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An electron paramagnetic resonance study of a manganese (IV) ion in a trigonal environment Byfleet, Colin Russell 1969

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AN ELECTRON PARAMAGNETIC RESONANCE STUDY OF A MANGANESE (IV) ION IN  A TRIGONAL ENVIRONMENT  BY  COLIN RUSSELL BYFLEET B.A., U n i v e r s i t y o f Cambridge, 1964  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  i n t h e Department of Chemistry  We accept t h i s t h e s i s as conforming t o t h e r e q u i r e d  THE UNIVERSITY OF BRITISH COLUMBIA March, 1969  standard  In p r e s e n t i n g an the  thesis  advanced degree at Library  I further for  this  shall  the  in p a r t i a l  U n i v e r s i t y of  make i t f r e e l y  agree that  permission  s c h o l a r l y p u r p o s e s may  by  his  of  this  written  representatives. thesis  for  f u l f i l m e n t of  be  British  available  for extensive  g r a n t e d by  the  It i s understood  financial  for  gain  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  Chemistry  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  Columbia  or  publication  allowed without  Colin Byfleet  of  thesis  Department  permission.  Department  that  Study.  this  copying or  for  my  ABSTRACT  An electron spin resonance study has been carried out at room temperature, on a magnetically dilute single crystal of ammonium 9molybdomanganate.  A general method for f i t t i n g E.S.R. results to  a spin Hamiltonian has been devised, and the results of the above study have been used as an example of this method.  The values of  the parameters thus determined wereJD| = 0.861 ± 0.001 cm"^; g^ = 1.9920 ± 0.0004 cm" ; g = 1.9880 ± 0.0004 c m " ; ^ ^ 0.00760 ± 1  1  ±  0.00004 cm" ;|AJ= 0.00684 ± 0.00004 cm" . 1  1  A review of previous theoretical calculations on d^ has been given, and a ligand field approach has been taken in an attempt to interpret the observed parameters.  This was successful for the  zero field splitting and hyperfine coupling constants, but not for the g-values. The experimental results from a study of an irradiated, single crystal of deuterated ammonium paramolybdate tetrahydrate have also been fitted to a suitable spin Hamiltonian. The principal axes of the  and A tensors were found not to be  coincident, and a method for treating the experimental results in this situation has been given. This study showed that the doublet splitting observed earlier, in the E.S.R. spectrum of the irradiated, undeuterated compound, was most probably due to a captured proton.  An interpretation of the  observed g-values and hyperfine coupling constants has been given, using a molecular orbital approach.  ii  TABLE  OF  CONTENTS  Part I : An Electron Spin Resonance Study of an Mn^ Compound in a Trigonal Environment. Page 1. INTRODUCTION  1  2. THEORETICAL  7  A. B. C. D. E. F. G.  Zeeman Interactions Spin Orbit Coupling Electron-Electron Dipolar Interaction Electron-Spin-Nuclear Spin Interaction Nuclear Spin-Electron Orbit Interaction Nuclear Quadrupole Interaction Smaller Interactions  7 8 9 9 9 10 10  3. SAMPLE, PREPARATION AND STRUCTURE  13  4.  EXPERIMENTAL METHODS AND RESULTS  15  5.  LIGAND FIELD CALCULATIONS ON THE GROUND STATE OF THE COMPLEX  35  A. Survey of Previous Calculations on d B. Hyperfine coupling calculations  36 62  6. DISCUSSION  systems  67  Part II : E.S.R. of Irradiated Single Crystals of Deuterated Ammonium Paramolybdate 1. INTRODUCTION  77  2. EXPERIMENTAL  77  3. THEORETICAL  80  4. DISCUSSION  85  I l l  Table of Contents (cont'd.)  REFERENCES  91  APPENDIX I - Perturbation Theory  95  APPENDIX II - Block Diagram of Program to Calculate Transition Magnetic Fields from Spin Hamiltonian Parameters  98  APPENDIX III - Example Calculation of an Electron Repulsion Integral  100  iv  Tables  (cont'd.)  Table  14  Matrix of a  Table  15  M a t r i x o f a_ i n the one e l e c t r o n s p i n o r b a s i s  Table  16A  The s p i n H a m i l t o n i a n H)^  Table  16B  z  i n the one e l e c t r o n s p i n o r b a s i s  ion i n their  parameters f o r the (M07O24  p r i n c i p a l axes systems  Values o f the j* t e n s o r and hyper f i n e t e n s o r f o r molybdenum complex  anions  interaction  vi  FIGURES  Energy l e v e l s o f a values  system w i t h d i f f e r i n g D-  Energy l e v e l diagram f o r a s p i n magnetic f i e l d  quartet i n a  - see F i g u r e 2 above -  Diagram o f m o l e c u l a r molybdomanganate  s t r u c t u r e of ammonium 9-  B l o c k diagram o f 100 k c . E.S.R.  spectrometer  A t y p i c a l c r y s t a l o f ammonium 9-molybdomanganate  A t y p i c a l spectrum o f the \ ~ I^ transi t i o n ( £ = 30°), f o r a magnetically d i l u t e s i n g l e c r y s t a l o f ammonium 9-molybdomanganate  A t y p i c a l s p i n - f o r b i d d e n t r a n s i t i o n ( & = 15°) for a magnetically d i l u t e single c r y s t a l of ammonium 9-molybdomanganate  The s p i n - f o r b i d d e n t r a n s i t i o n observed at & = 90° f o r a m a g n e t i c a l l y d i l u t e s i n g l e c r y s t a l o f ammonium 9-molybdomanganate  The  observed  fine structure  transitions  P e r t u r b a t i o n diagram  The observed h y p e r f i n e t r a n s i t i o n s f o r the - '.|-%^ ' t r a n s i t i o n w i t h the s o l i d l i n e i n d i c a t i n g the t h e o r e t i c a l curves  The  observed h y p e r f i n e s p l i t t i n g s f o r the 1+^> - '\-kK> t r a n s i t i o n w i t h the s o l i d l i n e i n d i c a t i n g the t h e o r e t i c a l curves 1  1  1  vii  One e l e c t r o n d - o r b i t a l s i n o c t a h e d r a l and t r i gonal f i e l d s  Complete c o r r e l l a t i o n diagram f o r a d^ system (not t o s c a l e )  S p l i t t i n g o f the gonal f i e l d s  s t a t e i n o c t a h e d r a l and t r i -  Diagram o f the s p l i t t i n g of s t r o n g o c t a h e d r a l f i e l d c o n f i g u r a t i o n s under the i n f l u e n c e o f e l e c t r o n r e p u l s i o n and t r i g o n a l f i e l d o p e r a t o r s  V a r i a t i o n o f the energy l e v e l s w i t h the comb i n a t i o n o f parameters which give a Z.F.S. o f approximately 1.7 cm~l  V i s i b l e a b s o r p t i o n spectrum o f ammonium 9molybdomanganate  A t y p i c a l c r y s t a l o f ammonium paramolybdate showing the chosen a x i s system  An E.S.R. spectrum o f an i r r a d i a t e d s i n g l e c r y s t a l o f d e u t e r a t e d ammonium paramolybdate t e t r a h y d r a t e , w i t h the magnetic f i e l d i n the a' - c plane  The v a r i a t i o n o f h y p e r f i n e s p l i t t i n g and g-value f o r an i r r a d i a t e d s i n g l e c r y s t a l o f d e u t e r a t e d ammonium paramolybdate t e t r a h y d r a t e f o r the o r i e n t a t i o n s where the magnetic f i e l d i s perpend i c u l a r t o the a' a x i s  The v a r i a t i o n o f h y p e r f i n e s p l i t t i n g and g-value f o r an i r r a d i a t e d s i n g l e c r y s t a l o f d e u t e r a t e d ammonium paramolybdate t e t r a h y d r a t e f o r the o r i e n t a t i o n s where the magnetic f i e l d i s perpend i c u l a r to the b a x i s  The v a r i a t i o n of h y p e r f i n e s p l i t t i n g and g-value f o r an i r r a d i a t e d s i n g l e c r y s t a l o f d e u t e r a t e d ammonium paramolybdate t e t r a h y d r a t e f o r the  viii  Figures  (cont'd.)  o r i e n t a t i o n s where the magnetic f i e l d d i c u l a r t o the c a x i s  F i g u r e 24  i s perpen-  An E.S.R. spectrum o f an i r r a d i a t e d s i n g l e c r y s t a l o f ammonium paramolybdate t e t r a h y d r a t e w i t h the magnetic f i e l d i n the a - c plane 1  83  87  ix  ACKNOWLEDGMENT  I would like to express my sincere thanks to Dr. W.C. Lin for suggesting this problem, for his untiring assistance and guidance, and for many hours of stimulating discussion. I am very grateful to Dr. CA. McDowell for making the excellent f a c i l i t i e s of the Department available to me; to Dr. D.P. Chong for clarifying many points in perturbation theory and, together with Dr. F.G. Herring and Mr D.E. Kennedy, for supplying computer sub-routines. My thanks are also due to Mr J. Sallos and Mr T. Markos for keeping the E.S.R. equipment in excellent condition, and to my wife for her help in the preparation of this manuscript. Finally, I would like to acknowledge the receipt of an assistantship from the Department of Chemistry, a University of British Columbia scholarship, and two scholarships from the National Research Council of Canada.  Symbols Used but Undefined in the Text  •ft  = h/27T = Planck's constant  V  • =  (d/dx., a / a y . ,  d/dz )  m  = electron rest mass  e  = electronic charge  c  = velocity of light  g  g-value of free electron  = e  ^  = absolute value of Bohr magneton  e  f3  ±  nuclear magneton  =  gjj  = nuclear g-factor  H  = magnetic field  s_, S^ = electron spin momentum vector I  •= nuclear spin momentum vector  1  = orbital angular momentum vector  S(r-^)= Q S« //  Dirac S -function  = nuclear quadrupole moment S^~> zz z-component of g-tensor  =  gi  =  ka  = A , z-component of hyperfine coupling tensor, A  A  J-  %(gx  x  + g  y y  )  22  =  P  ^ KK A  A  +  yy  )  = quadrupole coupling tensor  M  = eigenvalue of electron spin vector  Mj-  = eigenvalue of nuclear spin vector  s  e, a, t, etc., are one-electron group theoretical labels E, A, T, etc., are electronic state group theoretical labels NOTE:  the symbols e and e etc., are used interchangeably  E( A p the  e t c . , are the e n e r g i e s o f  ground s t a t e of the system.  A^ e l e c t r o n i c s t a t e s w i t h  respect  PART  I  1  1. Introduction Electron spin resonance spectra of transition metal complexes have been widely studied (1-4), and the information obtained by this method has led to a greater understanding of the detailed electronic structure of these compounds.  Three theoretical approaches have been used to  interpret the experimental results, these being the crystal field (5), ligand field (6) and molecular orbital (7,8) theories.  The latter  approach is most successful, although most applications have been to d  1  Q  or d systems, since calculations involving more than one free electron become very tedious. Ligand field theory is formally very similar to crystal field ideas, where the ion under study is considered to be co-ordinated by oppositely charged ions or dipoles; no allowance being made for covalency effects, which have been observed as, for instance, super-hyperfine interactions with the ligand nuclei (9).  Various parameters are allowed to vary  from their free ion value in ligand field theory, in a physically justifiable way, to account for covalency. Manganese occurs in several valence states, +2, +3, +4, +6 and +7. Of these the f i r s t four have the possibility of being paramagnetic and much E.S.R. work has been done on the d^Mn^ system. +  The major part of  this work is concerned with a d^ system in which the central transition metal ion is manganese in its +4 valence state;  namely, (NH^)^Mog032Mn.  8H2O,ammonium 9-molybdomanganate, a typical heteropolymolybdate. There are very few Mn ^ compounds and Mn02 is the only one commonly +  found.  This has led to a very small number of previous E.S.R. studies  being done on Mn+4 systems (11-22).  In fact, a l l the previous work  has been on crystals of compounds like SnOj or SrTiO^ having Mn+ as an 4  impurity ion replacing Sn"" or Ti+ in the lattice, and leading to a 1  4  4  2  magnetically dilute MnC>2 system. In the present system the Mn ^ +  ion is part of a well defined poly-  mo lybdate complex, a system which may have many different ions in the central position.  These have not been studied by E.S.R., but would seem  to offer a large field for further study, which would enable comparisons to be made between different ions in very similar environments. Magnetically concentrated systems give unresolved E.S.R. spectra due to dipole-dipole interactions between neighbouring ions ( 2 3 ) .  The poly-  molybdate ion under study is a very large unit, and i t was hoped that the pure crystals would have a sufficiently large distance between neighbouring ions to give fully resolved spectra.  In fact, this distance was  found to be just not large enough since the fine structure lines were easily observed, but these had a line width of about 350 gauss, which obscured the underlying hyperfine structure.  Accordingly a l l the experi-  ments were carried out on crystals of the isomorphic Ni^" " compound con1  taining a small amount of  Mn^ . +  The most commonly studied d also been investigated.  3  system is C r  3 +  although  (2-4)  has  Unfortunately ->3rj (the only isotope with r  nuclear spin) has a small nuclear moment and is only about 107o abundant in naturally occurring chromium, and these two factors have led to r e l atively few observations of its hyperfine structure. isotopes of both Mn^  +  and  The most abundant  both have f a i r l y large nuclear moments, and  this leads to much easier observation of hyperfine structure.  A des-  cription of the main features to be expected in the E.S.R. spectra of these d  3  systems w i l l be useful.  A free d  3  ion has a ^F ground state (10).  In a complex with regular  octahedral co-ordination, the angular momentum is "quenched" leading to a ^A  ?s  (6)  ground state, which is orbitally non-degenerate.  Only one  3  line would be observable in the E.S.R. spectrum of such a complex, since the levels would be split by a magnetic field in a simple linear fashion as shown below in Fig. 1(a) and transitions occur between levels having M  s  values differing by "tl.  Figure 1  H  —  1 (a)  H  4  H  -Ar (b)  -A  —1r () C  . There is however, the possibility of distortion of the complex from the Jahn-Teller (24) effect, and also of spin-orbit coupling with higher levels.  These two effects together lead to partial removal of the spin  degeneracy, producing two doublets as predicted by Kramers (25) theorem; the energy separation being known as zero field splitting (Z.F.S.). This situation is shown in Fig. 1(b) and i t can be seen that three lines are in general observable, i f the Z.F.S. is small in magnitude.  Large  splittings lead to the situation in Fig. 1(c) , where transition (i) of Fig.  1(b) is no longer possible with the available microwave quanta, and  transition ( i i i ) may be at too high a field to be observable; so only one transition (ii) is observed in general. Figs. 1(b) and 1(c) are shown with the direction of the magnetic field parallel to the direction of distortion.  For other orientations the  4  energy l e v e l s are c u r v e d , M  g  functions.  later  i s 1.722  The  s i n c e the e i g e n s t a t e s  zero f i e l d  crn"^, and  are m i x t u r e s of the  s p l i t t i n g of the Mn^  +  simple  complex d e s c r i b e d  graphs of energy l e v e l s versus magnetic f i e l d  v a r i o u s o r i e n t a t i o n s , u s i n g t h i s value  for  f o r the Z.F.S. are shown i n F i g . 2.  These were c a l c u l a t e d by n u m e r i c a l l y d i a g o n a l i z i n g the f i n e s t r u c t u r e matrix  given  later  (Equ. #18),  u s i n g chosen v a l u e s of D, The  transitions  was  not be  and  g  (0.861 cm" , 1  x  a v a i l a b l e microwave quantum was  the  to  f o r a s e r i e s of v a l u e s of the magnetic  1.992, 1.988 r e s p e c t i v e l y ) .  approximately  i n d i c a t e d were the o n l y ones observed.  0.3  cm" , 1  and  so  The magnetic  s t r o n g enough to enable the p o s s i b l e t r a n s i t i o n at  observed.  field,  field  20,000 gauss  4.000 r — THETA = 30 ••• T H E T A = 6 0 — THETA = 90  3.000  M  o G  pa M N5  >  2.000  7  i.ooo  o -  >o or  -ooo  UJ  2 -« .ooo -2.000  -3.000  •4.000 -OOO  _L  2.000  _L 4.000  6.000  8.000  IO.OOO  12.000  MAGNETIC F I E L D IN KILOGAUSS  14.000  J  16.000  7  Theoretical The calculation of electronic energy levels for paramagnetic molecules in a magnetic field requires the consideration of a Hamiltonian of the form below, which includes magnetic as well as electric interactions.  where  #  Q  = -^2/ m)^ V?  (£  +  2  Cpbeing the total electrostatic potential.  and  H  2  = #  n  + #  +K  c s  ...(1)  I r  The nine terms in iTt^are given in approximate decreasing order of magnitude, and their explicit forms are given below. A.  Zeeman Interactions The interactions with a magnetic field of electron and nuclear spin magnetic moments can be written as  Kzs+Kl  =  Se/3e«'iX  "  Aa^k-Ik  ."(2)  where the sums run over the number of electrons (n) and nuclei (N). There is a similar interaction between a magnetic field and the electrons' orbital magnetic moment which is of the form (6) ^zo  =  m"c E C V ^ ) i  +  2mc2 £  l  &  •••( ) 3  where £^ is the linear momentum operator and A^ is the vector potential of the magnetic f i e l d . %H x r. —  thus X o  where r. is the position vector of the i-th electron, and  —1  —TL  = £ H . E (L i  where U  For a constant magnetic field A^ =  t  e  * £i> + £• £} 8mc2 i  ( r  iE "  'S  is the unit dyadic i  The last term is very small and is neglected since i t is the same for a l l energy levels and thus contributes nothing to E.S.R.  However,  8  i t is important when calculating diamagnetic susceptibilities.  The  form given for <££ ,is correct only for systems having a well-defined 2C  axis, or centre, about which angular momenta can be calculated. B. Spin-Orbit Coupling According to the special theory of relativity an electron moving in an electric field "feels" part of this field as magnetic, and the interaction of the magnetic moment of the electron with this magnetic field gives rise to the phenomenon of spin-orbit coupling.  It may  be shown (6) using Dirac's theory that the contribution to the Hamiltonian is -e  J6s  d A  i^cJr  =  o  >  C <*>I-S.  .,.(5)  for one electron in a central f i e l d , AQ being the electrostatic potential of that f i e l d .  For a nucleus of charge Ze then A^ is Ze/r  spa  d A  6  and thus/C^ is positive since " j ^ S  -  w i l l be negative.  To generalise  this for a system of n electrons, including electron repulsion, two approximations are used.  Each electron is considered  to obey the  same formula, and then a summation is carried out over the total number of electrons.  Also since £(r)varies as  most of the contribution to  i t is assumed that  comes from regions close to the  nucleus, where the electric field should be nearly spherical. =E E  His  i  £k( ki> 1 ^ . r  k  Thus ...(6)  where the potential in which the electron moves is given by N \  °  = y  h L - TZ -s_  k ki r  ^  r  i j  It is conventional to set the radial integral M l  £ (r).r dr  equal to ^  2  where £ ^ is called the spin orbit constant for a  9 nl orbital. A very small contribution to  comes from the electron - other orbit  interaction, where the electron's magnetic moment interacts with the magnetic field set up by the orbital motions of other electrons besides i t s e l f . 'h ^ l ' s = ~~T 2mc C.  This may be written as (26)  V*  4-i.  i ~ ~ s^.(r^ x p.)  ..-(7)  4  Electron-Electron Dipolar Interaction This is the straightforward interaction of two magnetic dipoles a distance r ^ j apart, and is given by t  D.  = g B Y s..(r?.U - 3 r . r . ) r 7 5 e H e < —I ij= — I—y ij —j 2  ss  _  Z  &  s  v  1  ( 8 )  J  Electron Spin-Nuclear Spin Interaction Since there exists a finite probability of finding the electron at the nucleus, at least for s-electrons, then this produces a singularity which must be considered when writing the form for this interaction.  The explicit formula is  * §e%  # 8 1  / 3  e  £  N  EvU'ikll -  3r r. ]r:5 - S T T g ^ ik  k  }  ^  The f i r s t term is correct for p, d and f electrons, which a l l have zero probability of being at the nucleus and has the same form as since electrons also do not coexist at the same point.  ss  The  second term is due to Fermi (27) and is a first-order approximation which gives the same matrix elements as the terms arising from Dirac's equation. E.  It is called the contact term by convention.  Nuclear Spin-Electron Orbit Interaction #11  =  S e % Pe@N  Z  <Iik-Ik> Ik r  i,k  This term has to be considered sometimes (28) since i t can give, in  10  c o n j u n c t i o n w i t h s p i n - o r b i t c o u p l i n g , a second o r d e r c o n t r i b u t i o n t o the h y p e r f i n e i n t e r a c t i o n c o n v e n t i o n a l l y c a l l e d hyperfine F.  "pseudo-dipolar"  interaction.  N u c l e a r Quadrupole  Interaction  T h i s i s o f t e n a v e r y s m a l l term whose main e f f e c t i s t o i n c r e a s e the t r a n s i t i o n p r o b a b i l i t y o f " s p i n - f o r b i d d e n " l i n e s i n E.S.R. s p e c t r a . The e x p l i c i t form i s  [*.-<uE-%ufik>S-i]}  KQ"S{?|%  r  t  G.  R  •••< > u  Smaller I n t e r a c t i o n s The  t h r e e terms i n ^ t ^ , namely, o r b i t - o r b i t i n t e r a c t i o n , n u c l e a r chem-  i c a l s h i f t and n u c l e a r s p i n - n u c l e a r s p i n d i p o l e i n t e r a c t i o n s are a l l very s m a l l . f o r observed  I t has never been n e c e s s a r y  to i n c l u d e them to  account  E.S.R. s p e c t r a s i n c e e x p e r i m e n t a l a c c u r a c y would need t o  be improved by s e v e r a l o r d e r s of magnitude f o r them to be d e t e c t a b l e .  When w r i t t e n i n the form g i v e n above the g e n e r a l H a m i l t o n i a n i s not r e a d i l y a p p l i c a b l e t o the i n t e r p r e t a t i o n o f e x p e r i m e n t a l r e s u l t s . g e n e r a l , i t i s found  convenient  In  f i r s t to d e s c r i b e these r e s u l t s w i t h a  H a m i l t o n i a n i n v o l v i n g o n l y e l e c t r o n - s p i n and n u c l e a r - s p i n o p e r a t o r s , known as a " s p i n - H a m i l t o n i a n " , which i s not n e c e s s a r i l y u n i q u e l y d e f i n e d for  any g i v e n system. The  i d e a o f a s p i n H a m i l t o n i a n was  and Pryce  suggested  and developed  by Abragam  (29,30) and has r e c e n t l y been t r e a t e d i n a more g e n e r a l f a s h i o n  by K o s t e r and S t a t z ( 3 1 ) . The  s i m p l e Zeeman e f f e c t i s r e p r e s e n t e d by o p e r a t o r s ^ C  and *f^, zo . zs r  above and i s S  H.  ( 2 I-  + g  e  S  s.i)  (12)  11  whereas in the spin Hamiltonian this is written as J3 £-&-2.  -..(13)  e  where g is a second rank tensor and implicitly includes the orbital contributions to the Zeeman term. Pryce (30)  The derivation is shown clearly by  and involves calculating spin orbit coupling effects with  second order perturbation theory and comparing matrix elements of (Equ. #12 and #13) .  Kramers' theorem (25)  states that odd electron systems w i l l have spin  multiplicity in the ground state of at least 2 , which can only be removed by a magnetic f i e l d .  This result, together with the Jahn-Teller theorem  (24), ensures that ground states of non-linear molecules w i l l be prbitally non-degenerate with a maximum spin degeneracy of 2 in the case of odd electron systems. These lowest doublets are not pure spin states owing to the effects of spin-orbit coupling.  It is just this orbital contribution which causes  g to vary with direction in a crystal and to deviate from the free-electron value.  These deviations are most noticeable for metal complexes, spin-  orbit coupling in organic radicals being very small for atoms of the f i r s t period.  Since the excited states above the orbitally non degenerate  ground state may be only a few hundred cm"  1  away in metal complexes, then  g can deviate very considerably from 2.00232. a. A general spin Hamiltonian for one J3  + e  ^-'P---  ~ % /^N-'-  +  nuclear spin may be written as +  -----  The tensor D arises from one of two terms in Equ. #1. ward dipolar $C,  Is  •••(14) The straightfor-  term*j*i. or a second order spin orbit coupling term from gs  can contribute to D. =  For organic triplets the dipolar term is of  primary importance whereas for transition metal complexes this is usually less than 57o of the total (32) .  The axis systems in which g and D are  12  d i a g o n a l may different  be d i f f e r e n t .  In o r g a n i c  i n t e r a c t i o n s l e a d i n g to g and  D;  B  where a t r i p l e t The  different  tensor A a r i s e s  terms.  tensors.  I t may  A comes from term  dipolar hyperfine ^l-s  a  n  d  The  or may  and  two  i n some m e t a l complexes  MB  i n t e r a c t i o n i s observed  hyperfine  t r i p l e t s , because of the  a l o n g a metal-metal bond  (33).  i n a s i m i l a r manner from at l e a s t not  (34) be p a r a l l e l to the g or D  i n Equ.  i n t e r a c t i o n " (35)  two  #1  and  a l s o as a "pseudo-  from p e r t u r b a t i o n i n t e r a c t i o n s of  ^ir quadrupolar  sor A i f term  i n t e r a c t i o n P may  i s the primary  be  p a r a l l e l to the h y p e r f i n e  c o n t r i b u t i o n to the  g e n e r a l d i r e c t i o n determined by the  local electric  ten-  l a t t e r or i n some  fields  i n the mole-  cule . F o r systems w i t h S_ g r e a t e r than  1/2  i t may  be n e c e s s a r y  t h i s p e r t u r b a t i o n d e r i v a t i o n of the s p i n H a m i l t o n i a n Bleaney  (36) has  shown t h a t Equ. #2  but  f o r S = 3/2  and  g r e a t e r , terms i n S  order  are a l l o w e d .  f o r most S - s t a t e  In g e n e r a l , the t e n s o r s X  •••  g ) » but f °  = + A  S  S I z  z  + 8J  + A _(S I  This H a m i l t o n i a n w i t h i n the  H  J  x  x  S  x  orders.  ions  g, A, D and  1,  for S = 2  c o e f f i c i e n t of these S  4  higher  terms  (37) .  P have s i x components  ( g ^ * gj^  a x i a l l y symmetric system w i t h g, D and  3 1 1  can w r i t e i n the p r i n c i p a l a x i s c o - o r d i n a t e  /3e(S,, z z / /  r  z z  a l l c o a x i a l we  tt  The  and  w e l l be n e g l i g i b l y s m a l l i n many c a s e s , but  have been n e c e s s a r y  = gy )  i s allowed  3  terms may  to h i g h e r  i s always s u f f i c i e n t f o r S =  and h i g h e r , a term i n (S) H 4  to c a r r y  H  +  S  x  yV  }  +  D ( S  + S I ) + y  represents  z  "  ~ 1/3  y  1 / 3  S  (  S  +  1  1(1+1))  )  A_  system,  )  ...(15)  the Mn"*" complex i n t h i s work w i t h P = 0 4  l i m i t s of the experiments.  13  3.  Sample, P r e p a r a t i o n and S t r u c t u r e Ammonium 9-Molybdomanganate  (NH^gMnMo^O.^-SHgO  T h i s s a l t was r e a d i l y prepared by the method o f F r i e d h e i m and Samuelson (76) and a good y i e l d o f b r i g h t orange-red c r y s t a l s were o b t a i n e d . was  rhombohedral  The isomorphous, diamagnetic  nickel " " s a l t 4  1  prepared by the same method and mixed c r y s t a l s o f the two s a l t s  prepared by slow aqueous e v a p o r a t i o n .  The r a t i o Ni:Mn was 16:1 and  i n view o f the v e r y l a r g e s i z e o f the h e t e r o p o l y anion was s u f f i c i e n t to remove d i p o l a r e f f e c t s from n e i g h b o u r i n g paramagnetic The belongs  s t r u c t u r e was determined to the  - R32 space  of the complex i o n i s planes).  by Waugh e t a l . (77) and the c r y s t a l  group.  per u n i t c e l l and the s t r u c t u r e  centres.  Only one f o r m u l a u n i t  i s shown i n F i g . 3.  exists  The p o i n t group  ( d i f f e r i n g from D^j i n h a v i n g no r e f l e x i o n  14  F i g u r e 3 : Diagram of m o l e c u l a r s t r u c t u r e of ammonium 9-molybdomanganate. Oxygen atoms occupy p o s i t i o n s at the v e r t i c e s of the o c t a h e d r a , and molybdenum atoms the c e n t r e s o f these o c t a h e d r a marked w i t h a s i n g l e broken c i r c l e . The s o l e manganese atom i s at the midp o i n t of the c e n t r a l octahedron, and i s marked w i t h a double broken c i r c l e . The l e f t view i s exploded a l o n g the t r i g o n a l axis f o r c l a r i t y .  15  4. Experimental Methods and Results A standard 9.5MHz E.S.R. spectrometer was used for a l l measurements. A schematic diagram is given in Fig. 4.  The magnet was a Varian model  V-3927-3 with 12" pole tips and a gap of 2 3/4" controlled by a VFR 2501 Fieldial Mk. II.  A V-3921-3 rotating base was used in conjunction with  a V.4533 cylindrical cavity.  A maximum field of 12,200 gauss was  obtainable with the above equipment, and for the very high field measurements at about 15,000 gauss the sample was transferred to a similar spectrometer with a V.3600 magnet. ured with a Hewlett-Packard  The microwave frequency was meas-  5246L Frequency Counter, using a 5256A Plug-  In attachment and the Fieldial was calibrated with an N.M.R. probe and magnetometer constructed by the Electronics Group, Department of Chemistry, U.B.C. The crystal was mounted in a sample holder supplied by Magna Devices which enabled the crystal to be rotated in the vertical plane, whilst situated in the cavity.  Very precise alignment of the crystal was  then possible in the magnetic f i e l d . A typical crystal is shown in Fig. 5 and the axis system used throughout is shown.  In the case of the Mn^ molybdate the crystal axes are +  coincident with the molecular axesThe graphs of energy levels versus magnetic field shown in Fig. 2 indicate the possible field for transitions to occur given the selection rule^M '\+%^  1  s  = ±.1.  - '1-%^  1  A typical spectrum is shown in Fig. 6.  This is the  transition for Q = 30° (where the states are lab-  elled in the zero field notation), and the six hyperfine lines due to the 5/2 spin of the -'-'Mn nucleus are very easily distinguished.  The  small lines in the wings and between the hyperfine components are probably due to super-hyperfine effects from the surrounding Mo nuclei and  KLYSTRON POWER SUPPLY  A.F.C. UNIT TERMINAL LOAD  KLYSTRON POWER METER  ISOLATOR  20 dB COUPLER  FREQUENCY COUNTER  PHASE SHIFTER  n  20  dB COUPLER  ATTENUATOR  NMR  PROBE  CYLINDRICAL  CAVITY  MAGIC TEE  CRYSTAL DETECTOR  MAGNETOMETER  CR.  FIELDIAL POWER SUPPLY  MODULATION  COILS  HALL  PROBE  M A G N E T COILS — r  OSCILLOSCOPE  LOCK -IN A M P L FIER  PH/ XSE D E T E CTOR  —  IOO K c . OSCILLATOR  IOO Kc. MODULATOR  Figure  Block diagram  PREAMPLIFIER  PHASE SHIFTER  4  o f IOO k c . E . S . R .  Spectrometer  X - Y a STRIP RECORDERS  17  FIGURE 5 : A t y p i c a l  c r y s a l o f ammonium 9-molybdomanganate  C  (3)  Coordinate axis for Coordinate axis for spin manifold orbital manifold 4 fold axis of octahedron) (z = 3fold axis of octahedron)  IOO gauss 2800  \ -  FIGURE 6 : A t y p i c a l spectrum o f the <-!> |+i>> t r a n s i t i o n s i n g l e c r y s t a l o f ammonium 9-molybdomanganate  (  Q=  30°) , f o r a magnet  19  this possibility Similar  i s discussed  later.  s p e c t r a were taken e v e r y  10° around a l l three axes and  these showed the c r y s t a l to be a x i a l l y symmetric w i t h i n e x p e r i m e n t a l error.  Use o f the v e r t i c a l r o t a t i o n p r o p e r t y o f the Magna Devices  c r y s t a l h o l d e r , i n c o n j u n c t i o n w i t h the r o t a t i n g magnet now made i t possible  to a l i g n the c r y s t a l v e r y p r e c i s e l y i n the magnetic  As can be seen from F i g . 2 the  -  i t s h i g h e s t f i e l d when the m o l e c u l a r  axis i s p a r a l l e l  field,  and t h i s was the c r i t e r i o n used  The  I  t r a n s i t i o n occurs at  the magnetic f i e l d ,  to the magnetic  i n a l i g n i n g the c r y s t a l .  a c c u r a c y o f t h i s alignment was e s t i m a t e d  mean o f s e v e r a l attempts.  field.  to be 0.25° from the  With the c r y s t a l a c c u r a t e l y a l i g n e d w i t h  then because o f the s t r i c t l y  a x i a l symmetry,  a c c u r a t e s p e c t r a were o b t a i n a b l e at every 5° from the z - a x i s t o the x-y  p l a n e , s i m p l y by r o t a t i n g the magnet through The  '(+%>  1  - ' I-% ^ t r a n s i t i o n  the r e q u i r e d angle.  i s the o n l y one o b s e r v a b l e  o r i e n t a t i o n s w i t h the a v a i l a b l e x-band microwave quantum. i a t i o n o f the p o s i t i o n o f t h i s  at a l l  The v a r -  l i n e w i t h o r i e n t a t i o n provided the  d a t a from which most o f the s p i n - H a m i l t o n i a n parameters c o u l d be established.  The v a l u e o f D i s however, not found  from t h i s  s i n c e the p o s i t i o n s o f the t r a n s i t i o n s are v e r y i n s e n s i t i v e a c t u a l s i z e o f the zero f i e l d  s p l i t t i n g , when t h i s  data,  to the  i s larger  than  the microwave quantum. S i n c e the r o t a t i n g magnet was r e s t r i c t e d and no other t r a n s i t i o n s except  Q=0° ,  i t was decided  the  -  I  to mount the c r y s t a l  which c o u l d r e a c h somewhat more than  to about 12 k i l o g a u s s , were observed f o r i n a n o n - r o t a t i n g magnet  15 k i l o g a u s s .  After  the c r y s t a l by the method d e s c r i b e d above, a t r a n s i t i o n was observed  at around  15,000 gauss and was a s c r i b e d to the  realigning luckily  20  transition (see Fig. 2).  This enabled D to be calculated with cert-  ainty For  9=  0°  E(|-r%> ) = -D +  hg p tf  e  E(|-%>) = "D - %g, /3  e  E(l-3/2> ) = +D - 3/2  g /3 H  (  Then  H H )r  e  20-^/3^  E(l-3/2> ) - E(l-%>) =  = hi/ And  E(\+%>) - E(t-J>>) =  g„/S H e  2  = h V Whence D, knowing g , H^, lA^ and hi/ Since the horizontal rotation in this experiment was measured by a small 6" diameter protractor the accuracy of alignment was not as great as with the rotating magnet and thus D has a rather larger experimental error than the other measured parameters.  As the magnetic field is  moved away from the molecular axis, the |-%^  -\-3/2^  transition occurs  at an increased magnetic f i e l d , and since the magnet only just covered the$=0°  transition, i t was not possible to observe i t for any other  angle. At angles near to Q=0° the \+h} & |-3/2^ states have energy curves which make transitions between them possible at two values of the magnetic f i e l d .  These transitions have zero transition moment at 0°, and  are weakly allowed (23) for Q f 0°  due to mixing of the basis M  g  states by the magnetic f i e l d , as is indicated by the curvature of the energy levels in Fig. 2, for angles other than 0=0° .  At about (9 =  30° the energy separation is greater than the available microwave quantum and so transitions are not obtainable.  Weak transitions were observed  in the 7000 g and 11,000 g regions for values of Q up to 25°; these  21  lines were very weak and the hyperfine structure was not easily interpreted due to the low signal to noise ratio. this transition (Q=  A typical spectrum for  15°) is shown in Fig. 7.  Weak transitions are also possible (38) at angles near 90° between the upper two energy levels in Fig. 2.  The transition moment decreases  from a maximum at 90° to zero a.tQ= 0° and in fact this line could only be observed forQ=  90° and this spectrum is shown in Fig. 8.  This discussion has assumed the ±% states to l i e lower than the *.3/2 states, and the graphs in Fig. 2 were drawn on this assumption (i.e. D is positive).  For a negative D value the labels of the states  would be changed, although the E.S.R. spectra would be unchanged. A l l the observed fine structure transitions are plotted in Fig. 9 as open circles, and the continuous line indicates the theoretical positions in the regions where transitions were observed and the dotted line, the regions where the transition probability is not zero but too small to observe signals. The spin Hamiltonian V.  =/3  (g S HCos0 +g S HSin0)+D(S -l/3 S )+A„S I +A (S I +S I ) 2  |(  z  A  ...(16)  2  2  z  x  x  x  y  y  was found to reproduce the experimental results well when used in a basis of  IM  K y I/  s  product functions with S = 3/2 and I = 5/2.  Because of the complex inter-relationship between the five parameters, i t is not t r i v i a l to extract them from a given set of experimental data.  Bleaney (39) has given a set of equations describing  the field positions, using D and A as perturbations, and accurate to second order in both. These equations are not applicable to systems with so a different approach was taken. ft = Vi  Q  + %t  x  + JC,  &/3^>  a n d  We may write Equ. #16 as ...(17)  11350 FIGURE 7 : A t y p i c a l s p i n - f o r b i d d e n t r a n s i t i o n ammonium 9-molybdomanganate  ( Q = 15°) , f o r a m a g n e t i c a l l y d i l u t e s i n g l e c r y s t a l o f  FIGURE 8 : The s p i n - f o r b i d d e n t r a n s i t i o n observed at Q= ammonium 9-molybdomanganate  90°,  for a magnetically d i l u t e  single  c r y s t a l of  24  FIGURE 9 : The observed fine structure transitions  25  where  | ( Kl K  =D(S  o  =  2  z  /^e « z  H  ( 8  - ., z z A  2  - 1/3 S )  2  S  I  S  +  A  ^  C o s  x(S I x  Si i  +  x  s  s i n  ft  + S I ) y  ... (18)  y  J ^ c a n now be considered as the perturbing Hamiltonian with the magnetic field as a perturbation parameter.  Expressed in matrix form  in the l M ^ basis we have g  • -h  ( and  +k  -D  "3/2  - -  .  -h  +%  -A 2B 3B 0  ZB  A 0 V3B  +3/2  )  +D ' -3/2 V3B 0 -3A 0  +3/2 0 V3B 0 3A , ,  where A = %g Cos 0 M  B = ^g^Sin 6 A general introduction to perturbation theory is given in Appendix •I, and from this we see that since J^-, contains two pairs of degenerate levels, SL ^must be partially diagonalised in the 2 x 2 diagonal blocks before carrying out the perturbation, in order to find the correct zeroth order wave functions. A perturbation treatment was then carried out on each of the basis states in turn and the resulting energies and wave functions expressed as polynomials in the perturbation parameter H.  This was done to third  order in the wave functions and seventh order in energy.  A difference  polynomial in H was then constructed for any two chosen energies, and, with the correct field independent term, a root of this polynomial was  26  found by an extended Newton-Raphsbn (40) procedure.  In order to do  this numerically, i n i t i a l rough values of D, g and g from the experimental data. \~h.y  - \+h}  were chosen  For example:  transition = F (H) = ajH + a H + ... + a H  7  = F (H) = bj^H + b H + ... + b H  7  2  L  0"V|^ll+V^  2  y  2  2  2  .-.(19)  ?  where the primes indicate perturbed wave functions.  The fieId inde-  pendent (zeroth order) term of the difference polynomial is then A E  O  - <+%|tf |+%>- <-%|  K  0  0  - hi/  R >  ...(20)  where h j / i s the microwave energy. Thus, we have for the difference polynomial AE(H)  = A E  q  + F (H) -.F (H) =0  = A E  Q  + H(b - ap + H (b  2  1  2  x  and this can be solved for H. . transition provided a f i r s t guess at H  t r a n g  ^  - a ) + ... +H (b - a )  ...(21)  7  2  t i o n  2  J  7  ?  by the Newton-Raphson method, ' r  is supplied.  Correct wave  functions at this transition field are found by using the previous wave function polynomials and this new value for H . _ . transition v  J  The Hamiltonian ^d+^^was then treated as solved, with eigenvalues and eigenstates obtained in the above manner.  We have next  to consider the perturbation of these solutions by t^2'  ^p  t o n  o  w  a l l calculations have been done in the lM ^ basis, i.e. assuming no g  hyperfine effects.  In order to include these effects calculations  must be done from now on in the complete ]M Mj^ basis. g  When expressed in this basis we have for S.A.I the matrix shown in Table 1. are  However, to express this in the basis [ M^M-^  the solutions of M, + Q  w  e  n e e d  t o  where the |  transform the matrix by the  correct unitary transformation, which was obtained from the f i r s t  27 TABLE 1 : Matrix of the operator S.A.I S = 3/2 and 1=5/2  in the basis Mg-Mj- with  Si  3 S I  LO  sa s  28  perturbation procedure.  This transformation is effectively the co-  efficient matrix from above since C^HC = C^CE and so C^*AC is the correct matrix. After this transformation (^t + S^i) and t4*2 Q  basis and  w a s  applied  w e r e  both in the same  as a perturbation, solutions were obtained  again correct to seventh order in energy.  In order to find the field  for the transition between any two of the resulting 24 energy levels, a new field independent term /\E^ was defined for the difference polynomial. A  E' = Energy of level 1 - Energy of level 2 - 2hJ/ o  ...(22)  It is implicit in this definition that the hyperfine structure levels iMgM-j.^ are parallel to the | M ^ fine structure levels, or in other g  words a strong f i e l d approximation with respect to the hyperfine terms. Since g^H + D  lcm"' and A-—0.01 cm" 1  1  this is quite valid.  Fig. 10 shows this procedure in detail. Finally, this new difference polynomial was solved by the NewtonRaphson method and the hyperfine transition field was obtained.  This  process was repeated for a whole range of values of the parameters until the best f i t with experiment was obtained. The agreement between experiment and theory for the ' I -\ y  1  - '|+^^'  transition is very good and the results are shown in Table 2 together with the best f i t t i n g spin Hamiltonian parameters..  Figs. 11 and 12  show the orientation dependence of the six hyperfine components and the five hyperfine splittings respectively, the solid line being calculated using the above method.  A block, diagram of the computer program  used for the calculations is shown in Appendix II. A l l the computations were made in Fortran IV on an I.B.M. 7044 machine. This method should be very generally applicable to E.S.R. spectra  29  Figure 10  E  H,  Initially  A  E  = Q  E ( B )  H=0 '  E ( A )  H=0 "  h  H  V  = -hi/ for this case since E(b)^_Q = E(a) For a fine structure transition hi/ = E ( b )  H = H t  - ECa)^  For a hyperfine structure transition = E(2)  H = n  = E(2)  H = H t  -E(1)  'H=O  But this is  h = o  -E(l)  - hi/  H = H t  - (E(b)  i f the lines 1 and 2 are parallel to a and b. But we have seen above that E(b)H = H and so &E<  E(a) T  H=H  = hi/ T  = £ ( 2 ) ^ - E d ) ^ - 2hV  H = H t  -E(a)  H  — I  1500  1700  1900  2IOO  2300  MAGNETIC  2500  2700  FIELD  (GAUSS)  FIGURE 11 : The observed hyperfine transitions for the '|+^^' solid line indicating the theoretical curves  1 3500  2900  3IOO  - 'f-^^'  transition, with the  Q O  I  IO  i  i  I  1  1  20  30  40  50  60  1 TO  1 SO  ORIENTATION (0°) FIGURE 12 : The observed h y p e r f i n e s p l i t t i n g s f o r the '\+k} ' w i t h the s o l i d l i n e s i n d i c a t i n g t h e o r e t i c a l curves  1  1-k>  ' transition,  —' 90  A PARALLEL  0.00760CM-1.  ORIENTATION (DEGREES)  MEAN  A—PERPENDICULAR =  FINE STKUCTURE TRANSITIONS MS=(l/2) - MS=(-l/2)  0.00684CM-1, MI VALUES  0 = 0.861CM-1,  G-PARALLEL = 1.9920,  HYPERFINE STRUCTURE TRANSITIONS 5/2 3/2 1/2  G-PERPENDICULAR - 1.9880  (FIELDS IH GAUSS) -3/2 -1/2  -5/2  0.  3420.39  3209.47  3278.07  3352.54  3433.24  3520.58  3615.08  10.  3272.25  3344.31  3123.19  3205.68  3292.37  3383.91  3481.00  20.  2938.71  2696.18  2784.09  2874.36  2967.36  3063.58  3163.59  30.  2586. 13  2347.56  2435.46  2525.50  2617.94  2713.15  2811.55  40.  2291.56  2065.65  2149.39  2235.24  2323.43  2414.26  2508.09  50.  2068.56  1856.32  1935.20  2016.15  2099.40  2185.22  2273.94  60.  1909.60  1708.86  1783.55  1860.28  1939.25  2020.74  2105.06  70.  1804.19  1611.77  1683.41  1757.05  1832.88  1911.18  1992.26  80.  1744.17  1556.71  1626.53  1698.31  1772.26  1848.65  1927.77  90.  1724.69  1539.28  1607.97  1678.91  1752.28  1828.32  1907.31  ERROR =  ro  1.87 DEVIATIONS FROM EXPERIMENT 3.33  1.93  0.86  -0. 64  4.09  3.01  3.22  5.12  4.51  3-24  -0.08  -1.98  1.73  1.49  1.30  3.54  3.64  4.32  4.31  2.64  2.50  3.16  4.15  4.35  1.45  1.01  1.26  1.97  3.34  2.81  1.78  1.40  1.75  1.20  2.58  1.96  -1.46  -1.05  -0.68  -0.25  -0.44  -0.26  0.13  -0.31  -0.15  0.52  1.32  1.04  -0.91  -1.23  -1.41  -1.16  -0.85  -0.27  -1.48  -1.07  -1.01  -0.88  -1.22  -1.61  33  and i t is t r i v i a l to extend the program to include quadrupolar and nuclear Zeeman terms; forbidden transitions are also simple to obtain. One somewhat d i f f i c u l t point is the introduction of non-axial symmetry since this implies complex matrix elements.  However, this  problem has been overcome and a general program w i l l soon be available (41). In order to accommodate systems where g^H } D the following method As an example we w i l l treat the transition observed at Q =  was used.  90° at around 10.6 kilogauss, for which g^H ^ lcm" and D = 0.861cm"''". 1  Since, for the purposes of perturbation theory, there is no restriction on the way a Hamiltonian is divided into fC  Q  and  providing ^C*^  ^  j£ ; $t was taken as Q  %»)^-|S }+(g S Cosc? + g^S Sin#) /3 H' l  //  2  e  where H' is some field less than H  t r a n s  ^ ^ t  gauss (H = 10,000 gauss in our example). 1  o n  by about a few hundred w  a  s  solved by numerical  diagonalisation using Jacobi's (42) method and the resulting eigenstates and eigenvalues taken as new "zero f i e l d " parameters for the purposes of the calculation. The transition field was then found using the method described previously except that the resultant value of H now had to be added to H  1  to find the correct value.  This mode of calculation was used to cal-  culate the positions of the "forbidden" transitions observed ar around 7000 and around 11,000 gauss and also for the "allowed" transition at around 15,000 g as plotted.in Fig. 9. This type of calculation covers the whole range of possible values for D and is thus more general than the equations of Bleaney (24) which only apply for D ^ g ^ ^  #  Other methods (43,44) are equally useful  for the fine structure transitions, but these are not easily extended.  34  to t r e a t h y p e r f i n e n u c l e a r Zeeman or quadrupolar e f f e c t s , and these are o f t e n important.  For example, i n the p r e s e n t case i t i s not v a l i d to  take the f i n e s t r u c t u r e and  t r a n s i t i o n as p c c u r i n g midway between the t h i r d  fourth hyperfine structure  i n Table 2. used third  l i n e s , as can be seen from the r e s u l t s  T h i s i s a second o r d e r approximation which has o f t e n been  i n the p a s t , but i s not always  justified  and f o u r t h o r d e r terms are q u i t e  and i n the present  important.  system  35  5.  L i g a n d F i e l d C a l c u l a t i o n s on the Ground S t a t e of the Complex The  c e n t r a l manganese i o n i s surrounded  by oxygen ions s i t u a t e d  a p p r o x i m a t e l y at the c o r n e r s o f an o c t a h e d r o n .  Thus we may  the c e n t r a l i o n as b e i n g i n an environment which  i s the sum of  i n f l u e n c e s , a l a r g e o c t a h e d r a l f i e l d w i t h a superposed t r i g o n a l f i e l d , due ing  octahedra.  consider two  smaller t r i -  to the arrangement of the n i n e molybdenum-contain-  The one e l e c t r o n d - o r b i t a l s are s p l i t by these  two  f i e l d s i n the f o l l o w i n g manner ( 4 5 ) . F i g u r e 13  Free  p  Ion  Trigonal Field W h i l e the r e l a t i v e p o s i t i o n s of the e and t the o c t a h e d r a l f i e l d , the o r d e r of the a^ and trigonal field  2  orbitals is fixed in  lower e o r b i t a l s i n the  i s not unique but r a t h e r depends on the d i r e c t i o n of the  d i s t o r t i o n from 0^ symmetry whether by e x p a n s i o n or c o n t r a c t i o n a l o n g the t h r e e f o l d a x i s . §  and  There i s no a p r i o r i way  of f i n d i n g the s i g n of  i t i s not always p o s s i b l e to d e c i d e upon t h i s by  I g n o r i n g the t r i g o n a l f i e l d strong f i e l d  o  calculation.  f o r the moment, we can c o n s i d e r the o  2  configurations t ^ , t^e, t e , e  3  for a d  ^  J  system, each  36  separated by an amount /\ from the other.  Electrostatic repulsion  between the electrons splits these configurations into "strong f i e l d " states.  Viewed from the weak field standpoint the free ion states  of the d  3  configuration are split into various states by the octahedral  field and we may draw a correlation diagram of the form shown in Fig. 14.  The energy axis is not to scale, although the terms are in the  correct order.  From the available experimental results for optical  spectra (46,47) the d^ ions approximate intermediate  , Cr^+ and Mn  4+  positions indicated.  are found to occupy the In performing calcul-  ations on the ground state of the ions, i t seems reasonable to consider effects of the doublet states from t^ and the quartets from t ^ "  ^  e  calculations performed for d^ systems previously are reviewed below, and i t w i l l be seen that most authors have neglected  these lowest  doublet states. A.  Survey of Previous Calculations on d^ Systems Many calculations have been made on d  systems and i t is hoped that  this survey w i l l help to correlate a l l the results. Let us f i r s t define some terms:  V,  = A L . s X  so  ti  = r^efi-i-i  z  S w .  =  D  I  :  :  E  A  X  =  t  /3  Z.F.S. = 2D  X L . S | J ) 1  M  J  1  1939  C i h - ^ i  2  ij  Van Vleck (48) was the f i r s t to treat the problem using the 3+  data on the Cr  A  o  a d  J  ion is  J  ion in chrome-alum.  The ground state of  F and this is split by an octahedral and trigonal  field as shown in Fig. 15.  The ground state, ^A , 2  is split  by spin-orbit coupling in conjunction with the trigonal field  37 14 : Complete c o r r e l l a t i o n diagram f o r a d scale)  Weak O field  h  Intermediate  crystal  field  3  system (not to  Strong field terms  Strong field configuration  38  effect.  The d i f f e r e n c e i n energy between the two r e s u l t i n g  d o u b l e t s , 2D, i s g i v e n by 2D  = 8  w  t e )  " 8  w  O/2) =  +  7  2  K  9  T h i s comes f r o m s p i n - o r b i t l e v e l s o f ^^2' ^ t  s  ^  ( i . e . ± 3 / 2 lowest f o r K >  2  A  0) (23)  c o u p l i n g between  A^ and the s p l i t  l a t t e r s p l i t t i n g b e i n g due t o the t r i g o n a l  field. Figure  15  .  Octahedral  Field  Trigonal Field  Van V l e c k a l s o thought t h a t the s i g n o f 2D was c o r r e c t ,  since  K i s p o s i t i v e from c r y s t a l s t r u c t u r e arguments. 1942  Broer  ( 4 9 ) performed a s i m i l a r c a l c u l a t i o n w i t h a s l i g h t l y  d i f f e r e n t c r y s t a l f i e l d operator 88 2  1948  °  K \  2  9 - 7 ?  =  Weiss ( 5 0 ) used the same o p e r a t o r and found the same v a l u e as Van  Vleck g = 2( 9  1955  and o b t a i n e d  1 -__iA__) A  +2K/3  M e i j e r and G e r r i t s o n ( 5 1 ) performed the c a l c u l a t i o n  using  39  zero order functions for the spin-orbit coupling perturbation which were already eigenstates of the total crystal f i e l d . They suggested that Van Vleck's formula was derived using only octahedral field eigenstates. =  152K\  2  This is different in sign as well as magnitude from Van Vleck's result 1957  is lowest from this result, for K>0).  Jarrett (52) did not give a theoretical expression for 2D but only presented a graph of calculated splittings versus the ratio of the axial to the cubic components of the crystal field.  He appears to agree with Meijer and Gerritson in that  a negative value of K indicates that +3/2 states are lowest. 1957  Davies and Strandberg  (53) recalculated Van Vleck's result  without mentioning the original result.  They did however  mention Meijer and Gerritson's paper but without comment on the discrepancies. 1958  Sugano and Tanabe (54) made a very careful study of the opt i c a l spectrum of ruby and found that 2D = 0.36 cm"  1  was approximately  -134 cm"!.  and K  This difference in sign did  not agree with Van Vleck's ideas and so they suggested the use of anisotropic spin-orbit coupling, defining the parameters as follows )  (Author's approximations in parentheses)  state is split by a trigonal distortion and a l l the unknowns  40  are lumped t o g e t h e r  i n t o the s p i n - o r b i t c o u p l i n g parameters.  T h e i r e x p r e s s i o n f o r 2D was  2D = -• 9  The  WvT/  . « i— 9  12K 8 / 2 ^ ( Cx Cn  \ 2K  +2  W^ / 2  3  "A / 3  3  V E T T ^ T ! ) /  f i n a l c r o s s term comes from d o i n g the c a l c u l a t i o n by  u s i n g the t r i g o n a l f i e l d and the s p i n - o r b i t c o u p l i n g  together  as the p e r t u r b i n g terms. By a l i t t l e  a l g e b r a and some a p p r o x i m a t i o n s t h i s can be  b r o u g h t i n t o the same form as Van V l e c k ' s f o r m u l a t i o n . making the a p p r o x i m a t i o n reasonable 2D  By  t h a t E ^ T ^ ) = 2 E ( ^ T ) which i s 2  from the s p e c t r o s c o p i c r e s u l t s , we f i n d  \ _  (16 + 8 - 9 )  K  w h i c h i s s i m i l a r i n magnitude t o M e i j e r and G e r r i t s o n ' s r e s u l t , but opposite for  i n sign.  This p r e d i c t s a n e g a t i v e  2D s i n c e K i s i t s e l f n e g a t i v e .  value  I n t h e i r paper Sugano  and Tanabe s a i d t h a t s i n c e the l a s t two terms i n the e x p r e s s i o n are n e a r l y e q u a l and o p p o s i t e , then 2D i s p o s i t i v e f o r T \ f whatever the s i g n o f ition, L  L  K .  Unfortunately using their d e f i n -  i m p l i e s t h a t / \ > / \ and t h a t K  i s positive.  T h i s c o n t r a d i c t i o n a r i s e s , I f e e l , because o f the o b s c u r i n g of the e x a c t e f f e c t o f the t r i g o n a l f i e l d on the system by the use o f the parameters  and T .  f" ^1/  more s e n s i t i v e t o c o v a l e n c y  g-values are f a r  E l -  than i s 2D, and the e r r o r may l i e  i n the t r a n s f e r r i n g of g-value parameters d i r e c t l y t o those f o r 2D. S p i n - o r b i t c o u p l i n g p a r a m e t e r s , b e i n g r a d i a l i n t e g r a l s , may  41  well be anisotropic i f treated correctly.  However, the  assumptions involved in using this form of the spin-orbit coupling terms for n-electron systems have already been discussed in Section 2B.  One of these assumptions is that  since the radial integral varies as r " then the contrib3  ution is greatest near the nucleus, where the surrounding electric field is approximately spherical.  This spherical  field implies isotropic spin-orbit coupling factors for an electron in a nl orbital.  When the electron is in a mole-  cular orbital then ligand contributions to £ ^ must be conn  sidered.  Most common ligands (0, F, N) have small values  of £ and produce only small deviations from the free ion value for the central metal ion. 1959  Stahl-Brada and Low (55) measured the zero field splitting in spinel by E.S.R. and gave Van Vleck's result although in a different form  8\2  8\ :K 2  K  They were puzzled by the large value of 2D (0.990 cm" ) com1  pared with the results for ruby. 1961  Sugano and Peter (56) solved an 80 x 80 secular equation for ruby by a numerical method and varied parameters to get the best f i t with experimental results.  It is d i f f i c u l t to argue  with such a result, but i t gives l i t t l e insight into the physical situation and each problem has to be attacked separately.  However, they were the f i r s t to consider states  other than the T 4  2  42  1962  Kamimura (57) used the idea of distortion of the t„ molecular 2g orbitals through 7T-covalency in a complex. This distortion, represented by a parameter q ( */|/£)> gives a formula for =  2D which is very similar to Van Vleck's  A,  A„  where j3y and yQ^ are coefficients from configuration mixing. This is similar to Sugano and Tanabe's anisotropic spinorbit coupling but not as ad hoc. 1963  Lohr and Lipscomb (58) used the Sugano-Tanabe-Kamimura formulation but pointed out that a l l previous calculations had neglected the considerable contribution to the Z.F.S. from the x „ levels of t which occurs as well as the T„ from 2g 2g 2g 2 t geg. The formula that they arrived at is 2  3  4  2  \ 2D = -8 (  4  -\  V * -V" -  x  7 E( T )  2  )  "  6  (  ••  ) " smaller terms  4  2  with X  M  and Xj_(c lose ly related to /^and £ respectively) , def(  ined in terms of one-electron operators and molecular orbital functions.  The energy terms in the denominators are taken  from experiment, and the molecular orbitals are arrived at by the extended Huckel method.  The trigonal fields involved  are assumed to be small in this treatment and in this s i t uation they point out that the g-value can be isotropic even if a sizeable zero field splitting is observed. 1964  McGarvey (59) calculated 2D for systems with large trigonal fields and introduced the effect of distortion of the oneelectron d-orbitals by the ligands.  Both this, and a para-  A3  l l e l extended H u c k e l m o l e c u l a r o r b i t a l c a l c u l a t i o n were used to  get v a l u e s o f the d i s t o r t i o n parameters  v a l u e s f o r energy field  using experimental  l e v e l s i n the system, and f o r the zero  splitting.  E s t i m a t e s o f g-values were made, and i t  was found n e c e s s a r y t o i n c l u d e the e f f e c t o f i n c l u d i n g e x c i t e d s t a t e s o b t a i n e d by promoting a b o n d i n g e l e c t r o n i n t o the a n t i bonding 1966  d-orbitals.  Owen and T h o r n l e y (60) g i v e a v e r y g e n e r a l r e v i e w o f c o v a l e n t b o n d i n g and i t s e f f e c t on magnetic metal ions.  properties of t r a n s i t i o n  They d i s c u s s the e f f e c t s of Q- and 77- bonding on  the o r b i t a l magnetic moment, s p i n - o r b i t c o u p l i n g and Racah p a r a m e t e r s , and a f t e r c o n s i d e r i n g a l l the a v a i l a b l e  calcul-  a t i o n s and e x p e r i m e n t a l e v i d e n c e , t h e i r c o n c l u s i o n i s t h a t the e f f e c t s o f c o v a l e n c y on zero f i e l d  splittings,  g-values  and h y p e r f i n e c o n s t a n t s are a t b e s t i m p e r f e c t l y u n d e r s t o o d .  In view o f these p r e v i o u s r e s u l t s i t was d e c i d e d to perform a c a l c u l a t i o n on the d  3  system w i t h i n the l i g a n d f i e l d  framework, i n o r d e r  to o b t a i n more i n s i g h t i n t o the problem, and t o a r r i v e a t a l e s s e m p i r i c a l e x p r e s s i o n f o r the Z.F.S. than t h a t o b t a i n e d by McGarvey ( 5 9 ) . I t was found u s e f u l t o c l a s s i f y the t h r e e - e l e c t r o n d e t e r m i n a t i t a l f u n c t i o n s a c c o r d i n g t o the double group D3, s i n c e t h i s i s c o r r e c t f o r s p i n o r s w i t h i n an S = 3 / 2 m a n i f o l d , and i t reduces the a l g e b r a t o a reasonable l e v e l .  The c o m b i n a t i o n o f d - o r b i t a l s , f i r s t used by Pryce  and Runciman (61) , i n which the t r i g o n a l a x i s o f the o c t a h e d r o n i s the a x i s o f q u a n t i z a t i o n , seemed the l o g i c a l b a s i s t o u s e . i t a l s are t  D  « l/v/3(d  xy  + d  y z  + d  x z  )  The f i v e o r b -  44  t -TlA#(d e where w = e  27Tl  + w d il  ±  x y  = TlV2(d 2 ± z  ±  i d x  yz  + w  ± 2  d  x z  )  2. 2 >  ...(24)  y  ^ , in terms of the usual real orbitals having the four3  fold axis for quantization. A diagram of the way in which the strong field configurations, t^ 2 and t2"e, are split by electron repulsion and a trigonal field are shown in Fig. 16.  The classification is given in terms of the representations  of D-j as well as the more usual D^. Instead of using a l l the 120 functions of d , which would be a very 3  large task, i t was decided to use just the 20 arising from the t g con2  figuration and 24 from the 2g g configuration which gave rise to the t  4  T, and lg  e  4 T„ states of the 0, scheme: these latter were obtained by 2g h j+ + + I 3  the use of shift operators on determinants like I e_t_t J.  This sel-  +  ection is justified since the other states have considerably higher energies and would thus contribute much less to the Z.F.S.  Subshell  configuration mixing was neglected since the energy corrections are of the order of 50B //^ (62) where B is Racah's (63) electron repulsion 2  parameter and /\ is the strength of the octahedral f i e l d . free ion Mn"", B is estimated to be 1060 cm" 4 1  1  (6) and ^  For the is approx-  imately 22,000 cm"-'- from the optical spectrum (46,47) and previous work 4+ on Mn  (11-22).  So the correction is fairly small (-v 107» of the  energy of the lowest doublet). Table 3 gives the 44 determinants used for this calculation, and Fig. 5 shows the co-ordinate axes used for the spin and the orbital manifolds.  These were chosen so as to make the algebraic manipulation  as simple as possible. The complete character table for D3 is given in Table 4.  We  may  45  FIGURE 16 : Diagram o f the s p l i t t i n g of s t r o n g o c t a h e d r a l f i e l d conf i g u r a t i o n s , under the i n f l u e n c e of e l e c t r o n r e p u l s i o n and t r i g o n a l f i e l d operators. The l e f t s i d e of the d i a g r a m i s not t o s c a l e , b u t the r i g h t - h a n d _ s i d e i n d i c a t e s b o t h the l a b e l l i n g o f the s t a t e s i n the D3 group, and the c o r r e c t o r d e r i n g o f l e v e l s when B = 700 cm" , 5 = 7000 c m and A = 22000 cm' 1  - 1  1  etc.  4  Strong field configuration n U  h  4  Strong field terms. + Electron repulsion  +  D  3  D  3  „ Dj  46  TABLE 3 The 44 determirental basis functions  3 t  configuration  *3= l*oM-l  *4 = ivM-i * i^oMJ • *7 l*oMJ  *11  =  *12  =  *13  =  *14  =  *15  =  *16  =  *17  =  IW-I"  = *8  *18  =  *o = i t t 1 9 o o +' i|j, = It t t J 10 o o +  *19  =  *20  =  1  n  1  t e configuration  •  t t t  MA MA  ;  1  r  r  t ! t o o o + +  o - -  =  6 =  t"t t o o -  = 1//3[| ijE J  2  +  (quartets only).  | +1 e _ t _ l | + | e _ t _ t | ] +  +  <j> = l / / 3 [ | i _ t j | + | e _ t _ t | + |e_t_t |] 3  +  *5  =  *9  =  *13  =  |et 1  -  -  t  *21  =  where  The remaining components of the quartets  IMA 1  +  • +  +  0  IMA  <t>17 = i t t t  +  are found by operating with S_ on these +3/2 components.  -  0  +  0  stands f o r a normalised Slater determinant  TABLE -4  Character table for D*, together with the behaviour of spin and spatial basis functions E  C  3  C  3  CR  C  2  1  1  1  1  1  1  1  -1  -1  r-»°i 2  Lo-to]  I  i  e  a..  1  l t  o  t  a S  1  1  -1  -1  -1  1  T  1  1 2  o-ico ] f o-ico| f-1 o"l -itoo J L-iu) Q| L o-lj 2  2  1  -1  -1  -i  -i  -i  -1  1  1  t  t  t  t  cot  2  +  U)t_  to e coe  ,  +  -coa -co 6 2  - t ,  t0 t_  .  o  +  2  o  2  " + e  -ie  -ia _2Tri/3  +  t _  cot_  2  e  to e coe  -itoB -ico a  -ito B -itoa  -ct -6  coa to 6  2  2  +  2  2  +  2  a and 3 are spin functions  o  - t  C0 t_  -toe -co e  +  t  o  coe coe  +  -*+  -e  2  i  i  t  t  o  o -tot_  -C0 t_ 2  -cot  - 0 )  +  -co e -coe 2  +  2  cog  Lito^oJ -i  -i  i  t  o  -co e -coe 2  .-i  (Ot  -co t  -*+  -e  t  -cot  -to t  2  2  -to^a -coB where co  t  o  2  coe coe  t  r? H  2  1  o  3  Toico ] |ico o J  2  -1  o  U  Hi  [o°] [o °]  i  2  I ft  1  1  i  %  e  1  1  3  i  to t  +  1  CR  -1  o  e  CR  3  -1  o t  2  CR  [St Ell 1  2 CR  R  3  1  2  c 2 U  [too I  • "{  2  ia  icog ico a 2  2  t  +  -toe -co e 2  +  ito B itoa 2  4>  48  note here that two of the original C  2  operators in  one of the C R operators of D3, and the remaining C 2  another class with the other two C R operators.  C^R  and C ^ R  form a class.  2  operator is in  The particular C  2  ator depends on the axis system chosen.  form a class with  2  For the present system  oper»  The final six operators of D3 have char-  acters identical to the f i r s t six except for the E" representation where the sign is different for the two sets. Table 4 also gives the summary of the way in which the five orbitals together with the CL and /5 spin vectors transform in D3.  The transfor-  mation of the CL andyQspin vectors was derived using the formulation given by Ballhausen (64). e  ie  ;  i<  0 .0 ,  )/2cos  . - i 0 / / - < £ ) / 2s  n  The matrix i e  2  /2  e  1  ^ - ^  )/2  - K ^ ) / 2  S  i n Q /2  C  o  0  /2  s  is. given for co-ordinate system rotation and so for vector rotation for co-ordinates.  e.g. for a vector  / i ( - 2 7T/3)/2 e  C  since  = 120°,  3  =  ^  \ = 0°,  0  Q  v  0  e" " ^ Z ) / i(  2  3  2  )  = 0°  •(•:.:•) A l l the other operators can be treated similarly. By operating with the various operators in the group on the determinants in Table 3 we arrive at the symmetry adapted set given in Table 5. The f i r s t step in the partial removal of degeneracy from these 44  49  TABLE 5  Symmetry adapted determinantal  a  B  '  Hi  He  *13  *18  *19  6 configuration )< *10 t  -he  12  2Q  1//2(* +* ) 1  12  " *13 - *22  * 19  "*2  *18  " *11  * 2 i • - •a " *5  17  20  l//2(i|» -* ) 1//2(* +* ) 14  15  l//2(d> +d> ) 9  20  14  15  l//2(<j> -4> ) 9  20  1//2(4> +4. ) l//2(<j) -<f) ) 17  *15  *24  g  17  Ho  [ *7  l//2(i(/ -^ )  1//2(* -* ) l / / 2 ( * + * )  \  V  2  8  h  *i  P  l  l//2(^-iJ; )  H  configuration  p  functions  12  l//2(* +c(. ) 6  23  l//2(ct. +cf> ) 14  3  17  12  l//2C"* -4» ) 6  23  50  functions is to find the eigenvalues and eigenvectors of the electrostatic repulsion operator ^ ) e / r . . . i>j 2  Various quantities like the  1 J  Coulomb integral J ( t , t ^ are needed in the calculation, and this one is +  worked out in detail in Appendix III. The rest of the set of integrals in Table 6 may be worked out similarly.  The results are expressed in  terms of Racah's parameters. The matrices of ^ " l e / r . • for the E"(n ) and E'(Q) symmetry species i>j H are given in Table 7. E"(p) and E'(^3) functions w i l l have identical 2  1 J  matrices and these are not shown. On solving these matrices completely, we arrive at the set of eigenfunctions and eigenvalues given in Table 8.  It w i l l be noticed that  the eigenvalues and their respective degeneracies are identical regarding the electrostatic part, to those given by Ballhausen (64).  For con-  venience the eigenfunctions are renamed at this point. The second step in this calculation is to find the eigenfunctions and eigenvalues of the trigonal field operator.  This field is intro-  duced via the one-electron parameters § and /\ , defined in Figure 13. The original determinants have their energies modified as shown in Table 9. The matrices for the trigonal field operator in the basis of the newly defined functions are given in Table 10.  Only t r i v i a l  2x2  determinants have to be solved to find the eigenfunctions and eigenvalues and these are given in Table 11. In order to introduce spin-orbit coupling into these eigenfunctions it is necessary to calculate the matrix of / \ ^ l ^ ' S . ^ i 11+^  in the basis  etc., with the spin and orbital parts of the spinor referred to  the same axis system.  This was done by referring the t and e orbitals  defined in Equ. #24 to the spin axis system given earlier.  The result-  51  TABLE 6  Electrostatic repulsion integrals, given i n terms of Racah's parameters  J ( t o',to ) = A + '4B + 3C J ( t , t ) = A + B + 2C +  +  = J(t_,t_) = J(t_,t ) +  J ( t o .t.) + = A - 2B + C = J(t ,t_) Q  J(e_,t_) = A + C K(t o',t.) +  -  = JCe..,^) , - = ±,  = ±,o.  i  3B + C  = K(t ,t_) = %K(t ,t_) Q  +  K(ei,t^) = 2B + C ( o +l - t  t  t  t  )  =  0  =  ( t  , i = ±, j = ± or.o. o -l + + t  t  t  )  ( t t | t _ t ) = - (3B + C) Q  o  +  (e t |e.t ) = 4B - - + o 1  = " <e+t |e_t > +  0  = - (e_t |e t_) +  +  = (e_t |t_e ) +  +  = - (e t It e ) - - o where J(a,b) = / a*(l) b*(2) —  2 a  2 K(a,b) = / a*(l) b*(2) — r  b  d r  2  (ab|cd)= / a* (1) b*(2) — r  (l) (2)  12  a(2) b ( l ) d r c(l) d(2)dr.  52  TABLE 7  Matrices o f  Z e i>j r i j  3 i n t h e E'^CP^) and E'(a) b a s i s f o r t  2 and t e  c o n f i g u r a t i o n s , g i v e n i n terms o f Racah's parameters E""(P,) t  3  l//2(i|) -i|» )  l//2(4- -^ ) g  0  3A - 15B  3A-6B+3C  0  0  0  0  0  0  0  3A-3B+4C  0  0  3B + C  E^(a)  t  20  3B + C 3A-3B+4C.  d i v i d e s i n t o two 3 x 3 m a t r i c e s  V  16  *2 "3A-6B+3C  *3 -(6B+2C)  -(3B+C)~  -(6B+2C)  3A-6B+3C  -(3B+C)  -(3B+C)  -(3B+C)  3A-9B+2C  E  17  12  3A-6B+3C 0 0  '18  '10 0 3A-3B+4C  3B+C  3J3+C  3A-3B+4C  ( P ) t e d i v i d e s i n t o two i d e n t i c a l 2 x 2 m a t r i c e s 1  l/^2C*g+* ) 20  1//2^ +<|) ) 17  -6B  T3A-9B I  E^(a)  -6B  12  and  s i m i l a r l y f o r l//2(cf>g-<|> ) and 1//2(<(>-,—cf>^) lH 3 ' 23  y  u Y  3A-9B  t e d i v i d e s i n t o f o u r i d e n t i c a l 2 x 2 m a t r i c e s , each h a v i n g t h e same  elements a s t h e E  (P^) t e above.  Pairs of functions  I  *1 *21  \  *7  *io  *15  h&  53  TABLE 8  Eigenfunctions and eigenvalues of the electrostatic repulsion operator Eigenfunction l  x  x  1/72(^-1^)  =  3  x  p  a  =  l//2(ijJ +ij; +^ )  3  x  5  =  X  6  =  17  x  18  3  5  4  6  7  E" 3A - 15B  1//2(I(; +^ +^ ) 2  a  l//2(^g-i|; )  6  7  =  h ( *  + g  ^  < r  + 1  2  ,  + 1  7  < r  ,  ) 2  n  P  l  p  2  x  19  =  1/'^2(T|> +* )  a  x  20  =  l//2(t|;  6  x  x  x  8  9  x  10  x  ll  x  12  x  13  x  x  1 0  1 1  1 8  +^  1 9  )  4  J  =  s  B  J  L 2  L 7  2 0  2(^ +'l' -^ -* ) g  12  = l//2(^ -^ 0  17  1 8  )  = l//2(*  l u  1 4  X  20  p  l  p  2  a  3A - 6B + 6  -i(.  1 5  )  14 = l / / 2 ( ! j i + i | < ) 15  6  7  2Cip -^- -^- +^ ) g  =  3  M2i> -y -ii> )  1/  =  2  =  he  1 5  l  p  P  2  a  E' e  16  3A + 5C  a  1/^6(2* -T|) -I|; )  =  7  Eigenvalue  2  =  4  X  l  p  =  2  x  Symmetry species  54  TABLE 8A Eigenvalue  Eigenf unction E'(a) B  E ' ( e )  9 ': - l / / 2 ( *  = l/* 2(<J> -4» ) /  1  1  21  11 = - l / / 2 ( *  l//2(cf> -cf, ) 1+  2l+  -* )  1 6  8  1 3  -* ) 5  3A - 3B  13 = - l / / 2 C * - * ) 2 2  1 / / 2  ' *10-*18 C  i g  i : L  10 = -l//2(<t> +4> )  l//2(<fr +<f> )  12 = -l//2(<j> +(j) )  l//2(* +*  14 = -l//2(<). +(f» )  l//2(*  i 0  1 5  +*  ) 1 8  5  22  3A - 1 5 B  2  16 = - l / / 2 ( * + ( ( .  )  )  8  13  24  7  =  16  21  4  8  615 = - l / / 2 ( ^ - < t .  )  1//2(<|> +(|» ) 1  6  2  1 9  : L 1  )  E"(p ) 2  l  a  =  ^V^O""^ ^ -  a  3  = 2(* +* 3-<(> 4-<('3)  a  2  = %(* 'C20  a  3A - 3B  J  6  2  +  +<t>  9  17 *12 +  4 = ^ V*23 *14 V (  +  +  5(<r "*23-*14 *3  3  :L  +  )  6  )  J  sCl'9-*20 +i7-+12  3  +  sC* -+23 *14"*3 +  6  )  )  3A - 1 5 B  55  TABLE 9  Function  Crystal Field Energy  V  *1  0  - 6 *9  * 0  *13 " *17 *  +•6 * 2 0  • l *V *5 " *12 *13 " *16 *17 -  *24  26 A  +  A  -  A  +  A  -  3  X  3  26/ 7  3  X  3  56  TABLE 10  Z e i>j r i ]  Matrices of the c r y s t a l f i e l d operator and  (V t  E-  3  x  x  l ' 13 x  a  17  x  19  e  u n c  'ti  o n s  with unchanged eigenvalues  9  -6 3A--6B+3C  -6 3'  i§ ^  e e  x  3A + 5C  x  r  x  5'  and X  Xj ?  x  1 5  are  ll  T3A + 5C 3A - 6B + 3C P 2 E " ( D t e divides into two 2 x 2 matrices  [  ]L  73  T3A - 3B + A  73  3A - 3B + A 0  3A - 15B •+ A- 7 3  - /2 6  3A - 15B + A  -'/2  E' (a) t e divides into four 2 x 2 blocks  X+A+ /6 6  6  /2  5  /2  X+A+ /6  ][  Y+A+ /6. 6  5  6  /2  6  /2  Y+A+ /6J 6  X+A+ /6 6  - 6  17  _<S  "X+A-^3  /2 6,  Y+A+ /6  where X = 3A - 3B, Y = 3A - 15B  r  0  C  ]  Y+A'^  57  -TABLE 11 E i g e n f u n c t i o n s o f both t h e c r y s t a l f i e l d and e l e c t r o s t a t i c r e p u l s i o n operators E"  17  x  9  a x  a x  =  2  a  2  x  =  g  a  + b  a  a  7  °8  x  19 ll  =  = a  8£ = 3  2  8  3 =  =  S  + a  5  =  S  6  =  a  16  l ll x  2 U x  B  4  B  b  6  3 5  + b  2  3  + b  B  3 6 B  2  2  20 = a  x  4  2  i  3A - 6B +,3C 3A - 6B + 3C  l X 2 0  12 = a X  x  3A + 5C  2  l X l 2  2  1 2  F + ^ ( G + 6 )% 2  B  9=  B  10  e  ii  B  :  a  3 9~ 3 10 6  b  6  2  9  .9  a  3 ll 6  12 =-a e 4  _ b  il+  3 12 6  b B 4  F + h(G  + 6 )%  2  2  F - ^ ( G +6 )^ 2  12  2  F + %(G + S )^ 2  6  li  B  1U  V2  F - jj(G +6 ) =  =  a  3 13 3 14 B  + b  B  = a B +b e u  13  li  F - ^(G + 2  i4  2  S) 2  2  6  r  • !6 e  2  S j  3A - 3B + 4-C - ( E + 6 f 2  +b x  2 0  3A - 3B + UC + ( E + 5 ) 2  +b  3A - 3B + A - /3 *8  2  3A - 15B  x  3 3- S.  B  4  15  1 +  (G + S )^ + A  3A - 9B - ( G + 6 Y + A  7  x  4  /3  8  4  8  b  6  = a a +b a  :  x  6  fi  3A - 9B +  3  7  + b  2  A - /3  3A -'-3B +  7  X  19  3A - 3B + 4C - ( E + S )^ 2  1 Q  8  3  6  x  2 3  := a a + b a  X  2  fi ) *  2  3A - 15B + A-  5  - U 3 4 4  B  6  3A - 3B + UC + ( E +  E-(e)  3 l- 3 2  a  a  x  -a B b 6 + a  5 •  x  + b  2  a  3  x  b x  + 1 8  x  l 19  a  X  2  E'(o)  x  3A - 6B + 3C  4  a  2  3 3 3 U  3A - 15B  2  18 = ^ g ^ X ^  l  a  =  +b x  1 7  a  <*3  x  13  x  eigenvalue  2  l  x  x  E"(p )  (P-L)  3A - 15B + A - /3  where E = 3B + C, F = 3A - 9B, G = U B  S  7  2  58  ing  m a t r i x f o r the ten s p i n o r s i s g i v e n i n Table  12 i n u n i t s o f £  i s e q u a l to the r a d i a l i n t e g r a l f i ^ R ^ ^ (r) r d r .  Here R ^ i s the  2  r a d i a l p a r t o f the wave f u n c t i o n and £ i s thus p o s i t i v e . interested  i n c a l c u l a t i n g the  zero f i e l d  which  We are most  s p l i t t i n g o f the ground q u a r t e t  and  a second o r d e r s p i n - o r b i t c o u p l i n g c a l c u l a t i o n should be adequate  for  t h i s purpose.  A c c o r d i n g l y o n l y the top row elements o f the s p i n -  o r b i t c o u p l i n g m a t r i x have been e v a l u a t e d  and these are a l s o given i n  Table 12. E x p r e s s i o n s f o r the zero f i e l d  s p l i t t i n g can now be w r i t t e n and  .Xi»X F o r  these are g i v e n below, c o r r e c t t o second order i n £  2  the second o r d e r c o r r e c t i o n t o the energy i s given by  [  2  2  2 l  L4E+J l + _4E-J f2_ + a  and  4E+J XA  for  (2) E E  w  e  '-g/3 1  ^ A  4E-J have  2r 2 = -T _ 2 _ + l + ^ L15E 3(4E+J)  =  2  + 3(H+K) 2b 3 3(H+K)  2 2 + 3(4E-J)  a  a  1  where E = 3B+C ; G = 12B ; H = G/2 + ^ And  E  t h u s , the zero f i e l d  (2). (2) E  -t. \- + 2  =>  +  2bS 3(H-K)J  2  ,,, 2 3 + 9(H+K)  4  1  4  -.  I  !4b 9(H-K)  b  1 9(^-§/3)|  +  + § /6  '...'(25)  s p l i t t i n g i s g i v e n by  2*1 +  2  =  o  2  2_  2*1  +  8  ^  LL5E 3(4E+J) 3(4E-J) 9(  -  -g/3)  8b , 2  . _8b|_]  9(H-K)J  9(H+K)  ...(26)  * XA  AN<  comprise the  d o u b l e t and so we can w r i t e  e x p r e s s i o n s f o r the s p i n - o r b i t c o u p l i n g c o r r e c t i o n s to these wave f u n c t i o n s , c o r r e c t t o f i r s t order i n {* .  - r l'  2  X ' l\X m  3  N L  X  5 - l  ( 5^6E  3  2  /  f f ' 3 (F+K) 2  i  b  3  Y A  4  -  1  "W  5  a  i  X19  g  3(F-K)  6  0  Xn  2 i b ff.  +  3  /3(4E-J)  -  X <j (")X+(")X2+(") 4  - a l  V3(4E+J)  K  2  +  n  1 +  2ib  /6 (F+K)  4  ff  ff 8 _JJ vl 3 ^ -§  & 9  +  n  P  2  X6(F-K) (27)  \ o '  n  P 13-  (,,)  ^  W  - /5i ! (,,)  6  where  (E +g )% 2  2  =  J  and %  ( G  2  + §  2  ) %  =  K,  and N = n o r m a l i z a t i o n c o n s t a n t .  59 TABLE 12 M a t r i x o f l.s_ i n the one-electron s p i n o r b a s i s  _£.S t t  t  o o _i  t  t  o  t_  + V  0  0  0  -i  0  0  0  0  0  0  i  0  0  0  //2  0  i  t_ o  +  o  o  o  o  o  o  >s  0  0  0  0  0  H  0  0  0  0  //2  0  0  //2  i  //2  0  -V/2.  e  e  1  //2  o  V/2  +  o  o  0  0  -4  0  t_  o  o  0  0  0  -h  i  0  0  +  o  o  0  0  -i  0  0  0  0  i  o  0  0  0  0  0  0  + e  o  -i  0  0  0  0  0  0  e  o  //2  0  0  0  0  t  X  //2  //2  0 1  0  o  _1  //2  -i  0  0  _1  _1  1  -1  S p i n - o r b i t c o u p l i n g matrix elements f o r the lowest energy quartet and t h e higher energy f u n c t i o n s . For E * ( j ^ ) , (E* * ( P ) i n brackets !  2  X l  X l  0< ) 2  <X )  XjjCxJ)  2  (3A-15B)  Xg(x )  X (X  lu  -i?a  1  1 3  -ca 2 t  1 L +  )  a (a ) 2  0  0  "^"^  6  t,  2ib ?  2ib c  ~76  /6  3  4  and f o r E'a (E'3 i n brackets) x (x^) 3  X (x ) 3  4  x (x ) x (X ) 5  6  7  (3A-15B) . - 2 5  3  4  _ ib ^ 2  3  76  -2ib c 4  "V6  1 5  (x  0  B^lP X (x )  x  8  0  1 6  0  B  4 lP (g  0  )  x^Cxgp)  X l  a i?  (B  3  3  )  ?  B  2/2ib c  1 2  a i 5  1  *i lP  {(x  6 14 ( g  }  B  2/2ib ? 4  3  7  ( B  15  )  0  B  8 16 ( B  -?  )  /  3  60  In order to calculate g-values for this doublet we have to find eigenvalues of the Zeeman operator.  K  «  /3 H.(£li+ g £ s . ) e  e  i  1  If, for a given direction of the applied magnetic field we have eigenvalues of x ^ Q e H and y ^ S g H then g ^  = x-y.  For the axial system under consideration i t is sufficient to calculate only two principal values for g, g  and g , where g d  either g of l  z  x  or gy.  and l  Table 13.  x  equals  JL  In order to make these calculations, matrix elements  are needed in the t^, t , e^ basis and these are given in Q  The matrices of the Zeeman operator in the doublet for H^z  and H^c are  /  1  /  \  1  ,  <  >  K"l  - h  /  A  +  g  e  \  0  I  0  °  B  \  -(A + g B)/ e  C+gDV  (  e  where A, B, C and D are a l l real.  )  Calculations were taken to second  order in £V , but i t was found that only the f i r s t order terms were significant and that N was effectively unity.  B and D contain only  zeroth and second order terms, and these latter were found to be negligible.  Thus to f i r s t order A = -(4^)7(3 A  - 8  )  B = 0.5 C =  -<8£/3)(H+-K  + Trt>  D = 1.0 and the resulting g-values are g„  = 2(g B + A) = g + 2A £  e  = 2(geD.+ C) = 2g  e  + 2C  ...(28)  TABLE  Matrices of 1 z and 1  I  t  z  o o  t o *+  t_  e  I  +  e  e  +  • .  e o  o  -1  o  -/2  o  o  o  1  o  /2  o  -/2  o  o  o  o  o  /2  o  o  %  t_  "V/2  e  +  -i- •  e i  o  o  o  i  o  o  X  o  i  o  -i  6  o  -i  -i  o  o  o  _i  //2  t_ e  t_  o  o *+  v  o  o o  t  i n the one-electron o r b i t a l basis  x  o  t X  13  62  S i m i l a r c a l c u l a t i o n s f o r the "|*3/2^ " d o u b l e t formulae  f o r the apparent  g-values.  Since E.S.R. t r a n s i t i o n s f o r t h i s  a t Q = 9 0 ° , then o n l y the e x p r e s s i o n f o r  d o u b l e t c o u l d o n l y be observed g_^ was c a l c u l a t e d .  lead to e q u i v a l e n t  To second order t h i s was found  to be i d e n t i c a l l y  z e r o , i n d i c a t i n g no s p l i t t i n g o f the d o u b l e t f o r t h i s o r i e n t a t i o n . fact  t r a n s i t i o n s were observed  i n d i c a t i n g some s p l i t t i n g and t h i s  be d i s c u s s e d l a t e r w i t h the r e s t o f the n u m e r i c a l  B.  Hyperfine Term  results.  o f Equ. #1 may be t r e a t e d by the e q u i v a l e n t o p e r a t o r  and can be shown (6) to reduce to  = P ( L . I - kS.I + [ l / 7 ] ^ a . I )  ...(29)  k  (4.^)4 - 4<4'V g /3 /3 <r"3;>  where a ^ - 4 ^ -d  P =  will  Coupling C a l c u l a t i o n s  C$<,j)  technique  In  e%  -  e  N  We may w r i t e t h i s as *%l = P ( l + [1/7] a ) .1 - Pks.I k  ^  dipolar  ^  +  contact  where 1. and a are sums o f one e l e c t r o n o p e r a t o r s . becomes, on ^  dipolar -  Now G r i f f i t h  The f i r s t  term  expansion, P  {( x4/7]a )I +(a 4/7]api (l 4[l/7]a )I } 1  x  x  y  y +  z  z  (6) has shown t h a t f o r an o p e r a t o r p  c o n d i t i o n s yO = p  = p  a Kramers d o u b l e t  (  2  ...(30)  which obeys the  we have the f o l l o w i n g form o f m a t r i x w i t h i n  Q,y(3) p a  /3  a  /3  c  a + ib  a - ib  c  63  w i t h a, b and c r e a l . The  operator 2aS  + 2bS  x  y  + 2cS  Thus any o p e r a t o r p  \~h) •  z  it  form  i n the components o f S, h a v i n g  In p a r t i c u l a r  tf(s) = A S I and  ,  which s a t i s f i e s the above c o n d i t i o n s  may be r e p r e s e n t e d by a l i n e a r real coefficients.  has e x a c t l y t h i s m a t r i x w i t h i n  z  + BS I  z  X  + CS I  X  Z  - P ( l + (J-/7)aJ.I "  p  k  2  S.I  are covered  by t h i s  treatment.  We now wish  t o compare these three forms o f the H a m i l t o n i a n , and we  w i l l consider f i r s t 2<G/  p  the z components.  | Q >S  = 2cS  Z  = AS  z  p  i s g i v e n by Equ. #30  Thus 2P  <a|l +(l/7)a ja>  where  I  2  z  S z  z  z  = A S  z z I  and so A = 2P( < a u We have seen  + d/7)a |  z  z  t h a t 2 ( CM 1  a  ( G^  > ) A  =  g  t z  z  o  first  o r d e r , and so  A = P ( A S z z + (2/7) < a i a J G > ) z  I t may be shown s i m i l a r l y t h a t B  =  p  (A  §xx  (/X  +  2  7  (3\w  a  >>  = ( ASyy < X <* » V £ >  C  P  +  2i/7  For an a x i a l system w i t h B = C, A  g  x  x  )  = A  a > .)...•  A„  = P( ^ g  A  = p ( Asj.+( ></3KI a >)  M  +  (2/7)<G\a | z  g  1/7  x  In order to make comparison w i t h the e x p e r i m e n t a l l y observed we have t o e s t i m a t e the two m a t r i x elements above, w i t h  l°> -  X  3  •••<) 3 1  results  64  X,  1£>-  from our previous c a l c u l a t i o n .  Tables  14 and 15 give the matrices  of a and a_ w i t h i n the basis t , t , e , compiled with the help of z  t  ±  Table A.41, Appendix 2, of reference 6. To zeroth order a and a_ have vanishing matrix elements w i t h i n CL z  and  .  This i s to be expected since the zeroth order functions  are equivalent to one e l e c t r o n occupying each of the d , d y , d x v  Z  o r b i t a l s and t h i s s p h e r i c a l l y symmetric s i t u a t i o n should average  :ix  d i p o l a r c o n t r i b u t i o n s to zero.  To f i r s t order i n £ the matri  elements are  <0.|a|a>= z  ( £ /3)(2 /6/(4E+J) + (2P -P )/3 + ( P ; , ^ ) ) N  </3|a_|a>= ( where  4  1  A -8 )>  £ /3)(4P + 8 P + 1/(3 5  3  P = ^(a-i + b )/(4E + J) + a ( a L  P  2  1  =  a  l  " l  ( a i  b  ) / ( 4 E  2  + ) J  +  a  2  + b )/(4E - J) 2  2^ 2 " 2 ) / ( a  b  P  3  = a b / ( H + K) + a b / ( H - K)  P  4  = b (a  P  5  = a b / ( 4 E + J) + a b / ( 4 E - J)  3  3  3  1  1  4  3  4 E  "> J  4  - b ) / ( H + K) + b ( a 3  4  2  2  4  - b ) / ( H - K) 4  x z  TABLE 14 Matrix of a  z  i n the one-electron spinor basis  +  +  e. -i/  •2  -1 o  (V^l+i)  [V/Jl-i).  -72  ^(l+i)  /2  [°y/2]u+i :i+i)  -/2  -(1+i)  %(l-i)  /2  s(3-i)  i  /2  [-V/2}1+i i)  1+i  l-i  -^(3+i)  -M3-i)  _/2  -1  ^(3+i)  /2  -(l-i) -1  /2j  o  -/2i  1  V/2  -l  o - i / /2  -2  V/2  e  /2i  o  -JjCL-i)  [V^l-i) -/2i  b4  1+i)  o  TABLE 15 Matrix of a  t  a +  o  t  o  0  0  -2  0  t t  o  i n the one-electron spinor basis  t_  i -i//2  -2(l-i)  0  i//2  0  0  -/2i  0  2(l+i)  +  + e  +  e  i  -/2(l-i)  0  0  0  -i  0  o  h(l-i)  e  /2(l+i) 0  o  K  0  1  fl//2}(l-ri:  0  /2i  2(l+i)  0  2  [-l//2}(l-i)  o  0  -i//2  0  0  1  0  -/2(l-i)  0 t_ t +  i//2  0 +  e e  -2(1-i) (l//2j(l+i)  -i 0  0  0 /2(l+i) i  -(1+i)  0  /2  1+i  0  -2/2  0  0  |l//2}(l-i) o  o  -ha-i) /2  5s(3-i)  0  /2 0  %(3+i)  -Ha+i)  0  -2/2  1-i  0 -Jl(3+i) 0  0  0  /2  ^L//2J(l-i)  0  /2i  0  o  -/2i  0  |l//2}(l+i)  0  0  4  fL//2)(l-•i)  Js(l-i)  0  0  0  -(1-i)  0 |-l//2)(l-i)  67  6. Discussion Before comparing the experimental results with the foregoing theoretical calculations and the survey of previous work, i t would seem useful to establish the relationship between the principal g-values obtained from the spin Hamiltonian of Equ. #16, where the spin manifold was a quartet, and the apparent g-values obtained by considering just the  , |±3/2^ states as Kramers doublets.  From f i t t i n g the  1  experimental results to the quartet Hamiltonian, we find g = 1.9920 and g^ = 1.9880; whereas the apparent g-values, which are those calculated by the ligand field method are from Equ. #28, g'^~g and g_^ e  ~ 2g .  Thus straight away we can compare the two values for g , but  e  /(  we have to reconsider the situation for g . x  We may write the matrix of Equ. #18 in the basis |  , |±.3/2^  as •3/2  +3/2  0  D + 3G„/2  r  0  +1/2  -1/2  /3G /2  0  >  D - 3G„/2  I  0  •v/x/ 3 G  0  >/3G/2 4  -D + G„/2  0 where G  ff  2  -D - G,,/2  J  = gy6 HCos0 and Gj^= gyQ HSin£ e  e  Using the method of matrix partitioning (65) we may replace the element H  rs  by  ~ _ ^rs ~ A r s  +  2-i  t  ^rt^ts To c-r ~&t  Zfi  and this leads to the following matrix, correct to second order, where the basis is now two isolated doublets J*3/2^ and  68  +3/2  •• -3/2  "D+3/2G,, +3G /8D  0.  2  +1/2  -1/2  0  0  0  D-3/2G„+3G /8D  0  0  0  -D+l;G,+3G/8D  G  0  0  G  -D^G„+3Gf/8D  2  0 2  f  A  Thus the apparent splitting of the | +%^ > j  x  states is given  by the solution of the lower 2 x 2 determinant. (-D + 3G /8D - E ) - G /4 - G 2  therefore  2  2  =0  2  E = -D + 3G /8D±^G /4 + G 2  For E// to z then G = y/  2  g/5 H t  and G  2  ±  = 0; so E =iG„/2  =ig„/3 H/2 e  and the splitting is g„WH,i.e. an effective g-value of g . Hlz  then G^ =  gj(3  H  e  and G„ - 0 so E = ± G  ^gJ^H  a  X  is 2gJ^R, i.e. an effective g-value of 2g,. tive g-values of the other doublet |*3/2)and gj = 3g  n  But for splitting  d t h e  By inspection the effec|-3/2) are given by  and g^ = 0  With these results in mind, we may now proceed to compare the experimental results for D, g and A with the calculated values. The equations for zero field splitting and apparent g-values are expressed in terms of the parameters B, C,§, A approximation that C = 4B (62) and that A 22,000 cm" from other studies on the 1  m  a  QfaoJ  v b e  a n d  t a k  £ •  W  e  ma  ^ e the  e n as approximately  system (11-22).  leaves the parameters B, § and £ which are to be determined.  This For  a strict crystal field calculation i t would be correct to take the free ion values for B and  However, this would neglect a l l co-  69  valency effects and these are not negligible in complexes with oxygen as ligands (59) and are observed in the present system as superhyperfine lines, visible in Fig. 6 which w i l l be discussed in more detail later.  In view of the approximations (see Section 2B) involved in  using a sum over one electron operators for £  this parameter is taken  as isotropic with a value f a i r l y close to that of the free ion (Mn ; 4+  £  = 402 cm"l) (6).  The electron repulsion parameter B is much more  susceptible to electron delocalization effects and is thus taken as a variable parameter in the calculations, with values up to that for the free ion (  1060 cm"''") (6) .  § , the effective one-electron t r i -  gonal f i e l d , is left to be determined. The previous calculations of McGarvey mentioned, indicated that an approximate formula for 2D is given by (for small trigonal fields): 2D - 8 \  2  (1/E( E) - l/E(\))  = 8\  2  (1/A„  ~8\ K/ A 2  + 6X  4  2  "  1 1  A  >  x  - 1/E( E))  2  2  6\ (1/E( A ) - 1/E( E))  +  2  2  2  1  + 6\ (1/E( A ) - 1/E( E)) 2  2  ...(32)  2  1  which is very similar in form to that given by Lohr and Lipscomb (58). In fact, assuming their  X^"* Xx 1  to b  e  defined i -  n  a  similar manner  to those of Sugano and Tanabe (54), their formulae differ by a factor of two from the above.  It w i l l be noticed that the term 8 X K / 2  is that derived by Van Vleck and others. gives a value of about 15A K/ A// 2  ^  o r  t  Lohr and Lipscomb's ^  s  t e r m  >  anc  A  2  work  * this seems to  be in error for the same reasons as the treatment of Sugano and Tanabe. The second term of Equ. #32 is similarly half that of Lohr and Lipscomb. The detailed formula from the present work, Equ. #26, reduces to  70  the f o l l o w i n g f o r s m a l l g ~  1 : a  Thus  2  ~  when  0 : t> ~ 0 :  ^  3  2D ~ - ( 2 / 3 ) £ ( 1 / E ( E ) 2  1  (see Table  10)  + 4/3(E( Ap) - l/E( Aj) -  2  4  4/3(E( E)))  2  4  = -(2/3)£ (l/E( E) - 1 / E ( A ) ) - (8/9)£ ( 1 / E ( \ ) 2  2  2  - 1/( E))  2  4  X  =  ( 6 / 9 ) £ ( l / E ( A ) - 1 / E ( E ) ) + (8/9)£ ( 1 / E ( E ) 2  2  2  2  4  1  = 6 \ ( 1 / E ( A ) - 1/E( E)) + 8 \ d / E ( E ) 2  2  2  2  4  1  when ^  = 3  - 1/E( A )) 4  L  l/E(\))  \  w h i c h i s McGarvey's f o r m u l a , Equ. #32  above, f o r v e r y s m a l l  and i t  i s f e l t t h a t these are f a i r l y e q u i v a l e n t d e s c r i p t i o n s , a l t h o u g h t h a t of McGarvey d i f f e r s from the p r e s e n t work i n u s i n g more e m p i r i c a l p a r a meters. print  I t may  be noted  i n p a s s i n g , t h a t t h e r e i s an e r r o r or m i s -  i n McGarvey's paper.  s h o u l d i n f a c t be  The  l a s t + s i g n of E q u a t i o n #23,  -.  By s u b s t i t u t i n g v a l u e s of 400 cm"*/\  r e s p e c t i v e l y i n Equ. #26  Z.F.S. were o b t a i n e d .  (6) and  and v a r y i n g B  22,000 cm~^  and  §,  500  for £  8  shown below. Z.F.S.  5000  and  s e v e r a l v a l u e s of the  The e x p e r i m e n t a l v a l u e of 2D is±1.72 cm~*  n e a r e s t approaches to t h i s v a l u e were o b t a i n e d as B  Page 3754  -1.67  c 600  6000  -1.65  700  7000  -1.67  800  8000  -1.73  and  the  71  B  8  900  8900  -1.82  1000  9800  -1.79  Z.F.S.  A graph of the energies of the f i r s t eight excited states calculated using these parameter values is shown in Fig. 17.  The optical spectrum  of the manganese polymolybdate has been observed (46, 47) and this is shown in Fig. 18.  The f i r s t spin-allowed transition occurs at about  21,500 cm"*- and there is a small spin-forbidden doublet at about 14,500 cm  It can be seen from Fig. 17 that the energies calculated for  B = 700 cm"^ and § = 7000 cm"! f i t these observed values quite well. The small splitting of the line at 14,500 cm"-'- is probably due to sub2  2  shell configuration mixing of the two states labelled 2  A ( Tp 2  2  in the D3 classification.  E ( T^) and 2  The measured splitting is 640 cm~l.  We may compare this with the theoretical expression given by Jorgensen (62) which is 66B //^with the E ( T ) the lower. 2  2  2  1  and  For B = 700 c m  4  = 22,000 cm"l this expression gives a value of 1470 cm"-'-, which  is larger than the experimental value.  However, the graph on Page  261 of reference #6 for this splitting indicates a smaller value, more in line with that observed. The strong peak, at 21,500 cm"-'- is very broad and may consist of transitions to both the A ( T ) and ^E( T ) states, since both trans4  4  4  2  itions are formally allowed.  2  A more highly resolved spectrum taken  in polarised light would have been desirable, but unfortunately the crystals were very fragile at low temperatures and this could not be obtained (47). This interpretation is open to the objection that the E ^ ( T ) 2  2  1  level could well be the one to which the spin-forbidden transition  72 FIGURE 17 : V a r i a t i o n o f the energy l e v e l s w i t h the comb i n a t i o n o f parameters which give a Z.F.S. o f a p p r o x i m a t e l y 1.7 cm" . ( A d e f i n e d as 0) 1  4  2  40,000 r  35,000 h  J3 5 0 0  S  5000  600 6000  700 7000  PARAMETERS  800 8000  900 8900  1000 9800  U)  7300  7000  6500  5500  WAVELENGTH  5000  A  FIGURE 18 : V i s i b l e a b s o r p t i o n spectrum o f ammonium 9-molybdomanganate  4500  4000  74  o c c u r r e d as t h i s would be c o r r e c t f o r v a l u e s of B = 900 cm" 9000 cm . 1  and Q  1  However, t h e r e i s no e a s i l y seen mechanism f o r t h i s  -  state  t o g i v e r i s e to a d o u b l e t s p l i t t i n g and so the former d i s c u s s i o n seems valid. The v a l u e of  a p p r o x i m a t e l y 7000 c m  o f the v e r y s t r o n g a x i a l f i e l d produced  i s v e r y h i g h , but i n view  -1  by the molybdenum-centred  o c t a h e d r a i n the s t r u c t u r e of the i o n , t h i s h i g h v a l u e i s q u i t e a c c e p t able. There i s no a p r i o r i r e a s o n f o r c h o o s i n g  g  to be p o s i t i v e , s i n c e  the s p l i t t i n g of the one e l e c t r o n o r b i t a l s c o u l d w e l l be i n the o p p o s i t e d i r e c t i o n to t h a t i n d i c a t e d i n F i g . 13 and the t c o u l d w e l l have h i g h e r energy were a l s o made w i t h g  than the t  +  and t _ o r b i t a l s .  B and § .  found  must be p o s i t i v e i n  s t a t e s have lower energy  than  d o u b l e t , s i n c e 2D i s n e g a t i v e .  A p p l y i n g the two chosen v a l u e s of B and c u l a t i o n of g  /(  (g | = 1.955). observed  Calculations  s p l i t t i n g w i t h reasonable  I t would thus seem t h a t g  t h i s c a s e , and thus a l s o t h a t the | i 3 / 2 ^ the | ± % ^  orbital  assumed n e g a t i v e i n Equ.#26, but i t was  n o t p o s s i b l e to produce a l a r g e zero f i e l d v a l u e s of  Q  and gj_ i t was  found  that g ^  These d e v i a t i o n s from g  experimentally.  £  g  to Equ. #28 = 1.950  and g ^ =  are much l a r g e r than  McGarvey (59) has suggested  i m p o r t a n t c o n t r i b u t i o n to the g - f a c t o r , and which c o n t r i b u t i o n to g, i s the e f f e c t produced  f o r the  cal-  3.91 those  that a very  produces a p o s i t i v e  by those s t a t e s r e s u l t i n g  from the promotion of an e l e c t r o n from the f i l l e d c u l a r o r b i t a l s to the d o r b i t a l s of the m e t a l .  l i g a n d - m e t a l moleI t i s not p o s s i b l e to  c a l c u l a t e the s i z e o f t h i s e f f e c t u s i n g the c u r r e n t scheme, a l t h o u g h a m o l e c u l a r o r b i t a l t r e a t m e n t would be v e r y i l l u m i n a t i n g , c a r r i e d out a l o n g the l i n e s of Maki and McGarvey's (66) d  perhaps 1  calculation.  75  t h e o n l y drawback b e i n g the much l a r g e r amount o f b o o k - k e e p i n g for  a d  3  necessary  system and consequent d e t e r m i n a n t a l f u n c t i o n s .  We may a l s o use the f i t t e d v a l u e s o f B and 3  t o  m a  k e an e s t i m a t e o f  the s i z e o f the d i p o l a r c o n t r i b u t i o n t o the h y p e r f i n e c o u p l i n g t e n s o r . The m a t r i x elements i n Equ. #31 were c a l c u l a t e d and t h e i r v a l u e s t o second o r d e r are  <X |a |X > <XJa_ | % , > 3  z  S i n c e the g-values  =0.015  3  -0.016  p r e d i c t e d by the l i g a n d f i e l d  poor agreement w i t h experiment ment.  i t was d e c i d e d  t h e o r y are i n such  t o use those from e x p e r i -  S u b s t i t u t i o n i n Equ. #31 g i v e s A  /;  =±0.0076 = P(-0.010 + y (0.015)) - Pk  Aj_  =±0.0068 = P(-0.050 + y (0.016)) - Pk  There i s an a m b i g u i t y o f s i g n f o r the components o f A as measured by E.S.R. and t h i s i s i n d i c a t e d above.  S i n c e the v a l u e s are n e a r l y i s o -  t r o p i c then A ,j and Aj_ must have the same s i g n , b u t t h i s may be + o r -. The  f r e e i o n v a l u e o f P i s +0.024 cm"''' and t h i s w i l l produce a d i f f e r e n c e  i n the d i p o l a r terms o f A ^  and A ^ o f ±0.0010 c m . 4  T h i s compares f a v o u r a b l y w i t h the e x p e r i m e n t a l v a l u e o f ±0.0008 and so we c a n j u s t i f i a b l y s o l v e e q u a t i o n s of P and Pk.  Choosing b o t h A  P = +0.02 cm" ; 1  and f o r b o t h A  ((  ((  cm  (33) f o r approximate v a l u e s  and A_^ t o be p o s i t i v e t h e n , we f i n d  Pk = -0.0078  cm  4  and A_^ n e g a t i v e  P = -0.02 cm" ; 1  Pk = +0.0078 cm"  1  Now P, b e i n g a r a d i a l i n t e g r a l , must n e c e s s a r i l y be p o s i t i v e and so t h i s l a t t e r r e s u l t i s not a d m i s s a b l e .  The r a t i o Pk/P i s -0.39 which  i s c o n s i s t e n t w i t h t h e v a l u e s f o r most f i r s t row t r a n s i t i o n m e t a l s ( 6 ) . The  s m a l l l i n e s observed  a t the wings and i n the c e n t r e o f the  4  76  spectrum  i n F i g . 6 need to be e x p l a i n e d .  These cannot  form of " f o r b i d d e n " t r a n s i t i o n , s i n c e such w i t h i n the outer two  A good p o s s i b i l i t y i s  to d e l o c a l i z a t i o n of the mangan-  ese d - e l e c t r o n s onto the s u r r o u n d i n g molybdenum n u c l e i . not o b s e r v a b l e f o r a l l o r i e n t a t i o n s , probably due the wide manganese l i n e s , and  so i t was  ENDOR study of t h i s system c o u l d c l a r i f y  96 ( Mo)  has  lines.  s p i n of 5/2  isotopes  J  Mo  one  superhyper-  97 and 'Mo 7  both have n u c l e a r  and magnetic moments w i t h i n Tk of each o t h e r .  t r e a t them, e f f e c t i v e l y , as one  def-  (67).  three n a t u r a l l y o c c u r r i n g i s o t o p e s , of which  other two  are  I t i s hoped t h a t an  zero n u c l e a r s p i n and would thus not give e x t r a  The  lines  to o v e r l a p p i n g w i t h  this situation  • 95 fine  The  not p o s s i b l e to make any  i n i t e measurement of t h e i r v a r i a t i o n w i t h a n g l e .  Molybdenum has  any  l i n e s would n e c e s s a r i l y l i e  s t r o n g manganese l i n e s .  t h a t they are s u p e r h y p e r f i n e l i n e s due  a r i s e from  Thus we  can  i s o t o p e whose abundance i s approximately  25%. S i n c e there are n i n e molybdenum n u c l e i around the c e n t r a l manganese then i t i s a problem i n p r o b a b i l i t i e s  to c a l c u l a t e  the p o s s i b l e s p e c t r a  r e s u l t i n g from v a r i o u s p o s s i b l e o c c u p a t i o n of these nine s i t e s by s p i n or 5/2  spin  nuclei.  zero  PART  II  77  E.S.R. of X-Ray Irradiated Single Crystals of Deuterated Ammonium Paramolybdate Tetrahydrates  1. Introduction The E.S.R. spectrum of. ammonium paramolybdate tetrahydrate was measured recently (34) and this work was carried out to verify that the observed doublet super-hyperfine splittings were in fact due to a hydrogen nucleus near the central molybdenum.  2.  Experimental Single crystals were readily grown from D2O and a typical crystal is shown in Fig. 19 together with the chosen axes for the monoclinic class. Lindquist (68) determined the crystal structure and the present axis system differs from his in the choice of the a axis, since an orthogonal right-handed set is most useful for the purposes of experiment.  The  crystals were irradiated with 50 kv. x-rays at room temperature for one hour and a discolouration of the material after irradiation was noticed, this being typical of crystals with disrupted electronic structure. The crystal was rotated about the a', b and c axes and spectra were obtained at 15° intervals.  The method of mounting the crystal consists of a  small perspex device with three mutually perpendicular faces and each of these can be mounted with their plane parallel to the direction of the magnetic f i e l d . The spectra were obtained at room temperature  on a standard x-band  spectrometer described previously (69) and a typical spectrum is shown in Fig. 20.  This shows the single strong central line due to the moly-  bdenum isotope (^Mo) with zero nuclear spin and six outer lines due to  FIGURE 19 :- A t y p i c a l c r y s t a l _ o f ammonium paramolybdate , showing the chosen axis-system-. i n the c e n t r e i n d i c a t e s the e x t i n c t i o n p o s i t i o n s under p o l a r i s e d l i g h t  The c r o s s  79  FIGURE 20  : An E.S.R. spectrum o f an i r r a d i a t e d s i n g l e c r y s t a l o f d e u t e r a t e d ammonium paramolybdate t e t r a h y d r a t e , w i t h the magnetic f i e l d i n the a' - c plane  80  the isotopes  and Mo, both of which have a nuclear spin of 5/2. 97  The difference between the nuclear magnetic moments of these species is only 27o and so the two possible sets of hyperfine lines merge into one set with the observed line-width (the ratio of line-width to hyperfine splitting is of the order of 1:8).  In the normal hydrated  crystals the spectra showed a doublet splitting of a l l lines, of the order of 7 gauss in magnitude.  Because of the much smaller nuclear  magnetic moment of deuterium this splitting is reduced to about 1.2 gauss in the deuterated crystals and is thus not resolved. When spectra were measured in the b-c and a -b planes they were 1  found to consist of two superposed spectra since the monoclinic symmetry allows two non-equivalent sites for the molecules in the unit c e l l for these orientations.  Despite this superposition i t was quite simple  to trace the variation of the lines with orientation.  The variation  with angle of the observed hyperfine splitting and g-values for each of the three planes is shown in Figs. 2 1 - 2 3 . In order to obtain the components of the hyperfine coupling tensor the first-order approximation, of dividing the splitting between the second and f i f t h hyperfine lines by three was used.  The two centre  components of the hyperfine structure were often obscured by overlap with the strong central line, whereas the lines chosen were sharp and distinct for a l l orientations.  3. Theoretical Since Mo has a f a i r l y large spin orbit coupling constant and g and the hyperfine tensor are not parallel, i t is not justifiable to take term in the spin Hamiltonian as a good description of the hyperfine coupling.  Cross terms from Equ. #1 give  81 FIGURE 21 : The variation of hyperfine splitting and g-values, for an irradiated single crystal of deuterated ammonium paramolybdate tetrahydrate, for the orientations where the magnetic field is perpendicular to the a' axis  H J _ a'  Angle of rotation,deg  82 FIGURE  22  The v a r i a t i o n of h y p e r f i n e s p l i t t i n g and g-values f o r an i r r a d i a t e d s i n g l e c r y s t a l of d e u t e r a t e d ammonium paramolybdate t e t r a h y d r a t e f o r the o r i e n t a t i o n s where the magnetic f i e l d i s p e r p e n d i c u l a r to the b  axis  H±b  CO  to  O  c Q. CO  CD C  CL  X  CD O > I  150  Angle of rotation, deg  180  83 FIGURE 23  : The v a r i a t i o n of h y p e r f i n e s p l i t t i n g s and g-values f o r an i r r a d i a t e d s i n g l e c r y s t a l of d e u t e r a t e d ammonium paramolybdate t e t r a h y d r a t e f o r the o r i e n t a t i o n s where the magnetic f i e l d iB p e r p e n d i c u l a r to the c  axis  H±c  to D O  O  c Q. 0)  Q.  O > I  cn  1.89  80  90  120  Angle of rotation , deg  150  180  84  rise to a pseudo-dipolar hyperfine coupling which can rather be described by a term S.g.T.I,(70, 71) where A has been replaced by g.T .  The  phenomenological spin Hamiltonian used for this work is thus gH.g^.S Q^^jQ * S.jg.T.I which neglects the nuclear Zee+  l  e  man term and which reduces the more usual equation when g is very nearly isotropic and equal to the free electron value g .  Since the measured  e  deviation from gg is about 0.1, this, equation has been used. One can see from Figs. 21 - 23 that the principal axes of the g and hyperfine coupling tensors are not parallel and a method of determining the principal values of g_ and T has been given for this case (34) . The square of the observed g_value for a particular orientation is given by g  2  = y^Lij(g.g) -j n  where L i j = l i l j , l i being the direction cosines of the magnetic f i e l d . The square of the observed hyperfine splitting is given by (hfs) =I. 2  %  where K =  g.L.jg  l  i , j TF7K)  ^-  ^  A t  -  )ii 1  J  and A*- = transpose of A, A = g.T .  The derivation of these equations assumes that the electron spin s. is quantized along the direction H_.g_ and the nuclear spin in the direction of S.A (with A = g_.T) . The tensor components (.£•§) i j  a n  d  (A.A )^^ were evaluated using a t  least squares technique on a l l the experimental data.  The (g.g) tensor  was diagonalized to give elements whose square roots are the principal values of g.  Then from the definition of A we have  T.T = g" . (A.A*) - j f 1  1  and since the g_ tensor is already known, i t is possible to calculate the components of (J.TJ from the previous values of (A.A ). C  Diagonalization  85  of Qj.Tj) gives the diagonal elements which are the squares of the principal values of T.  In order to remove the ambiguity of the relative  signs for off-diagonal elements of (A.A*") in the a'-b and b-c planes, the "skew-axis" technique (72) was employed. These calculations were a l l carried out on an I.B.M. 7040 computer using programs written by Dr. F.G. Herring and the final results are given in Table 16.  The values for the normal hydrated crystal are also  given for comparison.  Discussion The E.S.R. spectrum of normal tetrahydrate crystal shown in Fig. 24 is for the same orientation as for the deuterated compounds in Fig. 20. It thus seems very probable that the doublet splitting is caused by the anion (Moy02 )^ 4  the  ion  capturing a hydrogen atom upon x-irradiation, giving  (M07O24H) ". 6  The undamaged crystal of (NH^)gMo70^.4^0 has discrete units which consist of seven distorted MoOg octahedra which are similar and nearly parallel to each other and which share corners and edges.  The dis-  tortion makes and 0 - Mo - 0 axes unequal and non-orthogonal. Since the principal hyperfine structure observed is that of an Mo nucleus, i t seems reasonable to suppose that the odd electron orbital has considerable contributions from the 5s and 4d atomic orbitals of the Mo.  Under S  2  symmetry these orbitals transform as Ag, as also do the  angular momentum operators l _ , ly, l _ . x  z  As a result of this, matrix  elements of the form  <C<£i  I X O O i a l ^ j } ^ (£j|lb|<£ t}  w i t h ^ . = 5s or 4d and a, b = x, y, z  which occur in the Maki and McGarvey (66) perturbation scheme for g-shifts would a l l be non-zero.  This would account for the large rhombic char-  86  D i r e c t i o n cosines w i t h respect to  P r i n c i p a l values* Deuterated  Non-deuterated  1.891 + 0.005 1.913 + 0.005 1.925 + 0.005  g  *  1.900 + 0.005 1.921 + .0.005 1.935 + 0.005 83.2 + 0 . 5 37.0 + 0.5 35.4 + 0.5  83.4 + 0.5 37.0 + 0.5 34.6 + 0 . 5  T  a'  b  c  0.917 -0.262 -0.300  +0.057 +0.658 +0.750  0.394 0.706 0.589  0.890 -0.308 0.331  +0.451 +0.514 +0.726  0.053 0.796 0.597  The h y p e r f i n e c o u p l i n g v a l u e s a r e g i v e n i n gauss  TABLE 16A : The s p i n H a m i l t o n i a n parameters f o r the (M07O24H) p r i n c i p a l axes system  Compound  8  2-  *e  T If u  I.  ion i n their  JtfA.U.)  Reference  1.874  1.918  99.8  48.2  0.97  -7.60  (14)  1.963  1.940  74.7  32.6  0.88  -5.61  (9)  2.090  1.945  66.0  30.0  0.60  -5.09  (MoO(SCN) ) " (Mo 0 H)6-  1.928  1.944  68.4  34.5  0.66  -5.44  1.90  1.90*  78.0  33.7  0.68  -5.25  T h i s work  (Mo 0 H)6-  1.891  1.9151)  78.0  33.7  0.67  -5.35  T h i s work  (MoOF ) 5  (M0OCI5) " 2  (MoOBr )25  2  5  7  7  24  24  _  "t Average ^ g-values ^Estimate // g t e n s o r FP T I, , T1  §  (10) (15)  1  taken o f x and y v a l u e s i n T frame n e g l e c t i n g l i g a n d s p i n - o r b i t c o u p l i n g from d a t a i n (10) not p a r a l l e l to T tensor i n u n i t s o f 10 4 cm-1 _  TABLE 16B : Values o f the g t e n s o r and h y p e r f i n e i n t e r a c t i o n t e n s o r f o r molybdenum complex anions  87 FIGURE 24 : An E.S.R. spectrum o f an i r r a d i a t e d s i n g l e c r y s t a l o f ammonium paramolybdate t e t r a h y d r a t e , w i t h the magnetic f i e l d i n the a' - c plane  88  acter of the £ tensor and, probably, for the fact that the p r i n c i p a l axes of the T and £ tensors are not  parallel.  The d i r e c t i o n cosines of T are nearly p a r a l l e l to the Mo-0 any one of the octahedra, whereas those for £ are not.  bonds in  This i s to be  expected for molecules of low symmetry since T depends primarily on the spin density i n the immediate neighbourhood of the nucleus i n question, but £ depends on the electron d i s t r i b u t i o n i n the molecules as a whole (see Section 2 ) . In order to make an estimate of the bonding around the molybdenum, the non-orthogonality  of the O-Mo-0 axes was  ignored  group for a single octahedron regarded as D2jj.  (34) and the point  The anti-bonding  orbitals  for an MoOg unit can then be written as follows  K>  -a|*y>  -  cr|",, ) l8  h >-£h> - /3'|s> K> • yI»•> g  2s  y'| B L  K>  -8|x2-y2>  where  Lj,  3 g  >  +' |z2)'+ e  6>>  -  g'|L > Ag  are the symmetry adapted linear combinations of oxygen o r b i t -  als belonging  to the  J" -irreducible representation of T>2h' 1  The previous work on paramagnetic molybdenum complexes (78-81) has been done on species which have C^ principally a d y X  orbital.  v  symmetry and a ground state which is  The hyperfine coupling tensor of the  species i s very s i m i l a r to those previously reported  (Table 16)  present  and  accordingly B-^g has been chosen as the ground state, and a reasonable ordering of the o r b i t a l s i n increasing energy would be B^  ^ 2g~ 3g ^ g' B  g  B  A  Following the method of Maki and McGarvey (66) the following expressions were obtained  (34) for the components of T  89  T  z  = P(-k - (4/7)a 2  " A§  T  x  = P(-k + (2/7) a  -  T  y  = P(-k + (2/7)Q "  2  2  2  " (3/14)( A§x  Ag  x  + (3/14) Ag  y  Ag  y  + (3/14) A g  x  where -Pk is the Fermi contact term.  +  Ag )) y  " ( 3/14) (  and T  v  as 34.6 gauss.  z  A  g) 2  S) 2  On substitution of the values of +  T  / 8 )  < 3/14) ( € / 8 ) A  +  the components of T and £ and a value of P for Mo^ , be solved for Q?2 and k.  €  the equations can  was taken as 83.4 gauss, T  x  as 37.0 gauss,  The problem of T and £ being non-parallel was  treated in two ways: (i)  The non-parallel effect was ignored and the g-values assigned as g  (ii)  2  = 1.891,  = 1.913, g  y  = 1.925, and  the £ tensor was transformed to the T-axes frame and these diagonal elements assigned to g , ^ z  and g .  The elements a l l turn out to  y  be 1.90 in this frame. The value of P was taken as -67.95 x 1 0 c m  - 1  , quoted by McGarvey (82).  The results of those two calculations are essentially identical. The computed value of Q  2  indicates a large contribution of in-plane  bonding of the oxygen atoms to the molybdenum.  There are two competing  effects which determine whether g^ ^ g^ or g ^ g^. l(  As Q  becomes  2  smaller in the halo-oxomolybdate series, the tendency for g ^ g^ in)(  creases (Table 16).  However, the spin orbit coupling of F, Cl and Br  increase in the same direction (84) and thus the effect of ligand spinorbit coupling is also important (78).  The spin-orbit coupling constant  of 0 or o" is smaller than that of the halogens (as is sulphur) and so we expect, in spite of the small value of Q , 2  that g^ ^ g^, as is ob-  served . The large value of ^ indicates that there is l i t t l e or no direct contribution to the contact term from the 5s orbital of molybdenum (82), and thus the hyperfine coupling arises primarily from core-polarization  a f f e c t i n g the  4d  orbitals.  91  References  1.  B.R. McGarvey, Transition Metal Chemistry, Vo1. 3 (R.L. Carlin, ed.) 1966.  2.  B. Bleaney & K.W.H. Stevens, Rept. Progr. Phys., 16, 108, 1953  3.  K.D. Bower & J. Owen, ibid., 18, 304, 1955  4.  J.W. Orton, ibid., 21, 204, 1958  5.  J.H. Van Vleck, Phys. Rev., 41, 208, 1932  6. J.S. G r i f f i t h , The Theory of Transition Metal Ions, C.U.P., 1961 7.  J.P. Dahl & C.J. Ballhausen, Quantum Chemistry, Vol. 4, 170, 1968  8.  J. Owen & J.H.M. Thornley, Rept. Progr. Phys., 2_9, 675, 1966  9. A.M. Clogston, J.P. Gordon, V. Jaccarino., M. Peter & L.R. Walker, Phys. Rev. , 117, 1222, 1960 10.  E.U. Condon & G.H. Shortley, Theory of Atomic Spectra, C.U.P., 1951  11.  S. Geschwind, P. Kisliuk, M.P. Klein, J.P. 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Davies, The Theory of the E l e c t r i c and Magnetic Properties of Molec u l e s , W i l e y , 1967  27.  E. Fermi, Z. Physik., 60, 320, 1930 .  28.  S.M. B l i n d e r , Adv. Quantum Chem., V o l . 2, 1965, Academic Press  29.  A. Abragam & M.H.L. Pryce, Proc. Roy. S o c , A205, 135, 1951  30.  M.H.L. Pryce, P r o c Phys. S o c , A63, 25, 1950  31.  G.F. Koster & H. S t a t z , Phys. Rev., 115, 1568, 1959  32.  R.M. P i t z e r , C.W. Kern & W.N. Lipscomb, J . Chem. Phys., 3_7, 267, 1963  33.  G.F. Kokoszka, H.C. A l l e n & G. Gordon, i b i d . , 46, 3013, 1967  34.  CR. B y f l e e t , F.G. Herring, W. C. L i n , CA. McDowell & D.J. Ward, Mol. Phys., 15, 239, 1968  35.  R. Lefebvre, i b i d . , 12, 417, 1967  36.  B. Bleaney, P r o c Phys. Soc., A73, 939, 1959  37.  W. Low, Paramagnetic Resonance i n S o l i d s , S o l i d State Physics, Supplement 2, Chap. 5  38.  R. Stahl-Brada & W. Low, Nuovo Cimerito, 15_, 290, 1960  39.  B. Bleaney, P h i l . Mag., 42, 441, 1951  40.  C-E. 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A., 230, 169, 1955  84.  G. Malli & S. Fraga, Theor. Chim. Acta., 70, 80, 1967  95  APPENDIX I Perturbation Theory Let us consider a Hamiltonian ti. which can be written as $4. + Xv Q  where the eigenvalues E*-°) and eigenstates \j/ (°) of  are known.  If  it is not possible, or not easy to find exact solutions to the total Hamiltonian  then, providing certain conditions are f u l f i l l e d , approximate  solutions can be found as polynomials in X •  The most important condition  is that the Hamiltonian Xv, called the perturbing Hamiltonian, should be small compared with «^ ' 0  Solutions for non-degenerate energies E^, and  wave functions V|/^ are given in terms of the solutions of $t  (E^and  Q  i j / f ) , by ( 7 3 , 7 4 ) : E  ±  E ?> + (  =  V//. where  +  XE<P +  X E ( 2 ) + ... + etc. 2  XV//(1) + X ^ 2 )  ...  +  +  e t c  .  ...(Al.l)  E^P = V,E  <2)  ^(o)|v'|^(J)>  =  E ?> -  <^<£>| V'| V|/(J)>  (  E  (2n)  E  (2n+1)  ^ ( n + D j . | ^ ( J ) ) - X T E ( k ) Z : < ^ ( f J-D|l//(fJ)>  =  V  =  "  <\//p|V'|v//^)> - £ E  k=2  and  ( V = V - E<J>)  ^ i" (  ^ Z :  (  1  "  j + 1  k )  j=o  l^ r (  3 )  ^  1  - -il a  fay  -£ k=l  :  c i - EI k=l  b V k  E  (2  l k  / ( ( f - E<°>> E  >a /(E ?(  i  1  96  When  \v  i s s m a l l these s o l u t i o n s are r a p i d l y convergent.  s o l u t i o n s to  c o n t a i n degenerate  energy  a p p l i c a b l e s i n c e E^°)  not immediately the summation.  I f the  l e v e l s then these formulae  -  would be  To remove t h i s d i f f i c u l t y the  there are no o f f - d i a g o n a l m a t r i x elements  zero f o r some terms i n are transformed so that  c o n n e c t i n g degenerate  levels.  T h i s t r a n s f o r m a t i o n i s found by a r r a n g i n g the m a t r i x V so that s o l u t i o n s to  w i t h degenerate  m a t r i c e s w i t h dimensions  e q u a l to the number of degenerate  s o l u t i o n s to t h i s d i a g o n a l i z a t i o n are c a l l e d ctions",  e.g.  X  Q  0  and  "correct  (3  Ik  o  o\  A 0  0] B/  are degenerate  2x2  /x  V =  i n $L0  and so we  z e r o t h o r d e r wave f u n c t i o n s  s o l u t i o n s may 2  :  2  l  = _ 2 . 2 % (x  The  o\  2z  y/  treatment.  states adjacent.  The  are found by d i a g o n a l i z i n g the  z  Z )  -X'  ^ l = jQ+k/3  .  }  v^  2  =  j  ( j + k = 1) 2  \o  T \VT = +  where p and  \  1o  /  ° E  \p*  q*  E  l  2  o  2  a-k/Q  t r a n s f o r m a t i o n m a t r i x T must thus be a p p l i e d  /j J [k -k and  2z ]  be w r i t t e n  E = +(x - z ) ^ 2  y  z -x  have to a p p l y the above  forXv  The  sub-matrix. K  E  (z \0  (X Z The  (3  a  The m a t r i c e s are a l r e a d y arranged w i t h the degenerate correct  levels.  Q,ff,y  y  = I 0 \0  sub-  z e r o t h order wave f u n -  Consider the two m a t r i c e s i n the b a s i s  a  states  are a d j a c e n t , and d i a g o n a l i z i n g the  Q  are  to\v  where T i s  o\  o)  1/  J  P\ q y/  q are the. o f f - d i a g o n a l elements  f o r m a t i o n , T, has made the m a t r i x element  after  transformation.  T h i s trans-  c o n n e c t i n g the two degenerate  states  97  o f j£  e q u a l to z e r o .  Q  f o r m a t i o n s i n c e any  Note t h a t the m a t r i x of  l i n e a r combination  o f Q and  i s unchanged by t h i s t r a n s ffwill  have the same e i g e n -  value . The d e g e n e r a t e case has been t r e a t e d i n g e n e r a l by D.P. he g i v e s e q u a t i o n s  s i m i l a r to Equ. #A  g e n e r a l case i s a v a i l a b l e  1.1).  Chong (75)  A program (DPERTl) (75)  from the Quantum Chemistry  Program Exchange.  ;The e q u a t i o n s A l . 1 d i f f e r from the normal p r e s e n t a t i o n o f p e r t u r b a t i o n t h e o r y i n t h a t they are f a r more amenable t o n u m e r i c a l c a l c u l a t i o n s , s i n c e s u c c e s s i v e o r d e r s o f e n e r g i e s and are found by p r o g r e s s i v e s u b s t i t u t i o n o f parameters.  wave-functions  and  f o r the  98  APPENDIX II Block Diagram of Program to Calculate Transition Magnetic Fields from Spin Hamiltonian Parameters 1.  Read parameters and experimental data Set up basis  matrices in | M ^ g  Equ. #18  If g^H ^ D go to 3 Diagonalize J£  Q  + iL-^ for chosen field  Go to 4 3.  Diagonalize \L  4.  Call perturbation subroutine DPERT1  blockwise  Construct difference polynomial Solve for H  Equ. #21  t r a n s i t i o n  Call DPERTl again to get correct wave functions at this field Set up $L 2 matrix and transform to correct basis set Set up (^ + matrix in this expanded \ M M-r^ basis 0  s  Call DPERTl to get hyperfine energy leveIs 2.  Define new field independent term and solve difference polynomial again  Equ. #22  99  Repeat  (2) f o r a l l h y p e r f i n e  Return  to (1)  f o r new  transitions  parameters  Compare c a l c u l a t e d f i e l d s w i t h e x p e r i mental v a l u e s  W r i t e out these f i e l d s from experiment  and d e v i a t i o n s  100  APPENDIX I I I Example C a l c u l a t i o n o f an E l e c t r o n R e p u l s i o n I n t e g r a l  Let  x y = d'xy  Then t and  +  d  v z  y z  = -(l/\/3)(xy +  J(t ,t ) +  =  +  =  :  xz = d  w.yz  +  x z  w?xz)  j"t*(l)t*(2)-.(e2/r ).(t (l)t (2).dr. 1 2  (1/9)J* | x y ( l ) . x y ( 2 )  +  +  + w2.xy(l).yz(2) +  + w2.yz(l).xy(2) +  w.yz(l).yz(2) +  + w.xz(l) .xy(2) + | x y ( l ) .xy(2) +  w.xy(l) .xz(2)  . x z ( l ) .yz(2)  yz(l).xz(2)  + w 2 . x z ( l ) . x z ( 2 ) }. ( e / r 2  w.xy(l) .yz(2) + w 2 . x y ( l ) .xz(2)  + w.yz(l).xy(2) + w .y (l).yz(2) +  yz(l).xz(2)  2  Z  + w 2 . x z ( l ) .xy(2) +  xz(l).yz(2)  w . x z ( l ) . x z ( 2 ) | .dr  +  = ( l / 3 ) ( J ( i , i ) + 2 J ( i , j ) + 2K(i,j) + (w+w )K(i,j)) 2  since  J ( i , i ) = A + 4B +3C J(i»j) = A - 2B + C  ( i , j = x y , y z or x z )  K ( i , j ) = 3B + C and  w + w  Thus  J(t ,t ) +  = -1  2  +  = (1/3) (3A + 3B + 6C) = A + B + 2C  1 2  ),  

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