THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING ON THE THEORY OF SUPER-HYPERFINE INTERACTION IN IRON GROUP ION COMPLEXES Inan Chen A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Nuclear Engineering... -1964.' IP-669,p f 66'

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ACKNOWLEDGMENTS The author wishes to express sincere gratitude to Professor Chihiro Kikuchi, chairman of the doctoral committee, for the suggestion of this problem and for providing guidance and encouragement throughout the course of this investigation. The assistance and concern extended by each member of the committee have been deeply appreciated. The author is indebted to Professor Geza Gyorey, a member of his committee, for introducing him to the language of electronic computers. Invaluable instructions and advice from Professor Hiroshi Watanabe of Hakkaido University, Japan, during his visit here are gratefully acknowledged. Many stimulating discussions with fellow students G. H. Azarbcayejani, R. H. Borcherts, S. H. Chen, and S. Karavelas have been very helpful. The following financial supports are recognized with gratitude: Research assistantship from National Aeronautics and Space Administration. Support of the author's study at Winter Institute in Quantum Chemistry and Solid State Physics (1962-63), from National Science Foundation and Institute of Science and Technology of the University of Michigan. ii

Partial support of the preparation of the manuscript by the Industry Program of the College of Engineering. iii

TABLE OF CONTENTS ACKNOWLEDGMENTS.............................................. LIST OF TABLES................................................. LIST OF FIGURES.............. CHAPTER I INTRODUCTION....................................... A. Theory of Transition Metal Ion Complex........ B. Survey of Studies on SHF Interaction.......... II MOLECULAR ORBITAL THEORY OF d-ELECTRONS.............. A. LCAO-MO Secular Equation..................... B. Simplified LCAO-MO Method..................... III HAMILTONIAN FOR THE MAGNETIC INTERACTION BETWEEN ELECTRONS AND NUCLEI................................. IV SHF INTERACTION IN CdTe:Mn+......................... A. Structure and Symmetry Orbitals of the Complex B. SHF Interaction Tensor........................ C. SHF Structure in EPR Spectrum................. D. SHF Structure in ENDOR Experiments............ V SHF STRUCTURE IN SnO2:V4........................... A. Structure of the Complex...................... B. Ground State of V4+ in SnO2............. C. Mechanism of SHF Interaction,,,......*.,,... D. Anisotropic Component of SHF Tensor.......... VI SUMMARY AND CONCLUSION........................... Page ii V vi 1 1 7 9 11 14 22 22 26 42 50 58 58 62 64 72 75 79 100 107 117 APPENDIX A OVERLAP INTEGRAS................................... B SOLUTION OF IMPROPER EIGENVALUE PROBLEM BY DIGITAL COMPUTER................ C SIMPLIFIED MO CALCULATIONS OF TETRAHEDRAL CCMPLEXES INCLUDING NEXT NEAREST LIGANDS...................... R-P-EIFE-RENCES..oES*#*0~~r ~ ~*~~~a~~~~ ~.. a * iv

LIST OF TABLES Table Page 4-1 Symmetry Orbitals of Nearest Ligands in CdTe:Mn++.... 27 4-2 Symmetry Orbitals of Next Nearest Ligands in CdTe:Mn++............................................. 29 4-3 Irreducible Representations of Impurity Orbitals..... 31 4-4 Relative Intensities of SHF Lines in CdTe:Mn+....... 49 5-1 Symmetry Characters of Orbitals in SnO2:V4.......... 61 5-2 Results of EPR Experiment on SnO2:V4+.. 63 5-3 V-Sn Overlap Integrals............................. 71 5-4 V-O Overlap Integrals.............................. 71 A-1 Transformation of Mn++ Coordinates.................. 83 A-2 Transformation of Cd and Te Coordinates.............. 84 A-3 Transformation of V, Sn and 0 Coordinates............ 85 A-4 Formulae for STO Overlap Integrals................... 86 C-1 Molecular Orbitals and Orbital Energies of CdTe:Mn++. 110 C-2 Molecular Orbitals and Orbital Energies of ZnS:Mn++.. 114 v

LIST OF FIGURES Figure Page 4-1 Nearest and Next Nearest Ligands in Zincblende Structure...................................... 24 4-2 SHF Levels and EPR Transitions............. 46 4-3 SHF Levels and ENDOR Transitions................... 51 5-1 Unit Cell of SnO2(TiO2)................ 59 4+ 5-2 Nearest and Next Nearest Ligands of V4+ in SnO2(TiO2).6).............................. 60 5-3 Splitting of d Levels in Crystalline Field of SnO2. 62 4+ 5-4 Schematic Energy Level Diagram for SnO2:V.......... 65 A-1 Coordinates for Overlap Integral Calculations....... 79 C-l MO Energy Level Diagram for CdTe:Mn1............... 111 C-2 MO Energy Level Diagram for ZnS:Mn++................. 115 vi

CHAPTER I INTRODUCTION The physics and chemistry of the transition metal ion complexes have been of considerable academic interest since the end of the last century. Along with the development of quantum theory, effort has been made to explain the electronic structure of these complexes with the new theory. The problem turns out to be one of the most difficult tasks common in many branches of physics - the many-body problem. The solution can only be obtained by successive approximations, and even with the high speed computational facilities available today, a first principle calculation is still difficult and the result unreliable. Consequently, the development of a semiempirical theory to exaplain observed phenomena is both necessary and appropriate. On the other hand, since the development of solid state maser, laser and other solid state electronic magnetic devices, interest has been stimulated in the physics of crystals containing transition metal ionso A better understanding of the electronic structure of such crystals would be useful in developing better devices. A, Theory of Transition Metal Ion Complex Transition metal ions are characterized by the partially filled shells of d-electrons. When a transition metal ion forms -1 -

-2 - a complex with a number of surrounding anions or molecules or substitutes the host cation in a crystal as an impurity, the d-electrons are no longer localized at the metal ion but move in orbits which extend to the whole complex.* The most direct evidence of this d-electron delocalization is the observed superhyperfine structure (SHFS),** stemming from the interaction of the electron spin with the magnetic moments of the ligand nuclei. Furthermore, the ligand nuclear moments act as a number of electron detectors embedded in the crystal and hence supply information about the motion of the electrons. Thus the study of SHF interaction is one of the most powerful tools for the understanding of the electronic structure of such complexes. Crystal field theory has been very successful in predict(2} ing the splitting of d-electron levels(2) in a complex and also fairly successful in interpreting experimental results(3) quantitatively. In this theory, the ligand ions are considered to be fixed point charges producing an electrostatic field having the symmetry of the complexo The d-electrons are affected by this non-spherical field but are assumed not to overlap the ligand ions and hence give no SHF interactiono Furthermore, in covalent complexes, the large discrepancies between experimental results and theoretical calculations served to emphasize the need of modifying the modelo The second approximation is usually called "ligand field T(4) theory"( ) In this model, the electronic structure of the ligand *Hereafter, we shall use the term "complex"' in a wide sense, ie,,, it includes the cluster consisting of an impurity ion and its ligands in a crystal. **Also cabled "Transferred Hyperfine Structure" by Marshall and Stuarto(12)

-3 - ions and the delocalization of the d-electrons are taken into account. Experimental evidences, other than the SHF interaction, which point to the need of modifying the crystal field theory are the reductions of parameters in crystal field theory such as g factors, Coulomb and exchange integral parameters(5) B, Survey of Studies on SHF Interaction The first observation of SHF interaction was made by Griffiths, et al.( in iridium complexes, IrC16 and IrBr6 In the electron paramagnetic resonance (EPR) spectra of these complexes they observed an anomalous hyperfine structure which can only be explained as arising from the interaction of d electrons with the surrounding halogen nuclei. Later, Tinkham(7) observed similar phenomena in the EPR spectra of ZnF2 (rutile structure, see Chapter V) containing iron group ion impurities. Assuming that d-electron orbitals are augmented by small amounts of ligand orbitals of the proper symmetry, he estimated that the magnetic electrons have a probability of about 6% each of being in fluorine n = 2 and n = 3 orbits, SHF interaction was also observed in nuclear magnetic resonance (NMR) experiments. Shulman and Jaccarino(8) observed the frequency shift of the NMR line of the fluorine in MnF2o This shift was explained as due to the mixing of the fluorine orbitals with the manganese orbitalso Later this problem of fluorine hyperfine interaction was re-examined by many investigators. Mukherji and Das(9) calculated

the interaction by orthogonalizing d wave function to the ligand wave functions. The calculated value of the hyperfine interaction is about half the experimental value. Keffer, et al.(0) considered both the orthogonalization of the d wave functions and the charge transfer from ligand to the cation. The results are in reasonable agreement with the NMR measurement of Shulman and Jaccarino. Clogston, et al.(ll) related Keffer's approach to the idea of covalent bonding, and introduced molecular orbital treatment of the problem. They noticed that for other than perfect octahedral symmetry, there will not exist a coordinate system in which the SHF interaction tensors for all of the ligand nuclei are simultaneously diagonal, and observed the effect of the off-diagonal components in EPR spectrum of ZnF2:Mn. From the neutron diffraction form factor measurements, (12) Marshall and Stuart( assert that in complexes the d wave functions are expanded over the free ion values and the SHF interaction in MnF2 can be explained by the Heitler-London model using this expanded d wave function. However, Alperin(13) reported that the neutron diffraction form factor measurement indicates a decrease in the Ni d wave function in nickel oxide. Also, Marshall and Stuart obtained their result by neglecting the j bonding. However, NMR measurement on KNiF3 and KMnF3 by Shulman and Knox(4) and Hartree-Fock calculation by Sugano and Shulman(l5) have shown that the 7 admixture is quite large. All the above mentioned observations are the SHF interaction with nearest ligands. The SHF interaction with next nearest ligands has been observed in cubic crystals of group II-VI compounds containing

-5 - S state ions of transition metals. The interaction constants are almost isotropic and have the following values: (16) -4 -l In CdS:Mn and CdTe:Mn, A = 2.6 x 104 cm; Cd (17) -4 -l. In CdSe:Mn(17 A 27 x 104 cm; Cdm IZnMn(17) 1 In ZnS:Mn 7, AZn 0= 75 x 10l4 cm-1. It was pointed out by Schneider, et al.(17) that the ratio of ACd to AZn is roughly the same as the ratio of the nuclear magnetic moment of cadmium to that of zinc. This means that the magnetic electron has almost the same probability of being in the next nearest ligand site in spite of the increase in the lattice constants of the above cited crystals from sulfide to telluride. This also indicates that the covalency increases in these crystals from sulfide to telluride. They also pointed out that no SHF interaction of nearest ligand has been observed although the nearest ligands S33(0o74%)> Se77(7,5%), and Te125 (7.03) have small but finite abundances. Most recently, From, Kikuchi and Dorain(18) and Kasai(l8a) observed two sets of SHF structure in Sn2:V4+ (which has rutile structure) associated with tin nuclei at non-equivalent sites. The interaction is anisotropic and ha: much larger value compared to those of Group II-VI compounds. SHF structures in TiO2 containing transition metal impurities are also observed by Yamaka (19) and Chang(20)o Observation of SBF interaction by electron-nuclear double resonance (ENDOR) has just been started. Ludwig and Lorenz(21) reported on the observation of cadmium hyperfine structure in CdTe containing Cr+ ion. The interaction is anisotropico

-6 -No ligand field theoretical (molecular orbital) treatment of next nearest ligand SHF interaction has been published. The purpose of this thesis is to present a theoretical investigation of SHF interaction due to next nearest ligands in two types of complexeso

CHAPTER II MOLECULAR ORBITAL THEORY OF d-ELECTRONS As mentioned in Chapter I, Section A, ever since the experimental observation of d-electron delocalization in transition metal ion complex, it has been generally accepted that the point charge model "crystalline field theory" should be replaced by the "ligand field theory," In ligand field theory, the wave functions of the delocalized d-electrons are approximated by the (anti-bonding) molecular orbitals *a constructed from the linear combinations of atomic orbitals (LCAO) of the impurity and the ligands, = CO 4 X C fr (2-1) where d represents an atomic d orbital of the impurity, 0 's represent atomic orbitals of ligands, and C, C 's are numerical coefficients usually known as "mixing coefficients"o The valence electrons of the complex are described by (bonding) molecular orbitals. These are also LCAO's of impurity and ligands where the ligand orbitals are the major constituents. The wave function of the many-electron system is represented by the aitisymmetrized products (Slater determinant) of all occupied molecular orbitals, ik= cM!) / Z;L-l) Pll(~) [ZLj) ~ - (-x^ (2-2) -7 -

where n is the number of electrons in the system; xl, x2,... are the space and spin coordinates of electron 1, 2,..; and P represents the permutation operator~ By applying the variational principle, i.e., minimizing the energy of the system, constraining the MO's to the orthonormality, we obtain the Hartree-Fock equation for the molecular orbital, ji: ^-VW(X) ^ a (E. - i ((2-3) with the effective Hamiltonian (for electron 1) ^ ^ (X) K- ^ ^ B^x, e(X^- ^-Z ( ',x2-F) (2-4) where (C.x3) - Lj i (Xl) ^ ^ ) (2-5) is the Fock-Dirac density matrix, and P12 is the "interchange operator" (22) with respect to the coordinates x1 and X20( Physically, this is an independent-particle model, according to which each electron in a many-electron system moves under the influence of the external field (nuclear charges) and an average field of all other electrons. Details of the Hartree-Fock process for LCAO-MO have been worked out by Roothaan.(23) This method leads to a secular equation with the self-consistent field (SCF) scheme for the determination of MO energies and the mixing coefficients. We shall discuss this scheme by a simpler but equivalent way in the following.

-9 - Ao LCAO-MO Secular Equation Let us represent the MO fi by LCAO as J= X.^ (2-6) U Cj= Q where Civ is the mixing coefficient of v-th AO in i-th MO, and cp 's are normalized but not necessarily orthogonal atomic orbitals. If we substitute this expression of ri in Equation (2-3), multiply by cp* from left on both sides of the equation, and integrate over all space, we get where ZA <2c 0 Z jK. - -SQ = o E Ie E ) a ~ H 5 9 H 1 5 d (2-7) (2-8) and $ S cp~~p ci (2-9) Equation (2-7) is the secular equation for the determination of MO energy Ei and the mixing coefficients Civ'So This is an "improper" eigenvalue problem. It is "improper" because the AO's cp are not orthogonal in general, i.e., S i,V and the unknown eigenvalue Ei appears not only on the diagonal of the secular determinant as in the usual eigenvalue problem, but also in the off-diagonal positions. This kind of eigenvalue problem can be solved by a combination of two successive diagonalizations as shown in Appendix B.

-10 - Further complication in the solution of Equation (2-7) arises from the fact that the matrix element ItI itself contains the unknown coefficients Ci 's through the term p in the operator Heffo Therefore, the solution of this problem must be done by an iterative procedure, starting with a set of first estimation on Civ's, repeating until self consistent results are obtained. This is illustrated by the following cycle: {Ci - > P - Heff > - it In principle, the secular equation (2-7) must be solved for all electrons in the complex, However, the inner shell electrons of the constituent ions are quite localized and have little to do with the bondingo Therefore, the secular determinant breaks into blocks, one for each inner shell of each ion, and one for the valence electrons of all the ions, Since we are interested in the valence electrons only, we consider only the block corresponding to the AO's of these electrons, Further reduction of the secular equation can be attained by taking into consideration the symmetry of the complex. Since the Hamiltonian of the system is invariant under the symmetry operations of the point group of the complex, the wave functions of the system can be classified according to their properties under symmetry operations giving one of the irreducible representations of the point group, The basis functions of the MO, cp can also be classified in this way. If we use such "symmetry orbitals" as basis functions, then since the matrix element of the Hamiltonian between two basis functions belonging

-11 - to different irreducible representations vanishes and since two such functions are orthogonal, the secular determinant breaks up into blocks one for each irreducible representation. In this way the secular determinant can be reduced in its order. However, even with these reductions, SCF-LCAO-MO calculation requires a tremendous amount of work for symmetry lower than spherical, and the results become less reliable as the number of ions increases, No such calculation has been done for a system more complicated than (24) B. Simplified LCAO-MO Method The closest approach to the SCF-LCAO-MO calculation which has been done for transition metal complex is the calculation of crystalline field splitting by Sugano and Shulman(15) Using the results of Hartree-Fock calculations for atoms(25), neglecting overlap and covalency effect in the Hamiltonian, they calculated the matrix elements of the Hamiltonian and overlap integrals. This is equivalent to the first step in the SCF calculation, They did not carry out the iterative procedure, but obtained good agreement with experimental results for the crystalline field splitting of KNiF3 Another simplified approach is the "semiempirical" MO calculation of Wolfsberg and Helmholz(26) In this method, the diagonal elements of the matrix H are approximated by some empirical energy values, and off-diagonal elements are calculated from an empirical formula

-12 - S A A+AK + HyS /A.) = + Ad ----— ^ ----- (2-10) or 14 v = -Hi S,4 /l A' Hvv (2-11) where f is a constant usually assigned a value of about 2. The (27) second formula is proposed by Ballhausen and Gray. The overlap integrals Sv can be computed by using either Slater radial functions or, if available, Hartree-Fock functions. The method of evaluation is given in Appendix A. Valence state ionization energies (VSIE) are used for the empirical values of the diagonal elements of H by Wolfsberg and Helmholz (26), and in an earlier paper by Ballhausen and Gray. (27) The method of evaluating VSIE is given by Moffitt(28) for the first short period elements. The applicability of this method to other (heavier) ions has not been justified0 Another approximation to the matrix elements H 's is the atomic one electron orbital energy obtained from Hartree-Fock calculations of atoms (e.g., Watson and Freeman (25)) This energy differs from the exact HW by the interaction with the electrons centered at other ionsO Sugano and Shulman's calculation shows that this difference is not always negligible, Jzrgensen(29) has pointed out that this semiempirical method may lead to a wrong ordering of MO levels with respect to experimental results.

-13 - In a recent paper Gray and Ballhausen(3) proposed a general rule for qualitative determination of the MO energies. They are: (1) The order of AO energies is taken to be a(ligand), ~b(ligand), nd(metal), (n+l)s (metal), *(ligand), (n+l)p (metal). (2) The amount of mixing of AO in MO is roughly proportional to overlap integral and inversely proportional to their AO energy difference. (3) Other things being approximately equal, a bonding MO is more stable than n bonding MO, and a antibonding MO correspondingly less stable than c antibonding MO. (4) The relative MO ordering is considered final only if it is fully consistent with the available experimental results, exact differences in the MO levels can only be obtained from experiment. Two simplified MO calculations for tetrahedral complexes including next nearest ligands are given in Appendix C. In one of them, spectroscopic data(31) are used for the matrix elements H, and in the other, Hartree-Fock orbital energies are used.

CHAPTER III HAMILTONIAN FOR THE MAGNETIC INTERACTION BETWEEN ELECTRONS AND NUCLEI The Hamiltonian for the interaction of an electron with a nuclear magnetic moment at the origin has been derived by Fermi and (52) others(32) as: Xi L2 4 9 t3N I (L3 L S) 5(l s r)+I '(rlS 3 (3-1) where Pe and PN are Bohr and nuclear magnetons respectively, and gn is the nuclear g factor.* I;, and S are, respectively, nuclear spin, electronic orbital, and electronic spin angular momentum operators in units of i. This Hamiltonian can be derived from Dirac's relativistic wave equation for one electron as the non-relativistic limit. A simplified alternate derivation is given by Blinder(^3) recently. In our problem, the electrons interact with a system of nuclei. The Hamiltonian for the interaction of one electron with a system of nuclei can be derived by generalization of the derivation of Equation (3-1). This is given in the following. Dirac's wave equation for an electron (with charge -e) in electromagnetic field (characterized by vector potential A and scalar potential 0) can be written as \-CO,, - - -A = E -R (3-2) *Nuclear g factor is defined such that nuclear magnetic moment Ln = gnInn -14 -

-15 - with a ( O i ( ~ -6 where a is the Pauli spin matrices 0 1 0 -i 1 0 x (1 0) i (i 0) / * 0 -1 The sources of the electromagnetic field are a system of point nuclei of infinite masses with charges Zne and magnetic dipole moments gnPNIn, n = 1,2,5,... Accordingly the scaler and vector potentials are, respectively, 1n = ~ Zt e(3-3) B =C E SW I xMn/t (3-4) where r is the radial vector from nucleus n to the electron. -n The wave function T is a four-component spinor, and we represent it as 2 = 1 ) (3-5) where 1 and p2 are two-component spinors. Since we shall be only interested in the non-relativistic limit of the equation, the relativistic energy ER differ very little from the rest mass energy mc2. It is convenient to introduce E = E - me2 (3-6) Note that E < mc2.

Thus, Equation (3-2) reduces to the following set of two spinor equations: (E + e> +2ecZ) l + C+r (j+ t A) A z ~ (3-7) ( E +t e 4 ) ir2. + C F ^r ( ) ^ -~ (3-8) From these equations, we see that P1 (positron component) is smaller than 12 (electron component) by a factor of order v/c. We can eliminate 1 between the two equations, yielding: {it ^ - [T." c(+ | (()[f(t +co (3-9) where we defined g. ( ) = M '/nt, L, 2 "'.) = [1 + J (3-10) Equation (3-9) can be rewritten in the form /e, ia = E t (5-11) where the effective Hamiltonian ef can be separated into three parts: e*^ = 3~ ( +;Z() ( ) (3-12) with + - * (') E ( +) 3( -.?) (3-13).7 =;,)c -{ (T ^) (^^d () (^ )le^ ^ t) )} (3.14)

-17 - (2) C2 M) l Ct ( [ Pye (_) t 8)Q (3-15) The first term (0) corresponds to the kinetic and potential energies with relativistic corrections plus the spin-orbit interaction. The third term is the self-energy of the nuclear magnet and is of second order in nuclear magnetic moment. The second term l(1) contains terms linear in nuclear moment and.gives the interaction Hamiltonian between electron and nuclei. We shall consider further reduction of this term only. Recalling the relations p)=K ( r) - V cr ( ) (3-16) and (a. a)(a. b) =a b + ia *(a x b) (3-17) where a and b are two arbitrary vectors commuting with a but not each other, Equation (3-14) can be rewritten as (X ) = o + i~ c(3-18) where iR =ft =Z^ (trif^ ti-Ar A*(P X + Axp)3 (3-19) 2 -. i V~(~).' c 4 7 - L-)K X A} (3-20) Using the expression (3-4) for A and the Lorentz condition div A = 0, we have

p - i+ 1 * -? 2 A- I1 = 2 2E, % i.3 in.xr.p = 2 F f, qn 3 l Q IV - 11 (3-21) where tn = r x p is the orbital angular momentum of electron with respect to nucleus n. Similarly, X, + Ax = - t Vx v but A = Cv x LZ^,\t Ir. x r, x = Z w In );C Q- I ])3f = S^t.I|Z r3,, + 3fln-L-r (n +^- t) In X'tn^ In rjln]} = 1niX ) + i C( IX t I~ VI%, (XJC)~ la~~~rb~~ = %?.- An ITnh +- 3. V )n (I CX n + In^, + 1 n ) } = l N r[-/~ In + Q ^ t r" * ^} Thus E x A + x E =P -T 2,,,, Iv + 3 n 5rrl (3-22) Introducing the electron spin angular momentum S (in units of 6) by a = 2S _._ (5-23)

-19 - Then from Equations (3-19, 3-21, 3-22, and 3-23), we have Pi 2 = K2 ( ) (r )^(^i —3 (3-24) Before reducing the second part h2, let us investigate the properties of K(r) as a function of tn, n = 1,2,.... From Equations (3-3, and 3-10) we have Kwm =[1 — 2mc)li 5 t j Z,,e )] e i/(1 t1 L Z2 z^' ) -^ 1 (3-25) in the region where 'n > Zne /2mc2 1.4089Zn x 10-13 cm. Furthermore, K(r) - (const.) x hn as hn -0 hence, the expectation values of hi will have zero contribution from the points hn = 0, n = 1,2,... i.e. only non-s orbitals contribute to the expectation values of hi, and for such orbitals the condition for the validity of Equation (3-25) is always satisfied. Thus we have, 2 3 2r-^- S)/) (3-26) Now let us consider the second part h2. Since 2V vr.- y T1 + en1 y 3^ = z AD,,(3-27) -- --

-20 - we have tLn ' A, + hF- 4 t * An (3-28) But since a3 rc Mh/an -?- r 2 ( )- ~ Y-t.. t\I ) A c ) c + 0 1 and 1 t e. r, _o t '^ i= t ) a - l a ( fn = o) - therefore 8 is essentially a delta function: a nn 'svG Za rt,"6*ILV - T(iL >) (3-29) Thus Equation (3-28) reduces to t Z = -A' I it1 (^.4 n) C t ) [ i x i t +A, 2S_ rn x I, x:. = 2k ele,,JjkrN -(,t)ZSI 4S(, ) (3-30) Noting that a,(t) O- 4ii ( ~:"), and the average over all asgls... s L I, - A 1 ( t f T^ )....... 0,a 2 3 - we have =o 2 2^ ( -,") s- - 4 VN 3~ (3-31) Summing Equations (3-26) and (3-31) we have for the hyperfine interaction Hamiltonian in the field of a system of nuclei, -\I ' =Z - Xtln), I-S s - 3-, rtr S)(rfitw), I',"a ) iv" o ^ ^-LS'^U^)^ 3/C^-X~i")l IT c~ s~> (3-32)

-21 - Our next task is to obtain a Hamiltonian for many-electronmany-nucleus interaction. Since an exact relativistic wave equation for many-electron system cannot be written in closed form, the manyelectron Hamiltonian cannot be derived in the same way as was done in Equation (3-1) or (3-32). However, it is reasonable to assume that to first order in fine structure constant, the many-electron Hamiltonian can be represented by a sum of one-electron operators, Equation (3-32), one for each electron. Thus we obtain the Hamiltonian which will be used for later discussion: 8 = ^S E a (2in- S o) At 3 S k)n amko ^^ )(rkt l)17 (3-33) + (tk) Sk In where rkn is the radial vector from nucleus n to electron k, -kn is the orbital angular momentum of electron k with respect to center n (position of nucleus n).

CHAPTER IV SUPERHYPERFINE INTERACTION IN CdTe:Mn++ The superhyperfine structure due to next nearest ligands has been observed by Lambe and Kikuchi(16), Dorain(16a), and Schneider, et al.(17) in cubic crystals of Group II-VI compounds containing S state transition metal ion impurities (Chapter I,B). In this chapter we shall derive the expressions of the superhyperfine (SHF) interaction tensor A (for the next nearest ligand n) from molecular orbital theory, and then discuss the SHF structures observed in electron paramagnetic resonance (EPR) and electron-nuclear double resonance (ENDOR) spectra. We shall take cadmium telluride containing manganese ion, CdTe:Mn++, as an example. However, the discussion applies also to other iron group S state ions in any cubic crystals of II-VI group compounds. A. Structure and Symmetry Orbitals of the Complex The transition metal ion impurity in cubic crystal of Group II-VI compound (zincblende structure) is surrounded by four anions tetrahedrally arranged at alternate corners of a cube, the edge length of which is one half of the lattice constant, a. This means that impurity-anion distance is 43 a/4. The next nearest neighbors are twelve cations situated at the centers of the edges of a cube with edge length equal to the lattice constant. Thus impurity-cation distance is a/ 2. (See Figure 4-1). Each nearest ligand is bonded to three next nearest ligands and the central impurity ion tetrahedrally. We shall number the nearest -22 -

-23 - ligands from 1 to 4, as shown in Figure 4-1, and denote the three cations which are bonded to anion i by ia, ib, and ic, i = 1,2,3,4. The coordinates at the ligands are chosen in the following way with respect to the coordinates (Xo, YO, ZO) of the central ion. xll 16 -21 1/ 6 Xo Y, I = 1/ 2 0 -1,2 Y (4-1) Z, J -1/ 3 -1/;3 -1/3 z 'X s f 1 0 0 X la! Y 0 1/ 2 -1/F2 Y (4-2) la 0 zl. Zo la 0 ~ -1/ 2 -11 2 Z la) Xlb 0 1 0 X y /2 0 1/02 Y (4-3) Ic o Zx -1 02 0 1 X~ Yc = | 1 /2 -1/;2 o Y~ (4-4) j ZlCJ -1lf2 -1/ 2 0 Z The coordinates of ligands i, ia, ib, and ic (i = 2,3,4) are obtained from the above set (i=l) by twofold rotations around XO, Y0, and Zo axes respectively. For the ligands, left-handed systems are chosen for the convenience of evaluating overlap integrals (see Appendix A). We shall consider the molecular orbitals formed from linear combinations of (i) impurity (manganese) 4s, 3d orbitals, (ii) nearest

-24 - ---- o - -- Figure 4-1. Nearest and Next Nearest Ligands in Zincblende Structure. I: Impurity Ion. 1, 2, 3, 4: Nearest Ligands ia, ib, ic (i = 1,2,3,4): Next Nearest Ligands.

-25 - ligands (tellurium) o(5s,5p ) and r(px, Py) orbitals, and (iii) next nearest ligands (cadmium) c(5s,5pz) orbitals. The symmetry of the four nearest ligands is that of the point group Td. Making use of the character table, we can construct the linear combinations of nearest ligand a and iT orbitals, which transform according to the irreducible representations of the Td group. The results are given in Table 4-1. The twelve next nearest ligands have octahedral symmetry. In Table 4-2, we present the linear combinations of next nearest ligand a orbitals, which transform according to the irreducible representations of octahedral group Oh. Finally, the 'classification of the manganese orbitals according to the irreducible representations of Td and 0h groups are given in Table 4-3. The wave functions of the five unpaired electrons are the manganese 3d orbitals augmented by ligand orbitals of the same symmetry. Thus they can be written as: -X)\= ^Cl$ t I, At bes W - ^ + <e l, (4-5) 2- ^^ e 'j + 1. t e t20 (4-6) cs^= "c^M-^V^Ms^32~%,~3, UP (4-7) D - C&6t+ SF,+thlz,+^lu^^.^^^.~ t tat it 'z (4-8)

-26 - K ' t^J +.ss Y S +- + T a + S c t m {', (4-9) These molecular orbitals are the antibonding MO's and hence the coefficients P, 7's are small quantities. Bo Superhyperfine Interaction Tensor The Hamiltonian for the interaction between the ligand nuclear spins and the unpaired electrons has been derived in Chapter III, Equation (5-55). It is in the form of a sum of one-electron operators, (4-10) Self k. -I where ~ l X +T( --- -- + il d^ l- r (4-11) The wave function of the ground state 3d5 6S, can be represented, to first approximation, by a Slater determinant of the five antibonding MO's, { i D:') )jr2) D3 L ) ")-(L4) 4 L-(4) (4-12) The first order perturbation to this state due to the Hamiltonian ' —SF can be evaluated as This pert ur bat-> i u e e as a1 spin Hainlti)> i(4-13) This perturbation is usually expressed as a spin Hamiltonian in the form

-27 - TABLE 4-1 SYMMETRY ORBITALS OF NEAREST LIGANDS Irreducible Representation A1 E(G) E(e) T1 T (x) 2 C2i 1 2 1 2 1 4 1 4 1 4 1 2 1 4 1 2 1 2 _ 1 2 = 2 =! 2 (si (zi [xi [yi [yi (Yi [Y, (si (Zi [Xi (s, (zi (xi + S2 + z2 + X2 + Y2 + Y2 - Y2 - Y2 + s2 + z2 + x2 + X2 - S2 - Z2 - X2 Symmetry Orbital + s3 + s) + 3+ Z4) + x3 4 - -3 (Yl + y2 + Y3 + Y4)] + y3 + y4 + 3 (x + x2 + X3 + x4)] 3- y-4 - (xi + 2 - x3 - X4)] Y3 - Y4) - y3 + y4 +3 (1 - X2- X3 + X4)] 3 - 54) - z5 - 24) - - x4 +' 3 (y + 2 - 3 - 4)] + S3 - S4) t z3 - Z4) + X3 - X4) cP3s %z CP3 z T4s 4z cP4t T2(Y)

-28 - TABLE 4-1 Irreducible Representation T2(z) P5s = 2 (s - 2 C5 = 1 (z - z 5 z 2 1 2 Cp5C = 1 x1 [ x2 5 -A 4-2 CONT'D Symmetry Orbital - s3 + S4) - 3 + 4) - x + x4 -3 (Yl - Y2 - Y + 4)]

-29 - TABLE 4-2 SYMMETRY ORBITALS OF NEXT NEAREST LIGANDS Irreducible Representation Symmetry Orbitals Alg 1 J12 Z (Sia i=l + Sib + Sic) Eg(G) g 1 = 1 xs= = 1 lz =t 1 X18 4 i=l 4 i=l 4 z i=l 4 i=l i=1 (Z. + z. + z ) ( ia + ib ic (Sia + Sib - 2Sic) (Zia + ib - 2z. ) 1C E(e ) (Sia - Sib) 1 X2z = ~s 8 4 i=l (Zia - Zib) T2g(e) T )3(s 2 + 5~s 2 la S2a - sa - 4a) T2g(n) Xz 1 (Zla X4 = 2 ( Slb X4z = 1 ( lb 2 Xs 1 (Slc 2 + Z2a - Z3a - Z4a) - 2b + 3b - z +2 I z3b - 4b ) T2g( ) - 2c - S3c + 4c) 5z = (lc - z2 - z + 4 ) 2c 5c 4c

-30 - Irreducible Representation TABLE 4-2 CONT'D Symmetry Orbitals T lu 1 (Slb - $8 -8 la '78 1f8 la and three 1 (slb + 18 1 (Sla + T8 8 (Sla + and three s1 + s - s - s lc + S2b 2c 3b sc - sa + sc + s7 lc - S2a + S2c 3a Slb - 2a + S2b - 3a similar combinations c Slc+ S2b + S2c - 3b + Sc - S4b + S4c) - 3c - 4a 4c + 3b + S4a - 4b) Df P orbitals. z - c - S4b - S4c Slc - 2a - 2c + 3a + 3c - S4a - S4c) slb - S2a - S2b - S3a similar combinations of S3b + S4a + S4b) P orbitals. z

TABLE 4-3 IRREDUCIBLE REPRESENTATIONS OF IMPURITY ORBITALS Orbital Irred. Rep. in Td Irred. Rep. in Oh 4s A1 Alg d- = 3dz2 E(@) Eg(Q) d2 = 352 _y2 E(e) Eg( ) 3 = 3yz T2(x) T2g() d = dzx T2(y) T2g(T) d5 = 3xy T2(z) T2g( )

-32 - / /..... By A 1W.l-lD> 9- By S By '1-^ (4-14) where S is the total electron spin operator. The program of this section is to obtain the expression of the n SHF interaction tensor A in terms of MO and geometric parameters, The Hamiltonian (4-11) consists of two kinds of interactions: (i) Contact interaction, and (ii) Dipole interaction. The former gives isotropic contribution to An, and the latter is responsible for the anisotropic part. (i) Contact Interaction The one-electron Hamiltonian for this interaction is KHsti) =*- Lt. 2 j 35.1). IL (4-15) According to Wigner-Echart theorem, we can relate the one-electron spin operator sk to the total spin operator S as, (within the manifold of fixed$ ) s = s/24 (4-16) where J (=5/2) is the eigenvalue of S. Thus Equation (4-15) reads, K 1. ). — r i (1 \ (4-17) and from L<^IHJ & l AS \>;J D ' ^AS ' - b Z T ". (4-18) L "

-33 - we have, for the isotropic SHF interaction, t i BT CtiH' l ( Li > v (4-19) The last factor i <DA I 8(L1)) I >) is the density of unpaired electron spin at the nucleus n. The contributions from the various AO's in D. to this density are estimated as follows: Using the Slater radial functions for Mn 3d and Te 5s, 5p, R3d(Mn) = zSS ( (') " x(-16t7 ) (4-20) Rsp(Te) - 9 /) (4-21) and Hydrogenlike wavefunction for Cd 5s, 5s R5s(Cd) (4-22) + Ci'';fG4^ j mx p _-, Etr67) we obtain 2 -9 R 3d(Mn)[ at Mn-Cd distance (8.58 at. units) = 1.09 x 10 a.u. 2 -4 R sp(Te)l at Te-Cd distance (5.25 at. units) = 1.08 x 10-4 a.u. 5 sP IR5s (Cd)l at Cd nucleus = 1.63 a.u. From this result we see that even if the probability of an electron being found in cadmium 5s orbital is as small as 0.1%, practically the contributions from tellurium and manganese orbitals are negligible. Thus finally we have the expression "Slater radial functions are used because they are more extended than Hartree-Fock functions and hence give the upper limit of the estimation. **This percentage is estimated from Lambe and Kikuchi's experimental value An = 2.6 x 10-4 cm-1 by comparing to Jonest ACd = 0.11 Cm1. 5 d=0.1C

-34 - cs = -IS S I>S\'It3 IKO$ lt c~ /$ s(4-23) where Rs(O) represents the value of cadmium 5s orbital at its nucleus, and ',.il are the mixing coefficients introduced in Equations (4-5) through (4-9). We have not taken into account the effect of spin polarization in deriving the above formula. This effect can be included simply by replacing IR (O) 2 by p (0), the density of unpaired spin at the s s nucleus when there is one electron in the orbit Rs. Thus A ~'"'~E a n 0()'i 1.Y v - ~ Hk}(4-24) p (O) can be-estimated from the hyperfine structure constant (isotropic part) ACd of cadmium, Ac& - 2 i3, T E CFo) (4-25) Jones(3) has reported the value Ad = 0.11 cm obtained from optical Cd measurement. (ii) Dipole Interaction The one-electron Hamiltonian for the dipole interaction is the first two terms in Equation (4-11) HD) a- g Zl: 3_iL _ - _ _ (4-26) The SHF interaction tensor due to this interaction, A D can be obtained from the equation

-35-.....Kvi ',,-L <1 )!^)jI;1' )> n i2: where _ is a tensor operator. )k = 2r, -AL - r, + j X (4-28) AD =t;- LI (4-29) Since D. is a linear combination of manganese, tellurium and 1) in manganese 5d orbitals, tellurium orbitals, and cadmium orbitals respectively. We have neglected the cross product terms in the expansion. The first two terms in Equation (4-50) are the contributions from the electron densities centered at manganese and tellurium atoms. cadmiumn consider thes we densities as concentrated at the nulement a calculate their interaction with cadmium nucleus as point dipoledipole in manganese 3d orbitals, tellurium orbitals, and cadmium orbitals respectively, We have neglected the cross product terms in the expansion. The first two terms in Equation (4-30) are the contributions We can consider these densities as concentrated at the nuclei and interaction. Further, in the second term we consider only the interaction with the electron density centered at the tellurium which is nearest to cadmium n.

-36 - Thus from the first term we have 2 n 57-W R-M I) L>Q ( -31) for the diagonal component A>, = x,y,z; and o -3 - 5 f RL(C ^ ( CoS es c ) 5; ZL (4-32) n for the off-diagonal component A, d, ', = x,y,z; where RMC is the distance from the manganese ion to cadmium n, and 0@ is the angle between.MC and 5 axis, Similarly from the second term, we have, 5 M3c - l 1 4: -i + 4 t } (4-5 ) for the diagonal component Al, and ImAW CTC(S ii Caij ) jc E ts + T s i (4-34) for the off-diagonal component A; where RTC is the distance from tellurium ion to cadmium ion n, and 0, is the angle between RC and 5 axis, ~ = x,y,z. In the third term of Equation (4-30), only the p orbital of the cadmium n has to be considered. By operator equivalence technique HD(k) can be rewritten as 2| (K)) I.1,:.+;Q i V i\ - ) l 2( ( Vr)t a id 1 id) 2 DIAXL Q (4-35) where = 2/(2i - 1)(2n + 3), and n is the orbital angular momentum of electron with respect to the nucleus n. Thus the pz orbital of the

-37 - ion n contributes M1 ^ < < ZI t 4 > )- 1 L B } (4-36) to diagonal component A, and (4-37) | Me " <% >5P< > l 3 ( 2, ti Pa i t )l> H b Ag + i,t, | (4-37) n to off-diagonal component A v In applying these general formulae to cadmium ions at different sites, we note that if the components of A tensor are referred to the coordinate system (Xo, Yo, Zo) of the central (manganese) ion, the expressions will be different for different sites of cadmium. However, if we refer the components to the coordinates (Xn, Yn' Zn) at the ion n as defined in Equations (4-2), (4-3), and (4-4), then the expressions will be the same for all the cadmium sites. Therefore we shall first derive the expressions for the components with respect to this set of coordinates and then transform the results into the coordinates of the central ion. The latter coordinates is the one which experimental results are referred to. For the simplicity of notation, we use (t, Ai, a) for (X,YnZn) and reserve (X, Y, Z) for (Xo, Yo Zo). The angles 9, 0 in the general formulae,Equations (4-31) through (4-34) are &Xi = 90o 6^= 1i (4-38) = 0~

-38 - cos20, = 1/3 cos2 0 = 0 cos2o = 2/3 cos0 cosO = 2/3 (4-39) for all of the twelve sites. Introducing the following abbreviations: A, A s AZ ~x / (4-4o) AM 1 5; \ 8t I 8\4 X 2 02 V;8 0(4-414) AT= N pK.k C + I't 4 47 t1? I (4_41) c -1 W N U < ->j i: < A - (4-42) -A,. -=- As - A - Ac (4-43) A - As- AM-Ac- AT (4-44) ATr- = AS + 2 AM -K2AC 1 AT (445) A.- A7x ~ {2 A~T\ (4-46) Other components are zero. The components of A tensor in the coordinate system of the central ion, (X,Y,Z) can be obtained by the transformation A - > -" f of. - A (4-47),, - L. A ~ A=7 /= /; - j, = N Y, -/ -

-59 - with the transformation matrix (ai) given in Equations (4-2), (4-5) and (4-4). The results are the following: For a-type cadmium sites AX= A1TT A,,- IGA Aj A%% = p ( j'' A..r) A$x =7 A)( -)( ( -I a-) (4-48) Axy. =- (+ t)- t-. ) (a-,-) A, X = )() )- - A(For b-type cadmium sites AXX = t + A.-C) Ayy= Az A ==1 + A +/~ (A~-..1 (4-49) A2 =()-)QtMT-) 4 ( ---,, t A0y) AYY 2 A7T A H H~~~~~~~~~~~~~~(~r

-40 - For c-type cadmium sites A ~(A8/ t As - Az7 A-TA, AYZ = ()T-)(-)+ (- A +. ) Axy =At(+) ( —)(+) ( )-)- ( 50) The four f: signs preceding the off-diagonal elements are for la, 2a, 3a, 4a, etc. respectively. Introducing the abbreviations A+ -- = (A 2 (A,+a Ac) (4-51) A_ = ). (Am- Aj) AT (4-52) we can express the components in more compact form as: For a-type cadmium, AYK = As + At A, = A +) -) -)(453 ) Air = () I) (-) t) (3A+ A ) ATyz = +) (+) )(-) A_ Azx = (+) - t)(A+ - A

-41 - For b-type cadmium, Ay= A, + At A/y= A, + A+ Ax = -) )(3 A+ A (4-54) Arz = (+)(+)(~-)l-) (3A+ -A-) A = -))()A_ For c-type cadmium, Ay, - As + A AYY= As t At A, = As-2A+ (4-55) Axy= (+)-)-)l-) A_ A, =- t+) l+ ) -) (o-)h 1 A - A-) Ay = (+-)Q )t-) (3 A+- A-) In total we have three independent parameters A, A and A + We shall discuss the relations between these parameters and the experimental quantities in the following sections.

-42 - Co SHF Structure in EPR Spectrum In natural cadmium only about 25% of nuclei (Cd, Cd ) have non-zero spin I = 1/2. The magnetic moments of Cd (-0.5922 PN) 115 and Cd (-0.6195 Nr) are nearly equal. We shall treat them as identical in the following discussion. The spin-Hamiltonian describing the interaction of cadmium nuclear spins In s with the unpaired electrons and external magnetic field H is. o a _S -Z A LI (4-56) where the summation is over the cadmium nuclei with non-zero spinso By introducing an effective magnetic field H]) = o- Al ^h (4-57) where h is the unit vector in the direction of H, and Ms is the projection of S along h, the Hamiltonian (4-56) can be rewritten as - X Hes 1(4-58) In EPR experiments we observe the transitions with AM = + 1, s -- MI = 0. Therefore, the direction of H changes after the transition. -eff Thus it is convenient to describe the nuclear spin states of the ligands with the direction of crystal Z axis as quantization axis. Consider the case Ho//Z axis, [001]: The effective magnetic field can be written as

-45 - 0 An An A n? 0 i * xx xy xz Thus the Hamiltonian (48) reads 7H = M0 - AAn A" 0 __ef iyx iyy yz n n n T o ec zx Azy azz = - (M~~%,k~~) -C^/^ A ^(4-59) Thus the Hamiltonian (4-58) reads Ms A VA +M-6 A) 7^ =Cln ) s Inv, AzIn n(-IPnHo+MsA')Zlz r3,= i'(AA,- '/Az +' nIn+ + (A:^^yM z 4-+ The electronic states specified by a set of quantum numbers (Ms, MIP mla, mlb, o.. m4b m4c) with the same Ms and MI (z component of impurity ion nuclear spin) but different mn's (z components of ligand nuclear spins) are degenerate before this perturbation is taken into consideration. Splitting due to this perturbation can be calculated by degenerate perturbation theory. However, for natural cadmium, even for the most probable case of three non-zero spin nuclei out of twelve, the perturbation theory leads to 8 x 8 secular determinant. Moreover, the probabilities of having four and five non-zero spin ligands are 3/4 and 2/5, respectively, of the most probable case and hence cannot be ignored. These cases will lead to secular determinants of 16 x 16 and 32 x 32 respectively. It is quite complicated to analyize such a spectrum. However, we can make use of the fact that each ligand nuclear spin is quite independent and first treat the

-44 - splitting due to each ligand spin separately, and then sum up the results. For a and b type cadmium, the 2 x 2 secular determinant has the form: I-u.no+ MsCAstA),-. 4 [(+(-As_4.l Solvi this equation, we have(At Solving this equation, we have AEob= i jllj5+At)Ms +A+yM- A -6=) [4t rM>]tM X A_].0 A+t)-AtE:4-61) (4-62) For -trpe ci- admm, s( As- A+) r For c-type cadmium, the secular determinant is ia WcE>+tSA5-2A+)] -a EC h S(eA-A-)(~c:t) hence, =0 -[1 t Ho s(As - 2 A)] -A C (4-63) AE= 2j iI-L(^NHo + MsQA+ t A,)j'+ [(M A,-A) _ Aj 2 V2 (4-64) l.- i- ~,H +- MN As-2A+) '" 2,~0 Let us introduce a set of new quantum numbers for the ligand nuclear spins, kla' klb,' lc' etc., An = + 1/2 for the state whose energy is shifted by +LAE of Equation (4-62) or (4-64) by the perturbation, and pn = - 1/2 for the state whose energy is shifted by - AE. The eight a and b type cadmium ions are equivalent with magnetic field in [001] direction, and contribute equal amount of

energy shift + Eab We can introduce the "total pi quantum number of a, b type ligands" gab by Cb =(Z_, b Depending on the number of odd cadmium nuclei and the values of individual Vnn fab can have the seventeen values pab = 0, + 1/2, + 1,.., + 3, + 7/2, + 4o Similarly the four c type cadmium ions are equivalent and all shift the energy level by + AE c We define the "total ki quantum number c of c type ligands" Ic by 8 = L-i A which can have the nine values pc = 0, + 1/2,..o, + 2. Thus the energy of the state specified by (Ms, MI, a b, pC) is E(Ms,M) A u A)= ECXMS, o 0,,)62,b E +2 A/AEc (4-65) and the frequency of transition AMs = + 1, AMI =O, is given by U a = o+ 2x )Eb- E) + 2C - At ) dhAo 4 ll As+A+) r t(As -zA) (4-66) where AE', ZE' are the values of E b, AE for M' = M + 1l A schematic ab c as s - diagram of energy levels and transitions is given in Figure 4-2. The above result shows that the HF structure line hvo is split into 17 x 9 = 153 lines, However, because of the high abundance of spinless nuclei, the higher values of gpab, Mp. are less probable and the intensities of these lines are not strong enough to be observable,

Ms MI Ia At AEab AEc ab c -- (1/2, 1) (I.0) (0,1) (1/2,0) (0,1/2) -- (0,0) -(0,-1/2) (-1/2,0) (0,-I) - (-1,0) - (-1/2,1) -- (1/2,0) -- (0,1/2) cv 4 l l L Z 2 MS= MTl, MI I AEab a^b T I AEc i (O,0) E(MS,MI, 0,0) E (MS, Mr.L, b c E (Ms, M,/. v IL ) Figure 4-2. SHF Levels and EPR Transitions.

-47 - The relative intensity of the line ( ab, Vc) can be expressed as Itrab IEs ' Y WCn)P(nkL) __i! ___-_ + b / 6 (6"-kti! —a)! (4-67) +21| where w\(n rCf) = f( 1 12 (4-68 ni (lZ-n)! is the probability of having n non-zero spin nuclei out of twelve, f is the natural abundance of non-zero spin cadmium, and 9( 7 i \~)=( 3 ) I! n-k! (4-69) is the probability of having k nuclei of ab type and n-k nuclei of c type out of n non-zero spin nuclei in total. The summation over k is to be taken from k = 21lab| to the lesser:c 8 and n-214 c, in steps of 2. The numerical values of I( ab, tc) are given in Table 4-6. If the anisotropic part of A tensor, A+ and A are too small to be observable, as in Lambe and Kikuchi's experiment, then 'Eab = AEc' and we can describe the SHF lines by "total p, quantum number of ligands" ALoL In this case the intensity ratio can be calculated by a simpler formula(16) \\ or\2 - i() 2 thi _____ (4-70) where W(n) is given by Equation (4-68), and the summation over n is to be taken from n = 2|1LI to 11 or 12 in step of 2. The same result can

-48 - be obtained from Table nations, for example: 4-6 by adding the intensities of possible combi L Vt ab Vt c Vt i(tab,c ) I([L) 0 0 0.202 + 1/2 T 1/2 2 x.070 + 1 +1 2 x.0066.3557 + 3/2 + 3/2 2 x.0002 + 2 + 2 2 x.000002 1/2 0 1/2.097 1/2 0.150 1 - 1/2.030 3/2 - 1.00172.2961 2 - 3/2.00003 5/2 - 2.000000 -1/2 1.016 1 3/2.00088 - /2 2.0000156 This result agrees with that calculated by Equation (4-70), and also with Lambe and Kikuchi's experiment. This shows that Lambe and Kikuchi's experiment is one special case of the general formulation given aboveo

TABLE 4-4 RELATIVE INTENSITIES OF SHF LINES IN CdTe:Mn ab. 0 + 1/2 +1 + 3/2 +_2 +5/2 + 3 + 7/2 4 +t 0 1/2 1 3/2 2.202.097.0235.0035.00037.150.070.016.0024.00023. o68. 30.0066.00ooo9.oooo8.02085.00865.00172.0002.00001.00 o45.0017.0005.00005 2 x 10-6.0007.0002.oooo4 31 x 106 1.6 x o10-7 8 x 10-5 2.5 x 10-5 3.2 x 10-6 2.2 x 10-7 8.8 x io-9 7.1 x 10-6 1.8 x 10-6 1.9 x 10-7 1.0 x 18 2.6 x 10-10 4.6 x 10-7 9.5 x 10-8 7.5 x 10-9 2.6 x 10-10 3.5 x l0-12 I \0 I

-50 - It will be shown in the next section that A+ is much smaller than A_, so that [001] is not a good direction to observe the anisotropy of A, since only A+, but not A, appears in the energy shift AEo Observation with external magnetic field in other directions (e.go [110]) may show the anisotropy. Do SHF Interaction in ENDOR Experiment In electron-nuclear double resonance experiment (ENDOR) we observe the transition AMs = 0, AMI = 1, where I may be either the impurity ion nuclear spin or ligand nuclear spin. For the latter case, the transitions between SHF levels (Figure 4-3) are observed. Since Ms does not change in such transitions, the direction of effective magnetic field Hfn Equation (4-57), unlike the case of — eff' EPR, does not change after the transition. Therefore, we can describe the ligand nuclear spin states by taking the direction of the effective magnetic field as the direction of quantization axis Zo1 Then the Hamiltonian (4-58) reduces into the form: X4~ - L N t$ Ig| -t - =~ Ct 1' \ 4~UI\ (4-71) 3"C e ff-~Fi| (4-71) Case I: H //[001] O The effective field has the components (4-59) and the magnitude: [lde l `t(W - ^;, 1)7 Av9)tt/.2 /: (4-72) Hv' (Ho- A

Line A: Line B: Line C: Central Ion ENDOR Ligand ENDOR Microwave Pumping Frequency + I ~1/2 m"rn= + 1/2 mn = - 1/2 A — 1/2. --- mn = + 1/2 -— ~ B mn = -1/2 IC mn = -1/2 -1/2 / n = + 1/2 /mn = -1/2 +1/2 \mn = +1/2 MI Figure 4-3. SEF Levels and ENDOR Transitions.

-52 - The frequency of transition between the states (M, MI, *.. mn.OO) and (Ms, MI, oo mA,,.O ) where ml = mn + 1, is given by the following formula cllj, = g.\hi Hc|H (4-75) Using the expressions for the components Aj, Equations (4-53), ij (4-54), and (4-55) we obtain two frequencies: K)ab=X, C= B [sitAs + A+)) + MXl Ah + -AA i,, Hoe - MzS(As + A+) (4-74) h9C0 = l:C ^- Ms A EA+f + ls (3A,-A_) " s1 -t s "Aa-2 ) (4-75) The first one corresponds to the transition Amn = + 1, where n is one of the eight a and b type cadmium. The second frequency corresponds to the change in the nuclear spin state of one of the four c type cadmiumo The intensity ratio of the two lines is 2:1, Case II: H //[110] The effective magnetic field is given by V W + — (A + /_' ) (A\ /A.)J J (4-76) In this case, there are four non-equivalent sets of ligands, and hence, four different frequencies, For the transition of the set(la, lb, 4a, 4b)

-53 - h 1 = ^1 ' j. ^t A4-A_)Ct(& tkh -A )+ A, - HoMs(sA ^-ZA)} = 2>HA - IA I S - (4-7) for the set (2a, 2b, 3a, 3b) ^h2 = i %, (3 %HYt+ tA^ -AA, + _Z At+(As-2 A )A2t+(-3A2)2 -. r, tMs 2 Rs — 7 r1 + 2 t\+ 1 1? /4%e - t 4 - ME(A(S- j 2 +A+A'_)| (4-78) for the set (lc, 4c) h j)3 = - -tMS^A + ^++A^- +2CA+^-A-)l 3-2^MB(As+-A+ Ah)i ' IjoH, - tIs As + A+ A (-79) and for the set (2c, 3c) h i4 -= | St,^H o- NsQ(A+s /A - I_) | (4-80) The relative intensity is 2:2:1:1, Case III: H //[111] In this case the effective magnetic field has the magnitude i H = I I4 OX/X ( x t A(Z831 +t -H3N(Aix+ t AZY + AZ )3 (4-81)

-54 - There are three sets of non-equivalent ligands, and the frequencies are, for the set (la, lb, lc) ph)1 = j kj- st lAst/0 3 (4-82) for the set (2b, 2c, 3a, 3c, 4a, 4b) Kh )[ H RtZAs _)j -t 2vl '( 2 A-) (4-83) for the set (2a, 3b, 4c) h-jl h (4-84) h -'3 = l U {i9w.- NI5(A- 6A~t2A )3+2<l p I1roMA~-24 9-)j (4-84) The intensity ratio is 1:2:1. Ludwig and'Lorenz(21) have observed the SHF interaction between the unpaired electrons and cadmium nuclear spins in CdTe containing Cr+ ion impurity by ENDOR. With magnetic field in [110] direction (which is equivalent to Case II above) they observed, for Ms = -3/2, four lines with relative intensities 2:2:1:1 as expected in the discussion under Case IIo From the frequencies they obtained the three principal values of the A tensor and found that the interaction is anisotropic. The three principal values are (using their notation) T1 = (5.82 + 0 x05) x 10-4 cm-1 4 -1 T2= (5.63 + 0.05) x 10- cm -4 -1 T = (5o61 + 0.05) x 10 cm Isotropic part (contact term) of the SHF tensor As is given s by

-55 - n A = (T1 T + T + T)/ = (5.69 + 0.05) x 104 cm. This value is of the same order of, but twice larger than, the value -4 -1 2.6 x 10 cm, obtained by Lame and Kikuchi for CdTe:Mn. This means chromium d electrons are more delocalized than manganese d electrons. This is in the right direction as nuclear charges are compared. From the relations (4-24) and (4-25) we have, for the average probability of an electron being found at the 5s orbital of cadmium ion ( Coe Yt) = 0~51 x 10-2 (4-85) which is comparable to what a simplified MO calculation gives (see Appendix C). Equating the observed transition frequencies to the expressions (4-77), (4-78), (4-79), and (4-80), with Ms = -3/2, hl) = n ~ a + - A+-S-A =_) 20.51 M /sec h)z = Nb t (CAs - A tA_) = 2. 2 )4= |As ( + =A, ) - (21. 4 hy= %" P H, t 2 (As + A - A_)- 2.H2 From the higher intensity lines (hvl and hv2), we have A_ -3 A = AT = 0.2 MC/sec and from the two lines (hv5 and hv4) of lower intensity, A = 0.21 MC/sec

-56 - The observed values of frequencies are very close to each other, hence the above calculations are subject to large errors. We can only say that the above results show that A+ = 1/2(AM + AC) is very small compared to A_ = 3/2(AM + AC) + AT which essentially equals to AT, the contribution from tellurium orbitals. From Equation (4-41) and Cl. 1.2, RTC = 5.25 a.u., we have =(o8)5 82 x+ i-Uf <t, MC/sec which gives for the probability of an electron being found at tellurium orbitals I - jz z (4-86) bt2 eTi \is+ 4 2 4 -\M on From the fact that no SHF structure of nearest ligand is observed, we can conclude that most part of this probability is due to the p orbitals and not the s orbital of tellurium. This probability is much larger than that obtained by simplified MO calculations as shown in Appendix C. The fact that AT is the largest component among the three (AM, AT, and AC) components of dipole-dipole interaction has an important meaning. It can be seen from Equation (4-46) that this component makes the a axis (Mn-Cd direction) not one of the principal axes of A tensor, and also it makes c axis not equivalent to. axis. The latter is one of the principal axes.

-57 -The fact that A+ is very small explains why the anisotropy of An is not observed in Lambe and Kikuchi's EPR experiment. Equation (4-66) shows that A+ appears in the expression of transition energy, but not A o

CHAPTER V SUPERHYPERFINE STRUCTURE IN SnO2V4+ Ao Structure of the "Complex" The crystal structure of tin oxide (Sn02) belongs to tetragonal D4h group~ In this structure atoms are located at the following positions: (Figure 5-1) Sn (0,0, 0), (1/2, 1/2, 1/2) 0: +(uu,0), +(u + 1/2, 1/2 - u, 1/2) The lattice parameters a, c, and u are given in Figure 5-1. The symmetry of the substitutional site in this crystal is, however, orthorhombic D2h, (Figure 5-2). Using the coordinate system of Figures 5-1 and 5-2, (following From, Kikuchi, and Dorain(18)) the six nearest ligands, oxygen ions, are located in x-y plane (1,2,3,4) and on z axis (5,6). Ten next nearest ligands, tin, are classified into three types: (i) two "a" tins, (15,16) which are closest to the impurity ion and lie on the y-axis; (ii) four "b" tins (7,8,9,10) which lie in the x-y plane; and (iii) four "c" tins (11,12,13,14) which lie in the y-z plane and at the same distance as "b" tins from the impurity ion. The distances and bond angles are given in Figure 5-2. Symmetry characters of vanadium orbitals and linear combinations of ligand orbitals in D2h group are given in Table 5-1.

-59 - [ool Y )15 8 11 C [I z 16 a 0 TIN 0 OXYGEN Lattice Parameter a(A) c(A) u c/a SnO2 4.757 3.185 0.307.6724 TiO2 4.594 2.959 0.306.6441 Figure 5-1. Unit Cell of SnO2 (TiO2).

-6o z 8 Y 15 13 14 Distance and Angles SnO2 TiO2 dl4 = do.15 = C 3.185 A 2.959 d = f2(1-2u)a 2.586 2.521 1, 2 d 8= 1,14 2a 6.699 6.497 7, 8 11, 14= do5 = 2 ua2.057 1.988 dol= [2( -u)2a2 + c2/4]1/2 2.051 1.944 1 c2 do7 = d11 2 (c2 + 2a2)1/2 3.709 3.569 o, 7:, 11: Cosa.7763.7612 Sina.6303.6485 Cosp.4294.4145 Sin5.9031.9102 Figure 5-2. Nearest and Next Nearest V4+ in SnO2 (TiO2). Ligands of

TA:BIE 5-1 SYMMETRY CHARACTERS OF ORBITALS IN SnO2:V Ir. Rep. V Orbital 0 in x-y Plane 0 on z Axis a Tins b Tins c Tins N1 4s s + s2 + s3 + s4 s5 + s5 6 s7 + s8 + s s + s + s + s4 53Z-r x1 - x2 - x3 x4 z5 - z6 15- Y16 x7 - x8 - x9 + xO Y1 - Y12 - 13 + Y14 x2-y - Z' + Z ' -Z -Z Z x"_y Y Y2 Y3 Y4 Y7 + Y89 0 11 12 15 14 N2 zx 1 - 2 - Z3 + z4 5 -6 - z 8 - Z9 + Z10 x1 + xl2 - 13 - x14 N3 xy x1 + x2 - x3 - x4 x15 - x16 x7 + x8 - X9 - x0 x11 - 12 - 13 + x14 Y1 - Y2 - Y3 + Y 7 Y8 - Y9 + YlO s - s + s s- s - s + s - s 1 2 3 4 7 8 9 10 N4 yz Z + z2 - z3 - z4 Y5 Y6 l5 + z16 z7 + 8 - 9 - zll - s12 + s13 - s14 Yll + Y12 - Y13 - Y14 Zl - 12 - z13 + 14 0\ I

-62 - 4+ Bo Ground State of V in SnOp According to point charge model crystal field theory, and using the coordinate system of Figure 5-2, the splitting of d electron levels in rhombic crystal field of SnO2 is as shown in the following figure (Figure 5-3).: z _ (Wn) 3Zt, Cx / (c/ sddxs~~j yx —y2 cl -Y.. Y iXl. \ dxr_____________________.. ). jiyz I14) Z dxx __ _N2) ckxZ-~y (__1___ ()I) Cubic field splitting Tetragonal field splitting Rhombic field splitting Figure 5-35 Splitting of Sn02. of d Levels in Crystalline Field The relative positions of dxz, dyz and dx2_y2 cannot be determined intuitively based on point charge model. In the EPR experiment on Sn02:V4+, From, et al.(18) observed (see Table 5-2): (i) large Agy; (ii) large AAz = Az - As, AAx and AAy have opposite signs to AAz; (iii) large superhyperfine interaction with "a" tins. These results suggest that dx2_y2 lies lowest. Thus the ground state consists of mainly d2_ y2and small amount of dnz admixed through spinorbit interaction.

TABLE 5-2 RESULTS OF EPR EXPERIMENT ON SnO2:V4+(18) g HFS SHFS SHFS A (gauss) a (gauss) b (gauss) x 1.939 23.3 -166. - 28 y 1.903 47.03 172.6 28 z 1.943 154.4 165.2 28

C, Mechanism of SHF Interaction ) In this section we shall discuss the mechanism of SHF interaction as inferred from the large and small SHF structure observed by From, et al. in the EPR spectrum of SnO2:V + (Table 5-2) SHF interaction is proportional to the density of unpaired electron at the ligand nucleus. We shall apply molecular orbital theory to obtain an expression for this density and compare the result with experimental observation, Consider three orbitals: (i) vanadium d orbital Uv, (ii) nearest ligand oxygen orbital, Uo, and (iii) next nearest ligand tin orbitals Uso We can construct three orthogonal molecular orbitals from the linear combinations of these three orbitals. They are: = U + Uo t ls (5-1) b ~ k U + +B Us (5-2) At = ~c UV + eU0 + U, (5-5) where the coefficients (assumed to be real) o,,, and the overlap integrals Svo (= -J- Jr,.,. (5-4) etco, are small quantities of the same order and if small quantities of higher order than this are neglected the MO's are normalized. A schematic diagram of the energy levels of MO's and AO's is shown in Figure 5-4.

-65 - Hvv Uv I/ /\ EC Ec Eb Eb \ \ \ \ \ \\ x\ Hss Us Hoo Uo Pb VANADIUM AO LEVEL MO LEVEL LIGAND AO LEVEL Figure -4. Schematic Eergy Level Diagram for Sn V4+ Figure 5-4. Schematic Energy Level Diagram for Sn02:V

-66 - From the orthogonality relations i 4f i t, ), L = kb, C (5-5) we have, I, - o, -:- ( S ( O C/b) (5-6) (5-7) ^?.- (\ SoS + The coefficients o, secular equation (2-7), (5-8) oc and t can be obtained from the. T -c -0= 0 / U- -- V, 1 S (5-9) where H v is the matrix element of the effective one-electron Hamiltonian between two AO's U and T; Civ is the coefficient of AO UV in MO iro Let Ei = Eb, the energy of MO tb, and [i = V, we have (II -- C +.(w- -AS + os- -)-O vv no b6 i \ W(5 b V0J 5- -boVs)~ ( (5-10) (26) In Wolfsberg and Helmholz's semi-empirical MO method, the offdiagonal elements Ht'v s are approximated by, (Equation 2-10), lH^. k IS S /I,,AV (5-11) where Kv is a factor depending on the energies of AO's U and U~ only. Hence the last term in Equation (5-10) is one order of magnitude smaller than the other two terms, By neglecting this term, we have,

-67 - N<'2 b ( Cv~l;t- p // v \ l *v) (512) Similarly, for Ei = Ec, i = V, we get i:.'. V ( Hv)/(HvV HK1 and for Ei = Ec, t = 0, we have Hi. ( <,>)t. F g~j t 0U 1/fc) U(5-14) The lowest and next lowest energy configurations of this threeelectron system are, respectively ~ -i )i^ it ' Al (5 15) and ^ V t l V (5^16) where { } represents Slater determinant and + signs superscript the MO represent the spin functions. In the second configuration an electron is transferred from the filled MO tb to unpaired MO tao The ground state wave function of this system can be written as the linear combination of these two configurations: 4-: = '31i -.. *1 14 H C \-l V g (5-17)

-68 - The unpaired spin density Ps can be obtained by tfQr) =S j * Z ( Sk ) ~ cLYck (5-18) k I where oz(k) is the third component of Pauli spin matrices for electron k, and Kd - d d7T& 3/I k= l 2, 325. (5-19) Carrying out the integration, we have b =) ^ ~() | | irt)2 4 ~)-X )j (5-20) At the nucleus of the next nearest ligand rs, C^ 5 - US(o), jrt s) z Yb s() (5-21) hence, (5-22) t(Ar) = Jo) l + Jb ( \ )} Substituting the relations (5-7), (5-8), (5-13), (5-14), we have Crs) - SO S + E' ')O (5-23) ^ -\l vvI ) As ( J-roo -I< }' vO "vv " -4 c V C1o o E J This result shows that there are two electron transfer processes which cause the SHF interaction. The first, which is represented by the

-69 - term ya in Equation (5-22), comes from the transfer of impurity d electron to the ligand orbital, or in other words, formation of antibonding MO *a, The second process, which is represented by the term Xyb in Equation (5-22), comes from the transfer of ligand electron into impurity ion orbital, or in other words, the mixture of higher energy configuration {2. The quantity X can be obtained by perturbation theory: <,el t <^1/ 2> _ __ _ (5-24) where X is the system Hamiltonian. If we assume that it can be approximated by a sum of effective one-electron Hamiltonian H, then we have Svc) 4o iSv ( vt l4vv ~- t vo ) I., --- "oo[) -I(- F r, " A) = Hvv- K vo..v.oo )(5-25) to first order in small quantities a, i, etc. We have used the relations (5-6), (5-11) and (5-12) in obtaining Equation (5-25). This equation shows that X is a quantity of the order of overlap integral Svo Thus the second process of electron transfer is less important than the first one. Also we see from Equation (5-23) that the first process is proportional to the square of the overlap integral Svs'

-70 - In order to compare this result with the experimental observation, calculations of overlap intergrals are made by using the following Slater radial functions and Hartree-Fock radial function obtained by Watson(25): Vanadium 3d ~t2Gi( / ^~3d0 = H3(1/ 43 (5-26) C 3) =,52 3 (/A1-) +.4q?1 (3,61) (5-27) 4+ v/-.//3/ 3(6,80) t.00.573(/2.4) V4 3d 4t/^ t L p(/6) (5-28) Tin 5s ~i>~ 5>( 3 god ^ (/ 4/24) (5-29) where n-1 -7/u A-, <^e gn- 2 (5-30) and. Xn _\nSn! 1J (5-31) The overlap.integrals between 3dx2_y2 orbital and the 5s orbitals of "a" tins and "b" tins are given in Table 5-5. We have used the radial functions of neutral atoms based on the electroneutrality principle of Pauling o6) However, we have also considered the 3d orbital of V ion. For all cases, the square of the ratio of overlap integrals is in

-71 - TABLE 5-3 VANADIUM-TIN OVERLAP INTEGRALS Svs (Slater 3d V~ t(ThTo +ir l. ) I Orbital) V4+ Svs (HF 3d >r~h +Q. 1 v \:,,.,, / v I. —L,, - J a Sn -0.1313 -0.0910 -0.04212 b Sn 0,0583 1 0.0379 o0.01640 S,(a) 2 S. (b) i f C \ 5.08 5.76 i 6.6o TABLE 5-4 VANADIUM-OXYGEN OVERLAP INTEGRALS* < 3dx2 2 2S > - 0.01932 < 3d 2 2 2Px > 0.06652 < -y < 3d2 -y2 2Po >.02805 X -y IPY> * Vanadium 3d orbital is the Hartree-Fock function given in Equation (5-27). Oxygen functions are R(2S) = 0.5459 02 (1.80) + 0.4839 02 (2.80) R(2P) = 0.6804 02 (1.55) + 0.4038 02 (3.43) obtained by fitting the numerical Hartree-Fock functions. (27)

good agreement with the ratio of the experimentally observed SHF structure constants, which is 6. The overlap integrals between vanadium orbital and nearest ligand oxygen orbitals are given in Table 5-4. The results show that for this complex, the assumption that all overlap integrals are of the same order is justified, Do Anisotropic Component of SHF Tensor The dipole-dipole interaction which contributes to the anisotropic part of SHF tensor can be treated in the same way as we did in Chapter IV, Section B. The result is A X = A-z s 2 (3 asO _ + A R- (3Cos0 - )tPjI> p j (5-32) where R, R are the distances from vanadium and oxygen m to tin n -vn -mn respectively: x, x, are the angles between Rvn, Rn and x axis respectively. 2, 2 are respectively the probabilities that the m np unpaired electron being found in orbitals of oxygen m, and 5p orbital of tin no. is given by Equation (4-3-5) With the subscript x replaced by y and z respectively, we can obtain AA and AAn. Offn diagonal elements Ax etc. vanishes in this structure. In SnO:V4+, the explicit expressions for the SHF tensor In Sn02:V, the explicit expressions for the SEF tensor components are, for "a" tins: AA^x - Av + o\ A o - A (5-)

-73 - AAa - 2 A -. og A, AA -Av- AA... Am6 (5-34) (5-35) where AQ= v AC T and for "b" tins: 4 1 A A - AAZ Z^nttBs (3.85X'o?) 3 ^ V^ C( >2, oslxo ) ) I, 2 S) l \ <a >5 )...) tp I -s? (5-36) (5-37) (5-38) (5-39) (5-40) (5-41) 1. W; A + - Ab l-.W7 A V A - Ay L, + 2 ATX- ATY ~2Ab Ab b b - A +Z ATy -* A;? TX ATY where A\ -= 2 V, ( o t lO-S)- (o5 —42) A' = (2S X )-3yZ (5-45) rATl=^ P <\X < (5-44) -ATY- 2 $7Y (5-45) 2 2 with 7x, Y7 respectively the fraction of 5PX, 5Py orbitals of b tin (e, tin-7) in the ground antibonding MO. (e.g., tin-7) in the ground antibonding MO.

Anisotropy is not observed in the SHF structure due to b tins. This can be expected as the parameters A b A and Ab are v 0 T smaller than the corresponding parameters for a tins. Also from Equations (5-33), (5-34), and (5-35) we see that AAa is positive Y and the largestj 6Aa is negative and the smallest. This agrees with z the experimental results of From, et al. (Table 5-2).

CHAPTER VI SUMMARY AND CONCLUSION The purpose of this thesis has been to study the delocalization of d electrons from their interaction with next nearest ligand spins, the so-called superhyperfine (SHF) interaction. The electrons are described by molecular orbitals (MO) formed from linear combinations of atomic orbitals of the central and ligand (nearest and next nearest) ions. The Hamiltonian for the interaction between electrons and nuclear spins are derived from the non-relativistic limit of the Dirac relativistic wave equation. This Hamiltonian is used to obtain the SHF interaction tensor An in terms of MO parameters (mixing coefficients) and geometry factors (interionic distances and bond angles). The details of derivation are given for the next nearest ligands in cubic A B compounds containing S state iron group ions. However, the formulation is quite general and can be easily applied to complexes of other structures. Electron spin resonance (ESR) and electron-nuclear double resonance (ENDOR) spectra are related to the components of SHF interaction tensor. An attempt is made to deduce the amount of d electron delocalization from these relations. Unfortunately, existing experimental results are not precise enough to give more than "of the order of magnitude" values.

The delocalization at next nearest ligand s orbital is found to be 0o2 and 0.5%, respectively, for iM+ and Cr in CdTe. The values are obtained by comparing the measured isotropic SHF structures with the isotropic hyperfine structure constant of atomic cadmium, Rigorously, we should use the value for cadmium in crystal. However, this has not been obtained either experimentally or theoretically, Watson and Freeman(37) have reported 20% increase in isotropic hyperfine structure constant for Ni++ in cubic crystalline field by unrestricted Hartree-Fock calculation. If this trend is also true for cadmium in crystal, it will lead to smaller value for the delocalization and in better agreement with the results of simplified MO calculations (Appendix C), which will be discussed in the following. Simplified MO calculation of CdTe:Mn++ using Slater radial functions and atomic spectroscopic data gives a value 0.1% for the amount of delocalization at cadmium 5s orbital. Similar calculation for ZnS:Mn++, using Hartree-Fock radial functions and one electron orbital energies gives 0.01% for the amount of delocalization at zinc 4s orbital, Experimentally, SHF structures for zinc and cadmium are proportional to their nuclear magnetic moments. Hence if the above mentioned amounts of delocalization (differ by one order of magnitude) are true, then the unpaired spin density at the nucleus of zinc (when there is one electron in 4s orbit) should be one order of magnitude larger than that of cadmium (when there is one electron in 5s)o Although this is in the correct direction a factor of ten is by no means obvious, Unrestricted Hartree-Fock calculation can be suggested

-77 - for further investigation of this point. The use of two different systems of radial functions is a defect in this analysis. A systematic Hartree-Fock calculations of Groups II and VI elements (for neutral, univalent, and divalent ions) are necessary for further studies of Group II-VI compounds. Anisotropic SHF structure data are needed to deduce the amount of delocalization at the nearest and next nearest ligand p orbitals. Experimentally observed anisotropies are, in all cases, small and subject to large experimental error. The development of ENDOR technique has already shown the possibility of refined measurement. The molecular orbital formulation of SHF structure as developed in this work, combined with existing experimental data, can be used to guide experimentalists to the best observation as has been pointed out in Chapter IV, Section C. In spite of the existence of small but finite abundance of odd isotope, no SHF structure due to nearest ligands has been observed. The overlap integral between Mn++ 3d orbital and Te 5s is comparable to that between Mn++ 3d and Te 5p, and even larger than that between 3d and Cd 5s (see Appendix C). Yet no observation of SHF structure means that the Te 5s level lies far below Mn++ 3d, and essentially can be removed from bonding orbitals. From Mossbauer experiment on MnTe crystal, Shikazono(38) also found that no internal magnetic field exists at the Te nucleus in MnTe. Further confirmation may be obtained by investigating the ESR of odd isotope enriched samples. The unpaired spin density at next nearest ligand nucleus derived in Chapter V has also included the mechanism of charge transfer

from ligand to central ion, although this mechanism has 1een showrn to be less important for SnO2:V4+o The more important mechanism: - direct interaction between central and next nearest ligand ions has been shown to be proportional to the square of overlap integral. This is a generalization of the Heitler-London model used by Marshall and Stuart (12) Further investigation of this proportionality can be attained by pressure experiments in which overlap integrals are largely affected while the second factor - energy difference -varies rather slowly, Equation (5=23)o As we have seen, the deduction of d electron delocalization from SHF structure data necessitates further information, both experimental and theoretical. The development of ENDOR technique and electronic computational facility casts delightful future on this approach.

APPENDIX A OVERLAP INTEGRALS In this appendix we describe the method of computing the overlap integrals appeared in the discussions of Chapter II, IV and V. Some formulae of the diatomic overlap integrals and IBM 7090 computer programs for the evaluation of the numerical values of overlap integrals are also given. The first step in the evaluation of the overlap integral between two atomic orbitals in a complex is to transform the coordinates of the two centers (ions) such that they are related to each other in the same way as the "a" and "b" coordinates of Mulliken, et al.(39) in their calculations of diatomic overlap integrals (see Figure A-1) X- -, -a.....__ — b z\. /. 2,....... X q Figure A-l. Coordinates for Overlap Integral Calculations, -79 -

-80 - In this set of coordinates, the z axes (Za and Zb) are pointing toward each other, and the two x axes and two y axes are parallel. Thus if the "a" coordinates is a right handed system, then "b" coordinates is a left handed system, and vice versa. We use the former choice. Examples of transformation matrices are given in Table A-i, A-2, and A-3. The next step is to introduce the ellipsoidal coordinates ( r,, cp) defined by. ~ (A-l) (A-2) - ( a- ^Vo, -I ~ a < I (A-3) do R()3 1 ct za)(A-4) where R is the interionic distance. Some other useful relations between a, b coordinates and (, Tr, cp) are: grocosJL Rt1 S+Xj/2 (A-5)!:&Sii p ^- 14 ( -1) (1 \(A-7) The two orbitals for which the overlap integral is to be evaluated are written as Slater type orbitals (STO) (in atomic unit), for example: itd6)= N4G ~ expK ^ZarWn* Y2a (6r<.) (A-8)

-81 - *btl6PQ- - N6 b 6-1 eKp - bV -,* ) 9 0 O0b ) ('b) (A-9) where Z* and n* are effective nuclear charge and effective principal quantum number respectively of the orbital. The normalization factor N is given by N (''/x). (2 n*) (A-10) Using the relations (A-l,...7), the integral <(nad(iGl p) can be written in terms of (, rT, cp) as follows: qVdc-lb pf where % z Pe - Z\lb - 0 xpZp- ^b ) Lt J;- (3 cos Z-G ) X rb 13/4% CoS Eb d -()( x * 1 /n, + ^/n } (A-12) /i4, - Z igns} I(A-13) A1~ P)- =; -e (A-14) 3a'.)-= \) e- J,l' (A-15)

i i and Cij is the coefficient of the term rj in the polynomial in the integrand. The integrals Ai and Bj can be evaluated by the recursion formulae: ~A~:,) = [y ^f^(A-l6) AL (.p' -" ~oC,.p) t- ~f A,_l(, ') (A-17) A1 Cf)= - i ) ^>_ C~T ) (A-18) = r cT - - r (A-19) In case lql is very small (~ < 0.25), the above recursion formulae (A-18), (A-19) break down because of the error introduced by the subtraction of two numbers of almost the same size. Hence in this case we use the following formulae: -I ei _ 2!.l 3! 3 < - s: X — j = even (A-20) -2 -o d j = odd. (A-201) du+ 2 3(+ 4) In Table A-4 we summarize formulae corresponding to Equations (A-ll) for two Slater type orbitals. Some of them have been given elsewhere(39, 40) but the formulae given here have the form directly connected with the computer programs which will be given later.

-83 - TABLE A-i TRANSFORMATION OF Mn+ RESPECT TO Tel COORDINATES INTO "a" COORDINATES WITH AND Cd la, lb, Ic COORDINATES Xo 1 o I Zoj xoJ L YO = oj 1/$6 - 2/5 1/-6 1/12 0 -1/12 1/53 Ya 1/13 Xa ~ f Xa " I 1/12 YI /~2 a z i aa 1 0 With -espect to Te-1 With respect to Cd-la 0 1/ 2 O -1/12 fx ) t Yo ly 1 - it Xo xo d,. 0 -1/12 11 1 0 0 0 1/12 1/12 1/12 rX ) 0 y ~ OYa I 1//2 za // K / 1/WF2 i /Xa\ +, -L y iya +~ I " V ^Yz 0 With respect to Cd-lb With irespect to Cd-lco w. r. t, Te 1 0 > -- d y - - x A ^, F3 ^-Y z dY >- n -L 4 -x- - Y _.2 d _ + C - x-y i z YzY > -d +r3y l -yZ -- F d -Z dy di- -. -^^-^ -^^ y -y^. r- t. C ab -, t, Ca Ic w, r, t Te J. r,-t CA IL,

-84 - TABLE A-2 TRANSFORMATION OF Cd COORDINATES INTO "a" COORDINATES AND Te COORDINATES INTO "b" COORDINATES WITH RESPECT TO EACH OTHER IN CUBIC CdTe tx~ [ -2/3 y1 - ~3 x1 1/3 Yl 2 Xla Yla = 3/2 yia -1/217 Yla 3 'xlb N- 2/3 lb Ylb I ~ "Ib L 1/I3 lxic 11 6 Yic = 3/2 ozle.-1/2C3 1/ 3 -2/3 I / xb o -27 yb W T -1/3 i ~ZbL 0 -1 0 2 /351 ' Xb 0 I Yb -1/3 i zb ^753 1 2 x zb -1/3 JI xa 0 ya ^^JYa, J2/3 -1/2 -1/2 With respect to Cd-la With respect to Cd-lb With respect to Cd-lc With respect to Te-1 With respect to Te-1 With respect to Te-1 0 1 L/15 FXa o I I 0 ya a 42/3 i z 0 -1/12 1/Is 1 -1/2 0 1/2 i! Xa Ya ".,

TABLE A-3 TRANSFORMATION OF COORDINATES INTO "a", "b" COORDINATES IN Sn02:V AND Ti02:V /xa i'fcosQ Ya- 0 ZaJ K sinQ -sing 0 cosQ I -1 y o z I 'I) \ ^ Q0 KSn for V - a Sn for V - 01 for V - b Sn d(z2)_ - _ 2 overlap ~ = 0 overlap 9 = tan-1 (1-2u) f2a/c = a overlap 9 = tan-1 f2a/c = d(z2) + \3/2 d(x2 - y2) d(x2 - y2)-> -_ 3/2 cos 29 d(z2) + 1/2 cos 29 d(x2-y2) + sin 20 d(xz) I xa coso Xa~ Iz jsino < a ) K -sin0 0 coso 0 x1 -1 Y Oj zj for 01 - a Sn overlap for 01 - b Sn overlap 0 = -tan-1 (1-2|4) f2 a/c 0 = -t/2

-86 - TABLE A-4 FORMULAE FOR STO OVERLAP INTEGRALS < 3da|1S > = NaNb 4 (R)5 {Ao(-3B2 + B4) - 4A1B3 + A2(53B - 3B4) 2 + 4A3B1 + A4(-Bo + 3B2)} < 3do 12S > = N N 5 ()6 {Ao(3B3 - B5) + A1(-3B2 + 5B4) + A2(-3B1 - 4B3 + 3B) ab 4 - + A3(3B - 4B2 - 3B4) + A4(5B1 - 3B3) + A5(-Bo + 3B2)} < 3dc 13S > = NaNb (R)7 42 {Ao(-3B4 + B6) + A(6B - 6B) + A2(9B -3B) + A3(-6B1 + 6B5) + A4(3Bo - 9B2) + A5(6B1 - 6B3) + A6(-B + 3B2) 6O 2 < 3a o14S > = N Nb (R)8 {Ao(3B5 - B7) + A1(-9B4 + 7B6) + A2(6B - 15B5 + 5B7) 4 2 + A3(6B2 + 9B4 - 9B6) + A4(-9B1 + 9B3 + 6B5) + A5(3Bo - 15B2 + 6B4) + A6(7B1 - 9B3) + A7(-Bo + 3B2)} < 35da2pc > = NN -1 a b 4 (R)6 {Ao0(-B2 B+ A(-B B) + + B) + A(-B - B) + A + B) 2 + A3(B1 + 3B5) + A4(-Bo - B2) + A5(B1 - 3B3)}

-87 - TABLE A-4 CONT'D < 3dr|3pa > = NN a b 4 (R)7 {A (3B3 - B5) + A1(-3B2 + 2B4 + B6) + A2(-3B1 - B3 - 2B5) + A3(3Bo - B2 + B4 - 3B6) + A4(2B1 + B3 + 3B5) + A5(-Bo - 2B2 + 3B4) + A6(B1 - 3B3)} < 3d 14pa > = NaNb - a b 4 (R )8 2 {Ao(-3B4 + B6) + A1(6B3 - 3B5 - B7) + A2(3B4 + 3B6) + A3(-6B1 - 3B5 + 3B7) + A4(3Bo - 3B2 - 6B6) + A(5B1 + 3B3) + A6(-Bo - 3B2 + 6B4) + A7(B1 - 3B3)} < 3dt 12prc > NNb 3 (R)6 {Ao(B2 B4) + A(B B) + A2(-B + B4) 4 2 + A(-B1+ B5) + A4(Bo - B2) + A5(B1 - B3)} < 3dit 13 pt > = NN -- ab 4 (R 7 () 2 {Ao(-B3 + B5) + A1(B2 - 2B4 + B6) + A2(B1 + B - 2B5) + 5-0 +22+B4 6 A4(-B + B + B) + A3(-Bo+ B2 + B4 - B6) + A4(-2B + B3+ B) + A5(Bo - 2B2 + B4) + A6(B1 - B3)} < 3dit14pt > - NaNb 4 ()8 {Ao(B4 - B6) + A1(-2B3 + 3B5 - B7) + A2(-3B4 + 3B6) 2 + A3(2B - 3B5 + B7) + A4(-Bo + 3B2 - 2B6) + A5(-3B1 + 3B3) + A6(Bo - 3B2 + 2B4) + A7(B1 - B3)}

-88 - TABLE A-4 CONT'D < 1S12S > = Nb 2 2 - A1B2 - A2B1 + A3Bo} < 2S 12S > = NaNb 2 (2) {AoB4 ab 2 2 0 4 - 2A2B2 + A4Bo < 3 12S > = (-)6 = Na 1 (R) 2 2 {AoB Lo 5 + A1B4 - 2A2B3 - 2A3B2 + A4B1 + A5Bo < 4S12S > = NaNb 1 ()7 {AoB6 2 2 + 2A1B - A2B4 - 4A3B3 - A4B2 + 2A5B1 + A6Bo} < 3s13S > = Na 1 R )7 2 2 {-AoB6 + 3A2B4 - 3A4B2 + A6Bo} < 4s 3s > = NaNb 2 ()8 {-AoB7 Na +A +B - AB6 + 3A2B5 + 3A3B4 - 3A4B3 - 3A2B5 + A6B1 + A7Bo} < 4s 14s > = NNb i (R)9 {AoB8 - 4A2B6 + 6A4B4 - 4A6B2 + A8Bo} " arJ 2 2 < lS12pa > aNb 2 (R) {-AB2 + A1B3 2 2 13 + A2Bo - A3B1} < 2S12pa > - i ab 2 R95 2 { -AB3 + A1(-B2 + B4) + A2(B1 + B3) + A3(Bo - B2) - A4B1}

-89 - TABLE A-4 CONT'D < 3S 2pa > NaNb 2 ()6 {-AoB4 + A1(-2B3 + 2 2 B5) + 2A2B4 + 2A3B1 + A4(Bo - 2B2) - A5B1} < 4S 12pa > =- N a (-)7 a b 2 2 {-AoB5 + A1(-3B4 + B6) + A2(-2B3 + 3B5) + A3(2B2 + 2B4) + A4(3B1 - 2B3) + A5(Bo - 3B2) - A6B1} < 2S 3pa > = NaNb ()6 {AoB4 2 2 - A1B - 2A2B - 2AB3 + A4Bo - A5Bj < 3S513p > N NN a b 2 ()7 {AoB + A1(B - B6) + A2(-2B3 - B) + A3(-2B + 2B4) + A4(B1 + 2B3) + A5(Bo - B2) - A6B1} < 4S 13ps > = Nb 3 (R)8 {AoB6 2 2 + Ai(2B5 - B7) + A2(-B4 - 2B6) + A3(-4B3 + B5) + A4(-B2 + 4B4) + A (2B1 + B3) + A6(B0 - 2B2) - A7B1} < 2S 14pa > Nab N 2 (R_7 (-) 2 {-AoB5 + A1(B4 + B6) + A2(2B3 - B5) + A3(-2B2 - 2B4) + A4(B1 + 2B3) + A5(B0 + B2) - A6B1}

-9o TABLE A-4 CONT'D < 3S 4pa > = Nab 3f (R)8 {-AoB6 + A1B7 + 5A2B4 - 3AB5 - A4B2 + 5AB 2 2 B + A6Bo - A7B1} < 4s14pa > ~3 aNb 2 (2) {-AoB7 + A1(-B + B) + A+ B8) AB + ) A3(5B4 - 3B6) + A4(-3B3 - 3B5) + A5(-3B2 + 3B4) + A6(B1 + 3B3) + A7(Bo - B2) - A8B1} < 2p 12pa > = Nb 3 (R)5 {-AoB2 2 2 + A2(Bo + B4) - A4B2} < 3pa12pca > = NaN 3 (R)6 2 2 {-AoB3 - A1B2 + A2(B1 + B5) + A3(Bo + B4) - A4B3 - A5B2 } < 4pc 12p > = NaNb 3 (R)7 a 2 ^ {-AoB4 - 2A1B3 + A2B6 + A3(2B1 + 2B5) + A4Bo - 2AsB3 - A6B2 < 3pa13pa > = NaNb 3 ()7 {AoB4 + A2(-2B2 - B6)+ (B + 2B) - A6B2} 2 6

-91 - TABLE A-4 CONT'D < 4p 135p > = NaNb 3 (R )8 {AoB5 + A1B+ A A2(-2B3 - B7)+ A3(2B2 - B6) + A4(B1 + 2B5) + A5(Bo + 2B4) - A6B3 - A7B2} < 4pa 14pa > = NaNb (2)9 {-AB6 + A2(3B+ B8) N,,b 2 24 ) + A4(-3B2 - 3B6) + A6(Bo + 3B4) - A8B2} < 2pT|:2pit > -,3 = N N a 4 (R)5 2 Ao(B2 - B4) + A2(-Bo+ B4)+A4(Bo -B2)) < 3p7 12prc > = Nb 3 (R)6 ^A(B - B5) + A1(B2 - B4) + A2(-B1 + B5) + A3(-Bo + B4) + A4(B1 - B) + A5(Bo - B2)} < 4pi 12pjt > NaNb (7 {(4 - B) + A1(2B - 2B)+ A2(-B + B) + A3(-2B1 + 2B) + A4(-Bo + B2) + A5(2B1 - 2B3) + A6(Bo - B2)} < 4pt 14pit > = Nb 3 ()9 ({A(B6 - B8) + A2(-3B4 + 2B6 + B) + A(B2 -B6) + A6 o - 22 + 3B4) + A(B B - B2)}

-92 - MAD Programs for Computing Overlap Integrals Program I: This program computes the integrals Ai(p) and Bj(q) of Equations (A-14) and (A-15) by the recursion formulae, Equations (A-16) through (A-20), and evaluates the overlap integral between two Slater type orbitals by summing the products of Ai and Bj according to the formula Equation (A-ll) (see also Table A-4). The effective nuclear charges (ZEFFA and ZEFFB), effective quantum numbers (NEFFA and NEFFB), interionic distance R and the coefficients Cij's are needed as input data. It is noted that the coefficients Cij's are either symmetric or antisymmetric with respect to the interchange of i and j, hence only "lower triangle" of the matrix (Cij) is read in, and the upper triangle is developed by the machine according to the value of the variable "CSYM". CSYM = 1. if C..'s are antisymmetric, and CSYM = 2. if Cij's are symmetric. In case the succeeding calculation uses the same set of ZEFF, NEFF, and R as the previous one, we set the variable RPC (relation to the previous calculation) equals to 2, and RPC = 3 if we use the same set of Cij as the previous calculation. If none of the previous data are used RPC = 1. A numerical constant coming from the angular function (e.g. 115/4 in Equation (A-ll)) is called NC in the program, and also needed as input data. The values of the integrals Ai(p) and Bj(q) computed by this program have been checked with the table compiled by Kotani, et al. (41) They agree with each other up to five figures or more. The following is the MAD program. The data are for the examples of computing the following overlap integrals:

< 3d52s > for R = 3.877 a.u. < 3dca12pa > for R = 5.877 au. and < 3do12pa > for R = 3.887 a.u.. $COMPILE MAD, EXECUTE PRINT COMMENT $1 OVERLAP INTEGRALS FOR SLATER TYPE ORBITALS$ DIMENSION A(20),B(20), C(400, V),FACTRL(20) VECTOR VALUES V=2, 0, INTEGER I, J, IMAX, JMAX, RPC LMAX=15 FACTRL (o.)=1, THROUGH LOOPl FOR L=l, 1., L.G. LMAX LOOP1 FACTRL(L)= L*FACTRL(L-1. ) PRINT RESULTS FACTRL(O.)..FACTRL(LMAX) START READ DATA PRINT RESULTS RPC, ZEFFA, ZEFFB, NEFFA, NEFFB, R WHENEVER RPC.E.2, TRANSFER TO CMTRIX V(1)=JMAX+2 V(2)=JMAX+1 MUA = ZEFFA/NEFFA MUB = ZEFFB/NEFFB NA=(2.*MUA).P. (NEFFA+0.5)/SQRT.(FACTRL(2.*NEFFA)) NB= ( 2. *MUB). P. (NEFFB+ o 5 )/SQRT. (FACTRL( 2.*NEFFB ) ) P=R*(MUA+MUB)/2. Q=R* (MUA-MUB)/2. A(O)=EXP. (-P)/P THROUGH LOOP2, FOR I=1, 1, I.G.IMAX

-94 - LOOP2 A(I)=A(O)+I*A(I-1)/P WHENEVER. ABS. Q. GE. 0.25 B(O)=(EXP.(Q) - EXP.(-Q))/Q THROUGH LOOP3, FOR J= 1,,J.G.JMAX LOOP3 B(J)=((-1. ).P.J*EXP. (Q)-EXP (-Q)+J*B(J-1))/Q OTHERWISE THROUGH LOOP3A, FOR J=0,2, J.G.JMAX LOOP3A B(J)=2./(J+1.)+Q.P.2./(J+3. ) THROUGH LOOP3B, FOR J=, 2, J. G.JMAX LOOP3B B(J)=-Q/(J+2. )-Q.P.3./(3.*J+12. ) END OF CONDITIONAL PRINT RESULTS P,A(O)...A(IMAX), Q, B(O).. B(JMAX) WHENEVER RPC.E. 5, TRANSFER TO CHECK CMTRIX EXECUTE ZERO.(C(O, ).. C(IMAX, JMAX)) READ DATA THROUGH LOOP4,FOR I=1, 1, IG.IMAX THROUGH LOOP4,FOR J=O 1, JE. I LOOP4 C(J, I)=(-1. ).P.CSYM*C(I,J) CHECK PRINT RESULTS C(O, 0).. C(IMA MAX) SUM=O. THROUGH LOOP5,FOR I=0, 1, I.G.IMAX THROUGH LOOP5, FOR J=O 1, J. G. JMAX LOOP5 SUM=SUM+A( I )*B(J)*C ( I, J) OVINT=NA*NB*NC* (R/2, ).P. (NEFFA+NEFFB+1. )*SUM PRINT RESULTS OVINT

-95 - PRINT COMMENT $1$ TRANSFER TO START END OF PROGRAM $DATA RPC=1, ZEFFA=4. 3, ZEFFB=4..55, NEFFA=3., NEFFB=2., R=5.8777, IMAX=5, JMAX=5 * CSYM=2.,NC=05590,C(2, )=-3., c(3, o)= 3., o., )-4., c(4, 1)=5., o, -3., c(5 o)=-1.,3. * RPC=2 * CSYM=1,, NC=0o96825, c(2, )=3.,C(3, 1)=1.,C(4, o)=-1., o,, -1.c(, C(5)=ioi 0*, 3 * RPC= 35, R=3.8877 *

-96 - Program II: Watson(25) and others have used linear combinations of Slater type orbitals for the Hartree-Fock atomic wave function. The overlap integral between two orbitals of this kind can be evaluated by simply introducing an iterative procedure in Program I, provided the effective quantum numbers of the orbitals in the combination are all the same. (If this is not the case, we can divide the combination into several parts each has the same effective quantum number). Instead of NEFF and ZEFF, we use their ratio MU(K) for k-th orbital, and FA(K), the fraction of k-th orbital in the combination A as input data. The following is the program with data which compute the overlap integral between vanadium 3do orbital 3da = {0.5243505(1.83) + 0.4989035(.61) + 0.113103(6.80) + 0.005505(12.45)} Y20(,cp) given by Watson (25), and oxygen 2s orbital obtained by Ballhausen(27) by fitting the numerical functions given by Hartree. *2s = {0.545902(1.80) + 0.483902(2.80o YO (G,p) where 0n( 4) = N rn1l exp(-pr) is the normalized Slater type orbital. $COMPILE MAD, EXECUTE PRINT COMMENT $1 OVERLAP INTEGRALS FOR WATSON TYPE ORBITALS $ DIMENSION A(20),B(20), C (400, V ),MUA(20), MUB(20 ),FA(20),FB( 20), IT( 400, W),FACTRL( 20)

-97 - VECTOR VALUES V=2, 0, 0 VECTOR VALUES W=2, 1, INTEGER RPC, I, IMAX, J, JMAX, K, KMAX, N, NMAX LMAX=15o FACTRL(0o )=lo THROUGH LOOP1, FOR L=1, 1,,LG LGMAX LOOP1 FACTRL(L)=L*FACTRL( L-1 ) PRINT RESULTS FACTRL(O. ),. FACTRL(LMAX) START READ DATA PRINT RESULTS RPC, NEFFA, NEFFB, R MUA(1)...MUA(KMAX),MUB(1)...: MUB(NMAX) V(1)=JMAX+2 V(2)=JMAX+1 W(2)=NMAX WHENEVER RPC.E 3, TRANSFER TO CHECK CMTRIX EXECUTE ZERO. (C(, 0)... C( IMAX, JMAX)) READ DATA THROUGH LOOP2, FOR I=1,1, I.G. IMAX THROUGH LOOP2, FOR J=0, 1, JE. I LOOP2 C(J, I )=(-l. )oP. CSYM*C( I, J) CHECK PRINT RESULTS C(,O 0).. C(IMAX, JMAX) OVINT=0o THROUGH INTGRL, FOR K=1, 1, K.G. KMAX NA=(2.*MUA (K)).P,(NEFFA+0,5)/SQRT.(FACTRL(2.*NEFFA)) THROUGH INTGRL, FOR N=,1,, N.G.NMAX NB=(2o*MUB(N)).P.(NEFFB+O.5)/SQRT.(FACTRL(2.*NEFFB) )

-98 - P=R*(MUA(K)+MUB(N) )/2. A(O)=EXP. (-P)/P THROUGH LOOP3, FOR I=1, 1, I.G.IMAX LOOP3 A(I)=A(O)+I*A(I-1)/P Q=R*(MUA(K)-MUB(N) )/2. WHENEVER.ABS. Q. GE.0. 25 B(O)=(EXP. (Q)-EXP. (-Q))/Q THROUGH LOOP4, FOR J=1, 1, J. G.JMAX LOOP4 B(J)=((-1.).P.J*EXP.(Q)-EXP.(-Q)+J*B(J-1))/Q OTHERWISE THROUGH LOOP4A, FOR J=0, 2, J.G.JMAX LOOP4A B(J)=2./(J+1. )+Q.P.2./(J53. ) THROUGH LOOP4B, FOR J=1 2,J.G.JMAX LOOP4B B(J)=-2.*Q/(J+ 2.)-Q.P. 3./(3.*J+12.) END OF CONDITIONAL SUM=0. THROUGH LOOP5, FOR I=0, 1, I G. IMAX THROUGH LOOP5, FOR J=0, 1, J. G. JMAX LOOP5 SUM=SUM+A( I )*B(J)*C( I, J) IT(K,N)=FA(K)*FB(N)*NA*NB*NC*(R/2. ).P. (NEFFA+NEFFB+1. )*SUM PRINT RESULTS IT(K, N) INTGRL OVINT= 0VINT+ IT (K, N) PRINT RESULTS OVINT PRINT COMMENT $1$ TRANSFER TO START END OF PROGRAM

-99 -$DATA RPC=1, NEFFA=3., NEFFB=2., R=3 8777, KMAX=4, NMAX=2, IMAX=5, JMAX=5, MUA(1)=1.8, 3.61, 6.80, 12.43,FA(1 )=.52435, 4989,. 1131,.0055 MUB(1)=1.80, 280, FB(1)=.5459,.4839 CSYM=2.,NC=.5590, C(2, 1)=-3., C(3 0)=3.,0.,-4., C(4, 1)=5., 0., -3, C(5, O0)= -1O.,0.,), * RPC = 3, R = 3.8877 *

APPENDIX B SOLUTION OF IMPROPER EIGETVALUE PROBLEM BY DIGITAL COMPUTER The improper eigenvalue problem we met in Chapter II Equation (2.7) can be reduced into a proper eigenvalue problem by successive diagonalization and unitary transformation. This will be shown in this appendix. The problem, stated in general, is to solve for the eigenvalues Xi and eigenvectors Xi of the equation: AX = BXA (B-l) where A and B are symmetric matrices and B is also positive definite. A is a diagonal matrix with the eigenvalues Xi as diagonal elements. X is the matrix with the eigenvectors Xi as i-th column. Since B is symmetric, we can find a unitary matrix U to diagonalize it, UtBU = D (B-2) Since B is also positive definite, the diagonal elements of D are all positive. Take the -1/2 power of the diagonal elements and construct another diagonal matrix R, symbolically: R =D-1/2 (B-3) then, Rt(uBU)R = RDR = D-1/2DD1/2 = I (Identity) (B-4) Let UR = S, multiply St from left on both sides of Equation (B-l) StAX = StBXA (B-5) -100 -

-101 - or, StASS-1X = StBSSXA (B-6) Let StAS = A', (B-7) -1X = Y (B-8) then using Equation (B-4), we have A'Y = YA (B-9) This is in the form of a proper eigenvalue problem. Since the matrix A' is symmetric, there is a subroutine (EIGN.) available in "Michigan Executive System Subroutines" for the solution of this problem The eigenvalues of the original equation are the same as those of Equation (B-9), and the eigenvectors of the original equation can be obtained by a matrix multiplication: X = SY (B-10) Since the eigenvectors are given in row form in subroutine EIGN., we actually do the multiplication Xt = yst (B-ll) For the convenience of later use we normalize the eigenvectors such that XikXjkB = 1 (B-12) i,j In order to check the calculation, we calculate the error matrix E, having the elements: Eki = (Ai - Bij)Xjk 130ij which must be zero if the solutions are perfect. In our calculations

-102 - all elements of error matrix are six orders or more smaller than Aij or XkBij The following is the MAD program which solve improper eigenvalue problems of order less than 20. $COMPILE MAD, EXECUTE, PUNCH OBJECT PRINT COMMENT $1 SOLUTION OF THE CHARACTERISTIC VALUE PROBLEM (A-LB)X=O $ PRINT COMMENT $0 WHERE A AND B ARE SYMMETRIC MATRICES, AND B IS POSITIVE DEFINITE $ DIMENSION A(400,V), B(400, V),X(400,V), APRIME(400,V),E(400,V) D(400,V ), R(400,V), ST(400,V), UT(400,V), S(400, V), YT(400,V ), LAMBDA( 400o V) EQUIVALENCE (D, R, ST, E), (UT, S, YT, X), (APRIME, LAMBDA), (V(2 ),N) VECTOR VALUES V=2, 1, 0 INTEGER N, I, J, K START READ DATA N PRINT COMMENT $1$ EXECUTE ZERO. (A(1, l).o A(N, N),B(l,l)...B(N, N)) READ DATA A(l,l)...A(N, N), B(l, 1)...B(N,N) THROUGH LOOP1, FOR I= 2,1, I.G.N THROUGH LOOP1, FOR J= 1,l,J.E. I A(J,I) = A(I,J) LOOP1 B(J,I) = B(I,J) PRINT RESULTS N,A(1,1)...A(N,N),B(l, )...B(N,N) IND1 = 5. IND2 = 5, IND3 = 5.

-103 - IND4 = 5o IND5 = 5o IND6 = 5o THROUGH LOOP1A, FOR I=1,1, I.G.N*N LOOP1A D(I) = B(I) SCFACT = lo IND1=E IGNo (D( 1 ), N, UT( 1 ), SCFACT) WHENEVER IND1.E.3o CONTINUE OR WHENEVER IND1.Eo1. PRINT COMMENT $0 B MATRIX NOT ACCEPTED BY SUBROUTINE $ TRANSFER TO- END OR WHENEVER IND1.E.2. PRINT COMMENT $0 CHARACTERISTIC VALUES OF B MATRIX SCALED BY$ PRINT RESULTS SCFACT TRANSFER TO END END OF CONDITIONAL THROUGH LOOP2, FOR I=1,1, I.GN WHENEVER D(I,I).LE. Oo PRINT COMMENT $0 B MATRIX IS NOT POSITIVE DEFINITE $ TRANSFER TO END OTHERWISE R(I2 E ID(OF I)P.-.5 LOOP2 END OF CONDITIONAL

-io4 - THROUGH LOOP3, FOR I=1,1, I.G.N THROUGH LOOP3, FOR J=1, 1,J.G.N WHENEVER I.E.J CONTINUE OTHERWISE R(I, J)=0. LOOP3 END OF CONDITIONAL IND2=DPMAT. (N, ST(1), UT(1)) WHENEVER IND2E. O., TRANSFER TO END THROUGH LOOP 5, FOR I=, 1, I.G.N THROUGH LOOP 5, FOR J=1,1,J.G.N S(I,J)=ST(J, I) LOOP5 APRIME( I, J)=ST( I, J) IND3=DPMAT. (N, APRIME ( 1 ), A(1)) WHENEVER IND3E. 0., TRANSFER TO END IND4=DPMAT. (N, APR IE(1), S(1)) WHENEVER IND4. E Oo, TRANSFER TO END THROUGH LOOP6, FOR I=2,1, I.G.N THROUGH LOOP6, FOR J=l, 1, J.E.I LOOP6 APRIME(I,J) = APRIME (J, I) SCFACT = 1. IND5=EIGN. (LAMBDA( 1 ), N 1, YT(1), SCFACT) WHENEVER IND5.E.* 3 CONTINUE OR WHENEVER IND5.E o1.

-105 - PRINT COMMENT $0 APRIME MATRIX NOT ACCEPTED BY SUBROUTINE $ TRANSFER TO END OR WHENEVER IND)5.E.2. PRINT COMMENT $1 CHARACTERISTIC VALUES SCALED BY $ PRINT RESULTS SCFACT END OF CONDITIONAL IND6=DPMAT. (N, YT(1), ST(1)) THROUGH LOOP7, FOR I=, 1, I.G.N XSUMSQ = 0. THROUGH LOOP8, FOR J=l,1, J.G.N THROUGH LOOPS, FOR K=l,1, K.G.N LOOP8 XSUMSQ=XSUMSQ+X ( I, J )-X (I, K )*B(J, K) ROOT = XSUMSQ.P..5 THROUGH LOOP7, FOR J = 1,1, J.G.N LOOP7 X(I, J )=X( I, J)/ROOT PRINT COMMENT $0 CHARACTERISTIC VALUES $ THROUGH LOOP8A, FOR I=1,1, I.G. N LOOP8A PRINT FORMAT F3, LAMBDA(I, I) VECTOR VALUES F3=$ S20, E20.8 *$ WHENEVER SCFACT.E. 1 CONTINIUE OTHERWISE PRINT COMMENT $0 ERROR MATRIX NOT COMPUTED $ TRANSFER TO END END OF CONDITIONAL

THROUGH LOOP9, FOR I=1,1, I. G. N THROUGH LOOP9, FOR J= 1, 1,JG. N E(I, J)=O. THROUGH LOOP9, FOR K=1, 1, K.G.N LOOP9 E(I, J)=E(I, J)+(A(J, K)-LAMDA(I, I)*B(J, K))*X(I, K) PRINT RESULTS X(1,1).. X(N,N),E(1, 1)...E(N,N) END PRINT COMMENT $0 INDICATOR VALUES $ PRINT RESULTS IND1, IND2, IND3, IND4, IND5, IND6 TRANSFER TO START END OF PROGRAM

APPENDIX C SIMPLIFIED MO CALCULATIONS OF TETRAHEDRAL COMPLEXES INCLUDING NEXT NEAREST LIGANDS As examples of simplified MO calculations discussed in Chapter II, Section B, cadmium telluride (CdTe) and zinc sulfide (ZnS) containing Mn++ impurities are treated in this appendix. In the first example (CdTe:Mn++), Slater radial functions and spectroscopic energy levels are used for the AO's. In the second example (ZnS:Mn++), Hartree-Fock radial functions and one electron orbital energies calculated by Watson and Freeman (25) are used. (i) CdTe:Mn++ Radial functions used are R3d(Mn) = N r2exp(-1.87 r) R p(Cd) = N'r3exp(-1.09 r) 5sp R (Te) = N"r3exp(-1.74 r) 5sp For T2 symmetry, the six basis functions are the AO's in D3 of Equation (4-7), and the matrices {Hij} and {Sij} are, -107 -

{H. ij}T2 (in units of K cm-1): d(yz) ) s zs xz -126. -25.25 -17.94 -12.57 -8.82 -11.15 -144. 0 0 -76.33 -65.539 -73. 0 13.02 -35.15 -735. -36.84 -32.70 -72. o -42. {Sii T2 1..08633.09355.06447.046532.07661 1. 0 0.3748.4204 1. 0 -.0898.0285 1..2541.29553 1. 0 1.

-109 - For E symmetry the four bases are the AO's in D1 of Equation (4-5), and the matrices are: {Hij}E (in units of K cm-1) ij E Xls Xlz {Sij}E -126. -21.425 -6.238 -7.881 -73 -22.56 -12.95 -72. 0 -42. 1..1117.03275.05417 1..1556.1169 1. 0 1. Solution of secular equation gives the following eigeneigenvectorso (Table C-1 and Figure C-1) values and

-110 - TABLE C-1 MOLECULAR ORBITALS AND ORBITAL ENERGIES OF CdTe:Mn++ T2 Symmetry: Energy K cm-152.7. * -122.2 - 80.3 - - 70.4 -. - 52.6 -. 81.5. 03s Mixing Coefficients 03z 03c X3s X3z 3252 9251 0055 2489 0018 0394.8197 -.5257 -.2512.0151 -.0950 -.8687.0166.1279 -.3006.8955.3243.0727.0043.0709.7872.3588 -.2019 -.7035.2011 -.1355.3678 -. 0434.6615.8749.oo66 -.0395 -.0299.1531 -.6640 1.015 E Symmetry: Energy K cm* -127.1 - 79.9 - 5901 - 39.2 Mixing 01c Coefficients Xls xlz.9727 -.2244.1338 -.0124.1320.6178 -.7431 -.3148.0433.6864 7258.1600. oo84.0377 - 2255.9816 *Molecular Orbitals of Unpaired Electrons.

-111- Kcm Kcm -40r E (-39.2) 5P(-42) 50 I T2 (-52.6) 60 F E (-59.1) -70 T2 (-70.4) 5S (-72) 5P(-73) E (-79.9) iii 80 F -90 -100 - T2(-80.3) T2 (-122.2) -110 -120 - 3d (-126) E (-127.1) -130 -140 - 5S (-144) -150 L T2 (-152.7) Mn LEVEL MO LEVEL Te LEVEL Cd LEVEL Figure C-1. MO Energy Level Diagram for CdTe:Mnt++

-112 - (ii) ZnS:Mn++ Radial functions are R3d(MN) =.467503 (2.025) +.534603(3.9754) +.137503(7.4822) +.005303(13.462) R4sp(Zn) =.020801(31.455) -.01132(28.027) +.0461502( 14.673) +.074703(13.652) -.028503(8.3237) -.371803(4.7860) +.164904(5.1559) +.53653504(2.5916) +.589304(1.4066) +.1677504(0.9130) R3s(S) = 0.035201(17.867) +.044901(13.924) +.o49102(13.753).064402 (8.9398) -.193702 (6.2464) -.191003(5.7842) +.o0005(5.04o31) +.7o46(2.49) +.13405(1.2872) R (S) = -.0130502(12.798) -.038602(8.1734) -.240602(5.0103) +.0871503(3.8107) +.379503(2.1976) +.572403(1.5528) +.0945505(0.7790). Watson does not give 4p function for Zn. In this calculation it is assumed to have the same radial dependence as 4s, and its energy is estimated from spectroscopic data. The basis functions are (as the case of CdTe:Mn++) given by Equations (4-5) and (4-7). The matrices are:

-113 - T2 Symmetry: {Hij }2 (in atomic units, 1 a.u. = 2 ryd) 03s 03z 03c X3s X3z -.6334 -.0850 -.0904 -.0389 -.0149 -.0173 -.8785 0 0 -.2864 -.2337 -.4363 0.0637 -.0171 -.4363 -.1803 -.1128 -.2855 0 -.1382 Isij T2 I--I --- 1..057 1..o86 o 1..037 o o 1..0175.286 -. 090.255.0293.335.035.230 0 1. 1. E Symmetry: {Hij}E (in atomic dl units, o1c 1 a.u. = 2 ryd) Xls Xlz {Sij E -.6334 -. 0674 -.0105 -.0122 -.4363 -.1104 -.0409 -.2855 0 -.1382 1. o0641.0124.0207 1..1564.0834 1. 0 1. i, O~~__

TABLE C-2 MOLECULAR ORBITALS AND ORBITAL ENERGIES OF ZnS:Mn++ T2 Symmetry Energy a.u. K cm-1 03s Mixing Coefficients 03z 03i X3s X3z -.8907 * -.6324 -.4563 -.4197 -.1977.0679 -195.4 -138.7 -100.1 - 92.1 - 43.4 14.9.1360 -.9557 -.0020.2800.0046.0436.9931.1712.0977 -.0271 -.1180 -.4957.0087 -.1774.2948 -.9304.1739.0573.0145 -.0744 -.8482 -.3162 -.2574 -. 5304.0510.o4o1 -.2662 -.0110.7798.7349 -.0893.oi64.oo46 -.0306 -.5927.9379 E Symmetry Energy a.u. K cm1 * -.6370 -139.7 -.4392 - 96.4 -.2580 - 56.6 -.1346 - 29.5 Mixing Coefficients 01T Xls d 1 xlz.9827 -.1934.0345 -. 0063.1310.9244 -.3882 -.1184.o114.2445.9800.0705 -.0050.0113 -. 0610 1.0018 *Molecular Orbitals of Unpaired Electrons.

Kcmr - 0r T2 (14.9) 20 F 4P (-30.3) E (-29.5) - 40 - 60 T2(-43.4) E (-56.6) T2 (-92.1) 4S(-62.6) - 80 F 3P(- 95.7) E (-96,4) - 100 r2 (-100.1) - 120 F 3d (-139.0) - 140 T2(-138.7) E(-139.7)I E (- 139.7) 160 - - 180 - 3S (-192.7) T2(-195.4) - 200 L Mn LEVEL MO LEVEL S LEVEL Zn LEVEL Figure C-2. MO Energy Level Diagram for ZnS:Mn++.

Solutions of secular equations are given in Table C-2 and Figure C-2. The average probabilities that one electron being found in the next nearest ligand S orbital, Equation (4-85) are for CdTe:Mn++ 1for ZnS Mn for ZnS Mn++ C The result is discussed in Chapter VI.

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