T HE UN IVER SIT Y OF MICHIGAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report AN MO THEORY OF THE CUBIC FIELD SPLITTING OF 3d5 6S IONS IN II-VI COMPOUNDS OF Td SYMMETRY G. H. Azarbayejani Chihrro Kikuchi ORA Project 04381 Supported by: NATIONAL AERONAUTICS AND SPACE ADMNISTRATION GRANT; NO. NsG:-,1.'WA'SHINGT N,, D administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1966

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TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii CHAPTER I. INTRODUCTION 1 II. EXPERIMENTAL DETERMINATIONS OF THE CUBIC FIELD SPLITTING OF THE 3d5 S-STATE IONS 7 III. THEORETICAL 19 1. Cubic Field Splitting 20 2. Hamiltonians 21 3. Wavefunctions 23 3.1 Ground State Wavefunction 23 3.2 Excited State Wavefunctions - Charge Transfer Wavefunctions 24 4. Matrix Elements of Spin Orbit Interaction 26 4.1 Reduced Matrix Elements 29 5. Coupling Coefficients of Spin Orbit Interaction KJJ (h' ht 32 6. Single Orbital Reduced Matrix Elements (1/2 alj'sl 11/2 b) 37 IV. CUBIC FIELD SPLITTING 3a 39 1. Determination of the Lowest Order of Perturbation by Hp -. ki si that 6 p i' can Split 6A1 39 2. Contribution to the Cubic Field Splitting 3a from Fourth Order Perturbation of 6A1 by i ii 41 V. DISCUSSION 44 1. Contribution to 3a from Watanabe's Calculation 44 2. Contribution to 3a from Powell's Calculation 45 3. Contribution to 3a from Low and Rosengarten Calculation 46 ii

TABLE OF CONTENTS (Continued) Page 4. Contribution to 3a from a-Bonding Charge Transfer States 48 4.1 Bonding-Nonbonding Charge Transfer 50 4.2 Bonding-Nonbonding and BondingAntibonding Charge Transfer 50 5. Contribution to 3a from x-Bonding Charge Transfer States 54 5.1 Determination of 3a (a,r, t1 + ea) 55 5.2 Determination of 3a (a,i, tb + ea) 56 5.3 Determination of Total 3a(a,r) 59 6. Comparison 60 7. Comparison with Measured 3a of Fe3+ in ZnS, ZnSe and ZnTe 61 8. Comparison of 3(a,t) of Td and Oh Cases 65 VI. SUMMARY AND CONCLUSIONS 68 1. Summary 68 2. Conclusions 70 ACKNOWLEDGMENT 71 APPENDIX A. DEFINITION OF SYMBOLS 72 APPENDIX B. SPIN ORBIT COUPLING IN MO SCHEME 78 APPENDIX C. SPIN ORBIT MATRIX ELEMENTS FROM THREE AND FOUR ORBITAL WAVEFUNCTIONS 84 1. Charge Transfer Wavefunctions 84 2. Matrix Elements of Hp = Zi i' si 85 3. Reduced Matrix Elements Rjk tSh-S'h') 86 4. Homoconfiguration Three Orbital Reduced Matrix Elements 87 5. Hetero-Configuration Three Orbital Reduced Matrix Elements 101 6. Hetero-Configuration Four Orbital Reduced Matrix Elements 114 APPENDIX D. COUPLING COEFFICIENTS OF SPIN ORBIT MATRIX ELEMENTS TO THEIR REDUCED MATRIX ELEMENTS 118 1. Determination of Kjj,' (SS'T1, h'ht) for h = A1 119 2. Determination of Kjjv (SS'T1, h'ht) for h = h' = T1 122 111

TABLE OF CONTENTS (Concluded) Page APPENDIX E. FOURTH ORDER PERTURBATION 125 APPENDIX F. SPIN-ORBIT MATRIX ELEMENTS BETWEEN Ix ShM0> and IX'S'h'M'e'> for S S' 130 APPENDIX G -- COVALENCY DEPENDENCE OF THE CHARGE TRANSFER CONTRIBUTION TO THE CUBIC FIELD SPLITTING 3a (a, 1) 133 REFERENCES 136 iv

LIST OF TABLES Table Page 2.1 Variation of separation of Mn++ ESR fine-structure components at e = 0 as a function of p = gBH/2a = E/a 15 2.2 ESR results of S-state ions in II-VI compounds 16 2.3 Comparison of ESR results with predictions of ionic theory 17 3.1 Double valued character table of group Td 25 3.2 Charge transfer configurations and terms 27 35. Reduced matrix elements I(X1Shl i i _.si IXiS' h I 29 3.4 Reduced matrix elements I X2ShlHLi.i s i Ix2S'h'>2 30 3.5 Reduced matrix elements I(X3ShlljiCi's XShX3 31 3.6 Transformation of 6E into the IR's of double valued group TA: I<ShMe0ShJtT)I2 33 3.7 Transformation coefficients of 6T1 into the IR's of double valued group T': I<5/2T1Me15/2TlJtO)12 34 3.8 Transformation coefficients of 6T2 into the IR's of double valued group T': I<5/2T2Me15/2T2JtT 12 35 3.9 Coupling coefficients KJJ for h = Al 36 3.10 Coupling coefficients Kjjfor h = T1 36 3.11 Coupling coefficients KJJ' for h = E 36 3.12 Coupling coefficients KJJ, for h = T2 36 3.13 Single orbital reduced matrix elements 38 4.1 The coefficients Ci (-2/6525 E63)-1 42 4.2 Calculated values of 3a for spin sextets 43 5.1 Calculated 3a in Mn2+ in units of 10-4 cm-1 46 5.2 Comparison of (3a)LR with (3a)p and (3a)Exp 47

LIST OF TABLES (Concluded) Table Page 5.3 Numerical values of 3a(o) in (18/625) (6nT)-3 d4 52 5.4 Measured 3a of Fe3 in 10-4 cm 62 5.5 Estimated (3a)p for Fe3+ in 10-4 cm-1 63 5.6 Calculated 3a(a,7) of Fe3+ 64 5.7 Measured and calculated values of 3a of Fe3+ 64 A-1 Definition of symbols 73 C-1 The values of (h) and (-l)h 95 vi

LIST OF FIGURES Figure Page 2.1 The octahedral coordination in cubic II-VI compounds (CaO:Mn) 8 2.2 The tetrahedral coordination in cubic II-VI compounds (ZnTe:Mn) 8 V2+ 3+ 2+ 3+ 2.3 ESR spectra of V, Cr, Mn and Fe in a single crystal of CaO at e = H A [100] = 0 and T = 3000K 8 2.4 Differentiation of ESR absorption spectra in ZnTe:Mn 9 2.5 (a) The splitting of MS = ~ and -~1 levels into six close lying levels and (b) the splitting of the MS = 1 +-+ MS = -~1 transition into six approximately equally spaced transitions. 12 2.6 Energy level scheme of 3d5 6S5/2 (Mn2+) in a tetrahedral field at e = 0 14 2.7 Assignment of ESR spectra of Mn2+ in both Oh and Td cases: (The spectrum belongs to Mn2+ in cubic ZnS) 15 2.8 Comparison of experimental and theoretical values of 3a 17 3.1 A schematic energy diagram of [ZA4]-n', 6A1) complex 23 3.2 Symmetry elements of a tetrahedron 25 5.1 c-bonding molecular orbitals in II-VI compounds of Td symmetry. 49 5.2 (a) Ground state of complex [A4]-n', (b) t2b+en charge transfer states and their schematic energy levels, and (c) t2but2a charge transfer states and their schematic energy levels. 49 5.3 Molecular orbital and energy levels of (a) the ground level 6A1 and (b) the t1 -+ ea electron transfer levels 6T1 and T2 55 5.4 Molecular orbital and energy levels of (a) ground level 6A1 and (b) the tb + ea electron transfer levels 6T1 and 26T 57 vii

ABSTRACT Molecular orbital techniques have been employed to find the cubic field splitting 3a of (3d) S ions in II-VI compounds of Td symmetry. The parameter 3a is calculated by perturbation analysis, through spinorbit interaction i i k s between the ground state t 3 ea A, and excited states taP S1 ~ ea 1 S2 h' Y S3 h3' S h >. Here ta,ea are the antibonding orbitals of the complex composed of the (3d)5 6S ion and its four nearest ligands and p and 4-p are the hole configurations of orbitals ta and e, respectively. The perturbation calculations have been carried out up to the fourth order which is the lowest order necessary for the splitting 3a to occur. Moreover, these calculations have been limited to the very small number of states which arise exclusively from those initial states t p 4,p a S1 h e S h......... with S S... having their maximum value. The analytical result is found as 4 3a c 4-i i a Ci d p i=o where Cd and p are spin-orbit parameters of the d-orbitals of (3d) S ion and p orbitals of the ligands respectively. The coefficients C. are functions of coefficients of linear combinations of d and p orbitals which give rise to the molecular orbitals ta, ea and y. They are also functions of energies Ejk required for promotion of a hole from a state IXj Sj hj > to another state Xk Sk hk >. The Xj and Xk in above states describe the hole configurations of orbitals ta, ea, y and their coupling scheme. ix

Numerical results, obtained for states IxS = 5/2 h> of Fe3+ in the series of ZnS, ZnSe and ZnTe compounds with a reasonable set of coefficients of linear combination of atomic orbitals and an average promotion energy of 32000 cm1, indicate that the term C4 4 contributes a large negative value to 3a in agreement with experimentally determined 3a of Fe3+ in ZnTe.

CHAPTER I INTRODUCTION The importance of the concept of spin Hamiltonian in electron spin resonance (ESR) is very well known.* The techniques of the measurement of the parameters in this Hamiltonian are also well developed. However, the attempts to interpret the measured values of the parameters have met with partial degree of success. A particularly puzzling discrepancy has been the ground state splitting of the iron group S-state ions in II-VI compounds of Td symmetry. The first ESR measurement of this splitting was made on ZnS:Mn by Matarrese and Kikuchi.1 This was followed by Watanabe's theory+ which predicted the 3a of a given S-state ion, in several compounds with the same formal charge, should decrease as the metal-ligand distance, R, increases. Predictions of this theory were given support by the measured 3a in II-VI compounds with Oh symmetry. Subsequent measurements showed that such is not always the case for every compound such as CdS:Mn and CdTe:Mn2. The 3a in CdTe:Mn was larger than that in CdS:Mn. This observation indicated that the point charge model is not adequate for the explanation of 3a in covalent Il-VI compounds and the covalency effects should also be taken into account. The purpose of this work is to explore the contributions to 3a caused by the above covalency effects present in such compounds such as CdTe by invoking the molecular orbital theory instead of the abovementioned point charge model. In order to obtain an insight into the sources of such contribution to 3a, as well as to the mechanisms causing *A. Abragam and M. H. L. Pryce, Proc. Roy. Soc. A, 205, 135 (1951). Ibid, 206, 164. Ibid, 206, 173 (1951). +See Reference 5.

the splitting to occur, a brief introduction to calculations based on the point charge model should be very helpful. Therefore, we proceed by giving a review of the previous work on 3a first, and then, we arrive at the possible covalency phenomena affecting this parameter. The ground state of the free ions Cr, Mn 2+and Fe3+ is six fold degenerate with the spectroscopic classification of (3d) S5/2. Substituting such an ion in the metal site of cubic II-VI compounds, such as Mn in the Zn2+ site of ZnS, one finds from electron spin resonance (ESR) spectra of the system ZnS:Mnl12, that the ground state of the Sstate ion splits into a spin quartet U' and a spin doublet E". This splitting is called the cubic field splitting of a (3d)5 6S/2 ion and is denoted by the parameter 3a = E(U') - E(E") with E(U') and E(E") as the lowest energy values of levels of symmetries U' and E", respectively. The crystalline cubic field can be expressed as:2 V = a (15)-1 (T4 + (5/14)/ (T44 + T4). The matrix elements of tensors T (k) of V for two states ly L ML > and lY' L' M'L > are:* L < L MLI T (k)ly, L' M' > = (-1)L-ML (Lk L' Lq M' (x) (y L'T(k) y' L') = 0. for L = L' = 0; k = 4 This result indicates that the ground state 6S5/2 is not split by a cubic field but that the splitting is caused from admixture of the ground state by excited states through perturbation by spin orbit coupling, spin-spin interaction, etc. B. R. Judd, "Operator Techniques in Atomic Spectroscopy," McGraw-Hill Book Company, Incorporated, New York, (1963), p. 42

A similar ground state splitting was manifested in an observation of the anisotropy of the magnetic susceptibility of paramagnetic crystals containing Mn2+ such as Mn (NH4)2 (S04)2 6H20. To explain this, Van Vleck and Penney (1934)3 considered various higher order processes involving the cubic field V and the spin orbit interaction, H = S.i 2isi p i - through intermediate excited states using the order of magnitude argument to estimate the resulting splitting. Later Pryce (1950),4 in explaining the same splitting for Fe3+, pointed out the inadequacy of mechanisms proposed by Van Vleck3and attributed the cubic ground state splitting of Fe3+ to a fifth order perturbation quartic in H and p linear in V. The work by Pryce was followed by Watanabe (1957),5 who based his calculations of the cubic splitting on the complimentary theorem in the crystalline field splitting of the transition ions. He argued that two ions with complementary electronic configurations, such 3+ 2+ as Ti and Cu whose ground level can be split by the first power of V, have always inverted splitting patterns with respect to each other when placed under the same crystalline environment. Based on this theorem, he concluded that a 3d5 ion is its own complementary and that any splitting arising from the first power of V should be both positive and negative, and hence identically zero. Proving, in this way, that linear contributions of the cubic field cannot contribute to the splitting, he extended the fifth order perturbation suggested by Pryce to the sixth order so that the crystal field contribution could appear in the second power and spin orbit interaction in the fourth power. In addition, he included contributions from fourth and fifth order perturbations by cubic field, spin orbit and spin-spin interaction. In these calculations, the excited states considered were spin quartets; 4P 4 4F and 4G r, D, F and G of the (3d)5 configuration with excited energies in the range of 30 to 50 x 104 cm1. The splitting 3a obtained from these calculations is positive, and -3 -4 -l varies from about 10 to 10 cm. It seems to satisfy the scant experimental data available at the time. (See Table 1, Ref. 5.)

Upon comparing the excited state energies of 5 x 104 cm-1 and the cubic field splitting 3a of the order of 10-3 cm-1 obtained from fifth and sixth order perturbations, there is an indication that none of the contributions which might arise from other excited multiplets of (3d) configuration can, a priori, be ignored. Indeed, there are spin doublets; 2S, 2p, 2D(3)' 2F(2), 2G(2), 2H and 2I lying in the region of 45 to 100 x 103 cm- Some of these such as I and H may be in the vicinity of 4D and 4F and can contribute to the splitting. Powell et al (1960)6 took all of the doublets S..... I into account and carried out sixth order perturbation calculations with and without spin-spin interaction. They found that the inclusion of doublets increases the predicted splitting by one to two orders of magnitude as compared to the predicted splitting arising from spin quartets alone. Their calculated results, for the particular case of MgO:Mn2+, agrees with experiments, provided that the spin orbit interaction constant, Cd of Mn 2+, is taken as 400 cml and the cubic field strength, 10ODq of MgO, as 10500 cm-1l Both of these are unreasonably high. Low and Rosengarten (1963, 1964)7,8 carried out calculations similar to that of Powell et al without spinspin interaction but they included the orbital polarization factor a, called Tree's correction factor.9 Their conclusion was that crystal field analysis is relatively successful in explaining the position of energy levels of the d5 manifold, but it is not capable of explaining the finer parameters such as the cubic field splitting, 3a, and the spectroscopic factor, g, both measured from ESR spectra of 3d5 6S ions. A comparison of the above theories with ESR measurements on Mn2+ in several compounds was made by Hall et al (1961).10 They observed that their measured 3a for Mn2+, in a number of fluorides and chlorides, could be accounted for by Powell's theory, whereas the agreement for ZnO got worse. For very covalent compounds, CdTeand ZnTe,ll a discrepancy of almost one to two orders of magnitude can be found. This indicates the inadequacy of Powell's purely ionic model for covalent systems. Another area in which both Powell's and Low's theories have failed is the spectroscopic g value. These theories predict a g value, for

3+ an S-state ion such as Fe, as less than the ge= 2.0023 of the free electron, in complete contradiction to experimental observations that the g parameter of Fe3+ is larger than 2.0023. Most of these investigators have attributed these irreconcilable discrepancies to the ligandto-metal charge transfer processes such as those suggested by Fidone and Stevens12 and by Watanabe 13-14 for the evaluation of Ag = g - geAn initial study for the determination of the charge transfer contribution to 3a, patterned after Watanabe's work, was carried out by Azarbayejani et al.15 These calculations included the construction of appropriate molecular orbital (MO) wavefunctions and the allowance of ligand-to-metal electron transfer. In constructing the MO wavefunction, a-bonding approximation was invoked and the cubic field splitting was obtained by a fourth order spin-orbit perturbation calculation. 15 4 6 E -3 It was found15 that 3a X 3al = 0.1728 X 6 (1 - E11/61) 61, where X = Cd is the single electron spin orbit parameter, 82 = 1 -a is the covalency of the d orbitals of 3d S ion and c11 and 61 are related to ligand-to-metal electron transfer energy. From free ion -1 optical spectra (Ref. 16, p. 437), an approximate value of 5d = 350 cm may be taken, and from a comparison of the hyperfine structure constant in crystals to that of the free ion,l7 12 may be estimated. For the particular case of ZnS:Mn where Cd = 350 cm 1, 2 = 0.22 energies 61 of the -1 order of 8000 to 10000 cm give qualitative agreements with the measured 3a. The most encouraging aspect of these 3a results is their correct trend for Mn in going from ZnS to ZnTe because 61 is expected to decrease as one goes from ZnS to ZnTe in accordance with Bube's conclusions on acceptor levels in II-VI compounds. In the present work, we have extended our previous analysisl5 to include rC-orbitals in addition to the a-orbitals. This has introduced R. H. Bube, "Photoconductivity of Solids," J. Wiley and Sons, Inc., New York (1960), p. 171 (Fig. 6.4-12).

extra orbitals in the charge transfer wavefunctions. Most of the desired spin orbit matrix elements for the determination of 3a arise from the above wavefunctions and contain three or four orbitals. Since no expression for the evaluation of these matrix elements is available in the literature, general formulae for obtaining such matrix elements have been found first, and then, 3a has been calculated. A brief introduction to the method of measuring 3a and the values of 3a for both the octahedral and tetrahedral II-VI compounds is given in II. Spin orbit matrix elements between excited spin multiplets is considered in III. The cubic field splitting 3a from these charge transfer states, tP e4-Py,is obtained in IV and is discussed in V. Concluding remarks are given in VI.

CHAPTER II EXPERIMENTAL DETERMINATION OF THE CUBIC FIELD SPLITTING OF THE 3d5 S-STATE IONS The purpose of this chapter is to give a brief introduction to the method of measuring the cubic field splitting, 3a, of the S-state ions such as Cr+, Mn2+ and Fe3+ The equipment employed consists of an electron spin resonance spectrometer such as the Varian V4502 EPR spectrometer in a 12-inch rotating electromagnet. Most of the measurements have been carried out at 4.2 and 770K with a few being performed at 3000K. The magnetic field, associated with a spectral line, has been obtained by first tuning a Varian F-8 Fluxmeter for the proton resonance at that field and then measuring of the proton resonance frequency by a Beckman 7370 electronic counter. The frequency of the microwave source used in the experiment was determined by first finding one of its subharmonics through Beckman transfer oscillator and then measuring the frequency of that subharmonic by the above mentioned counter. The ESR spectra of Mn2+ in CaO and ZnTe are given in order to 56 serve as representatives of ESR spectra of 3d S ions in octahedral and tetrahedral II-VI compounds. In the octahedral case (Oh), the paramagnetic 3d S ion is surrounded by six ligands or nonmetal nearest neighbors as shown in Fig. 2.1. These lie along the six crystallographic directions [100], [010], [001], [100], [010] and [001] with the paramagnetic ion at the origin of the coordinate system. On the other hand, in the tetrahedral case (Td), the paramagnetic 3d5 6S ion is surrounded by four nearest neighbors lying along the four crystallographic directions [111], [111], [111] and [111] as shown in Fig. 2.2.

The expression for the crystalline field of these ligands of the central ion is the same for both cases, provided the coordinate system is chosen as shown in Figs. 2.1 and 2.2. Denoting the angle between the d.c. magnetic field and one of the coordinate axes such as z by 0, we have shown the spectra at 0 = 0 for 2+ Mn in CaO (Fig. 2.3) and ZnTe (Fig. 2.4). As mentioned above, the tool 1 To l o ol!1 0011l0 /11111 X|tor lo0 ll Fig. 2.1. The octahedral coordination Fig. 2.2. The tetrahedral coordination in cubic It-VT compounds (CaO:Mn). in cubic II-VI compounds (ZnTe:Mn). ~ I. 1oTo Mn2+ P3O:3I v2+ Fig. 2.3. ESR spectra of Vat, Cr3+, Mn2+ and Fe3+ in a single crystal of CaO at 0 a H A t(001 = 0 and T 3000K

Zn Te; Mn 780K Zn Te; Mn 78OK a =00 4.2 K z o z 0 CC U) A m ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~80~ g o t 11 l i I i II a<~~~~~a a ct~~~~C o~~~~~~~~~~~~ MI F zI'13I I 4 1~~~~~~~~'r1, C 0 W a) C f() pi 0N1) f)e) fq) f( ) r() MAGNETIC FIELD, H MAGNETIC FIELD, H Fig. 2.4. Differentiation of ESR absorption spectra in ZnTe:Mn

10 proper choice of the coordinate system has allowed us to analyze the spectra in both Oh and Td cases by means of the same spin-Hamiltonian:* H =g ~ H-S + AI'S + a (S + S 4 + S 4)/6 + InA/'s - gN ~N H'I s e- x y z n (2.1) Here, g is the spectroscopic g factor, A, the hyperfine structure constant, 3a the cubic crystalline field splitting, A'n is the superhyperfine coupling coefficient and the last term is the nuclear Zeeman effect. The brief reports on the measurements of g, A and 3a of ZnTe:Mn and CaO:Mn obtained by using (2.1) are made previously. A brief introduction to the calculation of these parameters from the spin-Hamiltonian in (2.1) is as follows: For e = 0, the spin-Hamiltonian of (2.1) may be rewritten as Hs ( = 0) = g e HSZ + AI'S + a T40 + (T44 + T4_4))/15 (2.2) in which 40 35 S4 30S 2 6S + 253S ] 6S +3 /81 (2.3) and T4+4= s+4/16; S* = S(S + 1), S+ = (S + iS )/ (2.4) The H in (2.1) can be expressed as fs hfs H =H +H B. Bleaney and K. W. H. Stevens "Paramagnetic Resonance" Repts. Prog. Phys. 16, 108 (1953) p. 137.

11 where Hfs g e HSz + a [T40 +Vi7(TL44 + T4 4)j/15 (2.5) and hfs H hfs =AIS (2.6) s The energy of each MS level can be obtained by solving the secular equation corresponding to the fine structure Hamiltonian Hfs given in (2.5) I (H)MM, - E 6MMIII = o (2.7) where (Hs )MM = X MM + y 6MM +4 X = [2Me + a (14M4 - 95M2 + 184)/48] y = - 5a/2 and E = g e H/2 e Substituting for (2.7) one finds,: E (MS = + 1/2) = + c + a E (MS = + 3/2) = + 3c - 3a/2 + 5a /32e (2.8) E (MS = + 5/2) = + 5c + a/2 + 5a /32

1]2 mI MS - 5/2 - 3/2 5/2 1/2 1/2 3/2 ~~~~I - ~~~5/2 mX.5/2 -3/2 -1/2 11/2 3/2 5/2 MS n1/12 MS "1/2 (b) Fig. 2.5. (a) The splittings of Ms - ~ and -3 levels into six close lying levels and (b) the splitting of the M2S 1~ M 2 - transition into six approximately equally spaced transitions.

13 For many cases where the microwave frequency vo, used for ESR measurements is about 10 KMC and a II-VI compound is the host material, the ratio c/a = 0.01 - 0.1 for 3d5 6S ions Cr+, Mn2+ and Fe3. In such cases, one is able to use AILS as a perturbation on the first term of (2.5) which causes each MS level to split into six close lying levels. As an example, the splittings of the MS = 1/2 and -1/2 levels are given in Fig. 2.5. For the allowed ESR transitions (AMS = + 1, AmI = 0) each MS - 1 +-+ MS transition will split in 21 + 1 transitions. The number of MS - 1 4-+ MS transitions which can be observed distinctly is 2S, provided that the parameter A in (2.8) is large enough to offset the effect of the line broadening. The energy diagram of the MS levels of a 3d5 6S ion at e = 0 is obtained as a function of p = g lH/2a (Table 2.1 and Fig. 2.6). The numbers identify the upper MS values. Thus, the five transitions MS = -5/2 -++ MS -3/2..... M = 3/2 +- = 5/2 are designated by -3/2, -1/2.....5/2, respectively. When the lines are well resolved one expects to observe 2S(2I + 1) lines. This number for Mn with S = 5/2 and I = 5/2 is 30 (Figs. 2.3 and 2.7). These lines can be identified with the electronic and nuclear magnetic quantum numbers MS and mI by considering the fact that the intensity of the five lines 3/2.....5/2 belonging to any of the 2I + 1 quintets should vary as 5:8:9:8:5. Therefore, the following assignments are possible for both octahedral and tetrahedral cases (Fig. 2.7). ai,' i correspond to MS = + 3/2 ++ + 1/2 6,' 6i correspond to MS = + 5/2 ++ + 3/2 and i correspond to MS = + 1/2 +- + 1/2

14 50 M. 5/2 40 30 Wk5/2 3/Ms'3/2 20 Ms -3/2 00'~~~J 1 3/2 -501 L M, -1/2 Free Cubico Magnetic ion Field Field -20 Fig 26 Energy level scheme of (2) in a/2 tetrahedral f/2'40 "1/21 11/2,.) MI:-S/2 0 2 4 6 8 10 tettahedca.1 f:[el. at: 0 =3.

15 Table 2.1. Variation of separation of Mn++ ESR fine-structure components at 8 O0 as'a function of p = gBH/2a = c/a E-. E. _ - - 1 —E E -E P 5 /22 F E3/2 E F 1/2-F 23/F 3/ 2'EF 0 -2.00 -2.00 1.00 1.00 1.00 1.00 1 -4.71 -4.62 0.00 2.00 1.71 5.62 2 -9.59 -7.57 -1 3 4.59 10.57 3 -14.56 -10.55 -2 4 7.05 15.55 4 -19.54 -13.54 -3 5 10.54 20.54 5 -24.53 -16.53 -4 6 13.53 25.54 6 -29.52 -19.53 -5 7 16.53 30.52 7 -34.52 -22.52 -6 8 19.52 35.52 10 -49.52 -31.51 -9 11 28.51 50.5 100 -499.5 -301.50 -99 101 298.5 500.5 ml =-5/2 - 3/2 - 1/2 1/2 3/2 5/2 Yl 61 Al P6 Y6 66 Fig. 2.7. Assignment of ESR spectra of Mn2+ in both Oh and Td cases: (The spectrum belongs to Mn2+ in cubic ZnS) The next step to consider is the determination of the spin-Hamiltonian coefficients g, A and a of (2). For a fixed microwave frequency v, these coefficients can be measured as follows: g = hvo/ e [(Hyl + Hy6)]/2 IAI = g e [(hy6 - Hy1)1/5 (2.9)

16 and a| Be t (Hyl - Hal) + (HA6 - H~y6 /5 The signs of A and a can be determined relative to each other with the sign of a being determined independently by its measurement at low temperatures. The results of such measurements are given in Table 2.2 and 3a and g are compared with predictions of the present theories of these parameters in Table 2.3 and Fig. 2.8. The agreement is generally satisfactory for the case of MgO, CaO and SrS, whereas disagreement is observed for zinc and cadmium chalcogendies. These deviations from ionic theory which arise from larger covalency existing in the latter group compared to the former, have emphasized the need of a more Table 2.2. ESR results of S-state ions in II-VI compounds 0 Cr+ Mn Fe A 3a A g 3. A g 3a mo lcm 10' cm' 104cm 104cm 10 cm g0 Oh7 6 2.12 | -- -81.0 2.0014 55 2.0037 615 CaO Oh7 6 2.40 -- -- -- -80.7 2.0009 17.7 2.0052 191 SrS Oh 6 3.05. -77 2.0009 4.2 -- -- ZnO C 4 1.95 -- -- - -74 2.0016 18 2.006 123 6v ZnS Td2 4 2.36 13.4 1.9995 12 -64.9 2.0025 23.7 2.019 382 ZnS C 4 4 | -- - -- - -- 2.018 384 6v ZnSe Td2 4 2.45 13.3 2.0016 16.05 -- -- 144.9 ZngS C 4 | -. -61.7 2.0055 52.1 -. - 6V ZnTe Td2 4 2.66 12.4 2.0023 19.80 -56.5 2.0075 88.9 2.09 -7800 CdS C 4 A 2.32 --.. -65.3 2.003 11.7 2.01 285 CdSe C 4 2.64 | -. -62.7 2.005 4.3 -- -- C6v CdTe Td2 4 2.80 12.8 1,9997 9.3 -55 2.0078 83.1...

17 Table 2.3.* Comparison of ESR results with predictions of ionic theory Material Sym rA g- T + Cr Mn Fe Cr Mn Fe MgO Oh 1.000 1.000 1.000 <0 <0 >0 CaO 0h 0.287 0.32 0.311 <0 <0 >0 SrS 0h 0.026 0.07 <0 <0 ZnO C6v 6.635 0.76 0.316 <0 <0 >0 6v ZnS C4 & T 1.000 1.000 1.000 1,.000 <0 <0 %0 >0 6v d ZnSe C4 6 T 2 0.685 1.34 2.198 0.39 <0 <0 >0 >0 6v & ZnTe Td 0.301 1.65 3.751 -20 <0 "0 0 >0 CdS C64 1.000 1.000 <0 0 >0 CdSe C64 0.624 3.67 <0 0 CdTe T 2 0.345 7.02 <0 <0 0 *The rT and rE are the theoretical and experimental ratios of 3a respectively and Ag = g-2.0023. The ratio rT(1,j) = 3ai:3aj = (aoj! aoi)10 with a0 being the lattice conatant. 3- 50 3o(10'4cm'') C/8=3 - 300 3 COMPARISON OF EXPERIMENT -3o 0 WITH CALCULATIONS OF WATA- I (~) NABE, PGJ1 AND PGJ2 250 1 - 200I ""x.8 I /85 (I0OB+5C )/.23 Kk) 150 / PGJ! (DOUBLETS AND W,,)(b) / WATANABE (a) 100/ / ZnT CdTe / ZnSe M /1v CdSe - 50 ZnO ZnS dS Co. (e) N&CI -2000 -1600 -1200 -800 -400 06 400 800 1200 1600 200,fa (cm-') Fig. 2.8. Comparison of experimental and theoretical values of 3a.

18 realistic theory which takes these covalency effects into account. In the next few chapters the dependence on the covalency of the parameters given in (9) is pursued with a greater emphasis on calculations related to the cubic field splitting 3a.

CHAPTER III THEORETICAL As mentioned in the last section, we intend to obtain the cubic field splitting 3a by using the linear combination of atomic orbital molecular orbital (LCAO-MO) techniques. The wavefunctions constructed from these LCAO-MO's in a certain manner, serve as excited states which admix to the ground state wavefunction through spin orbit interaction and cause a contribution to the cubic splitting 3a. From this brief introduction, it is immediately evident that our task is twofold: (1) to construct the LCAO-MO (henceforth denoted by MO) and the desired wavefunctions and (2) to develop appropriate expressions for the matrix elements of the spin orbit interaction in the MO scheme. Since we are primarily concerned with the cubic field splitting, 3a, in compounds of Td symmetry, our effort will be directed toward the determination of the matrix elements of spin orbit interaction, H = i Q.s, between various wavefunctions of a complex, [Z A], ii- 5 A4] consisting of a 3d S ion Z and four ligands, A 10*0A4' the whole complex being located in a cubic crystal BA. For example, in the case of manganese doped zinc sulfide, (ZnS:Mn), Zn = B, S = A, Mn = E, n' = 6 and the complex is [MnS4] -6 In order to limit our analysis to those formulae affecting just 3a, we proceed by defining the cubic field splitting and the symmetry of the levels which give rise to that splitting. The excited wavefunctions considered here, are those obtained from an electron transfer from the ligand to the metal ion. ** A summary of the symbols is given in Appendix A. 19

20 1. CUBIC FIELD SPLITTING The following is a brief elaboration of the symmetry group of the states into which the ground state of the complex, (Z A4) splits ( = Cr+ Mn2+ Fe 3+A =S..Te ). The symmetry of the ground state of the above complex is of Al and has a total spin S = 5/2. Thus, the ground state may be given as [ A4] A1 S = 5/2)] or more simply by IX~ 6A1) where X0 denotes the MO's giving rise to the A1 state, their electronic configuration and finally, the total spin values and the irreducible representations, S. h i i, of each of these MO which comprise Xi. The symmetry group of the total Hamiltonian of the complex is 0 x U2 where 0 is the group of symmetry operations of a cube in orbital space and U2 is the group of rotation in spin space. The representation of 6A1, in the full rotation double group, G' = R3 x U2, is J = 5/2. The irreducible representations of J = 5/2 in G = 0 x Uo are E'' + U'. + 2 According to the irreducibility principle, the maximum number of levels created by the perturbation of |A1 S = 5/2)= 16A>)will be the number of irreducible representations of J = 5/2 in G which is two levels. The cubic field splitting is defined as the energy separation of these two levels: 3a = E(U') - E(E'') (3.1) where E (r') = E(0) (r') + E(1) (r') +....+ E(4) (r) = U' or E'' (3.2) Mulliken's notation (see Ref. 22) is used for all cases except when mentioned otherwise. The state symmetries and energy zeriis are identified by the irreducible representations Al, A2, E, T1, T2, E', E''" and U' or the cubic double group where the molecular orbitals are denoted by the small letters al, a2, tl and t2. V. Heine, "Group Theory in Quantum Mechanics,'" University of Cambridge Press, 1960, p. 45.

21 (n), th Here, E(n)(T') (n = 0, 1..., 4...) are the n order contribution to E(r'). The 3a will be positive or negative depending on the relative magnitudes of E(U') and E(E''). The Hamiltonians giving E(n)(r) will be examined in the next section. 2. HAMILTONIANS We wish to consider a Hamiltonian of the complex, [E A4], which includes a zero order Hamiltonian, Ho, satisfying HOPn = En n and a 0n nn perturbation Hamiltonain, H, from whose matrix elements Mn between pm and in, the corrections E( )(r) may be obtained. Denoting the above Hamiltonian by HO' one has Ho = H0 + H (3.3) In the present work, we limit out perturbation analysis to spin orbit interaction. Thus, H =Z.s (3.4) p - - i and the zeroth order Hamiltonian, HO is: nt n''t 4 n'' H = Pi/2mi - e r1 + e rir + V(rik) i-l i>j k=l i=l (3.5) where n'' = 37, refers to the sum of the 32 valence electrons in the -n molecular orbitals of the complex, (EA4), and the 5 electrons located in the d orbitals of the central ion E. The first term in (3.5), represents the kinetic and potential energies, the second one gives the %i acts as an operator, being (d when operating on d parts of the ith orbital and %p when operating on the p part of the ith orbital. (Appendix B)

22 Coulomb and exchange energies and the last term gives the effect of four ligands, k, separated from the central ion by rk. Za in (3.5) refers to the effective charge of the central ion. The eigenfunctions and eigenvalues of (3.5) are usually obtained by approximate techniques. One of these is known as the self-consistent charge configuration (SCCC) method. Ballhausen23 used this last technique to construct the eigenvalues and e igenvectors belonging to the [MnO4]-1 complex and very recently Basch et al24 extended the same method to the 32 complexes of transition ions in compounds with Oh or Td symmetries. The latter authors give a.n energy diagram for the [FeC14]2 complex. The levels lie from -220 x 103 cm 1 to about 90 x 103 cm-1 and they are classified according to their symmetry as follows: (al) (1t2) 6(2al) 2(2t2) (le)4(3t2) 6(tl)6(2e) 2(4t2)4(5t2) (3al) where the superscripts are the electronic configurations and lal, lt2....3al have the symmetry Al, T2....A1 of the cubic point group. The MO configuration for Fe, in tetrahedral complexes as well as Mn and Cr in such complexes will be the same as in (3.6) except the configuration of (4t2)reduces from 4 to 3. The orbitals we plan to use for the construction of the excited wavefunctions are le, 3t2, tl, 2e and 4t2. To simplify the notation, we label them eb, tb, tl, ea and ta, respectively. Here, the subscript b points out that eb and tb are bonding orbitals with E and T2 symmetries, respectively. Similarly, those with the subscript a are the antibonding orbitals, whereas tl, which does not have any subscript, is a nonbonding orbital. A schematic energy diagram associated with the above five orbitals; tb, eb, tl, ea and t and their corresponding electronic configurations characteristic of [Z A4]-n is given in Fig. 3.1.

23 ta 5 t2 / O \ 2 ~d~~~~ 2)..... i IFS -Pfil t1 ('" t2) I t2 b'21% t2'b 2t 2 Fig. 3.1. A schematic energy diagram of ([A4]-n', 6A1}complex. The p'i in Fig. 3.1 are linear combinations of the components of ligand p orbitals which are perpendicular to the interionic distance. The pa are the part of the p orbital projected along the interionic axis. Having defined the nature of the orbitals involved, we now proceed to construct the wavefunctions. 3. WAVEFUNCTIONS We want to describe the spin values Si and the irreducible representations, hi, of the individual molecular orbitals (MO) giving rise to the ground state and excited states. A knowledge of these is necessary for the determination of spin-orbit matrix elements as will be seen later (see 4). Therefore, we first consider the ground state and then, discuss the excited ones. 3.1 Ground State Wavefunction A description of the ground state wavefunction is being sought which emphasizes the symmetry, spin and irreducible representation of the molecular orbitals which constitute it. The radial part of the individual wavefunction will not be included for simplicity and the spin orbit interaction parameter, (i(r) of (3.4) will be considered as Cd for the d orbitals of' ion ~ and p for the p orbitals of ligands A.i in the complex [E A4]n (E = Cr+, Mn2+ 3+ - T Mn,Fe A = O...T ).

24 The orbital part of the ground state wavefunction can be deduced from Fig. 3.1, in the following form: X~e 6) ( A-n 6 4 6 2 3 a et36A Itb6e tl ea ta3 6A1 A1 A [(EA4 9c tb e tl e t A t e t e t A (3.7) where e 6 4 6 2 3 Xo = tb e t e t3 is electron configuration (3.8a) a 1 a a or h 0 o o 2 3 X = tb eb t1 ea ta is hole configuration (3.8b) The irreducible representations of the molecular orbitals tb, eb... in (3.7) - (3.8b) are r (tb) = r (ta) = T2 r (eb) = r (e) = E r (tl) = T1 (3.9) The symmetry of the irreducible representations T2, E and T of Td group can be deduced from the character table of this group (Ref. 25, p. 383) given in Table 3.1. The group classes C2, C3, ad and S4 of Table 3.1 are classes of symmetry elements of a tetrahedron as shown in Figure 3.2. 3.2 Excited State Wavefunctions - Charge Transfer Wavefunctions We wish to describe here, the excited states created exclusively by the process of promoting one electron from one of the three orbitals tb, eb or tl of Xo in (3.8a) to any of the two orbitals e and b' b 1 o ~~~~~~a

25 Table 3.1. Double Valued Character Table of Group Td Bethe Mulliken 1 R 8C3 8C3R 6C2 12 Cd 6S4 6S4R r1 A1 1 1 1 1 1 1 1 1 r2 A2 1 1 1 1 1 -1 -1 -1 r3 E 2 2 -1 -1 2 0 0 0 r4 T1 3 3 0 0 -1 -1 1 1 r5 T2 3 3 0 0 -1 1 -1 -1 r E' 2 -2 1 -1 0 0 Y T -~" r7 E" 2 -2 1 -1 0 0 -V V/ r8 U' 4 -4 -1 1 0 0 0 0 c2,4 /3 Fig. 3.2. Symmetry elements of a tetrahedron e t in X. All other excited wavefunctions arising either from multiple charge transfer or from the irreducible representations, hi, of terms of t a and ea5p which belong to spin values of Si = p/2 - 1 and 1/2 (5-p)-1 are ignored. A similar restriction is imposed upon hi after charge transfer (hole transfer) occurs, and, as a result of this, all excited stf t p 4-p' states arising from t a and e (after hole transfer, the sum of the a a hole configuration of ta and ea will be 4) which belong respectively to spin values of Si = p'/2 - 1 or (4-p')/2 - 1, are ignored. For

26 example, a hole transfer from t in (3.8b) gives t 2 as a new hole cona a figuration for this orbital. The irreducible representations, (IR) of t2 are a r (ta2) 3T1 + A + E + T2 (3.10a) and similarly, r (e2) = A2 + A1 + 1E (3.10b) From the IR's (3.10a - 3.10b) only those with the maximum spin of these two shells, namely, rmax (a ) 1 and rmax (e 2) A (3 11) are considered and all the remaining spin singlets are ignored. The electronic configurations of the complex, after charge transfer, and their corresponding terms constructed in the above scheme are given in Table 3.2. Now we consider the determination of the spin orbit matrix elements between spin sextets 6A1 of the ground state and the excited spin sextets and quartets given in Table 3.2. 4. MATRIX ELEMENTS OF SPIN ORBIT INTERACTION The matrix elements of the spin orbit Hamiltonian, H= p C. s, p i iwill be discussed in this section and Section 5. A few initial comments are necessary to point out the need for the development of new formulae for evaluations of the desired matrix elements. Considering Table 3.2, it is evident that a matrix element between the

27 Table 3.2. Charge transfer configurations and terms Hole Configuration * Spin Sextet Spin Quartet + No. ta ea t b eb tb E T1 T2 E1 T1 T2 2 2 2 1 0 0 1 1 1 1 1 1 3 2 2 0 1 0 1 1 1 1 4 2 2 0 0 1 1 1 1 1 1 5 3 1 1 0 0 1 1 1 1 6 3 1 0 1 0 1 1 7 3 1 0 0 1 1 1 1 1 +These are the spin quartets obtained from the spin sectets by allowing its total spin to add up to 3/2 instead of 5/2 *The MO's ta —— tb are linear combinations of atomic orbitals as will be seen later (Sec. IV). spin sextet of E symmetry from configuration 4 and the spin sextet of T symmetry from configuration 3 contain the four different orbitals, t ~6 6 ~a ea. t and eb, which participate in the construction of E and T1. Therefore, the final matrix elements depend on the coupling schemeeof the above four orbitals in E and T1. The behavior of the sublevels, S ihi, arising from t dtisp 4-p Sihi, arising from taP, ea and other orbitals tl, eb and tb is unique for spin sextets, but varies for quartets and doublets which in turn gives rise to several hundred spin quartets and doublets. The best technique for the determination of matrix elements of any operator between a huge number of states with the same spin S and IR, h, but with different configurations is the method of Reduced Matrix Elements. Griffith26 has applied this technique to calculate the matrix elements of the spin orbit interaction between various, Sihi of the cubic group. Our analysis follows his very closely and gives rise to new formulae for determination of the spin-orbit matrix elements between

28 pairs of the spin sextuplets arising from coupling of three or four orbits.* As in Griffith (p. 82), the matrix elements of spin orbit interaction, i Ci - *s, from a pair of states IX S h J t T > and IX'S'h'J't T > can be given as: (X S h J t TI i i S IXS'hJlt T)= Z (X S h J t TIX S h M 8> M M' 8 8' (x)(X S h M eI Ei isxhMle'>Kx'Slh'MxehIxS'h'J'h t T) (3.12) E <x S hIi i Qm sllx'S'h'> K, T1 (3.13) where S and h are the spin and irreducible representation (IR) of the state IXSh>; M and 8 are, respectively the components of S and h, t is an IR of the system in the cubic double group belonging to the resultant of the coupling of S and h; J is an identification number used wherever there are more than one t are, finally T is one of the components of t. ii The first term in (3.13) is the reduced matrix of Z. s from states IX S h > and Ix'S'h' > and the second one** is the coupling coefficient which is independent of X and X'. The study of the coupling coefficient will be reserved for Section 5. The reduced matrix elements will be elaborated further in the next subsection and new results, not found in the literature will be tabulated. Griffith26 has given all the formulae needed, for evaluation of the reduced matrix elements of spin orbit interaction, arising from two orbits t2 and e of cubic group. As a result of this, his book contains tables for spin quartets only (see Ref. 26 p. 126) ** KT-jt is exactly the same as the QJJ' defined by Griffith (p. 82)

29 4.1 Reduced Matrix Elements Here, the reduced matrix elements (RME), X S h i * s I x'S'h> (3.14) i of (3.13), will be discussed further with particular attention to the effect of X and X' on RME. There are three classes of RME depending on the nature of configurations X and X': (1) Both X and X' include three orbitals with the same configurations. (2) Both X and X' include three orbitals with different configurations. (3) X and X' include four orbitals with different configurations. The formulae for obtaining the reduced matrix elements, (RME), associated with these three classes of configuration are given in Sections 4, 5 and 6 of the Appendix C respectively. The numerical results are given here in Tables 3.3 through 3.5. Table 3.3. Reduced matrix elements i<XlShIl jZjii_'s'j ix t h>I2 I<nlshll~iCiiiaigsii I~iS'h'>I ni s |XlS'h ] = i 66T16T 4 6 6E T 4E 4 6 T4 4 4 4 6 4 4E -1 2 1 2 1 1 1 2 1 2 1 1 aa t3 4A2 (eatb) 3T2 tbtb 21/20 6/5* --- --- 6/5 3/10* --- --- a 2 2 b t 2 3T1 ea tb) T tbtb 7/20* 2/5 7/30* 8/30 2/5* 1/10 8/30* 2/30 t t 7/20* 1/40* 7/30* 1/60* 1/40 9/40* 1/60 3/20* ta 3T (e 2eb) E t t 21/20 3/40 --- --- 3/40* 27/40 --- --- ta3 4A2 (eat1) 3T2 tt1 21/20* 6/5 6/5* 3/10 ta (e2a tl T2 tata 7/20 1/40 7 1/40* 9/40 1/20* 9/20 ltit1 7/20 2/5* 7/10* 4/5 2/5 1/10* 4/5* 1/5 The sign of the square root of the numbers with asterisk is negative a. = <1/2a.1I ~ sl Il/2a.>

Table 3.4. Reduced matrix elements I(X2Shlml l s I il2S 1'h'2 < 1/2 b{t i si 1/2 c > Sh- S'h' X2 - X2 | bc 6 A-6T1 6A lT1 34 23 6 34 3 La A 2e A2 A1- ta A2 (eatb) T2 eat 1 7/* 9/40 3 4 2 3A 6 2 3 2 4 F A2e A2: A- ta T1 (eaTbO T1 tatb 7/5* 2/45 A2 ea A2: A- a 1 (eeb) E taeb 7/5 2/45* 34 23 6 34 3e/ 9/40 ta A2 ea 3A2 A - t A2 (e tl) T2 eat 7/5* | 9/40 a3 4 2 3a623 2 4a ta3 4A2 ea2 3A2:6Al - a 3T1 (ea t1) T2 tatl 7/5* 2/45 | 6 6T 6 6T 6 6T1 4 4T 6 4 4 4 64E 41-6T2 4 E T-T T-T T-E T-E T-T T-T4- E T11-4E 1 2 1 2 1 1 1 2 2 2 1 1 ta3 A2 (eatb) 3T2 - ta 1 a a a 7/20 1/40 14/15* 1/15* 1/40* 9/40 1/15 3/5* ta2 3T1 (ea2b) 41 _ 34 3 | a lT (eab t) 4T- ta A2 (eat) 3T2 etaa 21/20 2/40 - -- 3/40* 27/40 - - Lt T(e eb 4E - t A2 (e342e ) E eta -- 7/10 1/20* - - - 1/20 9/20 aT1 (ea a 42 eab aa 34 3 23 2 4 ta 2 (eatl) 3T2 - ta T1 (ea tl 2 taea 21/20* 3/40* - - - 3/40 27/40* - - ta 3T1 (ea 2tl) 4T2 t A2 (eat1) T2 b 1/20* 1/40* - -- 1/40 9/40 a 1 a h 2q a 2r a T2 a *The sign of the square root of the numbers with asterisk is negative

Table 3.5. Reduced matrix elements I(x3Shl I i i-'- -IIX 3S h 2 <1/2 cllC' Isj z1/2 d> Sh - S'h' 6 6 6 4 6 6 6 4 4 6 4 4 4 6 4 4 cd T-T T-T T-E T-E T-T T-T T-E T-E X~1~~ 3 3 1 2 1 2 1 1- 1 2 1 2 1 14 5- 2 (t 3 4 A e ) 5E (tb 0 ) 2 (t 3 4A e) 5E (t t 0) 2T t tl 7/20*t 2/5 2/5*r 1/10 a 2 a 1 2 a 2 a b I 1 b 1' (ta 3 4A2ea) 5E (teb%0) T2 (ta3 4A2e) 5E t0 2E tb0eb - - 7/10* 4/5 - - 4/5* 2/10 (ta2 3 T A 52 30) T2 (tb0 2T2 (t3 4A2ea) 5T2 (tc0eb) 2E teb 21/20 6/5* - - 6/5 3/10* - - (t e 23 52 0t% 2 34 / 311 a2 3 2 3 5 ) 22 (a3 4A2ea) 5T2 (0t) 2T tbt 7/20* 2/5 7/10* 4/5 2/5* 3/20 4/5* 2/10 a la A2 2 (tbt T2 (ta A2ea) T2 1 b 1 (t2 3T1e2 32) 5T2 ( 02E (ta34A2ea) 5T2 (ebOt) 2T2 eb0 ) 7/20* 2/5 14/15 16/15* 2/5* 3/10 16/15 4/15* 2 3 2 3 5 0 2 3 4 5 0eb ta23Tea A2) 23T5 ( 0)2 42ea) T2 (e.b t1) 5 2T1 ebtl 21/20 6/5* - - 6/5 3/10* - - (ta T3 ea A2) T ( T1 T2 %t1) (t34A2e2E) 5E (t 0) 2T 34A2e) 5E (t0) 2T 2 tltb 7/20 2/5* - - 2/5 1/10* - - a 2 a l b 1 (ta 2 a E 1 ) T2 (t3 48 2E 5 2 34 5 0 )2E (t3 4Ae 2E) 5E (tleb) 2T (t A2e) E (tie) E t - - 7/10* 4/5 - - 4/5* 2/10 a 2 a b I a 2 a lb 2 3 2 3 5 0 2 34 5 0 2 (ta Tlea A2) T2 (tl0) T1 (t3 4A2e) 5T2 (t1 tb ) T2 tltb 7/20* 2/5 7/30* 4/15 2/5* 3/20 4/15* 2/30 23 23 5 0 2 31 4 5 b0 ) 2 (ta Tiea A2 T1 (t1ea T1 A2ea) T2 (t1O) t1e 7/20 2/5* - - 2/5 1/10* - The sign of the square root of the numbers with asterisk is negative

5. COUPLING COEFFICIENTS OF SPIN ORBIT INTERACTION K SS'T ht Here, we want to obtain the coupling coefficients KJJ,, which were defined in (3.13). These coefficients couple the matrix elements of spin orbit interaction, from a pair of states identified by their irreducible representations IJ t > and IJ't > in the cubic double group, to the reduced matrix elements (X S hill Ci' s i iIX'S'h'> between the states IX S h >, and IX'S'h' > from which the states IJ t > and IJ't > are constructed. Following (3.12 - 3.13), we have SS'T1 (X S h J t TlI (i s XiS'h'J't T) = X hi C ti 9ii'Shht where (Griffith, p.82): K - SS' T1 Q, SS' T1 Kjj K h t JJ' h' h t (-1) +1[-1] V _s v )(h-'Te' (x) <S h ( t iT S h M i r t'h' d't'oS'h'J'S s) (3 15) (_1)a, and Vbb ci 2(b-c) The symbol, V la by in (3.15) is related to 3-j symbols by +b, and V b is related to V by (-1).2(b-c) The symbols, (a ~ i < S h J t tIS h M 0 > are coefficients of coupling S and h to obtain t of the cubic double group with occurrence number or angular momentum J. The latter coefficients are given by Griffith (Ref. 16, pp. 400-408) for 5 5 spin quartets, T2 and E only. Therefore, the coefficients U. Fano and G. Racah, "e->ducible Tensorial Sets," Academic Press, New York, (1959) p. 50

33 <S h M S h J t T> of Sh = 6E, 6T T <S h M S h J t T and, which are not found in the literature, are obtained and given in Tables 3.6 to 3.8. Having obtained the coefficients, <S h M eIS h J t T>, we now are able to calculate the coupling coefficients, KJJ, hh' t 1 for h = A1 and h' = E. T1 and T2. These are given in Tables 3.9 to 3.12. After substituting for reduced macrix elements and the coupling coefficients in matrix elements, (X S h J t TMI i-i!i I S xSthIJt ), in (3.13), we find this quantity as a function o< single electron reduced matrix elements such as tbtb9 t ata,... tltal e atb given in Tables 3.3 through 3.5. These matrix elements will be determined in the next section. Table 3.6. Transformation of 6E into the IR's of double valued group T'd: <ShM I ShJtT>) 2 E' E' 1U' 2U' S M l1-e a 6 a'.. K X " V K X V 5/2 5/2 E u 5/12* /12 1/6* 3/2 1/12* 1/12* S/6*5 1/2 1/2 _1/2* -1/2 1/2 1/2 -3/2 1/12* 1/12 5/6 -5/2 5/12* 5/12* 1/6 5/2 5/2 E v 5/12 5/12 1/6 3/2 1/12 1/12* 5/6 1/2 1/2 1/2 -1/2 1/2 1/2* -3/2 1112 1/12 5/6* -5/2 5/12 5/12* 1/6* The sign of the square root of coefficients with asterisk is negative

Table 3.7. Transformation coefficients of 6Ti into the IR's of double valued group T'd;..... __ 5/2TIMOf5/2TJte>j2 3/2 5/2 _ 7/2 1 \ t _ _ U' E' U' E''E U' M K O K v a | 6 k | i v | a' 8'' | X v' 8'' | A' v'' 5/2 1 5/12* 7/12 0 5/42 25/42 3/14 1/14 -1 2/3 5/21* 1/21*...... 1/84 1/28 3/2 1 1/21* _ 5/21 15/28 5/28 0 4/15* 3/14* 3/70* 5/42 5/14 -1 2/5 16/35 1/12 5/84* 1/2 1 1/15 8/21 8/105 5/42 5/14 0 2/5* 1/35 1/3 1 -5/21 i -1 1/5 18/35* 1/6* 5/42* -1/2 1 115 18/35* 1/6 5/42* 0 2/5* 1/35 1/3* 5/21* -1 1/15 8/21*... 8/105 5/42* 5/14 -3/2 1 2/5 16/35 1/12* 5/84* 0 4/15* 3/14 3/70* 5/42* 5/14 -1 5/21* 15/28* 5/28 -5/2 1 2/3 5/21 1/21* 1/84* 1/28 0 5/42* 25/42 3/14* 1114 -1 " 5/12 7/12 The sign of the square root of number with asterisk is negative

Table 3.8. Transformation coefficients of 6T2 into the IR's of double valued group Td 1(5/2T2MH j 5/2T2JtT > 2 3/2 5/2 7/2 2 U' E' U' E" E' U' M | X 1 v a' v'' B'' t' B' K V 5/2 1 5/12 7/12* 0 5/42 25/42 3/14 1/14 -1 2/3 5/21* 1/21* 1/84 1/28 3/2 1 1/21* 5/21* 15/28 5/28 0 4/15* 3/14* 3/70* 5/42 5/14 -1 2/5* 16/35* 1/12 5/84 1/2 1 1/15 8/21 8/105..5/42 5/14 0 I 2/5 1/35* 1/3 5/21 -1 1/5* 18/35 1/6* 5/42 -1/2 1 1/5* 18/35 1/6 5/42 o 12/5 1/35* 1/3* 5/21 -1 1/15 8/21* 8/105 "_ 5/42* 5/14 -3/2 1 2/5* 16/35* 1/12* 5/84 0 4/15* 3/14 3/70* 5/42* 5/14 -1 1/21 5/21* 15/28* 5/28 -5/2 1 2/3 5/21 1/21* 1/84* 1/28 0 5/42* 25/42 3/14* 1/14 -1 5/12 7/12* The sign of the square root of the numbers with asterisk is negative

Table 3.9. Coupling coefficients Table 3.10. Coupling coefficients KJJ'for h = A1 KJJt for h = T1 J J' S S t: Kjj3(SST1,A1T1t) J' S S S t KI'J(SST1 1TTlt /z 5/z 5/z 5/2 u'2 5/2 5/2 5/2 5/2 U'(1/18) 5/2 5/2 5 (1/15) (5/7) (1/18)1/2 Ell (1/15) (5/7)1/2 3/2 U' -(7/45) (5/7)1/2 E'' -(5/45) (5/7)1/2 3/2 3/2 U' (1/2) (1/10)1/2 E' -(1/2) (1/10) Table 3.11. Coupling coefficients Kjj, for h = E 3 3' S S t KI'JJ(SST1 T1Et) 1 3' S S t KJ,J(SST1 T1T2t) 5/2 1 5/2 5/2 U' (16/105) 5/2 1 3/2 5/2 U' -(2/15) 2 5/2 5/2 U' -(2/105) (10)1/2 2 3/2 5/2 U' (1/60) (10)1/2 5/2 5/2 E'' -(4/105) (5)1/2 5/2 2 3/2 5/2 E"l (1/30) (5)'1/2 3/2 U' (1/10) (3/7)1/2 2 3/2 3/2 U' -(1/10)(3/2)1/2 5/2 3/2 El (1/2) (3/35)1/2 3/2 3/2 E"' -(1/2)(3/10)1/ Table 3.12. Coupling coefficients Kjj, for h = T2 J T' S S t KJJ (SST1 TT2t) 3 3' S S t Kj (SST1 T Tt) 5/2 3/2 5/2 5/2 U',-(8/35) (1/30)1/2 5/2 7/2 5/2 3/2 (1/2) (1/105)1/2 5/2 5/2 5/2 U' -(35/35) (1/105)1/2 5/2 3/2 3/2 5/2 t' (1/5) (1/30)1/2 7/2 5/2 5/2 Ut (8/21) (1/35)1/2 5/2 5/2 3/2 5/2 U' (3/10) (3/35)1/2 5/2 7/2 3/2 5/2 U' -(1/3) (1/35)1/2 5/2 5/2 5/2 (8/21) (1/5)1/2 5/2 7/2 3/2 5/2 E''" -(1/3) (1/5)1/2 5/2 3/2 5/2 3/2 U' -(4/5) (1/105)1/2 5/2 3/2 3/2 3/2 U' (4/5) (1/30)1/2 5/2 3/2 3/2 U' (3/10) (3/10)1/2 5/2 5/2 5/2 3/2 U' -(9/10) (1/105)1/2 7/2 3/2 3/2 E" -(1/2) (1/30)1/2

37 6. SINGLE ORBITAL REDUCED MATRIX ELEMENTS (1/2 allj k s 1/2 b) The matrix elements of Z r. Qisi from a pair of states with electronic configurations X and X', (3.8), are related, among other factors, to these configurations. This.dependence on the electronic configuration of the orbitals giving rise to the above states, is manifested by the presence of single orbital reduced matrix elements of the type, ta ta = 1/2 t a IC Rs 111/2 taand taeb = (1/2 taIC P's 111/2 eb) which appear in Tables 3.3 through 3.5. To find t ta...., we should express them in terms of atomic symmetry orbitals. Since the determination of the energies of the molecular orbitals ta, ea of Fig. 3.1 is beyond the scope of the present work, no numerical values of the coefficients of linear combinations of atomic symmetry orbitals, d(e, t2) and p (tl, e, t2), (Fig. 3.1), are available. Therefore, we choose a set of arbitrary coefficients, a, B, K, X, V and v, to express the molecular orbitals, ta, ea, tl1 eb and tb, as follows: t =K Id t2)- j t2 - iioa t2 - -vs t2) ea d e) - 7re) t1 = I tl> tb = X'd t2)+ K'Tr t2)+?'la t2)+ vs t2) eb = Bld e)+ acr e) (3.16) where all coefficients, a, B,.. v', are real and positive.

38 Moreover, 2 + 2 2 + +2 2 =2 + X2 + 12 +,2, 1 (3.17) Substituting for ta....tb we find the single orbital reduced matrix elements as given in Table 3.13: Table 3.13. Single orbital reduced matrix elements No. a, b No. a, b <1/2a Ic'.18|1/2b> 1 eatl V B;p 7 tita 1/2 A C; p,,, _..'..p _._ 2 eata -3. a K Cd 3 B X Cp 8 tltb -1/2 K C p 3 | eatb -3'F a X Cd + 3 B K p 9 tatb -3 X K A Cd - 3/2 K C p 4 |-t1 - a| P| 10 tIt1 3/2 Cp p p2 5 ebta -3 T2 B K Cd + 3 a; Cp 11 t ta 3 K2 Cd + 3/2 K2 Cp 6 ebtb -3 8 0 Ad * 3 a K Cp 12 tbtb 3;2 Cd + 3/2 K2 Cp With the spin orbit matrix elements known, we can now proceed to formulate 3a in the following section.

CHAPTER IV CUBIC FIELD SPLITTING 3a In this section, we wish to obtain the contribution of spin orbit interaction to cubic field splitting 3a, with the intention of carrying the calculations through the lowest order of perturbation required for the ground state, 6A1, to split. Our task, therefore, is to establish the lowest perturbation order first, and then, carry on the numerical computations to obtain an estimate of 3a (under certain assumptions regarding the coefficients) for some special cases. 1. DETERMINATION OF THE LOWEST ORDER OF PERTURBATION BY H =.,i Ji'si THAT CAN SPLIT 6A P i 1 — 1 The first step in determining the lowest order perturbation required for the splitting of 6Al is to find those coupling coefficients, KJJ' (SS'Tl), which have different values for t = U' and t = E" levels h' ht of 6A1. Because the energy associated with level I6A1 J = 5/2 t = U' > must differ from that energy associated with the level, 16A1, J = 5/2 t = E" >, in order for the matrix element of Z Si i,'si to contribuite toward splitting 3a.. As shown in Appendix D, we have Kj SS'Tl 1 6 6 6 (4.1) JJ' hA1 t (H) (25 + 1) Sj S'J' h'T and K j SS' T1 = (-1) 6J+ W ( J (4.2) 39JJ' T T t JJ SS' 39

40 It is immediately evident from (4.1)-(4.2), that the matrix elements between 16A1) and 16T> and those between 16T1) and 16T'1) contribute the same amount to both levels t = U'and t = E'', and their contribution to 3a vanishes. The chains of the products of the matrix elements:* 6Al - 6T1 (T - A (4.3) and KAl 1 T' > T < T' - l 6A> (4.4) are the only nonvanishing products which give rise to terms for the evaluation of the second and third order energy contribution to levels with IR t = U' and t = E'' of the ground state. These energy contributions are the same, and consequently, both second and third order contributions to 3a, by spin-orbit interaction, vanish. Moreover, con- e tributions from higher than third order perturbation with excited states having T1 symmetry vanish too. The next perturbation order to consider is fourth order. Considering Tables 3.11 and 3.12, it is evident that for the excited states with T2 and E symmetries, the coupling coefficients, KJJ' Sh'T differ for t = U and t = E''. Therefore, a splitting will occur. This indicates that the fourth order perturbation is the lowest one which contributes to the splitting. <\A1-6T>/-<6A1 JtT< i i Q''s'6T Jt' >

L1 2. CONTRIBUTION TO THE CUBIC FIELD SPLITTING 3a FROM FOURTH ORDER PERTURBATION OF 6A1 BY ~ (i;i.s! i 1 — - Here, we formulate the 3a by considering the following relationships (3.1): 3a = E(U') - E(E") = E(4) (U') - E(4) (E'') (4.5) where E(4)(U') and E(4)(E'') are the fourth order contribution from spin orbit interaction to the levels U' and E'' of the ground state 6A1 of the S-state ion. The expression for E(4)(U') - E(4)(E'') is: E (4)(U) - E(4) (E'') - (EjEkE) RjkRk Ro jk2 mnp [ Kom(j)K mn(k)Kp (k t)K(po0(Ro -o(oJ) Kmn(k)K (kt)K (to)) where Rjk (xjSjhj Hp I1 XkSkhk) and K (jk) K T (4.5a) mn n hkhj~ The parameters Xj....*, X in (4.5a) represent the molecular electronic configuration and IxjSjhj> characterize the orbital part of configuration Xj.

42 Substituting for the various parameters involved, the expression for 3a will be of the form: 4 3a(4). C d 4-i i (4.6) 3a c (4.6) i=o where Ci are complicated functions of the coefficients; a, 0, K.... given in (3.16) and the promotion energies Ei, Ej and Ek of the excited states appearing in (4.6). The precise numerical values of A, S and K could be obtained from solving eigenvalue equations from which the energies; E*...Ej and Ek could be found too. As mentioned earlier, the determination of Ei...Ek is beyond the scope of the present work and as a result of this, we can use only a set of arbitrarily chosen numbers for both the coefficients and the energies involved. For the following set of coefficients:* 2 a 1 _ =2 0.7, K2 = 0.8 - A2 0.6, 2 2 2 + 0.2, and K X Kt A ", At; we find the coefficients C4 of (4.6) as given in Table 4.1. Table 4.1 The Coefficients Ci (-2/5625 E63)-1 C0 C1 C2 3 C4 87.17 -138.99 -164.94 1463 35.75 (See 3.16)

43 Substituting for Ci in (4.6), one finds 3a as a function of the ratio Cp/id. The result in units of 10-1d /E6 are given in Table 4.2. Table 4.2 Calculated Values of 3a for Spin Sextets -1 4 3 - 4/ 3 |p/d 3a (10 d /E6 ) |Cp/Cd 3a (10 d /E6 0.5 0.07 6 -152.30 1 0.59 8 -507.40 1.5 0.993 10 -126.30 2 0.573 12 -2645 3 -5.30 16 -8443 4 -24.90 20 -20562 A discussion of these results will be given in the next section.

CHAPTER V DISCUSSION We want to give a brief discussion of various models used to calculate 3a, first, and then apply the result of these models to the 3a of Fe3+ in the compounds ZnS, ZnSe, and ZnTe. As'was mentioned in Chapter I, Watanabe5 was the first to calculate 3a on the basis of the point charge model. His work was followed by Powell6 and by Low and Rosengarten.7,8 Azarbayejani, Kikuchi and Watanabe15 substituted the point charge model with the molecular orbital model and obtained the contributions to ground state splitting arising from charge transfer between a-bonding and a-nonbonding orbitals of the complex consisting of a central S-state ion and its four tetrahedrally coordinated neighbors. In the present work, the contribution to 3a arising from charge transfer between the fr-orbitals of the same complex has been found. To make an assessment of these various contributions to 3a and their relative importance, we are considering all of the above-mentioned calculations, in turn, as follows. 1. CONTRIBUTION TO 3a FROM WATANABE'S CALCULATION The cubic field splitting obtained by Watanabe is given in (15) of Ref. 5. The expression for 3a is as follows: (3a)w =3 (Dq)2 [2.015 + 15.9 M - 149.5 M2 - 5.937 (M - 8M2)2 - 0.388 (M 8M2)2 (Dq)2 10 ] x 10 1 cm- (5.1) where 10 Dq* is the cubic field strength of the host compound around the S-state ion and is about 3000 to 4000 cm for Mn and 5000 to 6000 cm S-state ion and is about 3000 to 4000 cm for Mn for Fe3+ See the first footnote on the following page. 4..

45 in II-VI compounds of Td symmetry. The coefficients M0 and M2 are (see Ref. 6, Part a) 0.204 and 0.0159 cm, respectively. Substituting for M0 and M2, one obtains: (3a) = 3 (Dq)2 [2.015 + 15.94 x 0.204 - 149.5 (0.0159) - 5.037 (0.024 2 2 2 -6 -10 -1 - 0.127) - 0.0388 (0.077) (Dq)2 x 10 6] x 10 cm = 3 (1jq)2 [2.015 + 0.87 - 0.029 - 2.25 x 10-9 (Dq) ] x 10-10 cm or (3a) 87 x 10-10 (Dq)2 _ 76.5 x 10-20 (Dq)4 cm1 (5.2) (3a) = - 7 x1 (Dq) cm (5.2) w 2. CONTRIBUTION TO 3a FROM POWELL'S CALCULATIONS The ground state splitting given by Powell et al (Ref. 6, part b) can be expressed as: (3a)p = Kp 4d (Dq)n; 3.5 n (6; Dq) 103 cm1 (5.3) The equation (5.3) was obtained by limiting their calculation to MgO:Mn where MgO is an octahedral II-VI compound for which Dq is large (Dq) 10 3 cm ). For the II-VI compounds of Td symmetry, Powell et al (Ref. 6a) give some numerical values of 3a as a function of (Dq) as given in Table 5.1. Ref. 16, Table 11.3 p. 310 gives 10 Dq [Mn (H20)6]2+ and [Fe (H20)6]3+ as 8300 and 14700, respectively. Pappalardo and Dietz (Phys. Rev 123 1188 (1961) have concluded Dq (CdS):Ni) =-0.85 x 4/9 Dq[Ni (H20)61. Thus, in an analogous way, 10 Dq [CdS:Mn2+] - -3100.

46 Table 5.1. Calculated+ 3a in Mn2+ in units of 10-4 cm-1 Dq (cm-l) 0 -200 -400 -600 -800 -1000 with doublets 0 3.56 11.0 23.8 45.1 81.4 (3a) | w0 0.115 0.338 0.668 1.09 1.59 without doublets (3a)w Eq. (5.2) 0 0.35 1.4 3.15 5.6 8.75 TThese values are obtained for the spin-orbit constant, = 400 cm1 and spin-spin interaction constants, MO and M2 as 0.284 cm-1 and 0.0159 cm'-, respectively. The first row of Table 5.1 gives 3a arising from all spin multiplets within the 3d5 manifold, whereas the second row is obtained without taking the spin doublets of the 3d5 manifold into account. Watanabe's calculations are based on spin quartets alone and are given in the third row of Table 5.1. The numerical values of the first row of Table 5.1 give the total contributions from excited states generated within the 3d5 manifold. Now, we consider the calculation by Low and Rosengarten. 3. CONTRIBUTION TO 3a FROM LOW AND ROSENGARTEN CALCULATIONS The cubic field splitting given by Low and Rosengarten, (3a)LR, was obtained from the same spin quartets and doublets of (3d) manifold considered by Powell et al. However, the techniques used by the former authors differ from those of the latter. Low et al diagonalized the energy matrices of E', E'" and U' levels which contain five parameters; B, C, Dq, Cd and a.* Powell el at, on the other hand, diagonalized the B, C are Racah coefficients, Dq is the cubic crystal field strength, a is Tree's correction factor and M0 and M2 are spin-spin interaction parameters.

47 energy matrices of Al, A2, E, T1 and T2 levels as functions of six parameters B, C, Dq, (d, M0 and M2, first. Then, they obtained the energies of levels E" and U' from the energy values of the above levels, A1....T2 by sixth order perturbation. The numerical values obtained by Low et al are given in Table 5.2. Considering Table 5.2, it is evident that (3a)LR and (3a)p are of the same order of magnitude, whereas (3a)W (Table 5.1), calculated by Watanabe, is much less than these two. This is expected because both (3a)LR and (3a)p have been found by taking into account all spin multiplets of (3d) configuration, whereas (3a)W is obtained from spin quartets of (3d) only. As for (3a)LR and (3a)p the latter gives 3a as a function of Dq and Cd. Therefore, it is more suitable for the calculation Table 5.2.* Comparison of (3a)LR with (3a)p and (3a)Exp. 2+ 3+ Mn Fe MnF2 MnC12 Mn(H20)6 MgO:Fe Be3A12(Si03)5:Fe Fe(H20)6 (3a) LR 4 -1 10 bO 10 cm Dq(cm-1 ) 750 1350 Cd(cm 1) 320 420 (3a)p (10 4cm- 11 325 (3a) (10 cm ) 12 6 20-30 615 450 350 Exp. (3a)LR is the 3a calculated by Low and Rosengarten, (3a)p is the 3a calculated by Powell and (3a)Exp is the experimentally determined value of 3a. +(3a)p are obtained from the relationship;(3a)p1 = Kp ~d4 (Dq)4 and from the numerical values of (3a)p at Dq = 1000 cm- and Cd = 376 cm1.

of the 3a of a certain ion in compounds of different Dq. Thus, we choose (3a) as the contribution to 3a from the excited states within 5 P the (3d) configuration and, discuss the charge transfer contribution in the next section. 4. CONTRIBUTION TO 3a FROM o-BONDING CHARGE TRANSFER STATES The contribution to 3a from the a-bonding charge transfer states was obtained previously.15 Here, it will be reviewed briefly in order to make a comparison between this and the contribution of the r-bonding transfer states given in the next section. The irreducible representations of the metal d orbital and ligand a-orbitals in II-VI compounds of Td symmetry are: h (d) = h (k = 2) = e + t2 (5.4) and h (a) = al + t2 (5.5) Considering (5.4)-(5.5), it is evident that the molecular orbitals consist of a d orbital of e symmetry, a a-orbital of a1 symmetry and a pair of orbitals comprised of metal d-orbital and ligand a-orbital of t2 symmetry. In the last two orbitals, the orbital with the higher energy is the antibonding, denoted by t2a, whereas the one with the lower b energy is called bonding and is denoted by t2. Thus, the molecular orbitals of interest to us, are (al is ignored): e>- = de), It2>)= T Idt2) and It2b) Idt 2) aTat2) (5.6) Ballhausen, "Introduction to Ligand Field Theory," McGaw-Hill Book Company, New York (1962), p. 53 [Eq. (3.34)], p. 171.

49 The energy diagram for such bonding is given in Fig. 5.1. The electronic configuration characteristic of the ground state, 6Ai, of a tetrahedral complex of 3d5 6S ion and its bonding-nonbonding and bonding-antibonding states are given in Fig. 5.2. The T1 and 6' 1 b T2 in Fig. 5.2(b) result from an electron transfer from the t2 orbital to e, whereas the levels; E, T and T result from the above procb 2 ess taking place between the t2b and t2a orbitals as shown in Fig. 5.2(c). t2 / \ I \ d 2) / 4/ (t2) 2''I \ A/ b / Fig. 5.1. a-bonding molecular orbitals in II-VI compounds of Td symmetry. tatw12 t #9 L 2 t2 6 6T; 6 n 1. T a2 T T a 6A A (a) (b) (c) Fig. 5.2. (a) Ground state of complex [EA ]-n', (b) t b-en charge transfer states and their schematic energy le4els, and (c, t2b-t2a charge transfer states and their schematic energy levels.

50 The contribution of these a-bonding levels to 3a depends on their stability for a given S-state ion in a given compound. In the case of 6Ea 6Ta 6a compounds where levels E, T1 and T2a may not be localized because 6 n 6 n of the small energy band gap of compound, only 6T and T2 can be taken into account. For the general case where anitbonding levels are also localized, the simultaneous effort of both antibonding and bonding levels on 3a must be considered. The contribution, 3a(a), to the cubic field splitting 3a, from the above a-bonding orbitals can be expressed as: 3a(a) = E(4) (U') - E(4) (E'') (5.7) We first obtain the 3a(a) for 6Tn and 6Tn alone. Then, we include the states Ea Ta and T2a 1 2 4.1 Bonding-Nonbonding Charge Transfer The contribution to 3a(o) from T and T will be identified by 3a(a;b-n). This can be obtained both from (4.11) or from the different techniques described in Appendix F. The result is:15 3a(a;b-n) = 0.1728 BT 4 (1 - EnT/6 T) (6nT)- 3 (5.8) T d~n n where T2 = 1- aT2 is the covalency of the d-orbitals of the S-state T T ion in the desired complex. Cd is the single electron spin orbit parameter and is the same as A in Ref. 15. 6 and e are as shown in n n Fig. 5.1. 4.2 Bonding-Nonbonding and Bonding-Antibonding Charge Transfer 114itng the same techniques as those employed for the bondingnonbonding process, one finds the contributions to 3a(a) arising from * For definition of U' and E'' see Table 3.1

51 6 n 6 n6a C6 a a T T n 6Ea 6 T2 1 and T2 as shown in Appendix F. The result is a function of the coefficients of atomic orbitals aT and T ( 2 T T T = 1- T ) in the molecular orbitals used, and the energies; 6 nT nT T n n 6T, CTT and C as shown in Fig. 5.1. For a particular case where a al a2 T T T E = C = O (5.9) n al a2 * and with the assumption that T T 6 r & (5.10) a n one finds that: 3a(a) = (108:625) (6 T)-3 (x) n (x) [T - 2 (1- BT4) r- + (1 2T)(3 - 5 BT ) r-2] T2 d (5 11) 6 a The parameter r and its power denote the presence of T1, 6Ta or 6Ea in the matrix elements from which 3a(a) is obtained. Thus, 2 the first term in the bracket in (5.11) represents contributions arising exclusively from 6T n and T2, whereas the last two terms give the contribution arising from the presence of both 6T n and 6T2n a 6a 6a 1 2' n T1 and T2a. An examination of (5.11) reveals that only for r + 0 or 6 T +a one obtains 3a(a)) 0. Numerical values of 3a(a) as functions a 2 2 of both 3 = 1 - aT and r can be found from the following relationships: T T 3a(c, T = 0.2) = (18/625) (6 )] [0.048 - 2.3r1 + 1.92 r 2] 4 (5.12) * T T r is a real number chosen as the ratio of the two energies 6 and 6. a n

3a(a, T2 = 0.3) [(18/625) (6 )3 [0.162 - 3.3 r + 1.89 ] 4 (5.13) 3a(o, BT2 = 0.4) = 18/625) (6T) 3[0.384 - 4.03 r 1 + 1.44 r 2 4 (5.14) The numerical values obtained from (5.12)-'(5.14) are given in Table 5.3. An examination of Table 5.3 indicates that a positive contribution to 3a(o) takes place only in very covalent compounds (T2= 0.4) and for 6 T:6 = 12. The latter condition is unrealistic because for T a n T 6 in the order of 1-2 e.v., 6 must be 12-24 e.v. which makes Ea n a Ta and T2a levels unstable. Thus, one can conclude that: 1 2 (1) 3a(a) is positive if only bonding-nonbonding states are localized (6aT/6nT +. (2) 3a(a) is negative when both bonding-nonbonding and bondingT T antibonding states are localized, and r = 6a:6n is 1-10. (3) 3a(a) depends only on Cd as shown in (5.12)-(5.13). Since 3a(7) depends on both Cd and Cp, it is desirable to elaborate further on the absence of C in 3a(a). The fact that 3a(a) does not p depend on rp is intuitively clear since o orbitals arise from atomic s and p orbitals, and since the matrix elements of spin orbit Table 5.3. Numerical values of 3a(a) in (18/625) (6nT)-3 Id4 r 1.2 1.6 2 5 10 12 14 16 1R 20 T 0.2 -0.54 -0.6 -0.60 -0.34 -0.17 -0.134 -0.106 -0.088 -0.072 -0.062 0.3 -1.25 -1.14 -0.94 -0.42 -0.15 -0.100 -0.064 -0.037 -0.015 0.002 0.4 -1.85 -1.55 -1.26 -0.37 -0.03 0.058 0.103 0.138 0.165 0.187 See Section 5.

53 interaction between such pairs of atomic orbtials, automatically vanish. To put this in a more rigorous language, we will consider the part of the matrix elements of H = isi between a pair of states of antibonding orbitals ta = aTIdt25 T- TIat2. The matrix elements arising exclusively from the ligand a orbitals have the general form of: Maa ( m, m ) /2 m'n') =t2 mt2 i ls/2 m' at2 n (5.15) where (Ref. 23, p. 108): t2 ) at2 yz = (1/2)(a +a3 -2 - a4) (5.16) lot2 n)= t2 zx) = (1/2)(a1 + a2 - 3 -a4) (5.17) and ak = a (k) + b pz(k), a + b 1, k = 1,.., 4 (5.18) Substituting in (5.15), we find that: Ma (ao m, m'n') =(1/4)[R1 - R2 - R3 + R4] where = a2 (112 m s(k) I sl 1/2 m' s(k)) + b2 (1/2 m p (k) I s 1l/2 m' P (k) resulting in M (a, ma, m'n) =(1/4)[R - R - R + R] = 0 + and n are the components of T2 irreducible representation behaving as yz and zx.

54 Thus, the off-diagonal elements, M a (a, mE, m'r'), vanish. For diagonal * aa elements, we have Maa (a, m=, m 1) =(1/4)[R1 + R2 + R3 + R4] where R = b2 (1/2 m p zlc _.sl/2 m' p) =(1/2)b2 p (Pzllp z 6 = (1/2)b2 rp\PltllIP) V 10001 6, 0 (5.19) Thus, we conclude that: (1) charge transfer from bonding to nonbonding a-orbitals gives a positive contribution to 3a, (2) simultaneous bonding-nonbonding and bonding-antibonding charge transfer give a negative contribution to 3a for 6 T: T varying from 1 to 10, and (3) a n these contributions do not depend on p, the ligand spin orbit interaction. 5. CONTRIBUTION TO 3a FROM x-BONDING CHARGE TRANSFER STATES The last contribution to consider is that of the:r-orbitals. This was included in the calculations of 3a in the previous chapter. From (4.17) we have: 4 3a(4) - 3a(4 ((,) Ci Cd (p (5.20) i=o The above result was obtained by substituting the promotion energies, for charge transfer among various orbitals tb, eb, tl, ea and ta of Fig. 3.1 by an average energy. To refine the above result further we consider the case of t1 -+ ea electron transfer first and then discuss the V ()11 in (5.19) is vector coupling coefficient of two vectors.

55 general case where all six transfers tb -+ ea' eb + ea, tl + ea tb + ta) eb + ta and t1 + t are taken into account. 5.1 Determination of 3a (a,r, tl + ea) The effect of t -+ ea charge transfer states on 3a (oa,) will be discussed in this section. The symmetry and electronic configurations of the T-bonding molecular orbitals of tl - ea charge transfer are shown in Fig. 5.3. tito f till, b X ti 1+1 / /\\ // -~ f,'6A (a) (b) Fig. 5.3. Molecular orbital and energy levels of (a) the ground level 6A1 and (b) the tl -+ ea electron transfer levels 6T1 and 6T2

56 3a (a,r, tl + e ) can be obtained from the following relation. 3a (a,1r, tl _ ea) + )] ( K (/2 5/2 T1' T2 T1 U K 5/2 (52 5/2 T1, T1 T2 Ut) - K52 J5/2 T1, T2 T1 Ei KJ' 5/2 (5/2 5/2 T1, T1 T2 Ell)]) Substituting for reduced matrix elements from Tables 3.3 - 3.4, and for KJJ, from Tables 3.9 - 3.12 one finds, a 1 (61 + (eat1)(tlea)(tit1)(tltl) Substituing for eatl and t tl from Table 3.13 we find b a The contribution from teb er ea charge transfer can be obtained in a similar fashion. The energy diagram is as shown in Fig. 5.4.

57 \ --- It~tt \ I d l! \..,_Afll 6'+ e" 6 6 A A1 (a) (b) Fig. 5.4. Molecular orbital and energy levels of (at ground level. 6A and (b) the tb + ea electron transfer levels ~T and 6T2 The effect of 6T and T' on 3a can be written as 3a(i, tb aea) 1- [6 (1 (e tb) 2(tbtb) (5.22) Table 3.13 gives eatb p= ad +- V'1 p (5.23) The fact that 3a (=, tb + ea) - 3a (op;tb + ea) and also 3a(T) - 3a(, i) is evident from the choic~. of K2 + V = 0.8 < 1 in (5 26).. The reason is that the only role played by a orbitals is to reduce the coefficient of K and A of dt2 and tt2 orbitals in molecular orbitals of t2 symmetry.

58 tbtb = 2 + 3/2 K p (5.24) SuDsti-uting for eatb and tbtb in (5.22) one finds 3a(a,n;tb + ea) = (18/625)[S 2(61 +' )1-1 (x) 6 a x d4 + (6 a 2K - 2 - aBcKX5)1d 3 2 2 4 2 2 4 2 2 (324 1/2 5 +(3/2a X K + K X d + (2K2 - (3/2) -K X) (x) d 3 + (4)-1 3K (5.25) J'The numerical value of 3a(rt, tb + ea) can be obtained from following coefficients = 1-2 = 0.7 2 = 0.8 - X2 = 0.6 (5.26) and it is found as 3a(o,;tb + e ) (18/625) (6 1 ~| + esl)]1 [034 Cd +.0674d P 2 2 37) +.0827 d p.0487 d 3 +.0251% (5.27) - p p

59 5.3 Determination of Total 3a(a,v) The 3a(o,r) representing the effect of all charge transfer states of transfers tb + ea, eb -+ ea, t1 + ea, and tb + ta, eb + ta and tl -+ ta, can be expressed as (4.6): 3a(o, ) = 3a(4 = C (5.28) i= o where Ci are functions of a,3,K,X and the charge transfer energies such as 61 and 6{ in (5.21) and (5.27). The numerical values of the coefficients C. are calculated for a2, 2,K2,X2 as in (5.26) and for =66 + e-6'= E 611 1 6+ 61 The results, given in Table 4.1, and 4.2 indicate that 3a(4)>0 c /d < 2 (5.29) p dand 3a(4) <0 Cp/Cd' 3 (5.30) For ligands 0 — and S — and S-state ions Mn2 or Fe3+ the C /Ad < 2 holds and consequently 3a(t,0 —), 3a(t,S —) > 0 (5.31)

60 whereas for Se and Te the condition p/Cd > 3 applies and one concludes that 3a(n, Se ), 3a(n, Te ) < 0 (5.32) Comparing sections 4 and 5 one concludes that (1) in both a- and n-bonding schemes t2 - e charge transfer gives a positive contribution to 3a, (2) tb -* e charge transfer seems to be the most probable in a-bonding scheme whereas the tl -+ e transfer seems to be the most probable in n-bonding scheme and gives a negative contribution to 3a(<) and (3) the 3a(a), for an average charge transfer energy*E6(a) is negative whereas 3a(w), under similar condition is positive if p/Cd' 2. Now we proceed to the next section for comparison of (3a), (3a), 3a(a) and 3a(x).t 6. COMPARISON The five separate calculations given in Sections 1 through 5 can be compared now. To simplify this comparison we ignore the effects of spinspin interaction on 3a which appear as small corrections in calculations of Watanabe and those of Powell. This enables us to describe their results as functions of d 4 and (Dq). The result is 4 2 (3a)W Kw d 4(Dq) (3a)P =KP Cd(Dq) n 3.5 < n <6 3a(a) = K (d 3a(a,n) -: Ci d4 i (5.33) i O Let 6aT: dnT r = 1 in (5.11) 3a(E) = 3a(a,^)

where (3a)W and (3a)p are contributions to 3a from calculations by Watanabe5 and by Powell6, respectively, and 3a(a) and 3a(o,i) in (5.33) are contributions from charge transfer excited states. Since (3a)p results from spin doublets and quartets of 3d5 manifold, whereas (3a)W results from spin quartets alone, one immediately concludes that (3a)W is included in (3a)p: (3a)W 3a)p (5.34) In a similar fashion* 3a(o)Ca( o,~) (5.35) Therefore, the total contribution from spin multiplets within 3d5 manifold and charge transfer states is (3a)p + 3a(O,ic) (5.36) The experimentally measured 3a can be affected by spin quartets and doublets which arise as a result of charge transfer. In this case, 3a can be written as 3a = (3a)p + 3a(o,K) + (3a)r (5.37) where (3a)r represents the rest of terms ignored in the evaluation of 3a(o,r). 7. COMPARISON WITH MEASURED 3a OF Fe3+ IN ZnS, ZnSe AND ZnTe We want to compare the measured 3a of Fe3+ in Zns, ZnSe and ZnTe with 3a in (5.37) on the assumption that (3a)r = 0. 3+ The measured 3a of Fe for above compounds are given in Table 2-2 and are repeated here in Table 5-4. *3a(o,i) = 3a(Ni) [See the footnote to Eq. (5.26)]

62 TABLE 5.4 Measured 3a of Fe3+ in 10-4 cm-1 ZnS ZnSe ZnTe 384 144.9 -7800 tReference 27 To find the contribution (3a)p to the measured 3a values in Table 5-4 we assume: (i) that the measured 3a of Fe3+ in ZnS arises completely from (3a)p, (ii) the power n in (Dq)n of the expression* (3a)p = Kp nd4(Dq)n is equal to 4 and (iii) (Dq) is proportional to inverse fifth power of interionic distance R. With these assumptions, the ratios of (3a)p of 3+ Fe in ZnS, ZnSe and ZnTe can be obtained as follows: (3a)p(ZnS): (3a)p(ZnSe): (3a)p(ZnTe) = 10.1:5.3:1. (5.38) The (3a)p obtained from (5.38) are given in Table 5.5. Kp, in (3a)p = Kp %d4 (Dq)n, depends on several parameters such as Racah coefficients B and C. For simplicity, however, both this and Cd are assumed to remain constant in three compounds ZnS, ZnSe and ZnTe.

63 TABLE 5.5 Estimated (3a)p for Fe in 10 cm ZnS ZnSe ZnTe 384 204 38 The contribution 3a(agr) can be obtained for the appropriate values of Tp/Id. The Cd* for Fe is 0.049 e.v. and Cpt for S, Se and Te are 0.06, 0.35 and 0.9 e.v., respectively. Thus, the Cp/Cd ratios are 1.09, 6.4 and 16.4 for Fe3+ in the three compounds ZnS, ZnSe and ZnTe respectively. The 3a(o,7~)at these ratios of CP/rd and for Cd = 0.049 e.v. and E6 = 4 e.v. is obtained from Table 4.2 as given in Table 5-6. The sum of (3a)p and 3a(a,c) is given in Table 5-7. - P3+ *Ref. 16, p. 431, ( Cd of Fe~ is chosen instead of Cd of Fe because the effective charge of Fe in ZnSe and ZnTe is expected to be close to zero). tJ. Dimmock et al "Band Structure of PbS, PbSe and PbTe," Phys. Rev. 135, A824(1964). g~ ~ 3a(4) _ E Ci 4-i i i= O

64 Table 5.6. Calculated 3a(a,i) of Fe3+ ZnS ZnSe ZnTe CP'/d 1.09 6.4 16.4 3a(a, t) (10-4 cm-1 0.564 -141.0 -6620 Table 5.7. Measured and calculated values of 3a of Fe3+ ZnS ZnSe ZnTe (3a)p+3a(a,r) (in 10-4cm I) 384.56 63 -6582 (10 cm1 A comparison of the calculated and measured 3a indicates that a ligand to metal charge transfer process is capable of accounting for the (3+) variation of 3a of the Fe in the series of ZnS, ZnSe and ZnTe compounds. A detailed examination of the coefficient Cq of 5p in the expression of 3a( )in (4.6)* indicates that the sign of this coefficient is insensitive to coefficients of the linear combination of atomic orbitals,,K and X in the molecular orbitals, whereas the coefficients of %d4... %drp are the sum of almost equal number of positive and negative terms. With small variations in such terms the sign and magnitude of these coefficients will change. Therefore, the spin sextet and ligand to metal charge transfer approximations are valid for metals of higher formal valency and ligands for which #p/ d"*10. ZnTe:Fe3+ meets both of these requirements. Hence, the agreement found should not be surprising. *See the footnote ~ on the preceding page tSee Ref. 27

3+ + 2+ In addition to Fe discussed above Cr and Mn, the other two S-state ions of 3d5 configuration, deserve a brief discussion. In case of these two ions, in addition to the ligand to metal charge transfer process, employed for Fe, another charge transfer should be taken into account. This latter charge transfer permits the transfer of an electron from the antibonding orbitals ta and ea to the higher lying antibonding orbitals localized in the vicinity of the next nearest neighbor metal ions such as Mn-+Zn charge transfer in ZnTe:Mn. For brevity, this is called the outgoing charge transfer whereas the former one is called the incoming charge transfer. The matrix elements arising from such processes can be obtained from general expressions given in Chapter III with slight modifications. The evaluation of charge transfer energies, however, would involve the next nearest ions Zn and Cd in (Zn, Cd) (S, Se, Te) compounds and more caution is needed for a correct assessment of such energies. The extension of present theory to these two ions has to be deferred to a later time when more accurate charge transfer energies are available. 8. COMPARISON OF 3(a,i) OF Td AND h CASES Considering Table 5-6 one finds that both the absolute value and the sign of 3a is determined by the presence of %p in the expression of 3a(a,:g) = C %d4-i pi. A question arises on the nature of the role i=O 1 3+ of Cp in 3a(o,n) of Fe in compounds of Oh symmetry. Before considering the above question it is worthwhile to give a brief remark on the 3a in Td case. Recalling (5.37) the total expression of the 3a is 3a = (3a)p + 3a(a,i) + (3a)r (5.39);*The orbitals ta and ea are the half filled orbitals which are localized near the S-state ion and in ionic case form the components of the d orbitals of the S-state ions.

66 where (3a)p is given in Table 5.1 and 3a(ao,) and (3a)r can be expressed as 3a(a,H) - klm EkE1Em) SS S SS SS Ss Ss SS Mok M im MmO] U' - [MOk Mkl lMm] E' (5.40) and L,~il.i- ss ss sq s 3a klm (EkElEmo, + kl'm' (EkE Em M0k Mkls' M m' Mm U -Ok Mkl M s Mm The E I ss sq qs s dq -OkMMkl' Ml m Mm'O E0' k'l'm' in (5.41) are the matrix element of H = -quartet and doublet, respectively and En and En(n k, L, m) referMmo,ml E l' lljE -1 M sq qdM dq qs to energies of these statest'+ Ek'll''m' Ek'Ei'm M Ok' Mk'l1' mU sq qd dq qs) -Ok' Xk'l'' X l''m Mm E' The Mok, Mk -,M1 dq in (5.41) are the matrix element of Hp Z~ Si.i si and the superscripts s, q and d refer to the spin sextet, quartet and doublet, respectively and En, En, and En, (n' k, 1, m) refer to energies of these states, An important distinction between II-VI compounds of Oh and Td symmetries lies in the fact that the band gaP energies in the former case

67 varies from* 4-8 e.v. whereas in the latter case it varies from 0.02-3.7 e.v. Therefore it is probable that the energies of quartets, Enw, and doublets, En,,, are below 8 e.v. and as a result of this the spin quartets or doublets can be localized around the complexes of Oh symmetry. Thus an a priori omission of (3a)r does not seem to be a reliable approximation for the Oh case. Another obstacle, in the Oh case is lack of experimental information on 3a of Fe3+ in such compounds as SrSe or SrTe where C becomes significant. Therefore it is impossible to assess the contribution to 3a(o,i) in the ocathedral case. In the case of Fe3+ in the tetrahedral compounds, such as CdTe or ZnTe where the energy band gaps are, respectively, 1.5 and 2.1 e.v., it is possible to assume that none of the charge transfer spin quartets are localized. As a result of this the (3a)r may be ignored and only 3a(a,A) taken into account. In case of ZnTe:Mn2+ where (3a> 0 one may conjecture that the charge transfer spin quartets also contribute to 3a(ao,) as well as spin sextets of outgoing charge transfer process referred to in section 7. *See R. Bube, "Photoconductivity of Solids" John Wiley and Sons, Incorporated, New York, (1960) p. 233

CHAPTER VI SUMMARY AND CONCLUSIONS 1. SUMMARY A calculation of cubic field splitting of S-state ions in II-VI compounds was planned. To achieve this, the following steps were taken: (1) Molecular orbital techniques were employed to construct the excited states of complexes (A4) n with Z as the S-state ion and A as 0, S, Se or Te. (2) A ligand to metal electron transfer process was taken into account and the excited states arising from such phenomenon were constructed with a and wr ligand orbitals. (3) The cubic field splitting 3a was expressed as the lowest order splitting of the spinor levels U' and E'' (Mulliken's notation) of the ground state as a result of perturbation by excited states through the spir orbit Hamiltonian,. i i H = I, s P i - - (4) Utilizing group theory arguments, it was established that (a), the lowest order perturbation, was four and (b), at this order of perturbation the contribution to 3a arises exclusively from the two groups of three excited states having symmetries of T1, E, T] or T1, T2, T1 respectively. (5) Utilizing reduced matrix techniques the matrix elements of H between any pair of states aik I X S h. Jk t T) and IZ) = Ix S1 J I' A

69 was obtained in terms of the reduced matrix elements R.. and the coupling 1J coefficients Kkg as follows: X Si hi Jk t >T IH I Xj S h. t | = (X Si hiIHp IXj S h. K k Rij Kk where Rij (Xi Si hiHp IXj Sj hj and kg KjkJ (Si Sj T1, hj hi t) and IJ tT > is the component of the irreducible representation of angular momentum J in the cubic point group as defined by Griffithl6 (p. 395).* (6) 3a was obtained as a sum of the products of the four matrix elements: KXo 6A1 5/2 tTIHp Xi Si T1 J tT) (Xi Si T1 J tIHpXj S h Jm t) hj = E or T2 xi S hj im tTH plXk 5k T1 in tT) and (Xk Sk T1 Jn tT H IX 6A 5/2 t) For example 15/2 Uv> = xT [f 15/2 5/2) + 15/2 3/2)]

70 with Xi.... Xk representing various electronic configurations giving rise to Si T1...* Sk T3+ (7) The numerical values of the 3a of Fe in Zn(S, Se, Te) compounds was obtained with restrictions of Si = Sj = Sk 5/2 Ei = Ek = E6 and Cd and Up as the spin orbit constants of metal and ligand orbitals. pd-1 For appropriate values of C for Fe, S, Se and Te, and with 32,000 cm for E6 it was found that the calculated 3a accounts satisfactorily for the difference between measured values and the ionic contributions to the 3a of Fe3 in the compounds ZnS, ZnSe and ZnTe. 2. CONCLUSIONS Most of the conclusions drawn from this study concern the effect of charge transfer states on the cubic field splitting 3a of S-state ions in II-VI compounds with tetrahedral symmetry. These conclusions are classified as follows: (1) The cubic field splitting 3a of S-state ions in covalent II-VI compounds of tetrahedral symmetry depends strongly on the excited states arising from charge transfer from ligand i orbitals to metal d-orbitals. (2) The effect of these E orbitals is relatively insensitive to the choice of promotion energies and coefficients of linear combinations of atomic orbitals. (3) To refine present theory, it is necessary to establish (a), the energy levels beyond which excited states are no longer localized, (b), the perturbation order beyond which the contribution to the initial splitting 3a is negligible, and (c), a search for a few parameters characteristic of charge transfer state energies.

71 (4) To verify the predictions of this theory with experiments, it is desirable to (a) determine the sign of the 3a of Cr+ Mn and Fe3+ wherever it is in doubt, (b) prepare single crystals of (Mg, Ca, Sr) (Se, Te) which have ocathedral symmetry and to measure the 3a of S-state ions, particularly Fe3+ in such compounds. ACKNOWLEDGEMENT The authors wish to acknowledge the useful comments of Professors H. Watanabe of Hokkaido University and T. M. Dunn of the University of Michigan. Continuous encouragement by Dr. R. K. Mueller, Manager of the General Science and Technology Laboratory of the Bendix Research Laboratories Divison is deeply appreciated. The secretarial work by Elsie Wells and the final typing and art work by the Graphic Arts Department is sincerely appreciated.

APPENDIX A DEFINITION OF SYMBOLS The frequently occurring symbols, in both the Latin and Greek alphabet, are defined in Table Al of this Appendix. The former group of symbols is given first and then the latter one. 72

73 TABLE A-1 DEFINITION OF SYMBOLS 6Al Term designation of a state of space irreducible representation, Al, and spin S = 5/2. 3a The cubic crystalline field splitting of a 6S level. Ci Numerical coefficients of the expression for the charge transfer contribution. 1P. Square root of the product of dimensions of space and spin representations hi and Si of a state ISihi). Thus, for a state ISihi>= 15/2 Ti one has = [(2S. + l)(hi)] = [(6)(3)] = [18]1/2 E'" An irreducible representation of cubic double group as defined in Table 3-1. ea Antibonding molecular orbital of symmetry E (Table 3.1). eb Bonding molecular orbital of symmetry E (Table 1.1). E(U') The lowest energy value of levels of symmetry U' (Table 3.1) E(e'') The lowest energy value of levels of symmetry E'' (Table 3.1) Ejk The energy difference of states ij and lk: Ejk Ek Ej. E. The energy of state lj from that of ground state: J E. = E. - E J J o H Perturbation Hamiltonian: H'= E.s = su(K) p p 1i K h An irreducible representation of single valued cubic group.

74 An identifying number of the irreducible representation resulting from the coupling of spin S and the irreducible representation h of a state ISh>such as UJ = U'/2 of the state 15/2 T1)>, and Uj = U' of the state 15/2 E). In the case of h = T1, T2 the index J behaves as total angular momentum associated with Russel Saunders level ISL> = ISL = 1> whereas for h = A2, E it is a designating number. KJJ'(SS'Tl,h'ht) Spin-orbit matrix element coupling coefficient between states I ShJtT >and IS'h'J'tT). M Magnetic quantum number associated with spin S. MO Molecular orbital. IR Irreducible representation. S Total spin associated with a total level or its sublevels. s Single electron spin operator. t An irreducible representation in the cubic double group of the coupling, the spin S, and space irreducible representation h of a given state ISh) such as U' of 15/2 T)>. ta Antibonding orbital of symmetry T2 (Table 3.1) tb Bonding orbital of symmetry T2 (Table 3.1) tI Non-bonding molecular orbital of symmetry T1. U' An irreducible representation of cubic double group (Table 3.1) V(abc,acy) Coupling coefficient of the components a and ~ of the irreducible representations a and b into the y component of the irreducible representation c such as V (ET1T2OxS) = 1/2. The components G, x, E of the

75 representations E, T1, T2... and their symmetry properties are defined in Table A.16 of Ref. 16. V (abc,BSy) Coupling of ~ and B components of spins a and b into y components of spin c such as V (5/2 5/2 1, 1/2 -1/2 0) = 1/2 (1/210)1. Tables of V are given by Rotenberg et al. (Ref. 26 footnote of p. 86). W(abc,def) An invariant product of four coefficients V(abk,fSy)...defined as W(abc,def) = CEaSy6c V(abc,cBy). V(aef,aEo) ~ V(bfd,36) ~ V(cde,y6c). The tables of coefficients W (abc,def) are given by Griffith (Ref. 26 p. 114) W(abc,def) An invariant product of four coefficients V(abc,aBy)...defined as W(abc,def) = ZEaSy6S a-a+b- +c-y+d-6+e- +f-4 (x) (-1) V(abc,a~y) (x) ~ V(aef,acx) ~ V(bfd,364) ~ V(cde,y6c). Values of W are the same as the 6-J symbols corresponding to a, b,...,f and the latter are given by Rotenberg et al. (Ref. 26 footnote of p. 86). (N iNjN0 NkN1Nm) Product of a W and W coefficient as tW(NiNjN,NkN1Nm) = W (SiSjl, SkSSm) (x) W (hihjTl,hkhlhm). X(abc,def,ghk) An invariant sum of the products of six coefficients V(abc,cSy),... V(cfk,yqK) expressed as X (abc,def,ghk) = Zacy6cQe OK V(abc, aBy). (x) V(def,6&t) ~ V(ghk,nOK) ~ V(adg,c6fn) ~ V(beh, e0) (x) V(cfk,yqK). These X coefficients are defined by Griffith (Ref. 26).

76 X(abc,def,ghk) This is similar to X(abc,def,ghk) defined above except instead of V(abc,~3y), one takes V(abc,c4y). Thus X(abc,def,ghk) = aZ 3 eK V(abc,cBy). (x) V(:def,b6E) V(ghk,nGK)...... V(cfk,~zK). The coefficients X are given by Howell (Ref. 26 footnote p. 86) X(N.N Nk NiN'N' Product of coefficients X and X related to N N N i j k j k, ij k' N N N ) NN'Nj', N NbN as follows: X (NiNjN kNjNk'NaN bNo) X N1j k a b b o j Mk' ij kNab 0o a b 0 = -X(S iSj SkSiSjSk,SaSl) X(hih hk,hihjhk,hahbT). A The ligands surrounding the metal ion of II-VI compounds such as S, Se and Te Z The S-state ion substituting the metal ion of a II-VI compound such as Mn in Zn site of ZnS single crystals. A-n (An 4) A complex formed of an S-state ion and its four nearest neighbors, with a formal negative charge of n'. For Z = Cr+ EMn or Fe3 the number n' is, 7, 6 or 5 respectively. Single electron spin orbit inter-action of an electron in the ith orbitals. Cd Cd of a d orbital of the S-state ion. P of a p orbital of the ligands S, Se or Te. T Component of t denoting an irreducible representation of the cubic double group. The properties of these components are given by Griffith (Ref. 16). X Electron configuration of five orbitals ta, ea, tl, eb and tb as defined in (3.8a). X Hole configuration of the five orbitals t, ea, tl, eb and tb as defined in (3.8b).

IXShjtT> T component of Jth irreducible representation t arising from coupling of spin S and space irreducible representation h of the state ISh> belonging to the X configuration. (XShIIHp IlX'S'h') Reduced matrix element of Hp between states IXSh> and IX'S'h').

APPENDIX B SPIN ORBIT COUPLING IN MO SCHEME This Appendix gives the appropriate form of the spin orbit interaction Hamiltonian HSO in the molecular orbital (MO) scheme. The expres sion of HSO for an n electron system is so H Z r.e E i i SO mc [ Za ria (r x p ) ~ s ri.- (ri x pi)~ (si + 2sj)] (B1) where a refers to all nuclei; r. is the distance between electron i and nucleus a, Z is the charge of nucleus a; i and j refer to all electrons in the complex and the remaining parameter have their usual meanings. The first sum in (Bl) gives the spin orbit interaction of each electron in the Coulomb field of all the nuclei in the complex whereas the second sum describes the interaction of each electron in the field of the other electrons and also the coupling of each spin with the orbital magnetic moment of the other electrons (spin-other-orbit interaction). The Hso can be rewritten as: HSO = a Hia - Hi (B2) H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Springer - Verlag, Berlin, 1957), p. 181. 78

79 where -3i i is i H. = (Be/mc) Z ri (r x p') s = (ria - s (B3) 1 s isx _- - - — (r.) and -1 -3 ij i i H.. = fe(mc) r. (r x p) ( + 2s (B4) ij ij Misetich and Buch+ have shown that for the molecular orbital wavefunctions io and in related to symmetry wavefunctions 4o and %n of a given term of the free central ion of the complex,one has ic 1 i ViHrc (n'c) cS + IL (riL) f Si>n) (B5) (r- ) iandL iL(1 The parameters c (ric) and EL (rcL) in (B5) give the spin orbit constants c and CL after being integrated over ri and riL respectively except for the fact that their numerical values depend on the coefficients of linear combination of atomic orbitals used. In the general case where the MO wavefunctions ~O and in cannot be related directly to free ion such as the charge transfer states in a complex [E A4]n, the spin orbit interaction can be considered in a slightly different way. Considering (Bl) - (B4) and denoting the single electron spin orbit interaction by H so(i), one has: Hso(i) = i HE H = Z (ri) i H.. j (+i) j(+i) -ic i i iL. - - Ec (ri) + (r) i H.. (B6) j (+i) A. A. Misetich and T. Buch, "Gyromagnetic Factors and Spin-Orbit Coupling in Ligand Field Theory," J. Chem. Phys. 41, 2524 (1964).

80 where the parameters c and L denote central and ligand ions respectively. The matrix element of Hso (i) between the ground state *o and an excited state of the system ln is (o IHSo(i) n= iHic + L HiLi') E (;il iji l j (B7) j (+i) The single orbital wavefunctions i and j can be described as ji) = KiliC)- X il ) (B9) I = K 1jC j I jL) (B9) where 2 2'" 2i + = 1 (B10).ubstituting in (B7) one finds: oIHS(i) Is n>= KiK'ci i (iC icl i c>+ XiXi iLI L HLi LL - j K.K i K iil K. H I i' j j ili \ iLjLc 1 j Hij liLJL + xi (iCl iC -'j(i cL H ij ci)Iit ) + ih'i; zIL~~iT T C'jTSIA;j 2Bii l;\i L) (B11)

81 If both i and i' have the same radial wavefunctions then the radial c c integration of the first sum gives the spin orbit constant of the central ion for the orbital i being corrected for a change in the electron density in orbitals j, measured by Kj. Since in this work, the molecular orbitals i are constructed from d orbitals of the central ion, c they have the same radial part. Thus the first term in (Bll) can be expressed as" KiK'iicH / K. H. i i< c ic E C IiC ic c j (+i) =..KiK i'i, 5 iCsi (B12) i i ckc - I c Similarly the radial parts of iL and ifL in the second term of (Bll) are the same. Thus (i j H. -~ (jIx.2 H j Iif) 1'i iLiL iL- JL|j ij L L i= i Q, i iL L i (B13) ~i i L-L.is i Now we define a spin-orbit interaction operator.Ci k s such that gi iKN >di i i Sli d)= d i s /i (B14) and.i~ s |iL = C _'s liL) (B15) The prime sign on <iclc iC'sili'c) in (B12) indicates it has been integrated over ric

82 The parameters d and p in (B14) - (B15) indicate that i and iL c L are constructed from d and p atomic orbitals respectively. Substituting in (Bll) one finds KldIHso(i)I in>= nX K'ic' (ii ~. i') + ~i iLi Q l si'L) Kii (iCI -a L i S Since Ho = Z HS (i) (B17) SO i SO then < o S > < o1i i ) resulting in H = - S iUM (B18) SO i i -i

83 where u(i) = (i R (B19) and su(i).= k. 2 s (B20) In (B16) - (B20) ki Q, behaves as operators defined in (B14) and (B15). The above definition of spin orbit Hamiltonian for the charge transfer states of a complex (Z A4)-n is certainly an approximate definition which will not be adequate for the precise evaluation of the matrix elements of HSo but is sufficient compared to other approximations made in construction of the molecular orbitals ii) and excited wavefunctions

APPENDIX C SPIN ORBIT MATRIX ELEMENTS FROM THREE AND FOUR ORBITAL WAVEFUNCTIONS The purpose of this Appendix is to give the spin orbit matrix elements between charge transfer states consisting of three or four types of distinct orbitals each having at least one electron such as those in Table 3-1. The spin orbit matrix elements between pair of states consisting of only two orbitals have been calculated by Tanabe and Kamimura* and by Griffith.+ The ligand to metal charge transfer process, in cubic complexes of S-state ions, results at least in three open shells of electrons two of these around metal and the third around the ligand. Thus the desired states consist of at least three orbitals. As a result of this the formulae by above authors should be modified and extended to be applicable for these wavefunctions. We proceed by giving a brief description of charge transfer wavefunctions first and then discuss the matrix elements of H = EiCi i between them. 1. CHARGE TRANSFER WAVEFUNCTIONS A description of the orbital part of the ligand to metal charge transfer wavefunctions, in complex [A4]-n, will be given here. Their radial part is omitted for simplicity; it must, however, be taken into account in a more refined analysis of this subject. Considering Table 3.1,one finds the electronic configurations p,q,..t and the representations ta, ea, tl, eb and tb of the orbitals in a charge transfer state. Denoting the spin and magnetic quantum number of the participating orbitals by SiMi and their space irreducible representation(IR) by hiei one can describe a charge transfer state of spin SM and irreducible representation he as follows. Y. Tanabe and H. Kamimura "C.:he Absorption Spectra of Complex Ions IV. The Effect of the Spin-Orbit Interaction and the Field of Lower Symmetry on d-Electrons in Cubic Field" J. Phys. Soc. Japan 13,394 (1958) +J.S. Griffith (Ref.26) 84

85 IXiShM>- [t S 1h1M1O 1 e q S h2M2..tbt hM;ShM] (Cl) For example one of the states arising from the first row of Table 3.2 is 1Xi5/2 T1 5/2Z) -ta2 1T1 lx, e 1A2la2 tl 1/2T1 l/2y eb OAOal', tb OA1 0al; 5/2 T1 5/2Z) h~t 2 2 1 a 3T1 lx, ea 3A21a2, t1 2T1 1/2y eb01Ala;6T1 5/2Z) (C2) where Xi in (C2), as before, denotes the manner by which the five orbitals ta, ea,.****, tb have coupled to give 6T in (C2). 2. MATRIX ELEMENTS OF H = C i p i iThe matrix elements of H between pairs of charge transfer states iXjShMO) and IXkS'h'M'e) will be obtained in this section. To simplify the notation the above matrix element will be denoted by Mjk(ShMe-S'h'M'9'): Mjk (ShMO-S'h'M'O')= xjShMe{Hp Xks'hM') a Rjk(Sh-S'h') Q(ShMO-S'h'M'8') (C3) where Rjk(Sh-S'h') = (jShl IHpI IXkS'h') (C4) is called the reduced matrix element and Q(ShMO-S'1IM'O') is the coefficient of the coupling of IShMe) and IS'h'M'O') through spin orbit interaction

86 and it is independent of j and k as will be seen later. Now we consider Rjk(Sh-S'h') and leave Q(ShMO-S'h'MO') for Appendix D. 3. REDUCED MATRIX ELEMENTS Rjk (Sh-S'h') Rjk depends on Xj and Xk. The Xj and Xk, in turn, depend on the configuration p, q,..., t of orbitals ta,ea..., tb as shown in (C1). Therefore the reduced matrix elements Rjk between a pair of states Ij) and Ik) can be characterized by the configurational numbers pj, qj,... tj and Pk' qk''*' tk in these two states. Considering this fact in mind and observing Table 3.2, one immediately finds that there are three classes of reduced matrix elements as follows: (i) Pj k' q qk = uqk = uk u = r,s,t (C5) (ii) j k'i qj q k + 1 uj = Uk u = r,s,t (C6) P(iii) = Pk q qk; uj uk u = r,s,t (C7) The numbers p,q, *.. t in (C5) - (C7) are given in rows of the hole configuration column in Table 3.2. In case (i) both states Ij) and 1k) have three open orbitals with the same configuration such as Ixj 6T1) and IXk6T2) of the first row in Table 3.2. Rjk in this case may be called homo-configuration three orbital reduced matrix element. In case (ii) the orbitals involved are the same but their configuration differ and therefore the Rjk of this case is called hetero-configuration three orbital reduced matrix element. In case (iii) only one of the five orbitals ta, ea, tl1 eb and tb remains closed in both states Ij) and 1k) such as tb in 6T1 of row 2 and 6T of row 3 in Table 3.2. The Rj of this case will be called 2 jk Of hetero-configuration four orbital reduced matrix elements. These three cases will be considered in the following sections.

87 4. HOMOCONFIGURATION THREE ORBITAL REDUCED MATRIX ELEMENTS Here we consider the case of Rjk between states Ij) and Ik) with both Ij) and 1k) containing three open orbitals of the same symmetry and configuration. The Rjk in this case can be expressed as Rjk(Sh, S'h') =(XjShl IHPl IXkS'h') (C8) where IxjSh>= I[aPSlhl(bqS2h2c S3h)S4h4] j;Sh> (C9) IXkS'h)= I[aPSh (bqS2h2c rS3h)Sgh] k; S'h') (ClO) The orbitals a, b and c, in (C8) - (C10), represent three of the five orbitals ta, e.., tb of Table 3.2 and the subscripts j and k denote the coupling of such orbitals. Since the perturbation Hamiltonian Hp* = =.u () i p ii in terms of single electron operators we must express the total wavefunction in terms of the single electron orbital which constitute such a wavefunction. To obtain this we rewrite (C8) as follows:+ Rjk(Sh-S'h') =(aPSlhl(bqS2h2 c S3h3)S4h4 Shj U + E s i) i=j i= +l p+q+r q,r + L s'u(i) aPS'h (bqS h' rS'' )S'h' S' h' 1 1h 2h2c 33 4 4' i=p+q+l = Rjk[p(qr)p,Sh-S'h'] + Rjk[p(qr)qSh-S'h' ] + Rjk[p(qr)r,Sh-S'h'] (cll) To simplify notation i!i is substituted by ui To simplify notation the brackets [....]j and [ k are omitted from IxjSh and IxkS'h' in (Cll).

88 the first sum operates on electrons in aP, the second on bq and the third on cr and Rjk[p(qr)pShS'h' = (aPSlhl (bq S2h2c S3 h3)Sh 4 Shl I s_ u(i)(l'l)(1l'l)1 (x) IlaPS' h1 (bqS'h' crSlht)Slht, S'h' (C12) (l, E 2 u | h h r h3 )S 4,Sh{ ) (C13) Rjk[p(qr)r,Sh-S'h'] = (aPSlhl(bq2h2c S3h3)ShSh (<1')x(l1)x s'u(k) klaPS.h'(bqS'h2,crsh')SthS'h') (C143 The symbols 1l1 represent double tensor operators of zero rank which operate on their respective part in (C12) - (C14). The -first symbol 1 of 101 acts as a spin operator with S = M = O and the second symbol 1 of 141 acts as the irreducible representation A1 of the cubic group. Rjk9 in (C12) - (C14), should be determined by the techniques of double tensor operators on coupled systems. This subJect is discussed by Griffith26 and will be given here as follows. Let a system n of electrons to be composed of two separate and independent parts t and m. Then

89 gives the Inv> state in terms of products of IRX) and Imp. Now the reduced matrix elements of a sihgle electron operator* Dd ope:~rating on the I|A) part can be described in terms of Inv) states as followst (<m n vIIDdl I'm'n'v'>) = (1) n+n+d Dd I') W'm (X) 6 (C16) Similarly an operator Ee operating on Imp) states has the reduced matrix elements (<mnhlEell'mm'n'> = (_l)z+m'+n+e (n(n,) ml lEemW n mel (C17) where (n) and (n)' are dimensions of these two irreducible representations and W coefficients behave as six j symbols. For spin orbit into-action both orbital and spin wavefunctions of each electrons should be taken into account. Considering, m,..., n' as space representations of the states and operators involved one will add Si, S2...S' for spin part. Thus (C16) can be rewritten as (S1LS2mSnl IDPdlS'S2m'S'n') = (-1)Sl+S'+P++m'+n'+d (x) [(2S+1) (2S'+l) (n)(n')] 1/2 (SI IDPI |s1z') W S )S'S WI nn ) (C18) (x) 6S2S2 6rm' Dd6 is the component of operator behaving as component of the irreducible representation d.'Ref. 26, p. 47

90 and pe IS)S 1+S2+S+q+h1+h2+h+e S S2mSn IEPel IS1'S2mS n = (-1)1 1 2 (x) [(2S+1)(2S+l)(n)'n') 1/2 (S2mlIEqel eS2m) S2S2) W (n me) (x) sls 8' (Cl9) Following (C17) the reduced matrix elements in (C12) - (C14) can be decomposed as follows: S +S 4+S++l+h +h4+h+T1 Rjk(Sh-S'h',p) = (-1) (x) (2S+1)(2S'+l)(h)(h')] 1/2 PShll (s(i) ~ u(i))lT11 aPSlthtl) 1 1 S SI 1 W h h' h 4 6 S (C20) The Rjk[p(qr)q,Sh-S'h' ] and Rjk[p(qr)r,Sh-S'h' ] must be obtained in two stages. First the part of the system represented by S4h4 should be decoupled from the part represented by Slhl and then the parts S2h2 and S3h3 in S4h4 should be treated as in (C20). Denoting part represented by S4h4 as Rjk(q,r) and considering (C16) - (C20) one finds S+S 4+S+l+h +h' +h+T Rjk~q~r) = (-1) 1 4 11 1 (x) [(2S+l) (2S'+l) (h) (h') 1/2 Rjk(q,r) = (-1) ( (bqS2h2, c S3h3 )S4h4 I s(K)'(K) ll(bqS2h2 crS3 h3)S4h4 < 29 3 3 4 4 K=i3'1 S4 1 h6h, ( S' S1 i h h' hi S1 1 (C21)

91 The term* q r T = <bqS2h2crS3h3)S4h41 s. u(K) + B sU(K) K=l K=1 ||(bqShcrS 3h3')Sh (C2) 22 33 4 4l (C22) in (C21), should be factorized in the same manner described in (C19) - (C20). T = (-) S2+S3+h2+h3 [(2s4+1)(2S'4+l)(h4))] 1/2 (xs)(bS2h2 | sus(K) | b h T1 ) K-1 S2S2 2 W S4S W 2hhh3 h (x) -S3 h3 Y S(K) CSh3 )] (C24) Substituting in (C21) one finds R k(Sh-S'h',q) and R k(Sh-S'h',qr) of (C13) and (C14) as the coefficients of (b2S2h2 1 I su(K)I bSlh)' ahnd (crs3h3 I su(K) I IcrS3h3 respectively. Thus, K Rjk[p(4'qr)q,Sh:-S'hT] 1S3 ++S h +h'+h S h hsh+ h h ) 1 4 h+++ +3+h2+h3+s4+h4 -. rb;rt HereaIfter ~s(K) u U(K) 3 su(K) 1hlh

92 (x) 6S3S3 (2S= [ 2 S +1) (h ) (h4+)(h) 1/2 s sZl S's 1 h'h4T h'h T ()W | S4S14 W 2SS2 W Z4 4 W 2 2 2 \ SS' SS hS3 h h h4h4h3 (x)(bqS2h21 I E su(K) I |b Sgh ) (C25) K andS l+S 4+h +h 4+h+S 2+S +h +h3 Rjk[p(qr)r,Sh-S'h'] = SS hh ( 1 4 1 4 2 3 2 3 S4+h4 (x)(-1) 2 6h2h, [(2S+1) (2S'+1) (h) (h') (2S4+1) (2S4+1) (h4) (h4) 1/2 22 22'2 S'S 1 S'S 1 hh4T1 h'h3 T Tq 4 4 W 3 3 W 4 4 1 W 3 3 1' S S S' S S4S'S3 h h'h h h442 (CrS3h3 I E su(K) I| crSlhl > (C26) K Substituting in (Cll) we have p+q+r Rjk(Sh-S'h') =([aPSlhl(bqS2h2c S3h3)S4h4] jSh I E SU(K) K=1 II [aPslhl(bqs2h2crs3h3)S4h4 k S'h' ) Rjk[p(qr)p, Sh-S'h'] + Rjk[p(qr)q,Sh-S'h'] (C27) + Rjk[p(qr)r,Sh-S'h' ]

93 The Rjk in (C27) are given in (C20) and (C25) - (C26). They are given here in simpler form as follows: S l+S4+S+hl+h'+h' Rjk[p(qr)p,Sh-S'h'] (-1) 4' 4 Si hl1 NN1No K SU(K) l aPSlhl > (/ N 6N4N1 (C28) S +S +S+h +h +h+S +S +h +h'+S'+h' Rjk[p(qr)q,Sh-S'h']= (-1)1 4 1 4 2 3 2 3 4 4 N~N2No N' N4No ( = (-1) 1 4Sh+h+2 2 2 32 4 4 4 4 jN4N4N3 i N N' N1 (x) <bqS2h21 E su(K) bqS2h2) 6N 1N'1 6N3N 3 (C29) K=i and S 1+S 4+Shl+hl+h+S 2+S 3+hh 1h3+S 4+h4 Rjk[p(qr)r,Sh-S'h'] = (-1) NNo N4N4 N' N (x) r, / 3 3 4 4 ~ 4 4 N4 N4N2 N N'N1 W c ~rs h'1'h'> SU(K)I IcrS'h 6N1 N 6N2N' where =[(2Si +1) (hi)] 1/2 NijNo = _ SiS (hihjT1 k Q m SkS Sl hkh h3

94 The coefficient W in (C31) is defined as l 1a b c) (-1) a-acb- B+c-y+d-6+e- +f- d e f a/ y6c a b c aa e f _( d b f d e c (x) I V V I a —B- a - -6 6-6 Y and Jnd 1J2J3- (- 1)2J2+J3-m 3 a = 7+ ((J1J2J33 JlmlJ2m2) (C33) V and W are used for spin coupling coefficiencts whereas for coupling of space irreducible representations the simpler V and W are used where (Ref. 26, p. 10 and p. 33) V(abc,aBy) =y <(abacBabcy> (C34) and W(abcdef) = Ea3y6c~ V(abc,acy) (x) V(aef,ac%) V(bfd,B%6) V(cde,y6) (C35) The numerical values of W are taken from Rotenberg's tables of 6J symbols** and W are obtained from Griffith's tables.t The dimensions of h. entering D. of (C31) are the same as their character under identity class in the character table. The (-1)i = + 1 as defined by Griffitht (p. 15). Both of these numerical parameter are given in Table C.1. V. Fano and G. Racah "Irreducible Tensorial Sets" Academic Press, New York 1959, pp. 50-54 M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten, "The 3-j and 6-j symbols." The Technology Press, Massachusetts Institute of Technology (1959). Ref. 26, p. 114

TABLE C-1 The Values of (h) and (-l)h h A1 A E T1 T2 (-1) 1 -1 1 -1 1 (h) 1 1 2 3 3 The last unknown in Rjk (P,p),...Rjk (r,r) in (C28) - (C30) is the single orbital reduced matrix elements of the type p p(pp) = (aPShll I su(K) llaPS'h'> K=1 = p (aPShl su(p) I aPS'h'> (C36) To obtain p(pp) we express it in terms of its matrix elements between pairs of iaPShMG> and JaPS'h'M'0'). (<aShl Z su(K) 1 T llaph,> (1)S-M V l) K=l MM -i 0 e i = <aPShM0 [ SU(K) aPs'h'M'O'> K=i J-ii

Multiplying both sides by (-1)S-M V (S,S'1, -MM'-i) and V(hh'T1 ee'i) and summing over the six parameter -M,...i one finds 1 T1 p(pp) = p (aPShMe [su(p)] aPS'h'M'e'> (x) MM'i -ii ee' S-M(x) (-1) V (SS'1, -MM'-i) ~ V(hh'T1,6e'i) (C37) In terms of the coefficients of fractional parentage (cfp) we have* aPShM = SM1m <aP lSlhl,a ) aPSh)(SllMlmI S~-SM) hlel 0a (x) (hlalahlahe>la P-SlhlM1>) lama > (C38) Substituting for laPShMO> and laPS'h'M'e'> in (C37) one finds <aPSh l a,a SaP- lhl>SSMI S1Mlm> (x) f(p,p) = p S1MlmMM iS'M'm' (hlahe IhlaelO>~ <aP- 1Slhl Me amlI h 6 alO'ih 0 lt' 1T I[su(p)] 1 lam'a'>)laP-ls'h'M'0'> -ii (x) <hlai' Ihlah''> (Sl-2M 1 lS' M)aP, 1S h, a aPS'h') (x) (-1) 1 1 V (SS'l,-MM'-i) ~ V (hh'T1, 0'i) (C39) Ref. 26, p. 62

97 The matrix element in (C39) can be abbreviated as follows: (aP- S lhlMll (ama Isu(p) am'a'). aP-lsihMX{e> = (am ajsulam'a' >' 6SlSl 6hh M1Ml 6e1 (1<l/2aI jsul 1/2a )(-1)1/2-m V(1/2 1/2 1, mm'-i) V (aa'Tl,ca'i) * 6. (S ~ * 6 (C40) SlS l MM hlh e11C The remaining coupling coefficients in (C39) may also be expressed in terms of V and V. Thus <Sll/2SMISll/2Mlm>= (2S+1)1/2 (_1)1+S- M V(S11/2S,Mlm-M) (C41) <hlahelhlaela ) = (h)1/2 V(hla h, ele) (C42) Substituting in (C39) and considering the effect of 6S SI,.. 6d, in (C40) one finds ( ) p S-M+l+S-M+1/2-m I' (-1)+I +S'-M' S M mM'm'im' hlO1 (x)( aPSh {la,aP-1Slhl>' (aP-1Slhlal aPS'h' )

98 (x) V(S 1/2S, Mlm-M)V(1/2 1/2 1,-mm'-i)V(S'1/2S',Mlm'-M')V(SSIl-MM'-i) (x) V(hlah,O18O) V(aaTl,aa'i) V(h1ah', Ola'O') V(hh'T1,ee'i) (x) (1/2a 1 sul 1/2a) (C43) The coefficients V and V may be rearranged according to the rules* abc cab ()ab+c vbac a+b+cv abc r = V ( = (-1) a+b ab+c7 V-)-B — (C44) V and (b)a+b+c V c a' (C45) Carrying through symmetry operations of (C44) and (C45),on coefficients V and V in (C43),one finds 1/2 1/2 1 1/2S'S 1/2s1S V(S 1/2S, M m-M)....= (-1)2(S1+S') V / r- 1_i m1M ) (i s's} 1 SS'1 i-MM' and V(Sll/2SM m-M) V...= (1)-S +S+S' +(m-m-i-M-M'-M 1) mlmMh' im' MmMm' iM' (x) (-1)+2+S-S'+S1-(m-m+i-M-M'-M1) Ref. 26 p. 77 and p. 15

99 1/2 1/2 1 1/2S'S 1/2S S' SS (X) V i v 1 -m( m-i m' -m-M MM i-M'M (C46) The first three powers of (-1) result into +S'-M'+1/2-m-S-S'-S1+ (m'-m+i-M-M'-M1) = (1) S'-M'+/2-m+S-'-S 1-M-M'- M1 =(-1)1/2-2M+SS1 =(_1)-(1/2+S+S1) +1 =(-1)1/2+S+S+1l (C47) Similarly the four V coefficients can be rearranged as follows aV). T +h +h'+a a aT 1 a hT (.1.. ( 4 )1 1 1 V ~ahl h'Tlh h' v l v( h (c48) a 010 i e 0' Substituting in (C43) we find P(PP) = P Z (-l)S+Sl+i/2+hl+h' aPSh a'aP- lSlhl) aP lSlhlal} Slh1 aPS'h') (l/2a| Isul I1/2a) (x)

100 (x) L'I) 1/2+1/2+1+S+S+S'+S 1- (m' -m+i-M-M-M1) MmM M'im' 1/2 1/2 1 1/2 S'S 1/2S S' 1SS' (x) V -m'm-i m'-M'M1 -m-M M' i-M'M ~aaTl ah'h alh' Tlhh' (x) V ) ) ( ) (C49) 8'aO'"a' a'a i a'8'8 a e't' iee' The second and third sums are identical to (C32) and (C35) respectively. Thus they can be substituted by their appropriate W and W coefficients. The final result is* p(p,p) = GP (Sh-S'h') < 1/2al1suIl1/2a) (C50) where GP(Sh-S'h') Z- (l)S+Sl+l/2+h'+hl+a (aPShi a,a lShaP-lh l sPhI',a 1 > IaPS'h') (X) p [(2S+l) (2S'+l) (h) (h')]1/2 (x) W (1/2 1/2 1, S S' S1)'W (a a T1, hh'hl) (C51) p, q and r must be less than half shell numbers. If not they should be substitutde by p' = 2(a)-p, q'=2(b)-q and r'=2(c)-r where (a), (b) and (c) are dimensions of a, b and c respectively.

101 Substituting in (C28) - (C30) we have* Rjk[p(qr)p,Sh-S'h ] = (-1)P.0,(NN N N'N4) (x) GP (S h-S'h') <1/2 al su| ll/2a 6dNN, (C52) Rjk[P(qr)q,Sh-S'h'] (-1)Sq D,)4W U4N(N 2N N4NN'N) U/(N' N4 N, N N'N1) (x) G (S2h2-Sh2) <1/2 bl suI11l/2b6NN, N3N (C53) and Rjk[P(qr)r, Sh-S'h' ]= (_1)Sr -,V'.)$(N N3N N 4N 2 ) 4 3o' 4N4N2 (x) W/(N4NNN NN'N1) Gr(S h3-S'h') (1/2cl sul 1/2c> 6N1N, 6N2N' (C54) c 3h3 33 \ 22 5. HETERO CONFIGURATION THREE ORBITAL REDUCED MATRIX ELEMENTS Rjk,(pqr Sh, pq'r'S'h') Here, we consider the reduced matrix elements RjkV between states Ij) and tk1) with both having three open orbitals of the same symmetry but different configurations. Sp, Sq and Sr are the sum of powers of (-1) in p(pp), p(qq) and p(rr) respectively.

102 The Rjk, in this case, can be expressed as Rjk(ShpS'h') (XjShl I Hp IXkS'h' ) (C55) where IXjSh) I[PS 1 h1(b lS2h2crS3h3)S h4] J;Sh ) (C56) XkS'h') [aPS h (bqSh' Cr-1S'h)S4h' ] kS'h' ) (C57) The orbitals a, b and c, in (C56) - (C57) represent three of the five orbitals ta, ea,..., tb of Table 3-2. Subscripts j and k' denote the electronic configuration and coupling scheme of the three orbitals a, b and c. The determination of Rjk, follows that of the Rjk defined in (C12) - (C14). Considering these equations, Rjk,can be written as follows: Rjk,(Sh-S'h') <aPSlhl1(bq-lS2h2cr S3h3)S4h4;Shl I (x) su(K) I aPSlh(bqS2h2c S3hS4h4; S'h' K1 Rjk(Sh-S'h',p) + Rjk (Sh-Sfh, qr) (C58) where Rjk(Sh-S'h',p) = (aPSlhl(bq- S2hh2crs3h3)4h4Shl I (K) (x) I laPsh (bq Shrc-l S;h;)Sh; S'h'h (C59)

103 and q-1 r pq+r Rk, h(Sh-S'h',qr) - aPSlhl(bq S2h2c S3h3)S4h4,ShIl SU(K)| K=p+p1 (x) IlaPS'h'(bqSh2cr shr-l,)Sh;Sh ) (C60) The Rjk(Sh-S'h',p) of (C59) is given in (C20), whereas Rjk, of (C60) is a new type of reduced matrix element to be examined in the following. Considering (C21) and (C31) one finds that Rk' (Sh-S'h',qr) = (-l)(S +S44 hl+h'+h) (x) 9) 2' (x) -1 \r 4 ~q+r (X) ((bq- S2h2C S3h3 )S4 hh411 SU(K) Ih(brShCc Sh)S h ) (x) UW(N4N4N, NN'N1) 6NN' (C61) where as in (C31) = -= [(2Si+l) (hi)] /2 (C62) and W/(NiNjN NkNNNm) = W (SiSjl, SkS2S) (x) (x) W (hihjT1, hkhkhm) - W [ij]W _ [ i _j1 (C63)

104 The dimensions (hi) of hi in (C62) are 1, 1, 2, 3 or 3 for hi = A1, A2, E, T1 or T2 irreducible representations of the cubic group, W and W are related to the 6J symbols as defined in (C32) - (C35). The last term to be determined in (C61) is: Rjik, q-l, r; q, r-l ](bq lS2h2, crS3h3)S4h411 U(K)l r-I~K1 (x) Il(bqs'h'c Sh')S h' (C64) 2 2 3 3 4h4 Before considering Rjk,(q-l,r,q,r-l), a preliminary investigation of the permutational part of the simplification of the bra (Z'| of bq-lcr and the ket IZ of bqcr-l is helpful. For any single electron operator, F = Zkf (k), the matrix element between(Z'I= ((bq-lcrl and Iz) = lbqcr l) may be described as follows: z>) [q!(r-l)' (q + r-l)!-1/ 2 (-l)Pb ). icr-l,>) (z'I = [(q-l)! r! (q + r-l)!] 1/2 (-l)V Pv (bq 1 IcrB V Then, (Z'IFIZ) [q!(r-l)!(q+r-l)!] 1/2 (Z'I (-1)" P Flbqa cr-l 1,) 1j,v (x) [qI (q-l)! r! (r-l)! (q+r-l) I (q+r-l) I] 1/2 (C65)

105 where(bq-la'l (cr B is a simple product. The next step is to express the jbqa) and IcrB> as function of their coefficients of fractional parentages: |bqa) =- E(b a# l 9i'b 3 bqa lb qbsla b, a) C66) I r c ( c crl'' Iccr,c, (C67) where a,B,a' and O' in (C66) - (C67) denote the characterizing symbols of bq ),.....cr), such as Sh' in (C61) for lb ). Substituting in (C65), we find: (Z' FIZ)= [q!(r-l)!(q + r-l)!)l P p (x) (c (Ic cr-l,). (c.cr-lt,qi-.(b'latlP Fbll a".t,a) (x) Ic r f) (bq a", b}b a) (q/r) Considering that q + r-l F = f(k) K=-1 one has Z P F = (q + r-l)! F uI

106 Substituting in (C68) one has V 2v'3'' v c ),C' (x) cco r-1,11,, (bq-1a" IF| b'-1a" b,) Ic r-'> (x bq- 1, *b b qa (q/r) 1/2 a''E [ q! (r~>b-,, q- q Z!, q] - B1 qZ~< r r-1q 1 (~ br-l6,, a'b I-1awlF(q)lb all,' b, a)Ic 6') (q/r) / = q!(r-1)! -1 6t P c (aB- 1, ( rb c-b') (x) cf b) (r-l)! 6b1811(q-l)!6aa'(q/r) /2,,) (qr )1 /2/2' rr <O, r|}bb (b cjfIb Thus <b,'aX crr -' f(K) lb, cr ) (q)112 (bq-1at bba)Kcr ccrQ) (clfib) r 69) \qr- -, b bqc f /. qI (r-l) I 8 Ic~~~~~~c 6'' a C6,

107 We now apply (C69) to obtain q-1 r q+r L = ((bq- ls2h2,cCS3h3) SU4h4I su(K) |j (x) ||(bqS'h', cr-l Sth') S'h' > (C70) S2h29 3 3 4s4 C= (_-1)S4-M4 bq-l S2h2c S3h3h S4h4Me I su(k) lbs;h; iM4M04 0'4 (x) cr- S'h', S'hM'o'> V (S4S 1,-M M'i) V (h4h' T, o0'-i). Here (bPq S2h2c S3h3;s4h4M4 1 =0 ( <S2S3M2M3S2S3S4M ) 2 2M3M 3 020203 11 (x)h 2h30203 I h2h3h4> )(bq-lS2h2M2021 rS3h3 c,c-l SxhNM0) K1/2S~NM I 1/2SS3M3) (chV0o lchh303 ) (cr-l s'h';N; (c muj (C72) 3raM3 3 3 3 Also, bqSh'c r-lsh3 4h.4. 4 4> < S(S;M'MI S2SM 4 )S M202M3of 2 2 38'

1o8 (x)(hh''e'O h'h'h ) (b l S 2h, b m''| bqS2h2 (S21/2M2m'' S21/2S'M' h2b>2' I h2bh') I>bm'u') Icr'l;h;M;>e) (x) Ib q-1S h2Me 2 Substituting in (C71), one finds: h4-84 S4-M4 iM 4 2 3 3 2M 3m' 044 2e33" 2 2 3 (qr) 1/2 V(S4SI1,-M4MI-i) (x) (S232M31S2S3S4M4)(1/2 SIm MI 1/2S3S3M3) (X) (SISMIM;I S2S3SM4)((S21/2M2m' tSI1/2SIM) (X) V(hh4Tli e46e i) (x) (h<h3 2 31h2h3h4 4 ( ch; o3l chlh3 0) (x) 2 3 2(3 Ihhh3h4) ( h2b32' I h2bh23 2 ) (x) (cr S3h3(C c Sh) bK S2h2b3b S2h2>

109 (x)(b q-1S2h2M22M (cr 1Slh'MO I (cmi|su| (x) Ibm'l' ) Icr S;h;M3 3) 3 b3 lS2h2M2 2) (C74) The individual coefficients appearing in (C74) can be simplified further as follows: (cmllsulb m'')= (_1)1/2-m V (1/2 1/2 1, -m m'-i) [-1]C-p (x) V (c b T1, -i. P'i)(1/2 c|IsuIll/2 b ( ssMiMj ISiSjSkMk) (2Sk+l)1/2(-l) 2+ sk-S (X) V (SiSjSk' MiMJ-Mk) and h ihhiJl |hjhkhk> = (hk)/2 V(hihjhk 00ijek) i ihjk i Substituting for coefficients in (C74), we find L S2-S + St -S3+ hl+h2+h2 (-3 3 2 2 iM 4M2M3MM'M'Mm' 4e2e3" 0' 3 02 3 (qr) l/2 4 p2 3 <crS3h3 IC c rl Sh3) <b 1S2h2bbqS2 h>

110 (X) V (S4S41,-M4M4-i) V(S2S3S4;M2M3-M4) V (1/2S3,-m-M+M3) (x) V (S'S'S,-M2-M3+Mt) V (S21/2S2,M2m-MX) V (1/2 1/2 1, -m m'-i) (x) V (h4h'Tle4e4i) V (h2h3h4) V (c h3h3, ee303) (x) V (h'h'h''0e' V (h2b h' 02' O') V (c b T,' 0 i) (C75) S2-S2 + S3-S3 + h2 + h2 + h3(qr)l/2 pp54, = (-1) 2 3 4 4' (x) q-lS2h2, b 3b S'h)cr 3h3(ccr 1Sh ) <1/2 cl Isul 1l/2 b) (x) Z V (S2.S3S4' M2M3-M) V ( S3St 1/2, M3-M3-m) iM4M2M3m 4M m' M4M4(x) V (S2S3S4,-M —M;M') V (S4S1i,-M4M4-i) (x) V (S2S2 1/2,-M]M'm') V (1/2 1/2 1, m'-m-i) 2~ ~~~~~~~~~~~~~~~~~~~~~~~ 2.

111 (x) V(h2h3h 4.020304) V ( h3h'c, 0COM) V (h2h3h', 081 ) P 020304i (x) V (hh'T1,848'i) V (h2h'b,2 e2) V (cbTl Vp i) (C76) The sums in (C76) are the same as the 9-j symbol* and defined as: X [abc, def, ghk] = CEay6Ebn0K V (abc,aBy) V (def,6de) (x) (ghk,nOK)-V (adg,a6n),V (geh,ce0).V (cfk,yK) (C77) Substituting for the sums in (C76) and recalling from (C64) and (C70) that, Rjk,(q-l,r; q, r-l) L, one has S -S'+S'-S +h2+h2+h3+h3' 2 2 3 3 2 2 3 3 1/2 Rjk,(q-lr;qr-1) - (-1) (qr)+S 3 K< blSh b3 ) c h(cS2 hc2 c S 3h3 ) (X) D ~ x [N N N, N'N'N',N N N 1 2 3 44 2 3 4 2 3 4 b c o (x) 1/2c Isu I 1/2b) (C78) where X[NiNjNk, N NNmNn NbNcNo] = X [SiSjSk, S SmSn, 1/2 1/2 1] (x) X [hihjhk, hghmh, b c T1] (C79) They are also called X coefficients (See Ref. 26)

112 and the remaining coefficients have their usual meanings. Substituting in (C61) and taking into account (C58) - (C60), one finds (aPSlhl (bq-S2h 2, crS3h3) S4h4, Shjl E < su(K) (x) |laPS'h' (bqSth, c Shr-l S ) S'h';S'h' = Rjk' (Sh-h) = Rjk(Sh-Stg, p-p) + Rjk,[ps(q-l,r,Sh)-(q,r-l, Sh)] - () S14S4+S+hl+Sh4+h Of' W/(N'N4N NN'N ) (x) GP (Slhl-S'hi) <l/2a|Isul|1/2a'> 6N N' S1+S4+S+hl+h4+h 4+ (-1) 1 4' W/(NN4No, NN'N1)6Nx N S) (-1) 2+S3-s3+h2+h2 +h+h3+h3 1/2 (qr) (x)<bq-ls2h2, bI b}qS) h crSh> (K( rcr S3h3) (x)'P~PP' X [N2N3N N'NN'', NbN N] ) 2 3 4 4 23U4 1/2 b) (C80) (x) (l/2cIIsulll/2 b)o (C80)

11.3 The complex conjugate of etero-configuration; three orbital reduced matrix elements can be obtained from (C80) by appropriate symmetry transformation on W and X coefficients in this equation. The result is: Ri. J(S'h'-Sh) = (aPSlhl (b Sh2 crl S'h) S'h1;S'h' Esu(K) (x) I laPslhl(bq-ls2h 2crs3h3) S4h4, Sh) = Rt(PS'h'-sh) + Rgj[p, (q,r-l,s'h')-(q-l,r,Sh)] (C81) where S'+S,+sh'{+h4+h Rkj (p, s'h'-Sh)=(-l),4 1 4 (N Nl,N'NN4) (x) Ga (Sh;-Slhl) l/2a llsull /2a) 6dNN (C82) and S'' S' +hi+h +h' Rkj [p, (q,r-1,s'h') - (q-l,r, sh)] = (-1) ) O' * J(N4N No, N'NN{) (qr)1/2 (x)bW S (bqS bb1 2h2j K<cc rlSf3h Sh3)Kl/2b Isul l/2c) (x)2P344 X2[N2N3N N 2NN4, NbN3N4 Nbn] 6. (C83)

114 Recalling (C56) - (C57), one finds that in both X and Xk the ap parts of the system appears first and then bq, bq-1 and c, c parts. Moreover, part bq and cr-1 are always coupled together, first, and then their results are coupled to aP. The desired matrix elements are not, however, arranged in this fashion and appropriate recouplings and couplings are needed to bring the three participating parts of the system in the above form. This has been done by using the following formula: [1 h' 2 S S3h3 h u() 1 2 2S C C 3 3' S +S 2+S3+S+S'+S'+S'+S-h +h 2+h +h +h' +h'+h+h' =?Z (-1) 23 123 123123 S S'h h' (W) E E' ~ U (N.N2N, NN ) ~ W(N.N'N'N3N'N' ) EE C 1 2 3 2 c3 (x) (<Shl(S2h2S3h3) S Shl, s su(K)s I'Sh'(Sh'S'h),Sh,S'h' ) h S S2 2 3h3 3,S, h I (C84) For the spin sextets, the sum reduces to one term because there is only one S hg and one S'h$ which results in the same Sh and S'h'. The values of Rjk' are given in Table 3-4. Now, we consider the problem of four orbital reduced matrix elements. 6. HETERO-CONFIGURATION FOUR ORBITAL REDUCED MATRIX ELEMENTS Rjk,, (pqrs-lSh,pqr-ls S'h') Here, we consider the reduced matrix elements Rjk,, between states

115 Ij) and 1k") both having four orbitals of the same symmetry. Two of these have the same configuration p and q in both I > and Ik">, whereas the remaining two have configurations r and s-1 in IJ) and r-l and s in Ik">. Therefore, Rjk,, [(pqr s-l) Sh- (pqr-ls) S'h'] <Xj(pqrs-1)Sh|| E su(k)||xkl,(pqr-l,s)S'h') k (C85) where IXj (pqrs-l)Sh> (aPSlhlbqS2h2)s3h3(crs4h4dS-ls5h5)s6h6;Sh (C86) and IXk.(Pqr-ls) S'h'> = 1(aPShbqSh2)Sh3(cr S4h4sS5h5)S6h6;Sh) - l 2 2 3 h3b 4h 4 4sS 5 6 (c87) All states can be arranged according to 1xj(pqrs-1)sh) and Xk,,(pqr-l,s)S'h' )by transformation similar to (C84). Hence the remaining calculations will be limited to the determination of Rjk,,[(pqrs-1)Sh-(pqr-ls) Sth3. Using (C16) - (C17), one decomposes Rjk,,[(pqr,s-l) h-(pqr-ls)sh'] in terms of Rjk(pq) and Rjk,(r,s). Rjk(p,q) and Rjk,(rs) are, respectively, similar to the Rjk(qr) given in (C21) and the Rjk,(qr) given in (C61), except for the subscripts of various spin operators Si and irreducible representations hi. Taking this into account, one can immediately write down the Rjk,, as follows: Rjk,,[(pqrs-l) h-(pqr-ls)S'h'] = Rjk,(pqrsSh-S'h')

116 (x)(aP Shb S2h2) S3h3 (cZqr 1 sapS h bqS h2) S h3(c S h d S h5) S h6;Sh[I I:su(K)[I (x)ll(aPS'h'b q S'h';Slhl Slhld sSSh;')Shl;S'h S 1S2 +S3 S 6+S 3 +S'+ hlS+hh+h 3+h ++h' 33 (x) /W(N'N1No N3N3N2) U(N3N3No, NN'N6) (x) GP(S lhl-Slh) (1/2alsul 1l/2a 6N2N' 6N6 a 2N2 6N6 S+ +S2+2 S3+S+SS'+hl+h+h'+h'.V'.3' (x) WN;N2No,N3N3N1)' /NN3NoNN'N6) (x) Gb(S2h2-Slh2) (1/2bl sull /2b) 6N N' 6N N 6N6 S3 S6+ S+h 3+h'+h 1 (x) (cc S'hj ~ crS4h4> d Sh ddSh)5 4 4 4 4 ~~~ ~~5h5 5

117 (x)W (N N N, NN'N ) ~[(N4N5N6, N4N5N6 dNo) (x),1/2cj suj 1l/2d', 6 N N' (C88) 33 The matrix elements R jklare given in Table 3.5.

APPENDIX D COUPLING COEFFICIENTS OF SPIN ORBIT MATRIX ELEMENTS TO THEIR REDUCED MATRIX ELEMENTS The purpose of this Appendix is to discuss the relationships between the matrix elements of spin orbit interaction between a pair or state and its corresponding reduced matrix element between the same states. The states which are suitable for calculation of spin orbit interaction are those behaving as the irreducible representations t of the spinor group. The spin orbit matrix elements arising from IXjShjtT> and IXkS'h'J't'T>) of the two states IxjSh) and IXkS'h')can be expressed as (3.13) (XjShJtl[zK su(K)IXkS'h'J't't') (XjSh IE SU(K) IXkS h')KJJ (SS'T1, h'ht) tt' 6TT (D1) Considering (5.22) and (2.20) of Ref. 26 one has (XShJtT I.[(..i) I (S) T] AzX'S'hJ''tT) (XShJrllz iaB[(i PIa T (sI) Tl All X'S'h'J't'T' (x) V(tt' A1, TT' y) XSh ia}[i aP T1 (si) T1] Al IX'S'h'J't'> -1/2 [(t)] 6tt, 6TT 118

119 The <XjShj K su(K) I IXkS'h') was discussed in Appendix C. Here we focus our attention on KJJ,. This coefficient may be also called the spin orbit matrix coupling coefficient or simply S-O matrix coupling coefficient. Moreover, it is written in several different forms as occasion demands. These are: KJJ, KJJ, (SS'T1, h'ht) - KJJ hSS'T) (D2) The coefficient KJJ, is obtained from the formula (Ref. 26, p. 82) S-M' +1 h+8 KJJ, (SS'T1, h'ht) = L (L1)S-M+ [-1] rMM' 06 (x) V (SS'1, -MM'r) V (hh'T1, -80'-r) (x) (ShJtlIShMO ).(S'h'M''IS'h'J'tT) (D3) The numerical values of KJJ, are given in Tables (3.9) - (3.12). The cases where h = Al and h = h' = T1 are of particular importance for evaluation of spin orbit matrix elements between ground state A1 of (3d)5 6S ions in crystals and charge transfer states and will be examined in more detail as follows. 1. DETERMINATION OF KJJ, (SS'T1, h'ht) for h = Al Here, we consider the coupling coefficient which relates the matrix elements of the spin orbit interaction between the ground state Al and charge transfer excited states to its corresponding reduced matrix elements.

120 We represent the above charge transfer excited state by IX'S'h'J'tT> where, as before, X', S' and h' are, respectively, the electronic configuration, spin, and irreducible representation of the cubic group of this state and J', t and T are pseudo-angular momentum, irreducible representation of the state in spinor group and its component, respectively. Instead of the ground state 6A1, 5/2, the state IS Al J T> will be used and the result will be applied to the particular case of 16A1, 5/2 >). Considering (D3), we have KJJ, (SS'T1, h'Alt) = (_1)SM +l [_1]h+e V S' rMM' 0' (x) V(l <S A1JtISAMi) <S'h'M''IS'h'JtT) (x) V i e-r 1 Considering Griffith17, p. 117, gives: A h'T1 1 also(p.r 77)T alsol (p. 77) SS) 2S'+S-M 1 (lS'rM'IlS'SM) -MM~r (2S+1)

121 Thus KJJ, (SS'T1, h'Alt) = (-1)S-M [-1]A+i (T1) (2S+1) rMM h 8'r eeMM' (x) (<S'rM' I1S'SM>)<S A1JtTISAM)><S'h'M'' IS'h'J'tT> *\1) 1 1(-1) 1S'rM'I1S'SM (T1) (2S+1) (-1) MM'r (x) (S A1JtrIlS A1M)><S'T1M'rlS'h'J't'T> T1 1 (/1) \S+LiE <SAJt A1Me\> JI (_JL)-M T1 (2s+l) M M"M' r (x) 1<S'rM' I1S'SM><l)(1S'rM' I1S'J'M' (J'M'J'tT = (T) (2+l (-1) 1 SAJtTIS A1 M)(JM" I'It' (-1) -M' IM'' M' r (x) <lSrM' iS SM S'rM' I1S'J'M''M ) 2(11 (2YK1 S <S A1JtT lS AM M)(J'M'J' 6MM 6SJ1' 6 (1) (T* (2S+1) SJh'T

122 Therefore SS'T1 KJJ' h'Alt 3(2S+) 6SJ 6SJ' 6h'T (D4) Several important conclusions may be drawn from (D4): (i) The spin orbit interactions couples the ground states only to excited states I 6T1 5/2t and Ix 4T 5/2t(ii) The matrix element is independent of t and, as a result of this, no splitting will occur from a second order perturbation. (iii) The matrix elements are independent of S' and thus, the energy shift resulting from Tli and Tli depends only on their reduced matrix elements. The next important coupling coefficient to determine is between lXiSh> and Ix'iS'h') where h, h' have T1, symmetry. 2. DETERMINATION OF KJJ, (SS'T1, h'ht) for h = h' = T The coupling coefficients relating a matrix element of the spin orbit interaction between a pair of charge transfer excited states whose irreducible representation in the cubic point group is T1 will be analyzed in this section. The importance of considering this coefficient is apparent from (D4): KjJ, (SS'T1, h Alt) = 1/3 (2S+1) J SJ' 6 (SS'TJ, ~ 1 ~SJ SJ' h'T1 which indicates that the ground state A1 couples to charge transfer excited states of T1 symmetry alone and is not split by that. Consequently, a splitting by spin orbit interaction of the ground state 6A1 of the complex [Z A4] may occur through higher than second order

123 perturbation and through the intermediary states, two of which, at least, must have symmetry T1. Therefore, determination of KJJ, related to such states of T1 symmetry deserves particular attention and we begin by studying KJJ, (SS'T1, T1Tlt) as follows (D3): JJrMM' co' (-1) M [_1] V I SS'T'\ I (1S M+ 1]Tl+ /SS',Sl i\ TlTlt k 1 1MM) =[-M' r -0e'-r (x) (ST1JtTIST1M0) ~ (S'T1M'O' IS'T1J'tT) where Ss'1 \ T1T1T1 ll - -r! (S2S+'lM'r|SM); V = V K-MM' r = S1 V -'r ee (ShJtlShMe> = I (ShJtTIJM''> (JM''IS1M>e M'T (S'h'M'e' Is'h'J'tT>= I (S'h'M'e' IS'J'MT'' J'M''' IS'lJ'tT Thus SS'Tl SS'1 (111 S'J' S1J (X) V M M'' M V M-OM'' W S (S1Jtrl:M") JlitT1 Jr ML

124 E Z Z l|-eer m VfmlMI V IM m-m M'M''' MM "r ee' (x) Y/ i2,+1) c2J+) <S1Jt IJM ><,J,, MI s1J, t~> 2J+1 } M M, SS' (2J'+1)(2J+1) (x) (<SJtT I|JM' (J'M''I J'tT) W SS'J <JtT IJM' i JM1 IJtT = W SSJ MT

APPENDIX E FOURTH ORDER PERTURBATION This Appendix gives the formulae necessary for the evaluation of the fourth order correction to the energy of the degnerate state 6A1 of the complex* [E A4]-n. In Chapter IV it was shown that the fourth order is the lowest order of spin-orbit perturbation of 6A1 by charge transfer states which can lift the degeneracy of 6A1 and contribute to the cubic field splitting 3a. Therefore, to determine 3a one must employ fourth order perturbation formulae. These formulae can be obtained from the general expressions of nth order perturbation given by Corson+ En(s) Hr = <HrsVs Hjk> HkIKrs; n> n-2 - E E (s) Hr<(HrslKrs; n-v> j + r (El) v=2 where IHr'S), IHJk)> are respectively the ground and excited states being involved in evaluation of E (s) Hr, the superscripts s and k in IHr'S> and IHj'k> designate the sth and kth degenerate states belonging to the energy levels Hr and Hj, See Appendix A E. M. Corson "Perturbation Methods in Quantum Mechanics of n-Electron," Hofner, New York (1951) p. 75 125

126 <Hjkkvr>(j klH mHKrss; n-l) Hi _Hr Q,m n- E (s)'r (HjklKr'; n-v> + V j' r (E2) v=l and <Hj,kKrs, 1>= (Hr - H)-1 Hj'kVIHr'sH > (E3) Substituting in (El) - (E2) one obtains the desired E (s. The A1 -n 1 ground state of the complex [E A4]n, behaves as angular momentum J = 5/2. The irreducible representations of J ='5/2 in the cubic double group are E'' and U'. Thus E (s) 6A E(n)(s); s = U' or E'' (E4) n 1 For determination of the cubic field splitting, 3a, up to fourth order perturbation, a much simpler formulation is enough as will be seen below. 3a is the difference of the energy corrections E(4) (U') and E(4) (E''), 3a = E(4) (U') - E( (E''), (E5) and fourth order perturbation is the lowest one giving rise to such splitting. Thus all terms containing E (s) Hr = E()(s) will vanish and the only contributing terms to (E5) are obtainable from the general formula

127 E(4) (s) - E(4) (s) = E4(S) Hr E4(s' )H r -= I [(HrSv'II k>HJ'kKr s,3) jk -<r s IVIHj,k)KH ~kIKr,s 3)] (E6) where <H'|jkKr'S(H's VIH IK S,2 Q m Hi - Hr Hi - H - n-i + ~ ~ Hr(HiJ k r~s; 3v) (E7) v=l and (H'm IKrs 2 )= <H'V HmlVIHP'q><HP'q VIH' s) P,q (HP Hr)(Hr H) + E.(s) Hr <HP'qKr, 1 (E8) 1 j r H HH

128 Substituting in (E6) and eliminating terms having E(s) one immediately finds that: 3a = - Z {(Hr'U IVIHJ'k>Hj'k IVIH km>HmlVIHPq> jk; Qm,pq (x) HP qlvIHrU'> [(Hi - Hr)(H- Hr)(HP - Hr)]l1 <Hrg'' IVI HJ k> <Hj,k IVI km) (HmlIVIHPq> (x) <HPqIVIHr'E)'>[(Hj _) Hr )(HP - H )]l} (E9) or more simply 3a = -(E.E )-lv(rU',jk) V(jk,Qm) V(km,pq) V(pq,rU') j p, kmq V (rE'',jk) V(jk,Qm) V(Qm,pq) V(pq,rE'') (E10) where V(rU',jk) (=HrU' IVHk) and i r E. H - H i = j,k,p (Ell)

129 Since spin orbit interaction is diagonal in U' and E'' we can substitute for k m and q in (El0) JU', J U'... J E'' (E12) k m q

APPENDIX F SPIN-ORBIT MATRIX ELEMENTS BETWEEN Ix ShM0) AND x'S'h'M'W' FOR S = S' In this Appendix we consider a different method of finding spinorbit matrix elements which is applicable to pair of states of the same spin value, S = S'. This technique is particularly useful for the evaluation of the contribution to the cubic field splitting 3a from the spin sextuplets of charge transfer states. Since S = 5/2 for all states it can be considered as a constant and integration to be carried out over the orbital part of spin-orbit Hamiltonian only. Thus, instead of bases of the spinor group, IJtT), we choose the bases Ih0) of the single valued cubic group for the evaluation of the matrix elements. Following (9.26) of Ref. 16 and considering the fact that for S = 5/2, there is only one state Ihe) for any of the charge transfer states given in Table 3.2 one can describe the spin-orbit matrix elements in the IShM0) scheme as follows: X ShMl ZE - I.s IXShime2 i Z x Sh K.liZiX'Sh'M)Ke Sh'Me'IslX'Sh'M'?'? (F1) The vector si in (F1) can be replaced as follows* S(S + 1) (X Sh'Me' _siX'Sh'M'e') =(X Sh'Me' S[X'TSh'M'e'e <'Sh'M' e' Ii'.SIX'Sh'M' (F2) E.U. Condon and G.hi. Shortley, "The Theory of Atomic Spectra," Cambridge University Press (1959) p. 61 130

131 where i i i i s S = s s + s S' = (1/2)(3/2) + (1/2) [S(S + 1) - (1/2)(3/2) - (S - 1/2)(S + 1/2)] = (1/2)(S + 1). (F3) Substituting in (F1) and (F2) one finds (X Sh'M' LsiIX'Sh'M''e) = (/2) (S 1) x Sh'Me'SLIX'Sh'M'e') = (X Sh'M0' (2S)-SISh'M'e) (F4) and X ShMe I i- s1IX Sh'M'?') = (2s) 1(X ShMeISix'ShM'e) *E(X ShM'OB ie iIShfM'e') (F5) To obtain the cubic field splitting from these matrix elements one carries a fourth order perturbation calculation on one of the components of S such as S and selects those states liie> which would result to a nonX 4 vanishing term bS. Comparing this term with the spin Hamiltonian given x in (2.1) one immediately finds that the contribution from the spin sextuplets of the charge transfer states to 3a is [3a (a, H)]s = 18 bj (F6)

132 where j covers all fourth order perturbation channels giving rise to terms of b.S 4. Our initial resultsl5 were obtained by this very simple j x technique. The disadvantage of this method is its limitation to a fixed manifold of spin S and, consequently, it is not applicable to spin quadruplets and doublets of the complex [E A4] which has a ground state spin S = 5/2. Moreover, in this technique an apriori knowledge of the spin-Hamiltonian is necessary which is in contrast to the method described in the text.

APPENDIX G COVALENCY DEPENDENCE OF THE CHARGE TRANSFER CONTRIBUTION TO THE CUBIC FIELD SPLITTING 3a (a,H) In this Appendix, we examine the dependence of the cubic field splitting, 3a(a,l) on the covalency of the molecular orbitals involved. Recalling (4.6) one has (4)-i 4~i i 3a(ca,f) = 3a(4) = 4 C i (G1) i=O C d p Here, Cd is the spin orbit interaction constant of the d orbitals of the S-state ion (e.g. Mn ), Cp is the spin orbit interaction constant of the p orbitals of the surrounding ligands [e.g. S in ZnS:Mn] and C. are functions of the coefficients of linear combinations of atomic d and p orbitals. Now, a question arises on the nature of the coefficients Ci in the limit of ionic approximation where the coefficients of ligand orbitals vanish. To investigate this we examine 3a(a,~;tl + ea) of (5.21) and 3a(a,f;tb + ea) of (5.25). The first one is 3a(a,;t + ea) = - (9/1250)[61 (61 + E1)] 2 P (G2) a 1 1 p Comparing (G2) with (G1) one finds for 3a(o,T;tl+ ea) C =C C2 C2 = 0 133

134 and C4 = -(9/1250)[612 (61 + E1)] 2 (G3) where 61 and 61 + e1 are energies required for a ligand to metal electron transfer as shown in Fig. 5.3 and 8 is the covalency of the de orbitals of the S-state ion as described in (3.16). For an S-state ion in a II-IV compound with higher ionicity 61 increases whereas f decreases. Thus at the ionic limit where B2 - 0 one has Lim C4 0+ (G4).2 + O and consequently Lim 3a(o,r;tl + ea) + 0 (G5) +2 + O Now we examine 3a(o,rr;tb e ). The coefficients Ci of this term are given in (5.25) as follows: CO= [(18/625)[61 (61'')] 6 a ] Cl = [(18/625)[6(1 + (1 6 a2 K 2 6 a K ) A C2 = (18/625) [612(61 +?1)] (3 a2 /2+ K 2 2 [ (1/62)[61 + 12) a

135 and C4 [(18/625) [61'2(61 + E1 (4) K 1 At the ionic limit both B and X approach to zero whereas 61' and 61' + s1' increase. Thus Lim C i+ 0; i = 0, 1,..., 4 (G6) B, X + 0 and consequently Lim 3a(o,r;tb + ea) + 0 (G7) $, +0 The vanishing of 3a(a,r,t + ea) and 3a(a,r,tb + ea) follows from the fact that in each term contributing to 3a(a,w) of (G1) there is at least one spin orbit matrix element of the type t tl, t aeb and tatb where ab = (1/2 afj'sJJl/2 b); a, b = ea. ta, tl, tb, eb The reduced matrix elements ab are given in Table 3.13. It is evident from this Table that all such reduced matrix elements contain 8 and or i. Both of these vanish at ionic approximation and consequently all contributions to 3a(a,r) of (G1) vanish at the ionic limit as expected.

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