ R^ i ^ r ^ „ = ^ [ G ^ + H - ^ - 2 L 3 ] (2-7) '"^40 " f i ^ 2 G ' -^ 2 H-W3 L^]

^ r : , ^ - ' - ^ L ' c o s 2 ,

^ r L = ^ L ^ ' 2 c o s ^ 2 ^ - 1 ) .

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The matrix elements of the d-orbitals 2-4 w i t h the operator 2-6 are: < Mz^ | V | M z ' > = r^^ -^ ^ (2 G ' -^ 2 H^ - 4 L ' ) -H - | - 7 ! 7 ( 1 2 G ' - M 2 H ' + 18 L ' ) 42 < M x ^ - y ^ l V I M x ' - y ' > = r ^ ^ - i - ^ ( - 2 G ^ - 2 H ^ -^-4L')-(-- l • -^-4L')-(-- ^ [2 G ' -^-4L')-(--I-^-4L')-(--2 H^ -^-4L')-(-- (32 -^-4L')-(-- 70 cos^ 2 V ' ) L 5 ] < M z ^ i V | M x ^ - y ^ > = - ( ^ L ^ + 1 ^ L 5 ) C O S 2 ^ < M x y | V | M x y > = T ^ ^ - H y ( - 2 G^ - 2 H^ -I-4 L^)-i- (2-8) -I- ^ [ 2 G^ -H 2 H^ -F (38 - 70 cos'' 2 i/;) L ' ]

< M z x | V | M z x > = r ^ ^ - i - i [ G ' - F H ' - 2 ( 1 - 3 c o s 2 . / ) ) L ^ ] - H

- l - T ^ [ - 8 G ' - 8 H ' - ^ ( 1 2 - I - 2 0 C O S 2 S C ) L ' ]

< M y z | V | M y z > = T^p + y [ G ^ iH^ 2(1 i3 cos 2 ^ ) L ' ] i

-- H r ^ [ -- 8 G ^ -- 8 H 5 -- ( 1 2 -- 2 0 C O S 2 I / ; ) L ' ]

All other matrix elements are zero. Now v? is assumed t o be near 4 5 ° so the matrix element < Mz^ |V|Mx^ — y^ > is small and can be neglected. Then the other expressions of 2-8 give the perturbed energies of the different d-orbitals of the

metal in the complex. As we are only interested in the differences between the sublevels of the d-level P is of no interest.

The formulae f o r frans complexes of the type MK L can be derived from the formulae 2-8 by putting G = H = K and formulae f o r complexes of the type M K L can be found by putting G = K and H = L.

2.2.2. The complex M G H L 2 K 2

The configuration discussed is that with G and H in trans position and the other ligands also in trans position as sketched in fig. 2-2. The symmetry of that configuration is C .

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4H

Fig. 2-2. The complex MGHL^K^ with C^^ configuration

|Mz^ > , |Mx^ - y^ > , IMxy > , |Mzx > and |Myz > respectively belong to the irreducible representations A , A B. and B,. The coordinates of the

effective charge: -b_e -bê -bê -bê -bê In the C ^ symmetry the crystal field operator again takes the form given by eq. 2-6. With equal. 2-2 the coefficients are now found to be:

^^.e' b^e^ 2 b, e^ 2 b ^ r '

r.

ligands are: G R = R^ H RH L. «L •-3 \ K, «K K, R.,

%

= 0 180 90 90 90 90 * G =

-0 180 90 - 9 0 0 0 _R,

b,e^ 2b^e^ 2b^e^ + -^— — r -I-R H R. R K rM^ = ^ ( G ' + H ^ - L - ' - K ^ ) 2 0 5

; ^ r ! = ^ ( L ^ - K ^ )

22 i) r" I \ „ = : ^ (4 G^ -^ 4 H' -I-3 L' -H 3 K=)

(2-9)

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Now the matrix element between both d-orbitals belonging to the irre-ducible representation A is:

_ 2 ^ ( L 3 - K 3 ) - ^ ( L ' - K = ) (2-10)

which is not negligible. So we now get two metal orbitals of A symmetry which are linear combinations of |Mz^ > and |Mx^ — y^ > :

IMAj 1 > = pIMz^ > + qlMx^ - y^ > |MA| 2 > = - qlMz^ > + p|Mx^ - y^ > with p^-I-q^ = 1 (2-11) such that < M A J | V | M A j 2 > = 0 = p q { y ( - 4 G ^ _ 4 H ' -H 4 L^ -i-4 K^ )-i-- i )-i-- ^ ( )-i-- 1 0 G ' )-i-- 1 0 H ' )-i--H I O L ' )-i-- H O K ' ) } )-i--H (2)-i--12) + (p^ - q ^ ) { - ^ (L^ - K^) - ^ ( L ' - K ' ) }

The diagonal matrix elements are:

< M A J | V | M A J > = r^phP ~ ^ ( 2 G ' F 2 H ' 2 L ' 2 K 3 )

-_ i P 3 ^ / 3 . ( L 3 -_ K 3 ) + - L [ ( 1 2 p ^ + 2 q ^ ) G = +

+ (12 p' -I- 2 q ^ ) H ' -I- (9p^ -I- 19q'' - 10v/3pq)L' +

+ (9p^ -I- 19 q^ + 10V3pq)K']

(2-13) < M A , 2 | V | M A , 2 > = r ^ ^ - H ^ ^ P (2G^ + 2 H^ - 2 L^ - 2 K^)-H

+ ^ ^ 3 V 3 . ( L ' - K ^ ) + 4 ^ [ ( 2 p ^ + 12q^)G' +

+ (2 p' + 12 q ' ) H ' -I- (19 p2 -I-9 q ' -I- 10^3 pq)L'

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< M x y l V I M x y > = T^^ y ( - 2 G^ - 2 H^ 2 L^ -H 2 K^) -i-- i -i-- T y ( 2 G ' -i--F 2 H ' -i-- 16 L ' -i-- 16 K ' ) < M z x | V | M z x > = r^jp - ( - y ( G ^ -1- H^ -h2 L ' - 4 K^)

-I-- H T y ( -I-- 8 G 5 -I-- 8 H ' -I-- 16 L^ -I-- I -I-- 4 K ' ) < Myz|V|Myz > = T^^ -H - (G^ -^ H^ - 4 L^ 2 K^)

-i-- i -i-- -i-- y ( -i-- 8 G 5 -i-- S H ^ -i--t-i--4 L ' -i-- 16 K ' )

In the case that G = L and H = K equations 2-11 and 2-12 are satisfied by 1

p = q = —p- and the matrix elements of the A representation in 2-13 become: < M A 1|V|MA 1 > = r - ? ^ ( L - ^ - K^) -I-1 1 00 7 - ^ ^ [ ( 2 1 - 5 V / 3 ) L ' - H ( 2 1 - H 5 V ' 3 ) K ' 1 (2-13) < M A 2|V|MA 2 > = r - I - ^ ^ ( L ^ - K^)-I-(2-14) - K ^ [ ( 2 1 - K 5 V ' 3 ) L 5 - H ( 2 1 - 5 V 3 ) K 5 ]

If L and K are identical then p = 1 and q = 0 is a solution of equations 2-11 and 2-12 and we get formulae that are the same as those of equation 2-8 with 1/5 = 4 5 ° .

2.2.3. The complex MG H_K with C symmetry

We discuss the configuration w i t h the pair K in trans position and the pairs G and H in cis positions. The symmetry of that configuration again is C . The configuration is sketched in fig. 2-3.

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Fig. 2-3. The complex MG H K in a configuration w i t h C symmetry |Mz^ > , |Mx^ - y^ > , IMxy > , |Mzx > and |Myz > respectively belong to the irreducible representations A , A , A , B and B . The coordinates and

1 ' 1 2 ' 1 2 effective charges of the ligands are:

K, R = R^ ^ 2 R Q 1 ÖK = 90 90 f

180-- ^

- ^

* K = 90 - 9 0 0 180 0 180 2 L

Again the crystal field operator has the form of eq. 2-6. The coefficients now are:

**bê^ bê* hê^**

r = 2 - ^ + 2 - # - + 2 ^

effective charge: -b,,e - b ^ e - b ^ e -b|^e - b , e K H

V 5

r' r = - V ^ [ - K ^ -I- ( 3 c o s V - 1)G-'-i-(3cos^i/^ - DH"* 2 0 5

r^r=

22

V15

- K^ - t - s i n ^ G ^ +sin^i//H^

r"

r

_{40 12}_1_

A

6

/:

44 12 (2-15) 3 K' -I- ( 3 5 c o s ' ' v ' - 3 0 c o s V + 3 ) G ' + (35 cos" i// - 30 cos^;//-^ 3 ) H ' ]

'" r^2 = ' ^ f ' < ' - F s i n V ( 7 c o s ^ V ' - 1)G^ -Fsin^i//(7cos^i// - 1 ) H ' I

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The oft-diagonal matrix element between the orbitals with A symmetry is: < M z ^ | V | M x ^ - y ' > = - ' ^ [ - K ^ - ( - G ^ s i n ' ^ + H^sin^

i//]-i-+ ^ [ K ' - H s i n V ( 7 c o s ^ i / ; - 1 ) G ' -I- (2-16) -i-sin^i//{7cos^;// - 1 ) H ' ]

This does not vanish when ^ and \j/ approach 4 5 ° . That is why we must make linear combinations of |Mz^ > and |Mx^ — y'^ > :

IMAj 1 > = pIMz^ > -I- qlMx^ - y^ > | M A j 2 > = - q | M z ^ > - i - p | M x ^ - y^ >

w i t h p^ -(• q^ = 1 (2-17) p and q must satisfy equation 2-17 and the f o l l o w i n g equation:

< M A J | V | M A j 2 > = 0 = y [ 2 p q - i - ( p ^ - q ^ ) V 3 ] K ^ - y [ 2 p q ( 1 - cos 2 v^) + H(p' q ' ) \ / 3 ( 1 Icos 2 ./))]G^ y [ 2 p q ( 1 c o s 2 i / / ) i -+ {p^ - q ' ) ^ / 3 ( 1 -+ c o s 2 ; ^ ) ] H ^ -+ ^ [ 2 p q - H ( p 2 - q ^ ) V 3 ] K ' -+ + : ^ [ 2 p q ( 1 - - c o s 2 i , ö - - c o s ^ 2 ^ ) - i - (2-18) -)- (p^ - q ^ ) V 3 ( 1 - ^ | - c o s 2 v 5 - | - c o s ^ 2 . , c ) ] G ' -i-b 5 + : ^ [ 2 p q ( 1 - • g - c o s 2 i// - - c o s ^ 2 ^ ^ ) -t--I- (p^ - q ^ ) V 3 ( 1 + | - c o s 2 i / ' - ^ c o s ' 2 i / / ) ] H ' 5 5

When i/j and i// are near 4 5 ° cos 2 ^ and cos 2 i// are negligible and eq. 2-18 becomes:

[ 2 p q + ( p ^ - q ' ) V 3 ] [ | K 3 - y G ^ - y H 3 +

^ 9R 9R (2-19)

+ ^ K = + - ^ G 5 - H - ^ H 5 ] = 0

42 168 168

From 2-19 and 2-17 we have p = % and q = V-isJZ independent of the crystal field parameters. These values are assumed to hold good approximately for angles ^ and i// near 4 5 ° .

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The diagonal matrix elements giving the first order perturbations on the energy then are: < M A j 1 | V | M A | 1 > = r ^ ^ - i - y ( 4 K ' - 2 G 3 - 2 H ^ ) - ( - T y ( 2 4 K=-H9G^ - I - 9 H ' ) < M A 2IVIMA 2 > = r F ( 4 K ^ I 2 G ^ | 2 H ^ ) | -l 1 0 0 7 - H - L [ 4 K 5 _ ( 1 6 - 3 5 cos^ 2^5) G ' - ( 1 6 - 3 5 cos^ 2 1//) H^]

< M x y | V | M x y > = r ^ ^ - F y [2 K^ - (1 -1-3 cos 2 sc) G^ - (1 + 3 cos 2 i//) H^

]-l-+ 4 y [ - 1 6 K ' - i - ( - 6 - H 10 cos 2 ^ ) G ' - ( - ( - 6 - i - 1 0 c o s 2 t / / ) H=] (2-20)

< M z x | V | M z x > = rgg-i- y [ - 4 K ^ - I - 2 G ' -1-2 H^]

-t-+ 1-[4K^ -t-+ ( 1 9 - 3 5 cos^ 2.^)) G ' -1- ( 1 9 - 3 5 c o s 2 2 i//)H^]

< M y z | V | M y z > = r ^ ^ - i - y [ 2 K ^ - ( 1 - 3 cos 2 .p) G ^ - ( 1 - 3 cos 2 1//) H^]-f

-I- ^ [ - 1 6 K^-^ ( - 6 - 10cos2</)) G^-i- ( - 6 - 1 0 C O S 2 i//) H ' 1

As a further approximation 1// can be put equal to 90 — 1/3. This means that the angle G—M—H can be assumed t o remain 90*^, when the octaeder deforms. In that case the equations 2-20 will become:

< M A | 1|V|MA_1 > = r ^ ( , + ^ ( 4 K ^ - 2 G ^ - 2 H 3 ) - i - T y ( 2 4 K ^ - i - 9 H ' - i - 9 G ' )

< M A | 2|V|MA| 2 > = r^^ -1- y ( - 4 K^ -I- 2 G^ -I- 2 H^)

-)-+ ^ [ 4 K= - ( 1 6 - 3 5 c o s ^ 2 ^ ) G^ - ( 1 6 - 3 5 cos^ 2i^) H^]

< M x y | V | M x y > = r ^ g - i - y [2 K^ - (1 -^ 3 cos 2 ^s) G ' - (1 - 3 cos 2 ./?) H^ ]-^

- i - ^ [ - 1 6 K ^ - i - ( - 6 - i - 10cos2v5) G ^ - ( 6 - i - 10 cos 2 ip) H^] (2-21)

<Mzx|V|Mzx>= r^^-i- y ( - 4 K^ -t-2Gn2 H^)

-I-- t -I-- T ^ [ 4 K^ -I--I-I-- ( 1 9 -I-- 3 5 c o s ^ 2i/)) G ' -I--I-I-- ( 1 9 -I-- 3 5 c o s ^ 2ip) H ^

< M y z | V | M y z > = r ^ ^ i y [2 K^ ( 1 3 cos 2 i/)) G^ (1 H 3 cos 2 ip) H ' ] 1 -- ^ -- y [ -- 1 6 K ^ -- ( -- i -- 6 -- i -- 10cos2i/)) G^--H ( -- 6 -- 1 -- 10cos2</j) H ^ ] .

These equations will be used in the chapters 4, 5 and 6 to interpret the spectra of the [TiCI (CH CN) ] " a n d [TiCI (CH OH) ] " complexes.

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CHAPTER 3. THE MOLECULAR ORBITAL M E T H O D

3.1. Introduction

In the M.O. approach the orbitals forming bonds between the metal atom and the ligands are studied. The molecular orbitals are usually approximated by a linear combination of atomic orbitals. For an accurate description many atomic orbitals are needed in the basis set. However, in that case many parameters appear in the calculation. From the U.V. spectrum and E.S.R. measurements usually only five quantities are known (two d-d-transitions in the U.V. spectrum and the three components of the g-tensor), so the number of parameters which can be determined from the experiment is limited. In order to restrict the number of parameters as far as possible, w i t h o u t too much loss of precision, only the d-orbitals of the central atom of the complex and the p-orbitals (in some cases s-p-hybrids) of the ligands will be considered. For the U.V. and the E.S.R. spectra o n l y the orbitals w i t h predominantly metal d-character are of interest.

The filled orbitals of the ligands generally have an unperturbed energy which is lower than that of the d-level of the central atom, while the energy of empty orbitals is higher. Therefore a metal d-orbital binding w i t h a filled orbital of a ligand will become higher in energy and a d-orbital binding w i t h an empty orbital will become lower in energy. The effect of the binding w i t h the ligands in a given configuration will be a splitting of the d-level into a number of sublevels analogous to the result of the electric field of the ligands in the crystal field theory. The approximations introduced in this chapter are of about the same order of magnitude as those in the preceding one.

In section 3.2 the approximations will be discussed in more detail and in section 3.3 the energies of the same configurations treated in section 2.2 will be calculated.

3.2. The bonds between the central atom and the ligands

As mentioned before only the bonds formed between the d-orbitals of the central atom and orbitals w i t h the symmetry properties of p-orbitals of the ligands are con-sidered. In figure 3-1 the positions of these orbitals are drawn relative to each other.

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Three bonds can be f o r m e d : one a-bond and t w o rr-bonds perpendicular to each other. The energy levels and wave functions are f o u n d in the usual way by solving the secular determinants, which have the same form for each of the three bonds:

H L M - S L M U H ^ ^ - U

0

where H . , . , , H, , and H , , , = H, . , are the matrix elements of the energy operator and S|y| |_ = S. „ is the overlap integral. Hj^jy. and H. . are the energies of an electron in the d-orbital of the metal atom {^.j.) and of an electron in the p-orbital of the ligands (E . for the o-bonding orbitals and E for the 7r-bonding orbitals of the ligand L). Usually the negative value of the ionization potential is taken for these energies. The determinant equation can be solved straight forward. Because S and H are small, terms in S"*, HS"^ and H^S^ can be neglected; in U this app = E M + roximation ( H M L -E M

s

-the ^.LE E L solution M ) ^ is:

Now, when other interactions are neglected, we have four parameters in this description: H|y|_ and S^^^ for the a-bond and H and S „ . for the t w o TT-bonds, which are taken to be equivalent. H , . , , however, can be approximated

, , , , , M L ' ^ ^ by the well known relation • • ' ' ' ' :

H^ =S,, . F ' ^ M M J A L ,3.1,

M L M L 2 F is a numerical factor usually c. 2.

Only t w o overlap integrals for each type of ligand remain to be determined from the spectroscopical data, when all overlaps of ligand orbitals between each other are neglected. Taking f o r F the value 2 the solution of the determinant equation becomes:

M ^ L

For the orbitals concerned, values for the ionization potentials are given by Hinzeand J a f f é " ' .

The treatment of a complex is now straightforward, if the configuration of the ligands is given. When more than one configuration is possible the ligand field splitting of the d-orbitals can be calculated for each configuration and the results of the calculations can be compared w i t h experimental data.

The procedure of the calculation is: w i t h the help of group theory the ligand orbitals are formed into group functions transforming under symmetry operations in the same way as irreducible representations; after that the overlap

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perturbation method for each irreducible representation. In appendix A the overlaps of the d-orbitals and a ligand in a general direction are discussed; in appendix B a general solution of the energy and overlap matrices is given.

In the following section some configurations are treated in detail.

3.3. Application of the theory to some specific configurations

3.3.1. The complex MGHL in a configuration with C symmetry

A complex of the composition M G H L , where G and H are two ligands equivalent or not, in trans position while L are four ligands of the same type situated in the corners of a rectangle in the equatorial plane.

In fig. 3-2 this situation is drawn; also the choice of the coordinates on the ligands and on the metal atom is depicted; the symmetry is C ; the Z-axis is a two-fold symmetry axis and the X Z and Y Z planes are mirror planes.

Fig. 3-2. The choice of coordinates in a complex M G H L with C symmetry The group orbitals of the irreducible representations of the pointgroup

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That there are non-bonding orbitals can be shown: e.g. j L A 3 > is a linear combination of | L x 3 > , | L x 4 > , |Lx5 > and | L x 6 > ; all these orbitals are antisymmetrical w i t h respect t o the X Y plane; |Mxy > , however, is situated in this plane and is symmetrical w i t h respect t o this plane, so the overlap between these orbitals must be zero.

Table 3-1. The group orbitals of the complex M G H L in a configuration w i t h C j ^ symmetry

Irred. metal repr. orbital A | |Mz^ > I M X —y'

group orbital of the ligancls

>

A | l V l x v > B JMzx > | M y z > LA LA LA LA LA 1 > = | G 2 l > 2 > = | H z 2 > 3 > = - ^ [ | L Z 3 > - ( - | L Z 4 > - I - | L Z 5 > - H | L Z 6 > ] 4 > = y [ | L v 3 > - | L y 4 > - i - | L v 5 > - | L v 6 > l 5 > = y [ | L x 3 > - i - | L x 4 > - i - | L x 5 > - i - | L x 6 > l ( n ) 1 > = JGxl > 2 > = | H x 2 > energy of the group orbital -TIL "77L L A ^ I > = y [ | L z 3 > - | L Z 4 > - I - | L Z 5 > - | L Z 6 > 1 E ^ ^ L A 2 2 > = y [ | L y 3 > - ( - | L V 4 > - I - | L y 5 > - ^ | L y 6 > l E^|_ L A ^ 3 > = y [ | L x 3 > - | L X 4 > + | L X 5 > - | L X 6 > 1 (n) E^^ 'ïïG '7TH 3 > = y l | L x 3 > - | L X 4 > - | L X 5 > - I - | L X 6 > 1 E ^ ^ 4 > = y [ | L z 3 > - | L z 4 > - | L Z 5 > - f | L Z 6 > ] (n) E^^ 5 > = y [ | L y 3 > + | L y 4 > - | L y 5 > - | L y 6 > l ( n ) E^^ L B ^ 1 > ^ - | G y 1 > E ^ ^ L B ^ 2 > = ^ | H y 2 > E ^ ^ L B 2 3 > = y [ | L x 3 > - i - | L x 4 > - | L X 5 > - | L X 6 > 1 E^^^ L B 2 4 > = y [ | L z 3 > - ( - | L Z 4 > - | L Z 5 > - | L Z 6 > ] (nl E L B 2 5 > = ^ [ | L y 3 > - | L y 4 > - | L y 5 > - ( - | L y 6 > l ( n ) E^^^

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express the overlap between |Mz > and the grouporbitals in the same interatomic overlap integrals as the overlap between |Mx^ — y^ > and the grouporbitals. The matrices are given below, the nonbonding orbitals have been omitted.

for the A representation:

|lVlz^> | l V l x ^ - l ^ > | L A | 1 > | L A , 2 > | L A , 3 > | L A , 4 >

< M z ^ l 1 0 4 r S „ ^ 4 r S „ ^ - 4 r S „ , 0 Use has been made of the relation: |Mz^ > = -R [ | M z ^ - x ^ > - i - j M z ^ - y ^ > ] t o

< M x ^ - y ^ | 0 1 0 0 2cos2(^S^L 2sin2i^S^|_ 2 2 „2 , . 2 | < L A I 1 | 4 r S „ ^ _ 0 1 0 0 0 < L A , 2 | - ; - S „ , , 0 0 1 0 0 2 < L A , 3 | - ^ S ^ 2 c o s 2 ^ S ^ ^ 0 0 1 0 < L A I 4 | 0 2sin2v3S 0 0 0 1 TTL

for the A representation:

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for the B representation: | M z x > | L B , 1 > | L B , 2 > | L B , 3 > < M z x | 1 S ^ S ^ - 2 c o s ^ S ^ ^ < L B , 1 | S ^ 1 0 0 < L B , 2 | S J ^ 0 1 0 < L B , 3 | - 2 C O S ( / 3 S , 0 0 1 ' TTL

for the B representation:

2 | M y z > | L B 2 l > |LB 2 > | L B 3 > 2 2 < M y z | < L B ^ 1 | < L B ^ 2 | < L B ^ 3 | 1 ^TTG ^TTH - 2 sin

_{^ V}

^VG 1 0 0 ^TTH 0 1 0 - 2 sin'PSjj.L 0 0 1

The matrices of the energy operator have the same coefficients. The solution obtained by the method explained in the appendix are;

Energies of the orbitals with mainly d-character:

U(A.1) = E ^ + ^ ' A ^ ^ S^,^ . ^ \ , , S ^ „ , 4 ( - p + q v / 3 c o s 2 ^ ) ^ A , , S ^ „ , + + 4 q ^ s i n ^ 2 ^ A ^ ^ S ^ „ ^

U(A,2) = E ^ + ^ ' A , ^ S ^ ^ ^ + ^ ' A , , S ^ ^ ^ + i ( q + pV3cos2^)^A^^S^^^ +

+ 4 p ^ s i n ^ 2 ^ A ^ ^ S ^ ^ {3-3} U(A,) = E^ + 4 sin^ 2 ^ A^^S^^+ 4 cos^ 2 ^ A ^ ^ S ^ ^

U(B,) = E^ + A ^ ^ S ; ^ + A ^ ^ S ^ ^ + 4 cos^- ^ A^^S^^^ U(B2) = EM + A ^ ^ S ; ^ . A ^ ^ S ^ , + 4 sin^ ^ A ^ ^ S ; ^

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Wave functions: ICM Aj 1 > = p I M x ' > - ^ q l M x ' - y ' > - F ? ^ B ^ Q S ^ Q I LAj 1 >-^ + 7 | B ^ H S a H l L A , 2 > + ^ ( - p + q V 3 c o s 2 ^ ) B „ ^ S ^ J L A , 3 > + -H2qsin2./;B^I_S^^|LAj4> | C M A , 2 > = - q | M z ^ > + p | M x ^ - y ^ > - ? | - B ^ 3 S „ ^ | L A , 1 > - ^ B ^ , S ^ | L A 2 > + + -/3-(q + PV'3 COS 2 V5) B^|_S^|_ I L A J 3 > -I--H2psin2i/)B^LS;rLl'-A,4> (3-4)

ICMA^ > = IMxy > -H 2 sin 2 ./> B ^ | _ S ^ L I ' - A 2 1 > + 2 COS 2 >/> B ^ | _ S ^ L | L A ^ 2 >

with

E^ E

Ai = r - ^ ^""^ ^i = H r r

M '^j M i and p and q the solution of the equations:

P ^ + q ^ = 1

pq[S^(3 + S^H + S^„L - 3 cos^ 2 ^ S^^^ - 3 sin^ 2 ^ S^^) + (3-5)

V 3 ( p ' - q ' ) c o s ^ 2 . / ) S ^ L = 0

If 1/5 is equal t o 4 5 ° , then equation 3-5 are satisfied by p = 1 and q = 0 independent of the values of the overlap integrals; so in the case that i/3 is w i t h i n a few degrees of 4 5 ° this solution can be used as an approximation.

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Equations 3-3 and 3-4 then are:

U(A, 1) = E^ 4 ^ G S a G

^^\HKH + | A „ L S ^ „ L U ( A , 2 ) = E ^ + 4 c o s ^ 2 ^ A „ ^ S ^ ^ L + 4 s i n ^ 2 ^ A ^ ^ S ^ L U(A^) = E ^ + 4 s i n ^ 2 ^ A ^ L S ^ L + 4 c o s ^ 2 ^ A ^ ^ S ^ ^ ^ (3-6) U(B,) = E ^ + A , ^ S ; ^ + A , ^ S ^ ^ M 2 + 2 c o s 2 ^ ) A ^ ^ S ^ ^ U(B2) = E ^ + A ^ ^ S ^ Q + A ^ ^ S ; ^ + (2 - 2 cos 2 ^) A ^ ^ S ^ ^ and | C M A , 1 > = | M z ^ > + ^ B „ 3 S „ ^ | L A J > + ^ B ^ ^ S „ J L A , 2 > -- ^ B , , S ^ J L A , 3 >

ICMAj 2 > = | M x ^ - y ^ > -H 2 cos 2 sP B ^ L ^ C 7 L " - ' ^ I 3 > + 2 sin 2 i/3 B J ^ L ^ 7 : L " - A I ^ > ICMAj > = IMxy > -H 2 sin 2 ip Bp,_S^Ll'-A3 1 > -^ 2 cos 2 i/: B^^^S^l_|LAj 2 >

ICMB, > = | M z x > + B ^ 3 S ^ 3 l L B , 1 > + B ^ ^ S ^ j L B , 2 > - (3-7) - 2 c o s ^ B „ , S „ , ILB 3 >

'^ TTL TTL 1

| C M B ^ > = | M y z > + B ^ 3 S ^ Q | L B ^ 1 > + B ^ ^ S ^ J L B ^ 2 > -- 2 s i n ^ B ^ ^ S ^ J L B ^ 3 >

The results 3-6 and 3-7 will be used in the following chapters to interpret the spectra of the complexes [TiCMCH^OH) J " ' (with G = CI and H = L = C H j O H ) , [ T i C l 2 ( C H 3 0 H ) J " ' ( w i t h G = H = CI and L = CH^OH) and [TiCI^ ( C H 3 C N ) J * (with G = H = CI and L = CH^CN).

3.3.2. The complex MGHL K In a configuration with C symmetry

In the complex M G H L K G and H are ligands situated on the Z-axis in

trans position; K and L are also in trans position: the two ligands K on the Y-axis

and the t w o ligands L on the X-axis. The symmetry is C w i t h the Z-axis as the twofold rotation axis and the X Z and Y Z planes as mirror planes. If K and L are equivalent this configuration is the same as that treated in the preceding section with 1/3 = 4 5 ° . In fig. 3-3 the choice of the coordinate systems is depicted.

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Fig. 3-3. The choice of the coordinate axes on the ligands for the complex M G H L K in a configuration w i t h C symmetry.

2 2 2 V

The group orbitals for the inreducible representations of the p o i n t group C are given in table 3-11; the nomenclature used is analogous to t h a t for table 3-1. Also in this case there are a number of ligand orbitals that are non-bonding; these have been marked w i t h (n).

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Table 3-11. The C2V Irred. metal repr. orbital A , | M z ^ >

group orbitals of the complex M G H L K in a configuration w i t h symmetry

grouporbitals of the ligands | L A 1 > = | G Z 1 > | M x ^ - y ^ > | L A | 2 > = | H z 2 > A | M x y > B | M z x > B^ | M Y Z > | L A , 3 > = 4 - I|LZ3>-I- | L Z 5 > 1 i L A , 4 > = 4 - [ | K Z 4 > - I - | K z 6 > l | L A J 5 > = 4 - [ | L x 3 > + | L X 5 > 1 (n) | L A , 6 > = - U [ | K y 4 > - H | K y 6 > l (n) | L A ^ 1 > = - ^ [ | L y 3 > - < - | L y 5 > l | L A ^ 2 > = 4 - [ | K x 4 > - t - | K x 6 > ] |LB 1 > = | G x 1 > | L B 2 > = - | H x 2 > | L B , 3 > = -J- [ | L x 3 > - | L x 5 > ] iLB 4 > - 4 - [ l L z 3 > - i L z 5 > l (n) 1 V 2 | L B | 5 > = - j - [ | K x 4 > - | K x 6 > l (nl |LB 1 > = - | H y 2 > | L B j 2 > = | G y 1 > | L B ^ 3 > = - V - [ | K y 4 > - | K y 6 > ] | L B ^ 4 > = - ^ | | K z 4 > - | K z 6 > l (n) | L B ^ 5 > = - t - [ | L y 3 > - | L y 5 > l (n)

For the calculation of the overlap matrices the overlap integrals S „ „ , S „ _ , S „ , . , S „ , and S„,^ are defined in the same way as in

TÏK CJo CJn CJL U K section. energy of the grouporbitals ÔG ÔH ÔL ÔK

^m.

^TTK ^TTL ^TTK ^TTG ^TTH ^TTL ^OL ^TTK ^TTH ^ « 5 E,rK ^OK ^TTL ^ T r G ' ^ T t H ' ^ f f L ' the preceding

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The matrices are: for the A, representation:

1 |Mx^> |lVlx^-y^> | L A j 1 > | L A , 2 > | L A , 3 > | L A , 4 > < ^ ^ ' l ' ° ^ÔG ^ÔH - ^ V ^ ^ O K < M x ^ - y ^ | 0 1 0 0 V2S„, \/2 S _{•OL " ' -ÔK} < L A , 1 | - ? - S ^ 0 1 0 0 1 ^ÔG < L A , 2 | 2 0 0 , 0 0 I ^/Ï^'OH < ' ^ . ' ' - ^ V ^ / 2 s ^ ^ 0 0 1 0 < ' - ' ' . ^ ' ^ ^ a K ^ 2 S ^ 0 0 0 1

for the A representation:

|Mxy> | L A 2 l > | L A ^ 2 >

<Mxy| I V 2 S ^ , V2S^K <LA^1| \/2S^I_ 1 0

< L A ^ 2 | V 2 S ^ ^ 0 1

for the B representation:

|Mzx> ILB 1 > ILB 2 > ILB 3 > <Mzxi 1 S ^ S ^ V 2 S ^ ^

< L B , 1 | S ^ 1 0 0

<LB,2| S ^ 0 1 0 <LBj3| V2 S^l_ 0 0 1

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for the B representation: | M y z > | L B ^ 1 > | L B ^ 2 > | L B ^ 3 > < W l y z | < L B ^ 1 | < L B ^ 2 | < L B 3| 1 S;,H ^TTG

V2S„^

^TtH 1 0 0 ^ r t G 0 1 0 ^^^nK 0 0 1

The matrices of the energy operator have the same coefficients. The solutions obtained by the method explained in the appendix are:

the energies of the levels with mainly metal d-character:

i i ( A 1 1 - P - u ' ^ r . ^ a c;2 . 4 ^ 2 . c2 , 2 ( - p - i - q V 3 ) ^ ^ 2 ,

2(p + q V 3 ) \ 2

3 %K^aK

i i / A o i - P +^r,^A <;2 - ^ ^ ^ 2 A c:2 , 2 ( p V 3 - i - q ) ^ „ 2 ,

U(A,2) - EM +3 q \G%G ^S"^ \H^aH ^ 3 A^^S^^ +

2(pV3-q)^ 2 ,3 o)

+ 3 %K^oK *-^'^' U(A2) = E M + 2 A ^ ^ S ^ , , + 2 A ^ ^ S ^ ^ U(B,) = E M + A ^ ^ S ^ ^ + A ^ ^ S ^ ^ + 2 A ^ ^ S ; ^

Ü(BJ =E„ + A - S ^ - + A^^S^^-i-2A,^S!.^

2 M TTG TTG TTH TTH TfK TTK

the wave functions:

ICMAJ > =p|Mz' > -hqlMx'-y^ > "^ M B^^S^^lLA J >

-1-+ ^ B „ ^ S ^ j L A , 2 > -1-+ ^ ( - p -1-+ q V 3 ) B „ ^ S ^ j L A ^ 3 > -1-+ (3-9)

+ ^ ( p + qV3)B„^S„^|LA,4>

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|CMA^2> = q | M z ^ > + p | M x ^ y 2 > ^ B ^ ^ , S ^ Q | L A j >

-- ^ B ^ , S ^ j L A , 2 > + ^ { p V 3 + q ) B , , S ^ J L A , 3 > +

^ ^ ^ ^ ^ B „ , S , , , L A . 4 >

|CMA^> =IMxy> + V2B^^Sj^Ll'-A2l> + V2B^^S^I^|LA^2>

|CMB,> =IMzx> + B^3S^^iLBJ> + B , , S ^ , | L B , 2 > +

+ V2B^,S^JLB,3>

|CMB^> = I M y z > + B ^ ^ S ^ 3 | L B ^ 2 > + B ^ ^ S ^ j L B j > + + V 2 B ^ ^ S ^ K I L B ^ 3 > with A = F ^ . B =•= 1^ ' E., - E. I E,. - E. M l M l

and p and q are the solution of the equation: p^ + q^ = 1

iP3.(c^ +<i^ - s ^ - 9 ^ ) I ^^^ (n^ n^ 1 r.^ - S ^ ) = 0 3 *^CTG \ H ^ O L ^CTK' 3 'P "^ '*^aL ^OK' "

(3-9)

(3-10)

In the case that L and K are equivalent 3-10 is satisfied by p = 1 and q = 0; then we get the same energies as in eq. 3-6 with ip = 45° (the configurations are identical under these conditions). In the following chapters we will need the case that G = L and H = K; then the coefficient of pq in the second equation of 3-10 becomes zero and we obtain the solution p = q = - x r . With these values for p and q 3-8 and 3-9 become:

U(A,1) = E ^ 4 ( 3 - V 3 ) A „ , S ^ ^ 4 ( 3 W 3 ) A „ ^ S ^ ^

^'<A,2) = E^ +I- (3+V3) A„,S^„^ +§-(3 -V3) A^^S^„^

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ICMA^ > = IMxy > + V 2 B^^S^^\\-A^ 1 > + V 2 B^|^S^|^|LA^2 >

The expressions 3-11 and 3-12 will be used to interpret the spectra of the complexes TiCI^(CHjCN)j and TiCI^ (CHjOH)^ .

3.3.3. The complex MG H K with C symmetry

The configuration of the complex MG H K is that with the ligands K in trans position and the ligands G and H in cis position on the vertices of a deformed octaeder. The positions of the ligands and the coordinates on the ligands are given in fig. 3-4. The symmetry again is C with the Z axis as the two-fold rotation axis and the XZ and YZ planes as mirror planes.

The group orbitals are given in table 3-111; the nomenclature is analogous to that of the sections 3.3.1 and 3.3.2. In this case there are also non-bonding ligand orbitals; they have been marked in the table by (n).

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Fig. 3-4. The choice of the coordinates on the ligands of the complex MG H K in a configuration with symmetry C,^,

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Table 3-111. The grouporbitals of the complex MG^ ^2 '^2 ' " ^^^ configuration of Ta

Irred. metal repr. orbital

fig. 3-4.

grouporbitals of the ligands

| M z ' > | M x ^ - y ' > ilVlxv> B iMzx > | M y z > LA 1 > = 2 > = 3 > = 4 > = 5 > = 6 > = LA 1 > = LA 2 > = LA 3 > = LB 1 > = 2 > = 3 > = 4 > = 5 > = LB 1 > = LB 2 > = LB 3 > = LB 4>-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Gzl > + |Gz2>] Hz3>-I-K z 5 > + Gxl >-!• H x 3 > + Kx5>-i-Gyl >-i^ H y 3 > + Ky5>iGzl > H z 3 > Gxl > H x 3 > K y 5 > Gyl > H y 3 > K x 5 > K z 5 > -H z 4 > ] K z 6 > ) G x 2 > l H x 4 > l K x 6 > ] (n) G y 2 > l H y 4 > ] K y 6 > l G z 2 > l H z 4 > l G x 2 > l H x 4 > l K y 6 > l In) G y 2 > ] H y 4 > l K x 6 > l K z 6 > l (n) energy of the grouporbital 'OG -OH 'OK -TTG TTH TTK -TTG TTH •TTK -OG -OH -TTG "TTH -TTK "TTG "TTH •TTK -OK

The overlap matrices for this configuration have again been calculated w i t h help of the appendix. The overlap integrals S „ , S , S , S , S and S_|, are defined in the same way as those in the preceding sections. The matrices

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for the Aj representation:

1IVI''> | M x ^ V > |LA,1> |LAj2> |LA,3> |LA^4> |LA,5> ^ ^ ' ' ' ^ ° ^|<3cos^^-1)3^,2 Vf-IScos'l/z-DSj^ " V^jS^K - VB sin ^cos>pS^ Vfe sin ,//cos l//S, < M , < ^ V l < L A , 1 | <LA,2| <UA,3| <LA|4| <LA 5| 0 v/|(3cas^V-l)S„Q V f - ( 3 c o s ^ ^ - 1 ) 3 ^

- V K K

- Vfesini^cosc^SjjQ VD sin li/cos l//S„,, 7TM 1 ^ s i n ^ ^ S ^ V2sin^.AS^ - v ' ^ s ^ ^ \/2 sin ^pcosi^S_^ - \/2 sin l// cos l// S-u

mH

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for the A representation: < M x y | < L A ^ 1 | < L A ^ 2 | < L A ^ 3 | | M x y > 1 - V 2 s i n ^ S ^ Q - V 2 s i n ^ S ^ ^ ^^\K ILA^

-V2

1 0 0 1 > s i n ^ S ^ ILA^

-V2

0 1 0 2 > sin \}j ^TTH | L A ^ 3 > V ^ S ^ K 0 0 1 for the B representation:

< M y z | < L B 2 l i < L B ^ 2 | < L B 2 3 | | M y z > 1 - V 2 c o s ^ S ^ ^ , V2cos-//S^^ -^^2 5 ^ ^ ILB^

-V2

1 0 0 > cosi/J TTG ILB

V2

0 1 0 2 2 > cos\p

^m

| L B ^ 3 > - V 2 S^K 0 0 1

The energy matrices have the same coefficients as the overlap matrices. The solutions for the energy of the levels w i t h mainly d-character of the metal obtained by the method explained in the appendix:

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U(A^ 1) = E^ + I { D ( 3 COS^ V ' - 1) + qV3 sin^ ^f A ^ ^ S ^ ^ + | {p(3 cos^ ^ - 1) + + q V 3 s i n ^ ^ f A ^ ^ S ^ ^ 4 ( p + q V 3 ) ^ A ^ ^ S ^ ^ + + l ( - p V 3 + q ) ^ m ^ 2 ^ A ^ ^ S ; ^ + l ( - p V 3 + q)^in^2 0 A ^ ^ S ; ^ U(A, 2) = E^ + I { - q ( 3 c o s V - 1) + P V 3 s i n ^ } ' A ^ ^ S ^ ^ + | { - ^ < 3 cos^\i/-1) + + p V 3 s i n ^ ^ } ' A ^ ^ S ^ ^ + | ( - q + p V 3 ) ^ A ^ ^ S ^ ^ + + l ( + q V 3 + p ) ^ - s i n ^ 2 ^ A ^ ^ S ; ^ + l ( + q V 3 + p ) ^ s i n ^ 2 ^ A ^ ^ S ; ^ U(A^) = E ^ + 2 s i n ^ ^ A ^ ^ S ; ^ + 2 s i n ^ V ' A ^ ^ S ; , + 2 A ^ ^ S ; ^ (3-13) U(B,) = E ^ + 2 s i n ^ 2 ^ A ^ ^ S ; ^ + 2 s i n ^ 2 ^ A ^ ^ S ^ ^ + 2 c o s ^ 2 ^ A ^ ^ S ; ^ + - 2 + 2 c o s ^ 2 ^ A ^ ^ S ; ^

U(B^) = E^ . 2 cos^ , A ^ ^ S ^ ^ . 2 cos^ ^ A ^ ^ S ^ ^ . 2 A ^ ^ S ^ ^

the wave functions:

ICMAj 1 > = p | M z ^ > - i - q | M x ' - y ' > - i - > ƒ - | p ( 3 c o s ^ - l ) + '2 + q V 3 s i n ^ ^ } B ^ g S ^ Q | L A J > + ^ | ( p { 3 c o s ^ ^ - 1 ) + + q V 3 s i n ^ ^ } B ^ ^ S ^ J L A , 2 > - ^ | ( p + q V 3 ) B ^ ^ S ^ ^ | L A , 3 > + 2 3 -p\/3 + q sin 2 ; p B „ ^ S „ ^ | L A 4 > " ^ ^ ^ ' " " ^ " T T G ^ T T G ' " ^ , - p \ / 3 + q - - ^ 7 J — s i n 2 v B ^ ^ S ^ J L A , 5 > |CMA_ 2 > = - q | M z 2 > + pIMx" - v' > + ^ | { - q(3 c o s ^ - 1) + + P V 3 sin=^}B^^S^^|LA, 1 > + ^ | { - q ( 3 cos^C^ - D +

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ICMA^ > = IMxy > - V 2 sin i p B ^ g S ^ g l L A J > - (3-14) - V 2 s i n ^ B ^ , S ^ J L A ^ 2 > + V 2 B ^ ^ S ^ J L A ^ 3 >

ICMB, > = IMzx > -H V 2 sin 2 ^ B^^S^^ |LB _ 1 > - V 2 sin 2 i// B^^S^^ |LB ^2> + + V 2 c o s 2 ^ B ^ ^ S ^ J L B , 3 > + V 2 c o s 2 ^ B ^ ^ S ^ j L B , 4 >

ICMB^ > = IMyz > - V 2 cos ^ B^^S^^ |LB^ 1 > + V 2 cos ^ B^^ S^H " " B j ^ > " - V 2 B , , S ^ , I L B ^ 3 >

E^ E. with A = p '-p- and B

^ M - ^ i ^ ' E ^ - E , p and q are the solution of the equations: P ^ + q ^ = 1

' l ^ a K - 2 " ' ' " ' ^ ^ ^^G - ^ s i n ^ 2 ^ s ; ^ ] [ 2 p q + ( p ^ - q ^ ) V 3 ] + (3-15) + ?r [ p g ( 2 - 12 cos 2 1/5-6 cos^ 2 i/3) -i- ( p ^ - q ^ ) ^ 3 (1 + 2 cos 2 v? - 3 cos^ 2i/!)] S^_ + -^ i [ p q ( 2 - 1 2 c o s 2 i / / - 6 cos^ 2 i//) -i- ( p ^ - q ^ ) V 3 ( 1 -^ 2 cos 2 i / / - 3 cos^ 2 i//)] S',^= O

6 OH

These formulae are too complicated to be used conveniently. The second equation of 3-15, however, becomes very easy to solve if i/3 and \p are 4 5 " . Then we have: p ' + q ' = 1

2 , . , (3-15) 3

S a K 4 s ; G 4 s ^ H 4 ' S a G " S ^ H ' l t 2 p q M p ^ - q ^ ) V 3 l = 0

The solution of 3-16 is p = ^ and q = ;r \ / 3 independent of the overlap integrals. Obviously these values for p and q are good approximations for angles ip and 4^ w i t h i n a few degrees of 4 5 ° . Furthermore, if the ligands H are more bulky than the ligands G i/5 must be smaller than 4 5 " and i// bigger; it is logical to assume as an approximation that 4 5 " — i/j = i// — 4 5 " . With these approximations 3-13 and 3-14 become after some rearrangements:

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U(A,1) = E M + K G S a G " f A ^ , S ^ , + | A „ ^ S ^ , U(A,2) = E ^ + 2 c o s ^ 2 v 5 A ^ ^ S ^ ^ + 2 c o s ^ 2 ^ A ^ ^ S ^ „ + + 2 s i n ^ 2 ^ A ^ Q S ^ 3 + 2 s i n ^ 2 ^ A ^ ^ ^ S ^ ^ (3-17) U(A^) = E ^ + ( 1 - C O S 2 ^ ) A ^ Q S ^ 3 + ( 1 + C O S 2 ^ ) A ^ ^ S ^ , - ^ 2 A ^ ^ S ^ ^ U ( B J = E ^ + 2 s i n ^ 2 ^ A ^ ^ S ^ ^ + 2 s i n ^ 2 ^ A ^ ^ S ^ H + + 2 C O S ^ 2 ^ A ^ ^ S ^ Q + 2 C O S ^ 2 ^ A ^ ^ S ^ H U(B^) = E^ + (1 + c o s 2 ^ ) A ^ ^ S ^ ^ + (1 - c o s 2 ^ ) A^^S^^^ + 2 A ^ ^ S ^ ^ and 1 , . . _ 2 ^ . V'3 ,.,..2 .2 v^ , / 2 ICMA, 1 > =-\Mz^> + \m'-s/'>+y/- B ^ Q S ^ Q I L A , 1 > + + | B a H S a H l L A , 2 > - ^ B ^ , S , , | L A , 3 >

| C M A j 2 > = - ^ | M z ^ > + i | M x 2 - y 2 >-K V 2 cos 2 v? B^^S^^jLAj 1 >i+ V 2 cos 2 ^ B^^S^^ ILAj 2 > >i+ ^ 2 sin 2 .^ B^gS^^^ |LA, 4 >

-- V 2 s i n 2 ^ B ^ ^ S ^ ^ | L A , 5 > (3--18) ICM A^ > = IMxy > - V 2 sin ^ B^^S^^ ILA^ 1 > - V 2 cos <^ B^^S^^^ |LA^ 2 >

-t-V 2 B ^ K S ^ J L A ^ 3 >

ICMBj > = | M z x > - H V 2 s i n 2 > / ; B p Q S ^ j Q | L B _ 1 > - v ' 2 s i n 2 ^ B ^ ^ S ^ ^ | L B , 2 > - h -I- V 2 cos 2 ^ B^gS^Q | L B , 3 > -I- V 2 cos 2 ip B^^^S^^ ILB, 4 >

ICMB^ > = IMyz >s/2 cos ^ B^^^S^^ |LB^ 1 > F V 2 sin >/> B^^S^^^ ILB^ 2 > -- V 2 B ^ ^ S , , | L B ^ 3 >

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CHAPTER 4. THE ORDER OF THE ENERGY LEVELS A N D THE G-VALUES

4 . 1 . Introduction

In the preceding chapters expressions were found for the energies of the d-orbitals. With tervalent titanium as the central ion there is one electron outside the closed shells; this electron w i l l occupy the lowest of the d-orbitals.

In a free Ti^"*" ion the d-orbitals are all degenerate and form the ^D level; this level is split into a ^T., and a ^E level, when the Ti-ion is surrounded by

2 q g

ligands in a pure octahedrafconfiguration. In the complexes studied here the ligands form a more or less deformed octahedron. In nearly all cases the symmetry is further lowered because the ligands are not all of the same type. In most cases the different d-orbitals have a different energy; however, because of the nearly octahedral arrangement of the ligands it is t o be expected that a group of three orbitals (corresponding w i t h the ^ T level of the pure octahedron) will have a lower energy than a group of t w o orbitals (the ^ E level of the octahedron). Then the t w o d-d-transitions observed in the near U.V. and visible regions of the spectrum will be the transitions from the ground state (the lowest d-orbital) to the t w o higher d-orbitals. The energy differences between the orbitals of the lower group are too small to be detected directly in the spectrum; they are in the region of the vibration transitions of the ligands which are much narrower and stronger. The g-values of the E.S.R. spectrum, however, strongly depend on these smaller energy differences. In this chapter the components of the g-tensor will be expressed in the energy differences of the d-orbitals and the spin-orbit compling parameter X.

T o make this possible the order of the energies of the d-levels must be k n o w n , so this will be discussed first.

4.2. The order of the energy levels in the different complexes

In this thesis the complexes [TiCI^(CH^CN)^ ^]^'^ and

[TiCI (CH OH)^ ] ^ ' y w i t h x = 2, 3 , 4 a n d y = 1, 2, 3 ' a n d 4 w i l t bediscussed in y 3 6 - y

detail, but before that some general arguments are given from which the order of the levels can be deduced.

In the M.O. picture the d-orbitals pointing to the ligands are the orbitals which form the a-bonds w i t h filled orbitals of the ligands. These d-orbitals are expected to have a larger overlap with the filled ligand orbitals than the d-orbitals that form only w-bonds, so they will be raised more in energy. The molecular orbitals of this type are those which form the E level in the pure octahedral situation. From the spectrochemical series''*' wnere CI stands before CH OH and C H ^ C N , it can be deduced that A ^ ^ S ^ Q and A^^^S^i^ will be greater than A a c i ^ a c i • ' A J Q ' A ^ N ^ " d A ^ „ | are the coefficients in the formulae 3-6, 3-11 and 3-17 when the appropriate identifications of the general ligands G, H, K and L are made with C H ^ O H , CH^CN and CI respectively; S^^Q , S^j^ and S ^ „ | are the overlap integrals of the a-bonding of resp. C H ^ O H , CH^CN and CI w i t h the Ti atom).

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The TT-bonding in complexes w i t h CH CN will take place via the anti-bonding orbitals of the 7r-bonds of the CN group; these antianti-bonding orbitals are empty and have an energy higher than the energy of the d-orbitals of titanium. So the coefficient A |^ will be negative. The 7r-bonding with the Cl-atom will take place via the filled 3p-orbitals of the CI atom, which have an orbital energy lower than the metal d-orbital, so A „ . is positive. Furthermore A „ , S i . p . is expected to be smaller than A „ . S ^ „ . . For the chlorocomplexes with acetonitrile we thus expect t o f i n d :

"aN^aN % c r a c i ^ '^rrcrwci TTN wN

If we assume that the oxygen atom is sp^ hybridized in methanol then there is only one p-orbital available for the rr-bonding between CH OH and T i . We assume that this orbital makes an angle of 4 5 ° w i t h the ligand x and y axes of the pictures 3-2, 3-3 and 3-4, or that the methanol molecule rotates rapidly round the T i - 0 axis, in order t o be able to use the same theory as for the acetonitrile ligand. However, the overlap integral in that case will be a factor \ / 2 smaller than the overlap integral of a full rr-bond. The p-orbital of oxygen is filled and has an

orbital energy lower than that of the T i d-orbital so A _ is positive. With this

argument it can be expected that

A .S^ > A 'S? ~> A <ï^ > A ^

^CTO^ao " 0 c r a c i ^ ^TrcrTrci ^ ^TTO^TTO •

The order of the energies of the orbitals w i t h mainly d-character can now be estimated in the M.O. picture.

In the crystal field approach it is clear that the electron in a d-orbital that points to a negatively charged ligand w i l l have a higher energy than the electron in an orbital that has its maximum density in a direction just between the negative charges. These orbitals form the E level in the octahedral configuration.

g _{5 v 5 ,.,u-,.-« v 5 •}

The crystal field splitting of an octahedral complex is ^ X where X is defined in eq. 2-5. So f r o m the spectrochemical series^'' we expect N ' and 0 ^ to be larger than C l ^ . I t is d i f f i c u l t t o estimate the relative magnitudes of the quantities N ' , O ' and C I ' ' . However, it can be remarked that the negative charge of the ligands CH OH and CH CN is concentrated in a sp^ orbital resp. a sp-orbital pointing in the direction of the Ti-ion, while in the C l ~ ion the negative charge is more smeared o u t . So the effective distances R_ and R are expected to be much

smaller than the effective distance R_. . N ' , O ' and CP are thus expected not t o

differ very much from each other. In the formulae 2-8, 2-13, 2-14 and 2-21 they mostly appear in the form ( C l ' — N'') and (CI'' - O ' ' ) , when the appropriate identifications of G, H, K and L w i t h N , 0 and CI are made, so the order of the levels will be governed by N ' , O'' and C l ' . When in doubt the results of the M.O. theory can be used, or the various possibilities can be studied and compared w i t h each other in view of the other complexes w i t h the same ligands.

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4.2.1. The complex [TiCI(CH^OH) J ^ *

The expressions for the energy of the 3 d-levels of the titanium can be found from the equations in the sections 2.2.1 and 3.3.1 by substituting G = CI and H = L = O, where 0 stands for methanol.

From 2-8 we find:

U(A 1) = r +l-[2C\^~2 0^] + 1 ^ [12C1' + 3 0 O ' ]

1 00 7 42

U(A_2) = r ^ ^ - i - ^ [ - 2 C P -H2 0 ' ] + ^ [ 2 C l ' - ( 3 0 - 7 0 c o s ^ 2 i p ) O ' ]

U(A ) = r - i - ^ [ - 2 C l ' - t - 2 0 ' l -I- -V [2CI= -H (40-70cos^2v3)O^] (4-1) 2 00 7 4 2

U(Bj) = r ^ g - i - i [ C P - ( 1 - 6 COS 2 ^ ) 0 ' ] +-^ [ - 8 C l ' - ( 2 0 - I - 2 0 cos 2 1^)0']

U ( B J = r g Q - H i [ C P - ( 1 -I-6 cos 2 ^) 0 ' ] +^ [ - 8 C I ^ - ( 2 0 - 2 0 c o s 2 i / j ) O ' l Keeping in mind that ^p is near 45" (cos 2 i^ is a small quantity) and O' > Cl' the estimated order of the energy levels is:

U(A| 2) < U(BJ < U(B^) < U(A J ) < U(A^).

In the case that ip = 45", U(B ) and U(B ) are degenerate; the point group of the complex is then C^„. From 3-6 we find:

U(A,1) = E ^ 4 V | S a C i 4 ^ o S ; o

U(A,2) = E ^ . 4 c o s ^ 2 ^ A ^ „ S ^ „ ^ + 4 s i n ^ 2 ^ A ^ ^ S ^ „ U(A^) = E ^ + 4 s i n ^ 2 ^ A ^ „ S ^ „ „ + 4 c o s ^ 2 ^ A ^ ^ S ^ ^ (4-2) U<B,) = E M + A ^ „ S ; „ M 3 + 2 C O S 2 V . ) A ^ ^ S ; O U(B2) = E M + A ^ „ S ; „ M 3 - 2 C O S 2 ^ ) A ^ „ S ; ^

The expected order of the energy levels is the same as in the crystal field approximation. The level schemes are depicted in fig. 4-1.

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A j A4 ATI A j I X y > A , ! p | z ' > . q | x = - y = > B, I zx> B2 I yz> A , 2 - q | z ' > . p | x ' - y ' > A 3

Ais]

A4 B 2 I X y > ( h x > I I y o B , x ' - y = > a: C symmetry (1,0< 45 ) b: C symmetry (1/3= 45 ) Fig. 4 - 1 . The estimated level scheme for the T i 3d-levels in [TiCI(CH^OH) ] '

4.2.2. The complexes [TiCI (CH CN) ] * and [TiCI (CH OH) V

2 3 4 2 3 4

The t w o chlorine atoms in these complexes may be in a trans position or in a cis position, so t w o possibilities must be studied.

a. the t w o Cl-atoms in trans position.

In the equations of the sections 2.2.1 and 3.3.1 now the substitutions G = H = CI and L = N or L = 0 must be made, the symmetry of the complex is

then D . in stead of C 2 h 2 V

From 2-8 we obtain for the methanol complex: U(A 1) = r + 1 [ 4 C I ^ - 4 0^1 + ^ [24 C I ' -1- I 8 O ' ;

00 7 42

U(A 2) = r + i [ - 4 CP -I- 4 0 ^ ] -I- - V [4 C I ' - (32 - 70 cos^ 2 ip) O ' :

0 0 7 ₄₂

ü ( B j ) = r ^ p - i - l [ - 4 C P + 4 0 ' ] + T ^ [ 4 C P - H ( 3 8 - 7 0 c o s ^ 2.^) 0 ' ] (4-3) U ( B J = r „ + l [ 2 C I ^ - 2 ( 1 - 3 c o s 2 i p ) 0 ^ ] + l - [ - 1 6 C l ' - ( 1 2 + 2 0 c o s 2 ^ ) O ' :

0 0 7 ₄₂

U(B^) = r ^ g + - [ 2 C P - 2 ( 1 + 3 cos 2 ^ 5 ) 0 ' ] + - ^ [ - 1 6 C l ' - ( 1 2 - 2 0 c o s 2 i / j ) 0

For the acetonitrile complexe the same expressieres must be used w i t h N ' and N ' in the place of O ' and O ' . The expected order of the levels is the same as that in section 4 . 2 . 1 :

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U(A,1) = E ^ + 3 V i S a c ^ g V S ^ o

U ( A , 2 ) = E ^ + 4 c o s ^ 2 ^ A ^ ^ S ^ ^ Q + 4 s i n ^ 2 ^ A ^ Q S ; ^

U ( B , ) = E ^ + 4 s i n ^ 2 ^ A ^ ^ S ^ Q + 4 c o s ^ 2 > , A ^ „ S ; Q (4^)

U(B2) = E M ^ 2 A ^ ^ , S ; ^ , + ( 2 + 2 C O S 2 ^ ) A ^ ^ S ^ „

U'Ba) = E M + 2 A ^ ^ , S ; ^ , + ( 2 - 2 C O S 2 ^ ) A ^ O S ^ „

Also from these expressions the same order of the energy levels of the 3 d-orbitals of the titanium is expected.

The level scheme is depicted in fig. 4-2 for the angle ip < 4 5 " and for

ip = 4 5 ° ; in the latter case the point group of the complex is D . .

A3 A2

M

a; D . symmetry ( i p < 4 5 " ) b: D , symmetry (ip= 45 )

Fig. 4-2. The estimated energy level scheme of the 3 d-orbitals of the complex [TiCl^ (CH^ 0 ^ ) 4 1 ^ w i t h the Cl-atoms in trans position

b. the t w o chlorine atoms in cis position

To obtain the expressions appropriate f o r this configuration now the equations derived in the sections 2.2.3 and 3.3.3. must be used w i t h the substitutions G = CI and H = K = O o r H = K = N. For the methanol complex we obtain from 2 - 2 1 :

An interpretation of the U.V. and E.S.R. spectra of some chlorocomplexes of tervalent titanium