ERRATA p. 4: Add the following note at the end: Note: The ground state is often referred to as being the x2-y2 In the context of the molecular orbital theory as used in this thesis, the term "ground state'" refers to the energetically highest occupied orbital in the Slater determinant. p. 29 1.9: Replace Table 2 by Table 3 p. 31 1.5: Delete the whole line and replace it by the following: are tabulated as well as some two-center integrals. For references see p. 31 1.6: Replace Ref. 35 by Ref. 24.?. 63 bottom: Replace Table 6 by Table 7 p. 103 1,5:Delete the word "nearest" p. 104: The level at the middle af the first colum is yz, not xzo p. 105 1.5: Replace Helmheltz by Helmholtz

THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report MOLECULAR ORBITAL THEORY OF VANADIUM IN THE RUTILE STRUCTURE CRYSTALS Sn02, TiO2 AND GeO2 Sophocles Karavelas Chihiro Kikuchi ORA. Project 04381 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NO. NSG-115-61 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR July 1964

This report was also a dissertation submitted by the first author in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1964.

TABLE OF CONTENTS Page LIST OF TABLES.......................... v LIST OF FIGURES......................... vi ABSTRACT............................. ix Chapter I. INTRODUCTION.....................1 II. RUTILE CRYSTAL STRUCTURE AND SYMMETRY.......... 5 1. Crystal Structure 2. Additional Assumptions 3. Oxygen sp2 Hybrid Orbitals 4. Reduction of the Secular Equation by Group Theory III. LINEAR COMBINATIONS OF ATOMIC ORBITALS TRANSFORMING 20 ACCORDING TO THE I.R.'S OF THE D GROUP........ at IV. COMPONENTS OF THE SECULAR EQUATIONS........... 24 1. Group Overlap Integrals 2. Two-Center Overlap Integrals 3. Energy Matrix Elements V. SOLUTION OF THE SECULAR EQUATION.......... 43 1. Energy Eigenvalues 2. Charge Self-Consistency 3. Detailed Charge Self-Consistency VI. THE ELECTRONIC C AND A TENSORS........... 61 1. The Electronic - Tensor as a Monitor 2. The Hyperfine Interaction as a Monitor VII. DISCUSSION AND CONCLUSIONS............... 77 1. Discussion 2. Conclusions iii

Page APPENDICES........... 84 A. General Theory B. Variational Method Applied to Linear Functions C. D2., Group Character Table D. Two-Center Overlap Integrals E. Energy Eigenvalues of SnO2:V, TiO2:V, GeO2:V F. Ground States of SnO2:V, TiO:V, GeO2:V G. Point Charge Crystalline Fie d Calculation of SnO2:V H1. Valence State, VSIP, Valency and Promotion Energy I. Effect of Ligand Wave Function Admixture in the Calculation of the Orbital Zeeman Matrix Elements J. MAD Program for Solving the Secular Equations K. Mo5+ and W5+ in TiO2 REFERENCES............................ 114 iv

LIST OF TABLES Table Page 1. Crystallographic Data of Sn02, TiO2, and 7 GeO2......... 2. Coefficients of the Oxygen Hybrid Orbital......... 13 3. LCAO Transforming According to the Irreducible Representations of the D2j Group.......... 22 4o Transformation of the Spherical Harmonics......... 25 5. Group Overlap Integrals.............. 32 6. VSIP of Vanadium and Oxygen Ions.......41 7. Eigenvalues and Eigenfunctions of Sn02:V for an Assumed Vanadium Charge of +.25e............ 50 8. Assumed and Calculated G' and 1T Orbital Occupancy............... 57 9. Experimental Deviations of e Tensors Components from the Free Electron Values for Vanadium in ~n02, TiO2 and GeO2 62 SnO2 TiO2, and GeO.................. 62 10. Observed and Calculated % Tensor Components of Vanadium in SnO2, TiO2, and GeO2. o..... 6 69 11. Experimental Hyperfine Tensor Components of Vanadium in SnO2, TiO2, and GeO2....... o...... 73 12. Anisotropic Part of the Hyperfine Tensor Components Deduced from Experiment......... 73 13. Calculated Relative Strength of the Magnetic Field at the Vanadium Nucleus Due to the Ground State Electronic Charge................. 75 14. Calculated Anisotropic Part of the Hyperfine Tensors Normalized to Az t.......... o.... 75 15. Reduction in /\; Values Due to Ligand Ojbital Parts at.40e Assumed Vanadium Charge and 35 Kcm TT -Electron VSIP Reduction o o............... 82 16o Crystal Field and Charge Transfer Contributions to the Tensor Components in SnOa2V in Units........ 83 v

LIST OF FIGURES Figure Page 1. Rutile-Type Crystal Structure.............. 6 2. Coordination of the Oxygen Ions in the Rutile-Type Structure...................... 12 3. Left-Handed Coordinate Systems Centered at the Oxygen Ligand Ions............ 15 4. Spheroidal Coordinate System............... 26 5. MO Energy Levels of Sn02:V................ 45 6. Central MO Energy Levels of SnO2:V............ 46 7. Central MO Energy Levels of TiO2:V........... 47 8. Central MO Energy Levels of GeO?:V............ 48 9. Fraction of the Electronic Charge of the First Thirteen MO's Assign to Vanadium............... 52 10. Calculated Vanadium Charge vs. the Assumed Charge.... 55 11. MO Energy Levels of SnO2:V for a -rr-Electron VSIP Reduction of 35 Kcm-l(left) and 45 Kcm-1 (right) ~. 58 12. Calculated Vanadium Charge vs. the Assumed Charge for a TT -electron VSIP Reduction of 35 Kcm-1 (left) and 45 Kcm- 1 (right)................... 59 13. Allowed Crystal Field and Charge Transfer Transitions.. 64 14. Calculated A Tensor Components of SnO2:V...... 68 15. Calculated A Tensor Components of TiO2:V... 68 16. Calculated ~ Tensor Components of GeO2:V.... 68 17. Calculated A Tensor Components of SnO2:V for a -1T-Electron VSIP Reduction of 35 Kcm-1 (lower) and 45 Kcm-1 (upper)........ 70 18. Calculated Tensor Components of TiO2:V for a -iT-Electron VSIP Reduction of 35 Kcm"1 (lower) and 45 Kcm1l (upper)............ 71 vi

Figure Page 19o Calculated l\ Tensor Components of GeO2:V for a -rr-Electron VSIP Reduction of 35 Kcm(lower) and 45 Kcm'- (upper).............. 71 vii

Abstract The purpose of this study is to investigate possible explanations of the observed ESR spectra in the rutile-type crystals of SnO2, TiO2, and GeO2, with vanadium as an impurity. The observed electronic 9, tensors cannot be accounted for by the simple crystal field theory, and there is ambiguity in the energy level diagram. Molecular orbital theory is used in a semiempirical calculation with the linear combination of atomic orbitals approximation (LCAO). The crystal region consisting of the vanadium ion and the six ligand oxygen ions is selected to be studied. The atomic orbitals to be used in the LCAO approximation are chosen. These are the nine vanadium 3d, 4s, 4p orbitals, and the twenty-four ligand oxygen 2s, 2p orbitals. It is argued that the hybridization of the oxygen orbitals is needed to account for the influence of the rest of the crystal, and these hybrid orbitals are constructed. Then it is shown how Group Theory can simplify the secular equation and formulate the selection rules. The use of group theory requires the construction of LCAO transforming according to the irreducible representations (I.R.) of the D2h group. These combinations are constructed by the method of projection operators. The group overlap integrals are calculated, and the valence state ionization potentials (VSIP) of the oxygen and vanadium ions are estimated. ix

Then the reduced secular equations are solved using the VSIP as parameters. The best solutions are selected by using as a monitor either the vanadium charge or the T tensors. The results are compared and the need of a reduction in the original estimate of the -<-electron VSIP is recognized. Also, it is concluded that charge self-consistency alone is not adequate in selecting the best solutions, but rather the detailed charge distribution among the orbitals must show selfconsistency. Finally, the following results are given: (a) The ground state in all cases is of the form - csi i - - where o10 cx <.20 and P.99. (b) The pertinent to the, tensor levels are in order of increasing energy, the xy (filled), xz (filled), yz (filled), x -y (ground state with one unpaired electron), yz (empty), xz (empty), and xy (empty). (c) The small admixture of the I|> state in the ground state is relatively important in calculating - q ( L (d) The ground state in (a) can explain the anisotropic part of the hyperfine tensors. (e) All obtained results seem to agree with the recently observed 5+ 5+ ESR spectra of Mo and W in TiO2. x

CHAPTER I INTRODUCTION The purpose of this study is to investigate possible explanations of the observed ESR spectra in the rutile-type crystals of SnO2, TiO2, and GeO2 with vanadium as an impurity. Gerritsen and Lewis were the first to study the paramagnetic spectrum of vanadium in Ti2o They attributed the spectrum to a single 3d electron of tetravalent vanadium occupying a substitutional site in the crystal. The experimentally deduced ~ tensor ( Q - 1.915, X Q9 1.913, 0 1.956) had almost complete axial symmetry about the z-axis, and the application of the theory of Abragam and Pryce was expected to give the splitting of the lower triplet, to lowever, calculation revealed inconsistencies, since the calculated splittings differed by a factor of two-and-a-half depending on whether - lor Qt c_ Q was usedo They commented that perhaps the rhombic component of the crystalline field was responsible / 40 for making the theory inapplicable. An attempt by Rei to account for 4+ the Q and A tensors of V in rutile by considering the rhombic com41 ponent of the field and the covalent bonding proposed by Stevens was 3 not successful either. Later Marley and MacAvoy observed the ESR 1

2 spectrum of vanadium in SnO2. Again the ESR spectrum was attributed to substitutional vanadium, but the tensor showed nearly axial symmetry around the y axis; they tried to interpret this by the improbable assumption that the doublet e. lies lower than the triplet t~. *In 1963, Kasai suggested that the rhombic part of the crystalline field has an important role in splitting the lower triplet t 1-~~~ ~% % i. It was also argued that the X - y state would be the lowest due to the stabilizing effect of the two tin ions lying on the 5 y -axis close to the vanadium ion. In 1964, From, Kikuchi and Dorain investigated the large superhyperfine structure in SnO2:V. They concurred that the ground state is X - - and that the next level is the state X E. The latter was arrived at by fitting the observed Q values to the usual crystalline field formulas: /Aq _ _x,/ =_, -. / _ However, a point-charge-crystalline-field-type calculation showed (see Appendix G) that the levels are in the order >X- yX y and < y 6 The same year, Siegel observed the ESR spectrum of vanadium in tetragonal GeO2 and obtained values comparable to that in TiO2 and equally difficult to interpret. Since SnO2 is an important material in the production of conducting glasses, and GeO2 exhibits both the crystalline and amorphous state related to glasses, the study of the electronic behavior of the

3 vanadium impurity may reveal important properties of conducting glasses. On the other hand, all three materials may have important quantum electronic properties. Theoretically, the interest is equally great because the spectra are due to a single unpaired electron, which can be thought of as the simplest case of magnetismo The principal results reported in this thesis are: (a) The ground state is of the form - o( | > +- P\ - yip where lO(O< <'.20 and pt.99. (b) The excited states are in the order of increasing energy: ji/^ |<X1> jIy) e^ad |9 The first two states are inverted with respect to the prediction based on the simple crystalline field formulas. (c) The observed tensors and the anisotropic parts of the hyperfine tensors A can be explained. (d) Results are applicable to the cases of Mod and W+ in rutile. 4+ 5+ 5+ (The electronic structures of V, Mo5, and W are similar, with a 3d, 4d, or 5d unpaired electron outside filled shells respectivelyo) The general theory is given in Appendix A. Chapter II states the assumptions needed to make the calculation feasible. The difficult problem of how to restrict the calculation to a limited number of ions surrounding the impurity ion and still get reliable results is con2 sidered. Only the nearest neighbors are taken into account for the sp hybridization of the valence electrons. Group theoretical methods are used to reduce the labor of solving the secular equation.

4 As described in Appendix A the secular determinant contains Coulomb integrals Hi and group overlap integrals i The latter are calculated by using self-consistent field radial functions. However, the Coulomb integrals H[- are taken as semiempirical parameters. This necessitates a trial and error method which is monitored by the self-consistency of the assumed and calculated vanadium charge. In addition, the Q tensor is taken as monitor of the calculation. Finally the results are discussed and conclusions are drawn.

CHAPTER II RUTILE CRYSTAL STRUCTURE AND SYMMETRY This chapter presents certain fundamental ideas essential for the development of the theory to be followed in dealing with the problem of vanadium in the rutile-type structures. First the rutile crystal structure is given and then the use of the symmetry properties is examined. 1. Crystal Structure The crystallographic data of SnO2, TiO2 (rutile) and GeO2 (tetragonal) crystals, all having the rutile structure, are given in Figure 1 and Table 1. The macroscopic symmetry is tetragonal, but the metal sites have the orthorhombic symmetry Dx, and the oxygen sites the orthorhombic symmetry. ~ p l represents collectively the following symmetry elements: three two-fold axes perpendicular to each other, three mirror planes perpendicular to them respectively, a center of inversion, and the identity element. Figure 1 shows that for the central metal ion, each of the >-, y- and )-axes is a two-fold symmetry axis and the planes perpendicular to them are the mirror planes. Similarly, %,j represents collectively a two-fold symmetry axis with two mirror 5

6 0 U4 -0 c::^ I — - )"^~~~~~~~~~~~~~~~~~~~~) r~*+ ~ ~ ~ ~ -4 rs4i~~~~~~~~~r4 I ---4-, ~ r. —I!

7 TABLE 1 CRYSTALLOGRAPHIC DATA OF SnO2, TiO2 AND GeO2 SnO2 TiO2 GeO2 c(A) 44.737 4.593 4.395 c(A) 3.185 2.959 2.859 r(a. vA,) 3.876 3.674 3.502 f(a. A.) 3.887 3.757 3.637 CoSel.63031.64815.63107 t(p ~ 51~ 49.5 50 9 planes perpendicular to each other, both containing the axis. For example, the site of the oxygen ion #3 has a two-fold axis parallel to the x-axis, the diagonal plane xy as a mirror plane and the plane perpendicular to the y-axis and passing through the oxygen #3 site, as another mirror plane. The unit cell has two types of metal sites, A and B. All considerations will be referred to type A. The axes of type B are rotated 900 about the crystal c-axis with respect to type A site. 2. Additional Assumptions Certain assumptions needed for a semiempirical molecular orbital calculation will be considered (see Appendix A for the general theory). It is assumed that the valence electrons move in orbitals v satisfying the Schr8dinger equation "ik (vi-) ^)= v cU^) w1

8 where H is the one-electron effective Hamiltonian. The HI is taken to be the same for.all valence electrons, even when they occupy excited states. In addition, the linear combination of atomic orbitals (LCAO) \/ L_ | I L(2) is used to provide a trial function for solving Eq. (1). The selection of the n atomic orbitals,^ to be used in Eq. (2) rests on intuition and experience, while the coefficients CI as well as the energy eigenvalues are given by solving the secular equation, Ml K~ v\ H E 5; j t.o ia.l-Li,... (3) where by definition 'LIL <i i H I >) (4) _~~;\4c)~~~~~ ~(5) The application of the above program to a crystal is hopelessly complicated by the great number of atomic orbitals needed in the expansion of Eq. (2). Therefore further assumptions are required to simplify the problem to a solvable one. For this, a region of the crystal around the vanadium impurity is defined, as small as possible, where there is a high probability of finding a specific number of electrons. The problem is confined to this region and these electrons. Also, the interaction of the rest of the

9 crystal must be considered. The smallest region then will be that containing the vanadium ion and the six ligand oxygen ions. The electronic configuration of the oxygen atom is L He3 s %, and that of vanadium atom LAj 3C 4As, where [ H and LAK represent the core of filled shells (see Appendix A) having the helium and argon configurations respectively. The set of nine vanadium 3d, 4s, 4p, and the twenty-four oxygen 2s, 2p, orbitals will be used as the trial solution for Eq. (1); i.e., 33 d):r 2_ C, CX. (6) [ L= t L to solve the problem H^ 6,A E V c(7) Equation (3) yields a 33x33 determinant. Group theory can be used to reduce this determinant to smaller 2x2 and 3x3 secular determinants. Then the calculation is simplified and the various states can be classified according to the irreducible representations (I.R.) of the symmetry group. This allows the formulation of new selection rules that replace the ones found in atomic spectroscopy. The lack of spherical symmetry in t\ renders classification into s-, p-, d-, states, etc., impossible. Another question must be considered: So far, the assumptions made do not allow for any interaction from the rest of the crystal. The vanadium ion and the six ligand oxygen ions are treated as a complex, i.e., as if they were isolated in space and were not part of an

10 extensive three dimensional structure. Complex-type calculations have 22 23 been made by Wolfsberg and Helmholtz, by Ballhausen and Gray, and by 24 Kuroda, Ito and Yamaterao What is needed here is a modification to account for the extensive crystal structure. To be more explicit, consider oxygen ion #5 of Figure 1 as an example. In the complex-type calculation all ions, except the central vanadium and the nearest six oxygen ions, are ignored so that only the bonding of oxygen #5 with the central vanadium is considered. The obvious bonding scheme is: (a) a hybrid orbital of the form sind'(2s)5 + cos a (2pz)5 which is directed toward the central vanadium ion, resulting in a greater overlap S () with the appropriate vanadium orbital than either one of the (2s)5 and (2p )5; (b) the orbital orthogonal to it costr(2s)5 -sin^'(2pz)5 directed away from the vanadium ion and therefore nonbonding; and (c) the two orbitals (2Px)5 and (2py)5 possibly involved in -tT - bonding. The angle T is determined by the requirement that F (w) = VSIP() be a minimum (VSIP is discussed in Appendix H). This is done for example in ref. 23. In minimizing the fraction FI(), a compromise is achieved between the tendency for greater overlapping by forming ligand hybrid orbitals and the promotion energy (see Appendix H) needed for the formation of these hybrid orbitals. However, if one considers the two metal ions, with which the oxygen ion #5 is to be bonded in addition to the vanadium ion, the bonding schemes just discussed is not appropriate since none of the considered orbitals is directed toward these two metal ions. Therefore a 25,26 small total overlapping will result in a less stable situation.

11 On the other hand, the sp2 hybridization scheme, for which the (2s)5 (2pz)5 and (2py)5 orbitals form three hybrid orbitals directed toward the vanadium ion and the two metal ions, will result in a greater total overlap and therefore is more stable. The (2Px)5 is not hybridized with a possible involvement in Th' - bonding. Hybridization requires an 39 increase in the promotion energy which is expected to be compensated by better and more numerous bondings. 3. Oxygen sp2 Hybrid Orbitals The sp2 hybridization assumption was made in an effort to account for the influence of the part of the crystal that is left outside of the region containing the vanadium ion and the six ligand oxygen ions. Now the problem of constructing these hybrid orbitals will be dealt with. These are linear combinations of the usual 2s, 2p functions of the same center, which are directed towards the neighboring metal ions, thus securing greater overlapping and therefore a larger binding 25,26 energy. Since the three metal ions lie in the same plane, as in Figure 2, only the 2s, 2Px and 2py atomic orbitals can be used. Furthermore the orthogonality condition is imposed on these hybrids, so that they can form bonds with the metal ions independently of each other. Such normalized hybrid orbitals along the x direction and along the directions of metal ions 2 and 3 are, respectively: sin r"(2s)+cos' "(2Px) sins '(2s)+cos (2p2) (8) sin`' (2s)+cos~&' (2p3)

12 METAL ION 3 OXYGEN ION *3 r, 180 1 1,- X 180 - 1 METAL ION VANADIUM ION Fig. 2. Coordination of the Oxygen Ions in the Rutile-Type Structures

13 where 2P2 and 2p3 are the 2p orbitals having the directions of atom 2 and 3 respectively. The same mixing coefficients are used in the last two hybrids due to symmetry. Orthogonality between the first and second hybrids gives <sinP" (2s)+cost "(2Px)lsin '(2s)+coss '(2P2)> = 0 (9) sin% " sin% '+cosi " cost ' cos12 = 0 (10) and between the second and third hybrids gives: sin2W'+cos2 ' cos 23 0 (11) From Eq. (10), (11), the known angles 8 12 = 180-? and 23 = 2, the mixing coefficients are calculated. Table 2 summarizes the results for the three crystals SnO2, TiO2, GeO2. The hybrid orbitals are written as 5 = sin (2s)+cost(2p). The longer metaloxygen bond is specified by ~ ", and the shorter one by ' TABLE 2 COEFFICIENTS OF THE OXYGEN HYBRID ORBITALS = sinv (2s)+cosr (2p) SnO2 TiO2 GeO2 sin o'1.413.371.411 cos '.911.929.912 sinY n".812.851.814 cos." o584.525.582

14 The construction of the hybrid orbitals ' effects an additional simplification to the trial function t) in Eq. (6) and therefore to the secular determinant Eq. (3), which is reduced from 33x33 to 21x21. At each ligand oxygen ion, three C orbitals replace one 2s and two 2p atomic orbitals so that the total number of functions in the expansion Eq. (6) remains thirty-three. However, of the hybrid C orbitals only those directed toward the central vanadium ion will be considered in treating the selected region of the vanadium ion and the surrounding six oxygen ions. Obviously the other e hybrid orbitals are involved in bonding with the neighboring metal ions. Therefore, referring to Figure 3, the following twelve ligand orbitals '^S;= \ X'(2@5)'CtOs W/(^p,L (Py), t =L,t,3,^ ^s^cos ( ) - (13) and the nine vanadium orbitals 3d, 4s, 4p will be considered in the expansion Eq. (2). Thus ~1 In^ _ X CK LCf (14) In Eq. (13) the numerical subscripts of the orbitals denote the oxygen ions to which they belong, and the coordinate subscripts refer to the ligand left-handed coordinate systems of Figure 3. The use of Eq. (4) as a trial function in solving Eq. (1) results in the 21x21 secular

15 a).) 0 0 ir ~ ~ ~ ~ ~ 4J-H rda c~ 0 o 4.r, r-4

16 determinantal equation: t1 it -ESj o =,, < 0.-.i (15) 4o Reduction of the Secular Determinant by Group Theory and Selection Rules This section will consider the use of group theory to reduce the secular determinant of Eqo (15) and to derive the selection rules. First a simplified argument will be presented based on the fact that the DP group is an Abelian group and then a general theorem of group theory will be stated. (a) Reduction of the Secular Determinant. The reduction of the secular determinant depends upon the following argument: The region under consideration is centered at a metal site of symmetry;f o The symmetry operators of this group leave the Hamiltonian H1 Ax invariant, i.e., they commute with it. Thus for every symmetry operator T of this group: YT T (16) Furthermore p is an Abelian group (the symmetry operators commute among themselves) If several operators commute among themselves it is possible to choose basis functions which are simultaneous- eigenfunctions of all the operators. Any two such functions are orthogonal if they differ in the eigenvalue of any one of the commuting operators. The character table of pa 4 group (see Appendix C) shows that there are only eight different types of basis functions for the symmetry

17 operators. Note that the characters are the eigenvalues of the symmetry operators. This is generally true when one deals with one-dimensional I.R. Any basis function must transform according to one of the eight irreducible representations. For example the function xy belongs to the B (or N 5 or I ) irreducible representation. In a matrix representation based on v simultaneous eigenfunctions of the symmetry operators of the f9 group, Eq. (16) becomes TI I-' X, H- iT7 (17) However, the T matrix is diagonal since eigenfunctions of the T operator are used. Therefore, Eq. (17) reduces to -L - \ - " -. i (18) or L Q A (19) Thus the off-diagonal matrix element Tik is zero if the eigenfunc' 1' Lions L. and K give different eigenvalues for the operator T. Since there is always a symmetry operator with different eigenvalues in two different I.R. (it is exactly this property that distinguishes the various I.R.s), all energy matrix elements l i are zero, if L and K refer to functions transforming according to different I.R. of the,, A group. The same applies to the matrix S in Eq. (15) because the whole argument can be repeated when the Hamiltonian operator is replaced by the unit operator.

18 Therefore, if instead of the 21 atomic orbitals C; in Eq. (14) combinations of them transforming according to the I.oR of the rhombic symmetry group are used, the secular determinant Eq. (15) will be reduced to a number of smaller determinants equal to the number of the different I.Ro's contained in the combinations of the 's, The order of each one of these determinants will be equal to the number of qc's belonging to an irreducible representation. (b) Selection Rules To formulate the selection rules one must find a way to determine if a matrix element of the form -M - < U,'>)1 kit), ^ ( )> (20) is identically zero or not. Let the functions (), 4 and the operator CD transform according to the, and ftf I.Ro's respectively. Since the matrix element Eq. (20) is a number it should remain invariant under the application of all the symmetry operators of the DJp groupo The application of a symmetry operator to Eqo (20) results in multiplying each one of the - ygj), and ( by the corresponding eigenvalue which is given in the character table in the Appendix C. Therefore the matrix element Eqo (20) is not zero only if the product of the characters (eigenvalues) which correspond to ) for every symmetry operation of the group is equal to unity. (c) General Theorem of Group Theory. In general, both the reduction of the secular equation and the selection rules are based on the following theorem of group theory:

19 The matrix element M — of Eqo (20) is nonzero only if the reduction of the direct product 9 I @~ P contains the I.R. D For a 20 proof see, for example, Heine. When the operator is the Hamiltonian H to or the unit operator, the I.R. D A is the identity I.R. Then the product D p ) is merely ), and for a nonzero matrix element the I.R. D /) must be the same as j. Thus all off-diagonal matrix element H.. and i:~ are zero for functions a( and belonging to different I.R.'s.

CHAPTER III LINEAR COMBINATIONS OF ATOMIC ORBITALS (LCAO) TRANSFORMING ACCORDING TO THE I.R.'S OF THE D2h GROUP Chapter 11-4 demonstrated that in order to simplify the secular determinant Eq. (3), combinations of atomic orbitals transforming according to the I.R. of the group are needed. The symmetry classification of the metal orbitals is as shown in the character table in Appendix C. For the ligand orbitals the method of projection operators is convenient in constructing the combinations which transform according to the I.R. In the case of one-dimensional representations the recipe is simply ' N ) (T) T (21) T where is any member of the original set of orbitals, rX T L is the character of the transformation T for the irreducible representation i, N is a normalization constant, and ) is the orbital which transforms according to the )\ th I.R. expressed as a linear combination of the (p 's. Occasionally the function C is zero. This happens when the function C, already has symmetry properties incompatible with the irreducible representation 0, i.e., its projection on ~ is zero. 20

21 The functions & to be used in Eq. (21) are the twelve ligand orbitals in Eq. (13). For a sample application of Eq. (21) consider Q. The symmetry operators T applied to it give: E C C}Y C I a,G 6X I G. a (22) (;i 6, ~6 3 6Lt 63 61 6 6 (22) Multiplying by the characters of the B I.Ro and summing one gets:,l+t 3 - ~6 - ~OS+ (3( + - 6 - b =( 6 - i 64 (23) or after normalization 3) = H- 6 + 63 6^ (24) The final twelve ligand combinations are listed in Table 3. The signs in front of these functions are chosen so that the majority of the overlap integrals with the metal orbitals is positive. This is done to facilitate programming for the IBM 7090 computer. Table 3 contains also the metal orbitals given in Appendix C. The last column indicates the number of metal and ligand functions which transform according to each I R. According to Chapter II-4, nonzero matrix elements 1Hbo, and, may occur only between functions of the same I.R. Therefore if the twenty-one functions of Table 2 are used in the expansion Eq. (14) instead of the functions Eq. (13), the secular determinant Eq. (15) splits into eight smaller ones. The dimension of these smaller determinants is exactly equal to the number

22:z LO N rO NC - cq J r 0o In Q L C H_ llb i. i lC r? 1' I( ~ -I- - + 0~ ~ ~ + + H Q0 W I I + I O H m L] N 1 I b I. ) 0 b + II I I I I I I b OO o <[ ro N F-4 + -o -S -^ a. a. O^ (n b1 01 01 + cq N 'ri O N + + m C 0 UJ N X tC) cU M cu~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ p to~~~~/ 0~~r <Zcq~L) ~

23 appearing in the last column of Table 3. Thus one gets eight secular equations of the form |etit H - E;lJ = o (25) where L & refer to the five functions of the A I.R., to the two functions of the B I.R., etc. The eigenstates belonging to the different I.R. are denoted by A1, A2, A3, A4, A5; B1, B2; C1, C2, C3;... Hi, H2, H3o For example, a 5x5 determinant corresponds to the I.R. A (or N [ or i ) with five eigenfunctions of the form 01 4ls>~j+ g 3+c\3&7^ + C~\i/+ C \ (26) For a particular case, the five coefficients C0 and the five eigenstates A, obtained by solving the corresponding secular equation, are given in Table 7 of Chapter V-2.

CHAPTER IV COMPONENTS OF THE SECULAR EQUATIONS In order to solve the secular Eq. (25), the overlap integrals S[ X and the energy matrix elements H- as defined in Eqs. (4) and (5) must first be determined. The indices refer to the functions of Table 2 and not to the atomic orbitals. This chapter describes ways of obtaining the - 1. Group Overlap Integrals When combinations of ligand functions transforming according to the various I.R.'s are used, the overlap integrals 1iT, are called group overlap integrals because they involve more than one two-center overlap integrals. Their evaluation is straightforward but tedious. For example, consider the D I.R. for which there are the metal orbital 3dyz and the ligand TT-orbital >, as shown in Table 3, The group overlap integral is given by 1= Jxf(3&ale 3 (27) By substituting X* from Table 3: 24

25 An integral like <(K ) 3 yL is called a two-center integral because the function ip s centered at the oxygen ion #1, and the 3dyz at the vanadium ion. Since there are many tabulations of twocentered integrals referred to a spheroidal coordinate system (see Figure 4) one must express the central metal orbital Y3yt (r)siA1G sliy cos9 (29) in four different coordinate systems which have their 2 -axis pointing towards the ligand oxygen #1, #2, #3, and #4 respectively, and their _, - axes parallel to the X, y axes of the left-handed systems at the individual ligand oxygen. Figure 3 shows that the abovementioned transformation can be accomplished by substituting for cos e sine ~ sin ~ and sinS * cos ~ the expressions given in Table 4, where 0K and p are angles in the rotated coordinate systems, which play the role of 6 and _ of the old system. TABLE 4 TRANSFORMATION OF THE SPHERICAL HARMONICS Ligand cosS Sine sin( Sine cos 1 -sincX sinr A-B - B - 2 -sin o< sin6 A + B + r 3 -sin O sin& -A + B + r + A 4 -sin o sin6 -A - B + L/ 5 -cosc( sin ( sin4 -sino< cost 6 +cos Cx -sin X sin6 -sin o< cos

26 v/ v/:*: tI II X0S02 0 LID.' --- —-------------— ' "~~~v-I -r4 0 P4 X -< --- ^ -------— (UO~~~v \Xoy7

27 where A cos? sin. cos, B = sin co., i cos., = sin * sin C,~ cos, A cos * cos C. Therefore -p _ _1 | COS ^(-^ itM|4 "a t_; f f Cx \ 5 ) - ( /P Qi) 8\.() X0 (S Q,u _ C 0 AK + L- S \Lv. COS 'X o > \ L \;'VY V wcx OScj' \. (30) The s, x s a two-center overlap integral. Following tt this method and neglecting any ligand-ligand overlapping, the following

28 nonzero expressions for the group overlap integrals were obtained for each IoR. A ( N,Q) cc \%' 1.,, 4 A4 —[5i>+ cosApl\31 (31) A 9vA k[LirVQt'<Z -> ~ c 4-.COSlt 3+ (32) (N'MPr =\lei 3 + c (33) D i (34)i Elt =O - S-its Cv\ rs +\- - Ca E (M3^ = % <t pf 1 4 pX (35) T ^ =.ffK[ -$ %si\ \ 4.ob4 \ ~^<X^4^ '

29 z 1 i^=s;LA S'L^ A s \ 4 pSt czo4l h b i (36) H ( i|4 U 19 1 =XJ coSp S sL / S C)s.\ 0co<4P 1 (37) where L stands for the shorter vanadium-oxygen distance and \ for the longer one. A, B, C, D, E, Z, H stand for N,, N%, N N, N, N2 and Nu, and the numbers denote the orbitals in N#~ 3L ' Z L the corresponding IoR. as listed in Table 2. For example, C13 denotes the group overlap integral of the first (i.e., 3dyz) and third (i.e., -'X ) orbitals in the Na I.R, 2. Two-Center Overlap Integrals It was seen that the group overlap integrals can be expressed in terms of two-center overlap integrals. In this paragraph the method of calculating the latter will be considered. For example, let the integral / | 4p > be calculated when the distances are in atomic units and the radial parts are given as (AsI= N,r,1 e R(B 4 V t'b b (38) Note that the normalization constants Na and Nb are given for the radial part only. Then

30 </5) A- -N Njr e r e V4ti " (39) Observing that in the spheroidal coordinate system of Figure 4 -= 2, ~ (g-vcoLs - _ 1,) oL=() Ko^)L~oLWd (40) and defining pr-.: x dx xi cb (41) the integral <ZS |4 p > becomes X<' ^N?/_I Trl ( < 4 > N1 o ( PI (( *.)(1i-l)IT- =a0t.

31 N N 3 (w R B i B + B AB 1 5y D 7b v^ sV 3 75 a ) (42) Similar expressions for the two-centered integrals needed in this work have been derived and are listed in Appendix D. The A and B integrals are tabulated in Refs. 32, 33, 34, and overall two-center integrals in Ref. 35. For the present problem the integrals A and B were calculated by a MAD program using the following SCF radial functions. ' For oxygen R (2s) =.5459' (1.80)+.4839 (i (2.80) 9V-,U~~~~~ -^~ ~(43) R (2p) =.6804 (1.55)+.4038 ) (3.45) For vanadium R (3d) =.52430836 4 (1.8289)+.49893811 4 (3.6102) +.11312810 ) (6.8020)+.00545223 ) (12.4322) R (4s) = R (4p) = -.02244797 1(23.9091)-.01390591 (20.5950) +..06962484 (10.16666)+.06773727 (9.3319) -.09707771 d (5.1562) -.024620956 d (3.5078) +.04411542 d (3.8742)+.36068942 (1.8764) +.608999600 (l 6(.1o462)+.14868524 (.e7800) q 'h (44)

32 where and Na~! (45) The final results for the group overlap integrals (see Appendix D) are given in Table 5. TABLE 5 GROUP OVERLAP INTEGRALS SnO2 TiO2 GeO2 A14.494.495.525 A15.425.445.456 A24 -.148 -.162 -.180 A25.217.233.255 A34 -.053 -.045 -.063 B12.251.278.305 C12.090.112.128 C13.099.113.126 D12.110.131.157 E12.604.627.630 E13.325.359.395 Z12.504.474.502 H12.409.404.408 H13.227.243.259

33 3, Diagonal Energy Matrix Elements The most subtle point in solving the secular Eq. (25) is the estimation of the energy matrix elements. Since these cannot be obtained from first principles, semiempirical methods to approximate them from known experimental spectroscopic data will be developed. Two types of energy matrix element are distinguished in the secular Eq. (25): (a) the diagonal elements Hii 5 called Coulomb integrals, and (b) the off-diagonal ones H S, called resonance integrals. The Coulomb integral i; gives the potential energy of an electron in the i-th orbital. It can be taken eaual to the free atom (or ion) ionization energy of an electron on this orbital to the zero-th approximation. For a hybrid orbital, the weighted average is taken, and for an orbital consisting of linear combination of ligand orbitals again the ionization energy of one of the similar ligand orbitals is taken. However, a better estimate of the Hi is obtained by means of the concept of the valence state ionization potential (VSIP). A discussion 29 of the procedure is given by Moffitt9 (see also Appendix H). 29 To use Moffitt's tables for the oxygen VSIP, a binding scheme must be adopted. Here it will be shown that the adoption of the ionic states 0 and V for the oxygen and vanadium respectively, can satisfy the symmetry requirements and the production of the ESR spectrum in a homopolar binding scheme with oxygen 5p hybridization. From Figure 1 the spatial arrangement of ions in the rutile structure suggests a valency of three (V3) for the oxygen ions and a valency of six (V6) for the metal ions. This implies that there are three and six

34 electrons with uncorrelated spins in the respective valence states (see Appendix H). The oxygen ion configuration giving valency of three is According to Figure 2 and the discussion in Chapter 11-3, the 2s, 2p, and 2py form hybrid orbitals with three spin-uncorrelated electrons. The other two electrons will occupy the 2pz orbital forming the socalled lone pair. Such an electron will be denoted as p. For an overall crystal neutrality the metal ions must have a double negative charge. This implies that for vanadium there are seven valence electrons outside the argon core. This is expected if one considers the six bonds with the surrounding oxygen ions and the single unpaired electron which produces the ESR spectrum, The bonding scheme fits the requirement of the symmetry and of the number of electrons except that oxygen has a much greater electronegativity than vanadium, and Pauling's electroneutrality principle1 asserts that the charge on each ion is in the range -le to +le. One can overcome these difficulties by assuming a partially ionic character of the bonds so that electronic charge is shifted towards the ligands, resulting in a small positive charge for vanadium and a correspondingly small negative charge for the oxygen ions. As it will be seen, the molecular orbital calculation will determine this ionicity of the bonds. In this sense the charges 0 V" will be considered from now on as nominal charges. Of course, none of the above problem arises if one considers a purely ionic bonding with V and 0 - The nearest noble gas

35 configuration is achieved for oxygen and metal ions except for V which is left with one valence electron producing the ESR spectrum. However, it is believed that a purely ionic bonding is generally rare outside the I-VII compounds. The VSIP's of 0+ calculated here are 416396 cm 1 277022 cm, and 253122 cm for an s-, p-, and p - electron respectively. The oxygen atom in the rutile structures is assumed to have the configuration o+: x Py with three unpaired spins. The corresponding valence state, according to Moffitt, is designated as y( ) with promotion energy 4 t 4 I,pI.+3 + C-. The fact that 2s, 2px, and 2py hybridize does not change this energy, as the configuration remains unchanged. This is called first-order hybridization. To estimate the ionization potential (IP.) of an sor p-electron, the valence state of the final configuration must also be considered. Thus, for an s-unpaired electron r a p-npa d p4(el n for a p-unpaired electron (46) sp (V ) > s p(V) for a p -electron of the lone pair 4p4-v.) > sp3(V)

36 Therefore: (s) + 4 VSIP of (p) - (I.P. of 0 ground state)-promotion energy to sp (V3)+ (P > p4 (V2) ++ +promotion energy to (sp3(V2) of 0 sp3(V4) /287472.15\ /416396 42cm1 (283550.9 cm 1) - (154626.63 cm-) +148097.82 = 277022.09cm1 (47) 124197.90 \253122.20cm1 as the promotion energies to p (Vs) Sp3(VSi ) and Sp3(V4l of 0++ are given by: ____ 3 (>O\( Lr.N0 3 '' 3(P)1(), At>)3 ~'K')H3rPO) P") 4 +~ g 1~ and - i~ gi+ 6 3 D ) respectively. The VSIP of 0 and are calculated next and found to be -1 -1l 222000 cm, 115300 cm, and 98968 cm for an s-, p-, and p - elecron of respecti, ad -1 -1cm -1 tron of 0 respectively, and (80000 cml ), 16200 cm, and 3710 cm for the corresponding electrons of 0. The procedures are as follows: It was seen that the polarity of the bonds is expected to decrease the positive charge on the oxygen center and most likely to reverse it. In such a case the VSIP will be different, clearly smaller, so that it needs to be re-estimated. Since the tables give spectroscopic data of the elements with integral electronic charge, the VSIP of fractional charge is to be obtained by interpolation. Next we need to

37 see how to estimate the VSIP of an oxygen which is effectively neutral (or with -l\e\ charge) because of the polarity of the bonds. For this it is assumed that the VSIP of the isoelectronic ion will be a good approximation. For an effectively neutral oxygen or for 0, the neutral nitrogen atom N~ and the carbon negative ion C are considered instead. All VSIP for N and C, except for the 2s elec30 tron of C are obtained from a table given by Skinner and Pritchard. A value of about 10 eV is not unreasonable for the C 2s electron. The procedure of calculating the VSIP is the same as in the previous paragraph. The values taken from the tables of Skinner and Pritchard are: (a) Valence state energies in eV. sp4(V3): C (9.38) extrapolated N 14.23 sp3 (V: C 8.14 rN 11.64 sp3(V,): C 9.69 N+ 14.03 p (V2) C N+ (27.2) extrapolated (b) Ionization potentials in eV. -23 2p 4S C (sp, S) C(s p P) 1.7 N (s2p3, 4SP ) N (s p P) 14.54

38 Finally the VSIP of the vanadium d electrons are estimated. The results are summarized in the following table. V (Co+) V+(Fe+) V~(Mn~) 4p 165658 cm1 102308 cm1 38722 cm1 4s 219465 127369 54762 3d 295997 141178 62516 Moffitt does not give tables for 3d electrons, so the following procedure was adopted. The isoelectric series for vanadium in the rutile structure with nominally seven electrons, but with an effective charge of zero, +le and +2e, is Mn~, Fe+, Co++, According to Moore's tables and notation, the I.P. of Mn~ (Q,, ) to Mn (L ) is 59960 cm. The average of the two states of Mn~, P (c t 4p'l) and P~o(s O + p d4 ), is 21257.74 cm; therefore VSIP of 4p electron = 59960-21237.74-0=38722.26 cm (48) The average, also, of the two states of Mn~, CUiX ( o L -+4 V5 and a (eJ +4-S4V), is 19784.41 cm1; the average of CL) of Mn is 14586.16 cm Therefore ro -1 VSIP of 4s electron = 59960-19784.41+14586.16=54761.75 cm (49) The same procedure is applied for Fe+ and Co++. The 3d VSIP for + I i 31 Mn~, Fe, and Co+ are estimated from the tables given by Slater and 27 Watson.

39 In Table 6 the valence state ionization potentials for the vanadium charge range 0 to +.65e and the corresponding oxygen range 0 to - 325e are tabulated. Interpolation is used to obtain the VSIP's of the electrons on the vanadium orbitals 4p, 4s, 3d and on the oxygen orbital of the lone pair p, for various fractional net charges of the vanadium and oxygen ions. For the electrons on the hybrid orbitals; -lL (25)t COeS(qp) (50) the VSIP's are the weighted averages: (VSIP) = sin2,'(VSIP of 2s)+cos29'(VSIP of 2p) (51) These are calculated for the integral values of the net ionic charge and then interpolated. The values of sink' and cost are taken from Table 2. These VSIP's present a weak point in all semiempirical calculations. However, there are two reassuring factors: (a) the use of isoelectronic-ion parameters does not affect the type and relative positions of molecular orbitals as the isoelectronic principle 32 asserts, and (b) since the VSIP's are used as parameters in solving the secular determinant even the numerical results will not be greatly different if the right parameters are chosen. 4. Off-Diagonal Energy Matrix Elements The off-diagonal energy matrix elements, or resonant integrals, are even more difficult to estimate. Mulliken's assertion Lb

40 that HI s is 1.5 to 2 times the quantity $.. ( H + H 1 / is often followed. Wolfsberg and Helmholtz used both 1.67 and 2. More 23 38 recently Gray and Ballhausen, and Lipscomb used 2. However, the geometric mean seems to give a better fit than the arithmetic mean, so throughout this work H[ will be approximated by l - - t vit (52) V - ^ IL b

41 o o o0 0 0 0 0 0 0 0 o0 0 0 0 0 0 0 0 0 0 00 O C ' -1 M O - n,. C, n,.I Ln. 00 M H cN Ln 'I N C Cn) IC x + in 00 00 H 0 r- rH _ - 'Do 0 0 0 0 H< r-< r- - H ro 4- -I O oHo o o 00 o 0n o L L r- ~n cM - o0- r 'r- cn O cN I"- o c, i n 0o I C O' o0 r -I 0 00 n 00 a' IC) I. 00 0, - o oo U. o cr o, - oo O + -- r - 00 - o r —I r- O O O O r r-J rO r lt c o I - Ln -< L ) O o r ) N CIA O 0 O c r 1- r, 0o cN IO I,r *n I ' o0 o H r IC 00 oo cN Cr) 00 0 cN C),a + 0 -. o00 c r- - 00 I r- o O O O O - r- - Hz 0 *-10 r< 00 00 4 00 00 r- 00 r s u %O r- 0 00 Hr Cn r- Cn) r H0H < 4 N- O rI I O r I c) -4 Z H c o00 Cu' O C oI -4 Lno + -t -- O CN - r 00 o g 0 00O O O O H H H H r- r- ~ 1< -H 0 *0; ~o> O C N 4 c' O < O < O H * 0C r-1 0 t r- 1 4O H- o- <0 ~ - ) CN r- < C 0. 00 or r r I CN r I O O PQ Ln i co00 n 00 0 o o r1 o00 r <c + I4 \S I' ON cl N 00 CN00 coN 00 O O O CO r-l r-l r- r- r —I 0 00 IC O O C) N C) N C) CN *In oo' \ 00 LC) C cq Cn). O 0 O ' Nc O Co ) I cn) I a ' I H j * H r- Ln '0 N 4 o00 r- rI - U. + -4 In oc 00 CA 0 N a 00 J> O CD0 0 0 0 H H H < I rO N c ON 0 00 I CN C cN I CN O N CC) H- P mc o c s -0 ' I4t o r- 00 Ln O I o0 ON I H N o00 N 00 r-i 4 In 00oo + C L C O O n 0oo C) 0C 00 O O CO O - v4 i - r-I t [ L i 42J, 9 Z - uz *E-l CD

42 0 0 0 0 o 0 0 0 0 I- Cl 0 0 0 0 0 0 0 ~0 00 o 'O 0 m Ln I O )0 I cn r r-<, L n L C CM o 0L Cn + 00 OI 0 0 O OC O O I O r^ 0 0 0 0 0 0 0 0 00 ^o o o o o ot ro oo %D. CI Ln O n 0 ) I C L I 00 e 00 4 C r| 00 0D I C) | s + r o O D O -t ON LC) o 0, —.i o 0 0 0 0 I"-. 0 4-4 Ol 0 0 O O 0 0 0 0 cI O 'O O O O O O O O0 L 0N 0 0 00 Lw r O O c O I0 I O I 0 Cn i 0 C) 00 0r- r- I ~ 0! I 0 l Hr O O O IC C O0 a+ 0 O O L O o O Ln CD0 0 0 0 4 -I O.- 4 r-4 O O O O O O O O0 11UO- O O O CO OO O O:,. O > oC) o 0 Cl OC I C D I 0 Co IJ 00 _I 0 C ') C O Cl O C0 CC0 + oo 00 r0' O C0 < O T O O O O r< r-l Or4 r- r4 rH * a5 X 0 > ( O CJ00 00 00 n O4 L g D n 0N r i9 0'o C-r N- 0' 0' IC) '0 00 o0 + v0 r- 00 - 0o LCn 0D D 0O IC 0 00 00 0 o o o O OO O C O rl0 0 0 o 0O 4 oo oo oo L r) o I n I T r + - o rN-c 00 -I 0 0 -o o0 *1 + o r\ oo r- O un O \ o o 3 O O O O I r- -r- r rCn i Ln o oo oo b I r I oo o c < n 4 ol c i3 oo o O Cs + v r oo oO 0-0a 0 O D< o D OOCL d lo 0 IJ c 9O d4- C Cl Cl C) 01 H 0

CHAPTER V SOLUTION OF SECULAR EQUATION 1. Energy Eigenvalues Chapter III showed that the original secular Eq. (15) is reduced to other smaller Eqs. (25) by using LCAO transforming according to the I.R. of the? L group. To each I.R. corresponds a secular equation whose order depends on the number of functions belonging to it. For example, Table 3 shows that there is a secular equation of fifth order corresponding to the irreducible representation A. The group overlap integrals ^j and the diagonal energy matrix elements ei- were calculated in Chapter IV (see Tables 5 and 6). The latter is found to vary with the assumed ion charge. The former, however, does not seem to 27 vary appreciably, according to the SCF calculations of Watson, so that no correction is applied. The off-diagonal energy matrix elements are found by the approximation Eq. (52). Secular Eqs. (25) are solved on the IBM 7090 computer, using a program written in MAD language (see Appendix J). The input consists of diagonal energy matrix elements Hl. and the group overlap integrals b. The off-diagonal elements U[T are calculated by the program following Eq. (52). The output consists of the one-electron eigenstates 43

44 and eigenvalues of the vanadium and oxygen valence electrons in the MO scheme. Also, the fraction of the orbital charge that can be assigned to vanadium is given (for more details on this see the next section, V-2). Thus in Figure 5 the electron eigenvalues of SnO2:V are shown. The VSIP of vanadium in the range 0 to +65e and of oxygen in the corresponding region 0 to -.325e are taken from Table 5 and used as parameters (see below). Only nineteen orbitals are shown: two others, A5 and E2, having energies around +90 Kcm-1, are omitted. The central part of Figure 5 is drawn again in Figure 6. The levels are designated according to their symmetry. The subscripts are used to distinguish the various levels of the same symmetry. Similar curves for TiO2:V and GeO2:V are drawn in Figures 7 and 8 respectively. Energy eigenvalues are tabulated also in Appendix E. The similarity of the three spectra for SnO2, TiO2, and GeO2 is striking, as well as the fact that the relative positions and values of the energy levels are sensitive to small changes in the ionic charge. Figure 5 shows that the A3 level crosses the four levels E3, H3, 1 X3 >, C2. The levels A2 and B2 cross some levels also. Similar results apply for the TiO2:V and GeO2:V. Now a justification is needed for treating the VSIP's as parameters. The difficulties in obtaining reliable values of the VSIP's were explained in Chapter IV. On the other hand, it is noticed that the energy eigenvalues depend rather critically on the VSIP's used. Therefore, any calculation based on a single set of VSIP's (i.e., on one

45 0 Z2 H2 -50 B,%B2 DI -150 Al Ei -200.65.55.45.35.25.15.05 Assumed Vanadium Charge, lel Fig. 5. MO Energy Levels of SnO2:V

46 -50 -60 C3 D2 C2 B2 -70 - H3 A2 3E3 -80 -90 -100.65.55.45.35.25.15.05 Assumed Vanadium Charge, lel Fig. 6. Central MO Energy Levels of SnO2:V

47 -50 -60^ Ix3! -70 T E E2 n -80 -90 -100 -110.65.55.45.35.25.15.05 Assumed Vanadium Charge, lel Fig. 7. Central MO Energy Levels of TiO2:V

48 -50 8, -70 H3 E2 -90 -100 A -110.65.55.45.35.25.15.05 Assumed Vanadium Charge, lel Fig. 8. Central MO Energy Levels of GeO2:V

49 assumed vanadium charge) cannot be expected to give results in quantitative agreement with experiment. Thus it is clear that to find the best set of VSIP another physical quantity is needed to monitor the calculation. Usually the ionic charge is taken as the means of achieving selfconsistency. This is explained below. 2. Charge Self-Consistency To select the best set of VSIP's and thus the solution of the secular Eqs. (25), the vanadium charge is taken as a monitor. A trial and error method is as followss (a) A vanadium charge is assumed. (b) The corresponding VSIP's of vanadium and oxygen are selected from Table 6 and the secular equations are solved. (c) This solution is then used to calculate the vanadium charge which is compared to the assumed value. (d) The above procedures are repeated with different assumed vanadium charges until agreement is reached in step (c). When agreement is reached one says that charge self-consistency is fulfilled. To carry out this program one must determine which MO's are occupied and then calculate the charge on the vanadium ion from the MO's. The charge is calculated (see Appendix J) as follows: The solution of the secular Eqs. (25) provide twenty-one eigenvalues and eigenfunctions. A typical set of eigenvalues and eigenfunctions is given in Table 7. The simplest normalized eigenfunctions, like the B1, are of the form t+)= C1 plV + s.4 (53) 1 ' VOL-\q ^ I1 tLQ

50 TABLE 7 EIGENFUNCTIONS OF VANADIUM IN SnO2 FOR AN ASSUMED VANADIUM CHARGE OF +.25e Energy in cm- Eigenfunctions 86743 E2 _- I. 3 l Lp>i +. 1 4 -. GX\>. 80660 A5._ 9 |^s)- -& )_ |!>4 —075> | ya>.+? |(i\+.675| -1182 Z2 _11 ->+I.75)i) -8235 H2 -_.0o -p> + * 3 1,>+.~ i \G, -58941 A2 |-A|L5qq > -/- 5. i > 5 -59396 B2 _ L4 > +-. 5qqo _%i> 4.- > -68666 03 _ 5 >.3 1.83 1 ) -71053 D2 -.U0(Iy,>+ (1o|,t -78298 A3 7 4s - _ * 1 l| 9q YA >+4-00 I -85328 C2 _. o )00 001 oXt > +-. - \ )>-. 669 \C 5> -85328 1{ nx> -87380 H3.nI|4p-. 14)^>+ ) 45Lr x(,> -88314 E,.17 p - 1 >+. 3 4 -90600 Di 3 Zy >+ S-17 ^ -92019 Ci 54 olX>t+.-5-t'lx>+, 57 j75> -122743 HI 1 |ll 4 > | b,,,+ 1 ).+ Ol X> -124155 Zi 1 5L4 P>)+.GO| 3 -125185 B1i 3 ) +1 7% e. J -130140 A4 |b I >s -. ZL > 510 xy+>+ 7|^i)-. > -171422 E1.07 lp> +.^ 4f) -174822 A1 | >> j ooooc A Y> o4\O1.J)+ j,............~~~~~~~4

51 The orbital charge normalized to 1 is if dj^^ =ct+ cc d l %t I +c L (54) 34 Following Mulliken's suggestion, the fraction of MO charge on the vanadium is set equal to the charge C2 found purely on vanadium, plus half of the overlap charge C C% v, i.e., the effective charge on vanadium due to orbital Qj) is taken equal to C + C - (55) The generalization of this procedure to more complicated orbitals is obvious. In Figure 9 the fraction of the MO charge assigned to vanadium is plotted for the first thirteen molecular orbitals of SnO2:V vs. the assumed vanadium charge. It is observed that this fraction does not change for some of the MO's such as A3, E3, H3, C2, and X3 >. This is exemplified in Appendix F for the A3 level. For the six MO's B1, A4, Z1, H1, A1, E1, there is a gradual decrease in the value of the orbital charge fraction assigned to vanadium from left to right, which corresponds to a gradual diminishing of the coefficients of the metal parts of the MO's. For the MO's D1, C1 this variation is larger. Next we need to determine which of the above orbitals are occupied. A total of twenty-five electrons need to be accommodated in the MO's. There are three electrons from each oxygen ion —two electrons on 2p orbital and one a (2 hybridized one —and seven electrons from the vanadium ion, as seen in Chapter VI-3. Following Pauli's principle, two

52 1.0 A3 DI CI.9.8 E d V BI Cu6 A4 Thirteen MO's Assign to Vanadium ZC.4 Al LL.2.1 E 0 _.65.55.45.35.25.15.05 Assumed Vanadium Charge, lel Fig. 9. Fraction of the Electronic Charge of the First Thirteen MO's Assign to Vanadium

53 electrons are accommodated in each MO starting from the energetically lower one until the number of electrons is exhausted. The above twenty-five electrons are divided into two groups, twelve electrons on the oxygen 2p orbitals and thirteen electrons on the metal and C oxygen orbitals. The nonhybridized oxygen 2p orbitals are distinguished from the rest because they have an almost symmetrical orientation with respect to the vanadium ion and two of the neighboring metal ions, so that their charge can be considered as belonging equally to any one of the crystal regions centered at the vanadium ion and the two neighboring metal ions. On the other hand, other orbitals assign their charge completely to the region centered at the vanadium ion. The assumption of having twelve electrons in 2p ligand orbitals and thirteen in metal or G orbitals is compatible with the situation that exists at the right-hand side of Figures 5 and 9 (i.e., for an assumed vanadium charge close to zero). In fact, the six orbitals Cl, D1, E3, H3, 1(3, and C2 accommodate twelve electrons on 2p oxygen orbitals. The rest are placed in metal and (' orbitals. When the twenty-five electrons are exhausted, it is seen that the ground-state is the A3 with one unpaired electron. However, on the left-hand side of the figures (assumed vanadium charge close to +.60e) the situation is different since twelve electrons go into the first six orbitals, and the next four into C1 and D1 orbitals. The latter are of metal character and not of ligand 2p. The next orbital A3 is also of metal character. This results from the fact that the metal orbitals are more stable than the oxygen 2p ones in this

54 region of the assumed charge. If all the twelve electrons of the ligand 2p orbitals migrate to metal orbitals only one-third of them would be attracted to vanadium orbitals and the other two-thirds into neighboring metal orbitals, provided no drastic energetic changes occur with respect to vanadium ones. Therefore in such a case only seventeen electrons (13+1/3,12) need be accommodated. It was seen that the left-hand part of the diagram can accommodate at least eighteen electrons before 2p ligand orbitals are used. This implies that only seventeen electrons have to be placed on the left, giving again, A3 as the ground state. The situation at the center of the diagram is not clear. Fortunately the slopes of the C1 and D1 curves in Figure 9 are quite steep at the center, so that the ambiguity region is reduced appreciably. In both cases, the wave function for the ground state A can be written as a Slater's determinant Al Al E_. C...- A,\3 (56) The unpaired molecular orbital A3 determines the transformation properties of the determinant. The same results hold true for TiO2 V and GeO2:V. The net vanadium charge can now be calculated using Figures 5 and 9 and the fact that seventeen electrons are placed on the MO when the assumed vanadium charge is greater than +.35e and twenty-five when it is less than +.30e. Figure 10 plots the calculated vs. the assumed vanadium charge in the region 0 to +.65e for SnO2:V. Charge self-consistency is shown to occur for an assumed value of about +027e. Similarly, +,26e and +.25e are obtained for TiO2:V and GeO2:V. Table 7

55 E: Assumed Charge / / ' -2 - / -4 -4I I I I I I.65.55.45.35.25.15.05 Assumed Vanadium Charge, lei Fig. 10. Calculated Vanadium Charge vs. the Assumed Charge

56 gives the eigenfunctions and eigenvalues for SnO2:V corresponding to +.25e, 3. Detailed Charge Self-Consistency Usually a semiempirical calculation stops when charge self-consistency is achieved. However, for an overall consistency of the calculation the detailed electron distribution and the VSIP's should be compatible. That is, not only must the assumed and calculated net charges agree, but also there must be an agreement at the assumed and calculated orbital charge distribution which determines the VSIP. As shown in Figure 9, at the assumed vanadium charge of +.25e the six ( -bonding orbitals give rise to the electronic charge distribution on the average as follows: 15% on the central metal ion and 85% on the six ligand oxygen ions, although it was assumed a 50%-distribution when the VSIP's were calculated in Chapter IV-3. Similarly, for the TT -orbitals C1, C2, D1, E3, H3, and | I<X, a 100% distribution on the ligand ions was assumed in contrast to the calculated distribution which shifts roughly 35% of the electronic charge to the metal ion for the orbitals C1 and D and 7% for the orbitals E and H3. On the average, 14% of the TV -orbital charge is shifted towards the metal ion. Therefore, relatively, the 6 -orbital VSIP's should be increased and the iT-orbital VSIP's decreased. In order to see the effect of such a correction, the calculations were repeated in the region from +o60e to +.40e by reducing the VSIP's of the XT -electrons only. The results for SnO,2:V with a

57 -l -1 reduction of 35000 cm and 45000 cm are shown in Figures 11, 12, and 9. In Figure 9 the dotted lines represent changes produced by the TTreduction of 35 Kcm. Charge self-consistency occurs at about +.40e. It is observed that there is a change in the energy level position of the C, D, E, H, and |3> symmetries as well as in the electronic charge distribution. Table 8 summarizes the assumed electronic charge occupancy of the 6 and Tr oxygen orbitals in the bonding scheme of Chapter IV as well as in the calculated one. TABLE 8 ASSUMED AND CALCULATED 6 AND NT ORBITAL OCCUPANCY Orbital Assumed Calculated no Ir VSIP reduction 35 Kcml1 TT VSIP red. < _ 50% 85% 76% -f a100% 86% 94% It is observed that the calculated values in the second column of Table 8 imply a correction to the VSIP calculated in Chapter IV. The smaller charge of the TT orbitals (86%) with respect to the assumed one (100%) indicates a reduction of the corresponding VSIP due to the decrease in the electron-electron repulsion energy. Similarly, the VSIP of the C orbitals should be increased. In the last column the results are listed when the TT electron VSIP is reduced by 35 Kcm 1

58 -60 B2 A2 B2 -70 A2.80 - i -90 C3~ C3 5 1 ~~D2 -100 A3 3 IX3> C2 C2 3" A -120 C A C:.60.50.40.60.50.40 Assumed Vanadium Charge, let Fig. 11. MO Energy Levels of SnO:V for a <1-electron VSIP Reduction of 35 Kcm-1 left) and 45 Kcm-1 (right)

59 L. C) U) E l/3 C ) m ~~~~~~~VE r - \ E 03:0 4I C) + > <, a, X a) ia 'a~Jeqo wn'peueA paleln3eo> 0: $4 C\J4J I^~ ~ ~~~~ \ =3U~~~~~~~~. I \ L < EE Q, \2 _ O r 4 G.I ~~~~~~~I I I I I i t3 rb ~~~~-a; W 1 O- i -I_____, ) 0 CC U, c* 00 l3 wU P \) C) \ E I E~l '8fi~W~j9 Wflip~Ue/\ p81e~nJI9>

60 The charge occupancy of the -r- orbitals increases to 94% while that of the C orbitals decreases to 76%. These changes are in the direction that requires smaller reduction in the Tr electron VSIP. Therefore, in principle, consistent values of VSIP should exist with respect to the occupancy of the Tr and A orbitals by the electronic charge. The determination of these consistent VSIP does not seem feasible without additional information (see also Chapters VI-1 and VII-2),

CHAPTER VI THE ELECTRONIC Q AND A TENSOR In this chapter the electronic o tensor is used as a monitor instead of the ionic charge. The solutions that satisfy the Q tensors are singled out and compared with those found in Chapter V. The hyperfine tensor A provides additional checking. 1. The Electronic Q Tensor as a Monitor As stated in the Introduction, the purpose of this work is to attempt an explanation of the observed ESR spectra in the rutile-type crystals having vanadium as an impurity. In Chapter V-2, the-valence electronic levels were found using self-consistency. Since these solutions can be used to calculate the electronic tensors, the next step would be to compare the experimental results on the Q tensors with the calculated ones. However, due to the approximate nature of the semiempirical methods, the set of VSIP's which gives the best charge selfconsistency is not necessarily expected to give the best fit for the Q tensor. Furthermore, in Chapter V-3, the need of changing the VSIP's to obtain detailed charge self-consistency is pointed out. The need of -rr-orbital VSIP reduction was determined but not its amount. In view of these facts it is felt that the experimental tensors have to be used 61

62 as monitors in selecting the best solutions out of the many ones found in Chapter V. This procedure is followed in this section. The experimentally found deviations of the Q tensor from the free electron value are given in Table 9. TABLE 9 EXPERIMENTAL DEVIATIONS OF - TENSORS COMPONENTS FROM THE FREE ELECTRON VALUES FOR VANADIUM IN SnO2, TiO2, GeO2... sn02:V -.061 -.097 -.057 TiO2:V -.085 -.087 -.044 GeO:V -.079 -.079 -.037 The theory for the 0 tensor when the ground state is a singlet 35 (orbital) has been worked out by Pryce. The components of the most general 0 tensor are given by -i- X(t- - ~ /V.) (57) where A. - ~7(o L, <m L o0> "/.\ E o L _ ------- r -- r. -- -- -- -- -(58) is a real, symmetric, positive, definite tensor and \ is the spinorbit coupling constant. Excited states are denoted by |n)>. Using the

63 transformation properties of the eigenfunctions and the operators as shown in Appendix C and recalling that the ground stat.e belongs to the identity I.R, one observes that: (a) All off-diagonal elements /A- (iL ' in the relation Eqo (57) are identically zero. The reason is that in Eqo (58) the excited state |-r) should belong to the same IoRo with the corresponding A ON operator LL or L for a nonzero matrix element and each one of the L L, L L transforms according to a different I oR,, namely, D, C and B respectively (see Appendix C) (b) The only nonzero diagonal matrix elements \iN occur with excited states belonging to the B, C, or D I oR The ESR spectra of vanadium in Sn02, Ti02, and Ge02, reveal an electronic spin of S = 1/2 Therefore, the excited states can occur in two ways: In the expression (56) of the ground states as a Slater determinant either the orbital A3 is replaced by one of the higher lying orbitals of symmetry B, C, D or one of lower lying orbitals of symmetry B, C, D is replaced by the A3 orbital (see Figure 13)o The operators L = t are one-electron oper- I c ators, so that <ARA E,^ — E -..DI.AAj L|{A.-ID- -Ai> <,|(X|A3> ec (59) There are two eigenfunctions of type B, two of type D, and three of type C that must be considered (the B15 B29 D1D D2 C2 and C3 of Table 6) Using the following relations:

64 In 0 CI + + + + S4 -ON #^% 4 U — H I+. __ EN - 0 A " — ^ + -- A1A "0 O44 NN NA __ N >' >,Q (.> -I- - ~ P%4 =-i r-I ~~'-J4 -J r x. -J C~ 0 C0 ~Q0

65 A -y{Xt~-r>=-L (;)> (60) 4I -- A A and neglecting conttributions from the nonmetal parts of the orbitals (see D Lscussi on) one gets' - -\x _L I x > - -lx C7L% \^^ ^3+ Lw^ \Ib4i (SC2J cx^ yL A XYr' E ~,_SL P -Y S[-i3+ (61) Ep~~~- ER~~~~~ FRI~~~~ ED~~Z j ia.~ I-,~) - c~ ~4~ {',,~-) _ %~ ~ i_~~~ ) ~a

66 XA E aJ, E Ep (62) E^ L I " 3E_ - E E Lx3F ^64 B A (64) where - are the efficiets of e of t rrespnding metal parts in the molecular orbitals A, B. C, aad D rspes4itely, and > is the eo-electron spimn orbit coupling constant, The minus sign before the second term in the first brackets is due to the fact that the charge transfer transitions affect electrons with opposite spin. The coefficients 0.;, b - and the energy terms EA EB. were - 3 83 1 obtained in Chapter V for the range of the assumed vanadium charge +.65e to 0. The irange is extended to -o40e in this chapter. The needed energy matrix elements in the interval 0 to - 40e ar taken from Chapter IV-3 with the necessary int erpolatios o

67 The calculated values of Ai are plotted in Figures 14, 15, and 16 for SnO2:V, TiO2:V, and GeO2:V respectively. A significant point to note is that Adj is always larger than X< From the Eq. (57) one gets LA;L LL = /, (65) Therefore, the calculated % is always absolutely larger than X X Q although the experimental results (see Table 9) show the opposite. The spin-orbit parameter \ is taken as constant in the range 140 cm1 to 250 cm-1 according to Moore's spectroscopic tables. The results so far indicate that none of the sets of the VSIP derived in Chapter IV can be compatible with the observed Q tensors. In Chapter V-3, the first need for a change of the VSIP's used was seen. Now an additional factor is added. It is interesting to see what changes in the values of the VSIP's are needed to account for the observed Q tensor and how these changes compare with the results found in Chapter V-3. Consider first and. The calculated and experi~X mental values are tabulated in Table 10. The calculated values were obtained by selecting the points (see arrows in Figures 14, 15, and 16) giving the best agreement between experimental and calculated values of \[ (i =- ) /). The spin-orbit coupling parameter j was taken -1 to be 250 cm So far ( was not considered. For this problem one notes that the calculation of \c involves the energy levels A3, B1, and

68 E ' 80 | Fig 14. Calculated \ Tensor c. 60; Components of SnO2:V E zz 0 40 - X 248 C20.65-. 55.45.25.15.05 -. 05 -. 15 -. 25 -. 35 -. 45 -20 -/ Assumed Vanadium Charge, l el -20 E k I-40 - / / Fig. 15. Calculated / Tensor Components of TiO2:V ~-60 E A 40 / A248 _ 0 I 1. —"? II C20 E.65.55.45.35.25.15.05 -.05 -.15 -.25 -.35 -.45 C _ Assumed Vanadium Charge, lel |: ~ Fig. 16. Calculated \ Tensor 60 Components of GeO2:V 10 0 | 40 - X=248 20 - OI I II I I I.65.55.45.35.25.15.05 -.05 -.15 -.25 -.35 -.45 Assumed Vanadium Charge, le

69 B2 (see Figure 13)o These levels remain unchanged by the TT electron VSIP reduction mentioned in Chapter V-3, in contrast to /\y and /\y which involve the levels D, and C. The last two sets of levels are both affected by the T —electron VSIP reduction. TABLE 10 OBSERVED AND CALCULATED ~ TENSOR COMPONENTS OF VANADIUM IN SnO2, TiO2, AND GeO2 rx rY ___ obso 1.939 1.903 1.943 SnO:V cal. 1.943 1.903 obs, 1.915 1.913 1.955 TiOo'V cal. 1.928 1.898 obs. 1.921 1.921 1.963 GeO2'V cal. 1.949 1.819 The components of A/ tensors were calculated with a t-r-electron VSIP reduction of 35 Kcm1l and 45 Kcm in the interval +.60e to +.40e of the assumed vanadium charge. The results are plotted in Figures 17, 18, and 19 for SnO 2V, TiO2 V, and GeO:V respectively. The upper part-of each -1 figure corresponds to the 45 Kcm reduction. In the case of SnO2:V -1 agreement is achieved at +.40e with 45 Kcm reduction. A charge selfconsistency calculation requires a vanadium charge of +.41e (see Figure 12). Therefore, an almost exact coincidence of the two methods is reached. Similar results can be found with TiO2oV and GeO2:V, although -1 -1 a greater TT —electron VSIP reduction is needed (10 Kcm to 20 Kcm more).

70 20- y E tOU 10 C I FUzz M O E 00 10.60.50.40 Assumed Vanadium Charge, lel Fig. 17. Calculated / Tensor Components of SnO2:V for a -ff-electron VSIP Reduction of 35 Kcm(lower) and 45 Kcm-1 (upper)

71 20 E Axx 0 c A yy 10 zz 0 I 1.60.50.40 Assumed Vanadium Charge, le I Fig. 18. Calculated / Tensor Components of TiO2:V for a -f-electron VSIP Reduction of 35 Kcm1 (lower) and 45 Kcm-1 (upper) E 4= A yy 10 C I C:) 0 qAzz.60.50.40 Assumed Vanadium Charge, lel Fig. 19. Calculated A Tensor Components of GeO2 V for a -rt-electron VSIP Reduction of 35 Kcm-1 (lower) and 45 Kcm-1 (upper)

72 The result of the previous paragraph is that a choice of the TT-electron VSIP can be made so that the calculated value of Q is the same as the experimental valueo Thus Table 10 is completedo Whether this choice of the VSIP represents also the crystal reality and not just a mathematical device is not known. It is observed, though, that the change needed by the % tensor in the VSIP of -fT-electrons coincides with the similar need of the detailed charge self-consistency (see also Chapter VII-2). 2. The Hyperfine Interaction Tensor A The hyperfine tensor A can also be used to check the results found in Chapter V. The anisotropic part of the hyperfine tensor A depends on the form of the ground state wave function, and the discussion that follows is limited to this state. The ESR spectra of vanadium in Sn02, Ti02, and GeO2 reveal a strong hyperfine interaction of the unpaired electron with the vanadium nucleus. Experimental values of the hyperfine tensor in units of 10-4 -l cm are given in Table 11 Since the relative sign of the hyperfine tensor components cannot be determined, they are assumed to be all of the same sign so that the isotropic part becomes maximum. Subtraction of the isotropic part leads to the following components of the anisotropic part listed in Table 12.

73 TABLE 11 EXPERIMENTAL HYPERFINE TENSOR COMPONENTS OF VANADIUM IN SnO2, TiO2, AND GeO2 Ax AA Az SnO2:V 21 44 144 TiO2:V 31 43 142 GeO2: V 37 38 134 TABLE 12 ANISTROPIC PART OF THE HYPERFINE TENSOR COMPONENTS DEDUCED FROM EXPERIMENT.A- Is AQ —'V LS A 'tx y z SnO2oV -49 -26 75 TiO2:V -41 -29 70 GeO2'V -33 -32 65

74 In Chapter V-2, it was found that the A3 level was the ground state Its form is given in Table 7 of Chapter V. In Appendix F the coefficients of the metal parts of the A3 level are listed for the interval +o65e to -.50e of the assumed vanadium charge. The variation of these coefficients is negligibleo Therefore any result based on these coefficients does not depend critically on the assumed vanadium charge. For the discussion of the anisotropic part of the hyperfine tensors only the parts of the ground state wave functions that contain the metal | IX- y% 7 and | ^>) states are needed. These are taken from Appendix F. The vanadium charge is taken as +.40e because of the results found in Chapter VI-lo Using the relation / y^ Y >- Go) > =- it \3> (66) and the coefficients in Appendix F, one gets SnO2oV.912 1x-y2> +.154 1 X262 TiO2V.929 x2-y> +.126 | x2-z2 (67) x2 Ge0O2V.907 Ix-y'> +,160 | x-z The field produced at the center by an electron in the orbital I Ky/) has a relative strength of 1, 1, and -2 when an external magnetic field is applied along the x, y, and z axes respectively, while for the orbital |X - > it is 1, -2, and 1. When the occupational probability for an orbital is less than one, the relative strengths have to be multiplied by that probabilityo For example, the coefficient of the |Xy > state of SnO2V is o912 in Eq. (67), the occupational

75 2 probability is (.912).82 and the field produced at the nucleus will be proportional to.82,.82, and -1.64. In this way, one gets fromEqs. (67) the following relative strengths of the magnetic field and therefore of the hyperfine interaction. TABLE 13 CALCULATED RELATIVE STRENGTH OF THE MAGNETIC FIELD AT THE VANADIUM NUCLEUS DUE TO THE GROUND STATE ELECTRONIC CHARGE x y z Sn02:V from yky0'.820.820 -1.640!x^-9>.026 -.052.026 Total.846.768 -1.636 TiO:V.881.833 -1.714 GeO2:V.854.782 -1.636 Normalizing them to the A of Table 11 one gets the results in z Table 14. TABLE 14 CALCULATED ANISOTROPIC PART OF THE HYPERFINE TENSORS NORMALIZED TO A-")5$z AA Aouis A CL is SnO2V -39 -36 75 TiO2:V -36 -34 70 Ge02~V -34 -31 65

76 The agreement with the experimental data of Table 11 is good. For Sn02 V and TiO2:V, it is noted that a greater contribution of the t> state is required than indicated by the coefficients in Appendix F.

CHAPTER VII DISCUSSION AND CONCLUSIONS In conclusion, a summary of the obtained results is presented with some additional discussion. 1. Summary of Results The main results found in this work are: (a) The ground state is found to be A3, which is mainly X-y.4 5 as Kasai and From, Kikucht, and Dorain indicated earlier, but with a small admixture of 2- and an even smaller admixture of 4s orbital. These admixtures are caused by the rhombic component of the crystalline field. In a tetragonal or axial field this admixture would be symmetry forbidden. As evident from Appendix F, the admixture coefficients are relatively constant over a wide region of the assumed vanadium charge from +.65 to -.50. The relatively lower admixture of the ' state in TiO2 can be attributed to the smaller rhombicity that this crystal presents with respect to the other two. (b) The ordering of the levels involved in the Q tensor is found invariably to be Bl(xy), Cl(xz), Dl(yz), A3(x2-y2) D (yz), 42 C3(xz), B2(xy) in increasing energy (see Figures 5, 6,7, 8, and 11). The levels D2 C,, and B2 above the ground level A3(x2-y2) correspond to 77

78 the levels of the crystal field theory. It is observed that D2(yz) is below C3(xz) whereas the simple crystal field theory predicts the reverse order. The formulas (62), (63), and (64) derived in Chapter VI-1 show the importance of the. % admixture in calculating the L~ values. For example, in SnO2:V the relative importance of adiL mixture is given by the fraction _____ - ____t [j3q+9*.'3T] % 5 [o^-W YX+ a^t [.W8 -.133VP ((68) This fraction shows that the admixture of -.1331 ) in the ground state introduces a factor of 2.5 in the value of Q with respect to dyy jAt. The numerical values of the coefficients are taken from Appendix F. (Note that this admixture gives an occupational probability of the state J A> of less than 2%.) (c) The small admixture of i function is important also in explaining the features of the anisotropic part of the hyperfine tensor components (d) Although a tetragonal symmetry implies a L tensor of axial symmetry, the converse is not always true. The same ordering of levels can be compatible with very different C tensors, as found in SnO2:V and TiO2:V. (e) In a heuristic way, assuming a ground state of the form -Cl o< + ~v J t-y">, 'the coefficients 0C and a that satisfy the anisotropic parts of the hyperfine tensor are found to be:

79 2 2 -.2931 S'> +.958 x -y > for SnO2:V -.232| C> +.972 x2-y > for TiO2:V -.083? > +.996 Ix-y 2 for GeO2:V (f) Using the above ground states and neglecting the effects of any charge transfer transition and of any admixture of ligand functions in the excited states D2(yz), C3(xz), and B2(xy), the following sequences of levels satisfy the observed Q tensors. SnO2:V TiO2:V GeO2:V B2(xy) 31900cm'1 42400cm"1 53200cm-1 C3(xz) 11000cm-1 10700cm"1 8160cm-1 D2(yz) 1650cm-1 1895cm 1 4550cm-1 A3 0 0 0 For the sake of comparison, the calculated levels of SnO2:V are given below when a reduction of the 1i -electron VSIP of 35 Kcm1 and 45 Kcm are used (see Chapter VI-1) Reduction in 'T -Electron VSIP 35 Kcm-1 45 Kcm-1 B2(xy) 24202 24202 C3(xz) 7599 6767 D2(yz) 5315 4662 A3 0 0

80 (g) Finally, the discussion in the next section shows that the ligand parts of the MO, which were neglected in the calculation of the A, in Chapter VI-1, may have absolute contributions of from 5% up to 25%. 2. Discussion The freedom in choosing the VSIP of the <'-electrons independently from the G -electrons is enough to bring the values in the right region although the necessary reduction in the above VSIP is found to be somewhat large. Further study of this matter is desirable. For the moment one can observe that even within the framework of this calculation a smaller reduction of the -<-electron VSIP is really needed. The i.. values are due to the interplay of the spin-orbit coupling Ail and the orbital Zeeman perturbations. In formula (58) one of the matrix elements is due to ~ L' and the other to ' L o Due to i/r3 dependence of the S - O coupling parameter, only metal-metal and ligand-ligand terms need be kept in the first matrix element. In the second matrix element, though, the metalligand terms may become appreciable depending on the overlapping of metal and ligand functions. Since the A3 level consists almost exclusively of metal functions, no correction is needed in the S - O matrix elements The correction due to the orbital Zeeman term amounts in substituting

81 b[I+ L<$ x/>] or L1 (69) in formulas (62), (63), and (64) respectively, where the orbitals Bj (L= l, ), C (i=1,3), and j ([1%1 ) are written as follows: B: b; I xy) + > C-: C, X(I)+C I, -hC/5l i;: clLly >)i diL)> L. L +/. (70) (see Appendix I) Since the coefficients b2, c3, and d2 of the corresponding antibonding orbitals are negative, a reduction is implied in the calculated values of Al by using the Eqs. (62), (63), and (64). Similarly, the positive coefficients bl, cl, and d1 of the bonding orbitals imply also a reduction in the values of A/\ A numerical calculation gives the

82 reductions of Ai5 listed in Table 15, when the assumed vanadium charge of +.40e and the -fT-electron VSIP reduction of 35 Kcm are used. TABLE 15 REDUCTION IN ALL VALUES DUE TO LIGAND ORBITAL PARTS AT +.40e ASSUMED VANADIUM CHARGE AND 35 Kcm-1 TT[-ELECTRON VSIP REDUCTION SnO2:V TiO2 V GeO2~ V 4.5% 6,0% 8.5% 5.5% 8.5% 10.5% 25.0% 45.0% 47.5% From Table 15 one observes that the inclusion of the ligand part has the greatest effect on -A.. In Chapter VI-1 the reduction of the TT-electron VSIP was used for the purpose of reducing A. Now a more careful calculation of the orbital Zeeman matrix elements with the inclusion of the ligand part shows that the required — r-electron VSIP -i -i reduction is less by about 10 Kcm to 20 Kcm than the original estimate. However, Table 15 is somewhat misleading for the following reason. The much greater percentage in the reduction of the A~ with respect to the AyX and A\ is partly due to the greater contribution of the charge transfer process., The latter point is made clear in Table 16 where the contributions from crystal field and charge transfer transitions are shown. If one decides to consider the reductions in the

83 \A-5 due to the ligand orbital part, then the calculated values of the A.-s become too small to fit the observed S. Thus a LL LL point corresponding to a smaller vanadium charge than the.40e is needed, according to Figures 14, 15, and 16, for such a point the charge transfer contribution to becomes much smaller than the value listed in Table 16. A rough estimate gives 18%, 22%, and 25% for the last row of Table 15. TABLE 16 CRYSTAL FIELD AND CHARGE TRANSFER CONTRIBUTIONS TO THE A TENSOR COMPONENTS IN SnO2:V IN UNITS OF 10-5cm (.40e METAL CHARGE AND 35 Kcm-1 REDUCTION IN -r -ELECTRON VSIP) A__ AYy A-. Crystal field 10.014 17.797 13.853 Charge transfer -.170 -.538 -2.529 Total 9.844 17.529 11.325 This thesis has been concerned with the properties of vanadium in Sn02, TiO2, and GeO2o Recent ESR spectra of Mo and W in TiO2 44 have been reported. Comments on these experimental results are given in Appendix K.

APPENDIX A GENERAL THEORY The Hamiltonian for a system of TO nuclei and N electrons is L^ L^ ^ % ~K LI To this Hamiltonian one should have added terms depending on the electron spin, the nuclear spin, quadrupole moments, etc., but due to their smallness in comparison with H they are neglected. Thus the Hamiltonian H in Eq. (A-l) is spin independento The Schrodinger equation for a stationary state is H A E 4 (A-2) totWt The Born-Oppenheimer approximation simplifies Eqo (A-2) to t-^v 84 L b; j (A-3) +-L;,Z ~, 4= 84

85 Equation (A-3) is simplified to k( Afro; ^ r; ^ )U ^.<i;' (A-4) by using the definition E - _.y-E z (A-5) Unfortunately, only approximate numerical solutions of Eqo (A-4) can be obtained by the use of high speed computers in the very simple cases of small molecules. The difficulty comes from the last term giving the Alectron-electron interactions The Electron-Independent Model. The simplest (and crudest) approximation in solving Eq (A-4) is to neglect completely the electron-electron interaction, i.e., to solve the equation: ~L (b Em V; L I f (1,> )=S49 ( 1 (A- 6) As the Hamiltonian is a sum of one-electron operators the solutions of Eqo (A-6) are of the form of product functions: tQ (i,.-. ) 4) (14 ^(I) () 19) (A-7) "a~~-.)$ "Ii \LL~ ''*

86 where the S are solutions of the equation ( Aim 7 L ) ([) ( ) = D i(A-8) L K A wave function O {(i) which depends on the spatial coordinates of one electron only is generally called an orbital. If the one-electron wave function depends on the spatial and spin coordinates it is called a spin-orbital. When the Hamiltonian is independent of spin, a spinorbital is a product of an orbital and a spin function like C\;) ~<(-L) or tRLPk i) ) ) Thus, every product function of the form (A-7) can produce % product functions, if spin is included. However, not all of them are necessarily possible, on account of the Pauli's principle and the indistinguishability of the electrons, which are both satisfied if a product function like ' (1 CX (1) f (t) p (a) ---- W cx CX ( (N ) 'K*-~~ 't<~~~~ ~(A-9) is replaced by the normalized Slater determinant, which will be abbreviated usually as K(I ) (~1 ) % ()_(K(N (A-10)

87 Self-Consistent (SC) o The previous approximation, in neglecting the electron-electron interaction term, brought a great simplification of the problem, but one does not expect to get anything like the true energy eigenfunctions and eigenvalues In the SCF approximation, each electron is considered to move in a fixed effective electric field which is obtained by averaging over the positions of all the other electrons, in addition to the field produced by the nuclei. Therefore, each electron is expected to be described by an orbital (or a spin-orbital) and the Hamiltonian becomes again a sum of one-electron operators with product functions as solutions of the Schrodinger's equation. Using a trial function of the form: and applying the variational principle to minimize 5 (1) |H i| )> (A-12) 8 where H is the Hamiltonian in Eq. (A-4) one gets the following N Hartree equations N L~ LLL E,$;L u) c(A-13) If, instead of a product function, a Slater determinant like Eqo (A-10) 9 is used one gets the following N Hartree-Fock equations:

88 '-rL< (L t LL LI ItLL (A-14) The term self-consistent field is appropriate since each Si depends on every B[e and whichever orbital one chooses, it must come as a solution of the Schrodinger equation in which the potential energy due to all the other orbitals has been calculated by means of the The self-consistent orbitals are obtained by iterations. In general, the results of SCF calculations are good but the calculations are quite complicated and lengthy, and the wave functions are expressed in a numerical table or at best as sums of many analytical functions. Atoms o If there is only one nucleus, ioe,, k 1, and the potential JLl L is, if necessary, averaged (approximation) over all directions so as to be always spherically symmetric, then the Hamiltonian of the Hartree

89 equations (A-13) becomes spherically symmetric and the solutions can be 8 expressed as hydrogen-like orbitals. t = A } ( ) g 9) (( (A-15) 10 This is in agreement with an empirical method that Slater had suggested earlier Molecules (Complexes, Solids) The presence of many nuclei does not allow spherical symmetry (even approximately), and the problem of solving Eq. (A-13) or Eqo (A-14) becomes extremely difficult. Only for the hydrogen molecule have 12 SCF-molecular orbitals (SCF-MO) been obtained. An approximation that is widely used in "small molecules" like HF, H2, CH, CH2.. is to consider a linear combination of atomic orbitals centered on the nuclei of the molecule (the term "molecule" will be used collectively for molecules, complexes, and solids) iOe,o. :c. (A-16) See Refs. 13 to 19. Semiempirical Methods. The result of the Hartree SCF method was to change the Hamiltonian of Eqo (A-4) into a sum of one-electron operators of the forma l'; L (A-17>+ L74 (A-17)

90 The last sum of integrals is the operator whose expected value expresses the Coulombs potential energy of the i-th electron (strictly speaking of an electron in the i-th orbital) due to the average field of the rest of the electrons, and it is different for different orbitals. However, if two orbitals are approximately in the same relative position with respect to the others, one anticipates almost the same expectation values. This idea is reflected, also, in the Slater's rules which give the same screening constant G for all the orbitals of the same group. The case of complex molecules is certainly more involved as there is no-spherical symmetry in the Hamiltonian, As a more complex situation needs more drastic measures, the following assumptions are made: (a) Electrons are divided into core and valence electrons. (b) Core electrons form closed shells that affect the motion of the valence electrons only through the screening of the corresponding nuclei. (c) Each of the valence electrons moves on an orbital ~Lsatisfying the Schrodinger equation: (A-18) where is the effective charge of the -th nucleus and \V(V,) is the average potential energy of the valence electron due to the rest of the electrons. (d) The function Vt \) is the same for all valence electrons even if they occupy orbitals corresponding to an excited state of the group of valence electrons.

91 (e) \ 5 and _ are not to be used explicitly. Suppose that one knows the He and let p be a complete, but not necessarily orthogonal, set of one-electron functions that obey the same mathematical restrictions as the valence orbitals. Then one can always expand o in an infinite series: ^ LV L fi ( (A-19) Substituting in Eqo (A-18) 00 H -C, (fj jE 5 1j- ik; (A-20) and multiplying on the left by P ( \ and integrating one gets an infinite number of equations 00 'I L L Er48" i0=o a=1,> (A-21) Since there are mathematical and practical difficulties in dealing with an infinite number of equations, one generally restricts the expansion (A-19) to a small number 'y of functions i. hoping that with the proper selection of (p S and the best coefficients C S the approximation mn i= V L Tl.rene e i (A-22) will be adequate. The selection of the proper Xi S rests on intuition and experience, but the best coefficients C;[ 5 are determined

92 rigorously by the variational method, i.e., one minimizes r ^ ^ ^ ct-c E ---------- 'Jfv~ H vA(A-23) Thus (see Appendix B), the following secular equations and secular determinants are obtained: ~n L_ CHK X - ESi)z= o k= 1,- j.- (A-24) L-, t at Hi -E. l, bLL=l, — (A-25) where by definition A J fL H (i Tl JL Sf \ d (A-26) Since the Hamiltonian operator is hermitian I L - i and 1L - [; if the functions ( 5 are real, as is almost always the case, then He - hi. As for the set of orbitals (A to be used, generally, atomic orbitals centered at the different nuclei are selected so that c '>- C. C is a linear combination of atomic orbitals (LCAO)o The selected atomic orbitals are the energetically 14 lower valence orbitals as the atoms-in-molecules method implies. At an infinite separation of the nuclei, the valence electrons are rigorously on atomic orbitals. However, at a smaller separation of the nuclei the LCAO is only an approximation.

APPENDIX B APPLICATION OF THE VARIATION METHOD TO LINEAR FUNCTIONS Substituting relation (A-22) into (A-23) and using definitions (A-26) -\ Ezc; cL HL c. c $j '(B -1) T, L Normalization of z, requires that,^c..c.$..=l <i(B-2) Bringing the denominator on the left side and differentiating with respect to C, one gets % E ^I- c S E / c.+L~C. z: — C (E Lt c; E^i i =L L LC LC (B-3) For a minimum in energy, necessarily =E =o =t,%._ ~. (B-4) Therefore _E _ (~ J Lc S, cc... (B-5),,,. -93 93

94 or E AZL c $ - AZt c H. or n F. tH. - ES )-o -.l,.- -_, L= I K ) go (B-7) For a nontrivial solution, necessarily -— it - E L | = O (B-8) The coefficients CG S are determined by solving the equations: [ - ] —H +E (B-9) for the ratios and then using the normalization condition (A-2). 1

APPENDIX C GROUP CHARACTER TABLE FOR THE SINGLE-VALUED IRREDUCIBLE REPRESENTATIONS _____ 2 - Y = E C IC I A 2 2 i I i iu ~i r - i| A r1 N 1 i 1 11 -L y * c jl iNi 7-1 1 -1 i -' 1y D i:r N 1 1 1 -1 -1 -1 " LAt~ lLi; I 11 1 1 -1 i_ -1 -1 E r 1 1 1' 1 -1 -1 -1 1 1 1 7 I r,,, 1 1 l | -l -l 1 -l i1 E j 4t 1 -1 -1 -1 1 1 -1 I X Nz (x N 1 -1 1 -1 - -1 i -i | N4/ __ i M4 i i - - _ _ _ -1 1 - _ ___ N ~ N% i% -1 -1 - 1 - 1-i i N4,E~ 1t M4 1 ' 1/ 1 t1 % 1|l 95

APPENDIX D TWO-CENTER OVERLAP INTEGRALS The two-center overlap integrals are calculated using the following formulas: K=-aslc3L 0^=N6LN ( A -3B + B )+ A3- B +6)+ + A~, 3B y+ 6,- 3 6, + A, l,- A4-364+ +'(-5 bt63 + t(-6,+3b>9 L -=<Z sl 3 >Q.N \F-3 [A6 - Ab5- 2 SA A +63Bt A A56;l I;, 0 ~3pd)~'l~ -Nb' 1 6 % %, a& 3 r Li —<as)5p)= N l^U tRt)\7I-% FoAQ cA 1+0t Ft^(Zt- BD j3r(Rl L 8 N AN (-S 1 +A 6+ Ast +6 )- M6 - 6 5A^ (S; ^6,)bA (+ [- 6*`t AAt +A+6- F36 A6 \lA=<tsl|3s>=l IN,N(7Tk[B —AB,+ A BA4t 9^-6^ ^ A BIt 9 oRNbk - I1 3 t 1 s -Nu I 3 = Nb1 qA ( ( A [(-36tF 6 )t ( B + e)+ + A83t+ e6) - fPrf,+ 36)- A (Bo +6X)+ At PS-> 3 B3) 3=pzol >=Na^( A- )"A.B i6)tA(-B t)+ +-fFc+(t^t)+t86-et8( -6a+2-+,) Pi=<%pj 1 3p >= NMj'8, ) [-LerO3- A18,- FA,(t RWto+ 6l+) o6-<e,~..,>- A, N 3 6^-A B("[ ~ -%1',-%-tt,'%)9 96 PR'S~~ap,\ Lt~)=N~f~-l9L \al b 1 5 ~~ Lt, 1 4 f" 1 C

97 Q1 <xp, ~3t>= N N 3,'B -~ (~l~c + l3- l5)+ ~ 4- t- 3 - lt GJ bL4J 'B -5 + A (6+ - 46 - e lB~| A+ — (- s>q(+ )3a)- 3) -8B +<, |- 6 5>^NNo A Mt * (- t1+ +3)+ 5f(0 ~ ) X\ '*f=tlPIQT Nl$bv 6L [c t-If ' 0 5+14 ( 5 1 1 v B Y Po ~. 5L 3 The radial parts are given by f (pl)_- ry-1 e-~Lr. However, many times the radial part of an yqS p oVrhd/function is given with a smaller exponent as in (38) or as a sum of functions of different exponents as in (44). In such a case one should be careful to look for the proper principal quantum number. For example the computed expression of 5 4P in the text will appear as S p> in the tables since the radial part of is given as ( p ) -NrT e-' and not as f (p )-N 3 Vei. Values of the two-center integrals are given below. SnO2~:V Ti02:V GeO2:V KL o109.128.146 K-.108.120.132 L1- -,002 -,002 -.003

98 Sn02 V TiO2:V GeO2:V Li- -.002 -.002 -.002 L2i.403.429.450 L2-.402.419.434 M11 -.002 -.002 -.003 Ml- —.002 -.002 -.002 M2i.265.286.304 M2-.264.277.290 n -.113.124.131 N-.113.120.125 0-.071.086.101 0-.070.080.089 P1- -.004 -.005 -.006 P1- -.004 -.004 -.005 P2L.178.170.159 P2-.179.174.168 Ql- -.005 -.006 -.007 Q1i -.005 -.005 -.006 Q2-.156.157.159 Q2-.156.158.159 Rl -.001 -.001 -.001 R1- -.001 -.001 -.001 R2.162.181.198 R2-.161.173.184 1

APPENDIX E ENERGY EIGENVALUES IN Kcm 1 FOR SnO2:V.65.55.45.35.30.20.10.00 111. E2 106.5E2 101. E2 94.5E2 90.5E2 82.5E2 73.5E2 67. A5 93.5A5 91.5A5 88. A5 84.5A5 83. A5 78.5A5 73. A5 63. E2 1.5Z2 1.5Z2 1.5Z2 1.5Z2 1. Z2 1. Z2 1. Z2 1. Z2 -10. H2 -10. H2 -9.5H2 -9. H2 -8.5H2 -8. H2 -7. H2 -6. H2 -61. C3 -64.5C3 -64.5B2 -62.5B2 -61. A2 -57. A2 -52.5A2 -48. A2 -62.5D2 -66. B2 -65.5A2 -62.5A2 -61. B2 -57.5B2 -53.5B2 -49. B2 -65.5C2 -66. D2 -67.5C3 -69.5C3 -69.503 -67.5C3 -63.5C3 -58. C3 -65.5 -69. A2 -70. D2 -72. D2 -72. D2 -69.5D2 -65. D2 -59.5D2 -66.5B2 -70. C2 -75. C2 -80. C2 -81.5A3 -75. A3 -69. A3 -62.5A3 -70. H3 -70. -75. -80. -82.5C2 -88. C2 -93.5C2 -99. C2 -71. A2 -74. H3 -78.5H3 -83. H3 -82.5 -88. -93.5 -99. -71.5E3 -75.5E3 -79.5E3 -84. E3 -85. H3 -89.5H3 -94.5H3 -99.5H3 -107. A3 -99 A3 -91. A3 -84.5A3 -86. E3 -90.5E3 -95. E3 -99.5E3 -108. D1 -101. D1 -94. D1 -91. D1 -90. D1 -92. D1 -95.5D1 -100. DI -108.5C1 -101. 501 -95. C1 -92.5C0 -91.5C1 -93. C1 -96. CI -100.5C1 -115. H1 -115. HI -116.5H1 -118.5H1 -121.5H1 -125. H1 -130. H1 -135.5H1 -118.5Z1 -118. Z1 -119. E1 -121. Z1 -122.5Z1 -126. Z1 -130. Z1 -135.5Z1 -122. B1 -119.5B1 -119.5B1 -125B -15B1 5B1 -127.5B1 -132. B1 -137. B1 -127. A4 -125.5A4 -126. A4 -127. A4 -128.5A4 -131.5A4 -135. A4 -139. A4 -159. E1 -161. E1 -163.5E1 -166.5E1 -169. E1 -174. E1 -180.5E1 -188.5A1 -164. Al -165.5A1 -167.5A1 -170. A1 -172.5A1 -177. A1 -182.5A1 -189. E1 99

100 l1 ENERGY EIGENVALUES IN Kcm FOR TiO2:V.65.55.45.35.30.20 10.00 153.5E3 147. E3 139.5E3 130.5E3 125. E3 114. E3 101. E3 87. E3 105. A5 103. A3 99.5A3 95. A5 93. A5 88. A5 82. A5 75.5A5 -8.5Z2 2 -8. 2 -8 Z2 -7.5Z2 -7, Z2 -6.5Z2 -6. Z2 -5. Z2 -9. H2 -8.5H2 -8.5H2 -8. H2 -7o.52 -7. H2 -6.5H2 -5.5H2 -59.5C3 -60. B2 -59. B2 -57.5B2 -56. B2 -53.5B2 -50. B2 -45.5A2 -60. B2 -62.5C3 -61.5A2 -59. A2 -57.5A2 -54. A2 -50. A2 -46. B2 -61. D2 -64.5A2 -65. 3 -6.53 -66.5C3 -65. C3 -61. 03 -56.5C3 -65.5C2 -65. D2 -68. D2 -69.5D2 -69.5D2 -67.5D2 -63.5D2 -58.5D2 -65.5 -70, C2 -75. C2 -80. C2 -81.5A3 -75. A3 -69. A3 -62.5A3 -66.5A2 -70. -75. -80. -82.50C -88. C2 -93.5C2 -99. C2 -70.5H3 -74.5H3 -79. H3 -83. H3 -82.5 -88. -93.5 -99. -72.5E2 -76. E2 -80.5E2 -84.5E2 -85.5H3 -90. H3 -94.5H3 -99.5H3 -107. A3 -99. A3 -91. A3 -84.5A3 -86.5E2 -91. E2 -9552 -100 E2 -108.5D1 -101.5D! -95 D1 -92. D -91.5D1 -93. D1 -96. D1 -100.5D1 -109. C -102.5C1 -96.5C1 -94. C1 -93.5C1 -94.5C 1 -97. C -101. C1 -112.5H1 -112. H1 -113.5H1 -115.5H -117.5H1 -121.5H! -126.5H1 -132. H1 -115. Z1 -114.5Z1 -115.5Z1 -117. Z1 -119. Z 1-122.5Z1 -127. Z1 -132. Z1 -122. B1 -119.5Bp -118.5B1 -120. B1 -121.5B1 -125. B1 -129.5B1 -134.5B1 -126. A4 -124. A4 -124. A4 -125. A4 -126.5A4 -129. A4 -132. A4 -136. A -165.5E1 -167.5E1 -170. E1 -173. E1 -175.5E1 -181. E1 -188. E1 -195.5A1 -170.5A1 -172.5A1 -174. A1 -177. A1 -179.5A1 -184. A1 -189.5A1 -196.5E

101 ENERGY EIGENVALUES IN Kcmn FOR GeO2:V.65.55.45.35.30.20.10.00 176. E3 170. E3 161.5E3 151.5E3 145.5E3 133. E3 118.5E3 102.5E3 137.5A4 135.5A4 130.5A4 125. A4 122.5A4 116. A4 108.5A4 99.5A4.5Z2 ~5Z2 ~5Z2.5Z2,5Z2.5Z2.5Z2.5Z2 5.5H2 -4. H2 -5. H2 -4.5H2 -4.5H2 -4. H2 -4. H2 -3.5H2 -56.5B2 -56.5B2 -54.5B2 -53. B2 -52. B2 -49.5B2 -45.5A5 -41.5A5 -60.5A5 -59. A5 -56. A5 -53.5A5 -52. A5 -49. A5 -46. B2 -42.5B2 -57.5C3 -60.5C3 -63. C3 -64. C3 -64. C3 -62.5C3 -59. C3 -55. 03 -59.5D2 -62.5D2 -65.5D2 -67. D2 -67. D2 -65. D2 -61. D2 -56.5D2 -65.5C2 -70. C2 -75. C2 -80. C2 -81.5A3 -75. A3 -69. A3 -62.5A3 =65.5 -70. -75. -80. -82.5C2 -88. C2 -93.5C2 -99. C2 -71. H3 -77. E2 -79.5H3 -83.5H3 -82.5 -88. -93.5 -99. -73. E2 -90. H3 -81. E2 -84.5A3 -85.5H3 -90. 1H3 95. H3 -99.5H3 -107. A3 -99. A3 -91. A3 -85. E2 -87. E2 -91.5E2 -95.5E2 -100. E2 -109. D1 -102.5D1 -96.5D1 -93.5D1 -93. D1 -94. D1 -97. D1 -101. D1 -109.5C1 -103. C1 -97.5Ci -95. C1 -94.5C1 -95.5CI -98. C1 -101.5C1 -115. HI -119.5Z1 -116.5H1 -118.5H1 -120.5H1 -125. HI -129.5H1 -135.5H1 -118.5Z1 -125. H1 -119. Z1 -120.5Z1 -122.5Z1 -126. ZI -130. Z1 -135.5Z1 -125.5B1 -124. B1 -122. B1 -124. B1 -125.5B1 -129. B1 -133. B1 -138. B1 -129.5A2 -129.5A2 -128. A2 -129. A2 -130.5A2 -133. A2 -136. A2 -139.5A2 -160. E1 -162. E1 -164. E1 -167. E1 -169.5E1 -174.5E1 -181. E1 -189. A1 -166.5A1 -168. A1 -169. A1 -172. A1 -174. A1 -178. A1 -183. A1 -189. E1

APPENDIX F VARIATION OF THE 2 AND 4S ADMIXTURE IN THE GROUND STATE A3 SnO2:V TiO 2V GeO 2V Assumed Vanadium 4s 1 X-y 4s 2/ R y% Charge~ _.65.073 -.139.986.068 -.114.992,082 -.140.987.55.071 -.135.988.059 -.110.992.078 -.136.987.45.069 -.133.989.058 -.108.992.077 -.135.988.35.068 -.132.989.057 -.108.992.076 -.134.988.30.068 -.132.989.057 -.107.993.076 -.134.988.20.067 -.131.989.056 -.o106.993.075 -.133.988.10.067 -.130.989.056 -.106.993.749 -.133.988.00.066 -.130.989.055 -.105.993.074 -.132.988 -.10.065 -.130.989.055 -.105.993.074 -.132.988 -.20.065 -.129.989.054 -.105.993.073 -.132.989 -.30.064 -.129.990.054 -.104.993.073 -.131.989 -.40.064 -.128.990.053 -.104.993.072 -.131.989 -.50.063 -.128.990.053 -.103.993.072 -.130.989 102

APPENDIX G POINT CHARGE CRYSTALLINE FIELD CALCULATION OF THE ELECTRONIC LEVELS OF SnO2:V Using Watson's radial functions the following splitting of the nearest 3d vanadium electronic levels is obtained. The number of the nearest ions, which have been considered in each case, is written in parentheses. The cases with 8 and 32 nearest ions are considered closest to reality because the corresponding total charge of the complex is zero. 105

104 Appendix G Table 4 -Z2 Z2 3 ~~~~~~Z22 2- XY XY XY XZ z2 2 - 2x YXZ xz xY E ' XY wi0 0- XZ XZ 0 xz, XZ 0-I Yz~ 2V L 22 Xr!^C — X-Y w 7~z yz~~~ XZ Y YZ -Xv2y22 1 22 YZ XZ X-Y X-l 2 2 2 X z x-Y ' -31 - — 4 (6) (8) (12) (20) (24) (32)

APPENDIX H VALENCE STATE, VALENCE STATE IONIZATION POTENTIAL (VSIP), VALENCY AND PROMOTION ENERGY A simplified discussion of the water molecule is given to clarify the above ideas, The ground configuration of oxygen is Is ' S 2?p~, p p, and it is a P with the two unpaired electrons having parallel spins. 28 -1 The ionization energy is 109836.7 cm (13.614 eV). The water molecule is formed by pairing each of these two electrons with the electrons of the two hydrogen atoms (neglect hybridization)o The two oxygen electrons are randomly oriented with respect to each other. If by some imaginary process one could remove the hydrogen atoms and still keep the spins of the two oxygen electrons uncorrelated, the state of the oxygen atom would be a valence state. The number of the spin-uncorrelated electrons determines the valency of the state. In this example, the oxygen is divalent. The ionization energy of one of the valence electrons is less than in the case of the single oxygen atom by the amount that is needed to uncouple the two electrons from the triplet Jl^ 15867.7cm-1 state. Since the singlet state is 15867~7 cm higher than the 3 A ground state P the valence state ionization potential (VSIP) is 105

106 less than in the free atom by the amount 3 (0)+ - (15867.7), where the t 4 factors - and 1 take into account the spin multiplicity. From this 4 L example, it is obvious that the valence state is not a spectroscopic state and excitation to it is not physically possible. It is a nonstationary state. The amount - (o)+-1 (15867.7) is usually referred to as the promotion energy. In general, the orbital part of the valence state of an ion can be a linear combination of the form ocIXts+p ip>, for example, which is referred to as a hybrid orbital. In this case, the promotion energy might have, also, a contribution from the hybridization.

APPENDIX I EFFECTS OF THE LIGAND ORBITAL PART IN THE CALCULATION OF THE ZEEMAN MATRIX ELEMENTS Using the notation in Eq. (70) for the calculation of AX one has the matrix elements: / " \ ~ ci(y-,>+-K IjX>L o >+a^> z1Y > = /54& l Xt- Y tI 2 iv29i4 / |Y >+ \ I | (- ) al 4 (-L) y = oL (a l+t ou^ y, ) [CL -La L a-) -t s-iCt-Xi>fi - i ~ zcLcA~ L\5~+~%, %/LO\A,y 107

108 ct1 ^ + OV Xy l X y1~c > | L t (D < )i X>+ CL| CA> + C l 19>|LY |C > +.y| K= y) [CLCL< C <'A) |L Y> A >L -y L /.Iy>+^ J GU Ix ya L b> X - e a ' " "L+ i <, ly:Y>. '~~. - Y 'k h ii ~ )~l%. 'y%

APPENDIX J COMPUTER PROGRAM IN MAD LANGUAGE FOR SOLVING THE SECULAR EQUATIONS (25) SCOMPILE MAD. EXECUTE. PUNCH OBJECT PRINT COMMENT $1 SOLUTION OF THE CHARACTERISTIC VALUF PROBLEM 1 (A-LB)X=O $ PRINT COMMENT $0 WHERE A AND B ARE SYMMETRIC MATRICES. AND B 1 IS POSITIVE DEFINITE $ DIMENSION A(400,V), B(40n0V) X(400*V). APRIME(400,V) E(400,V) DIMENSION D(400,V), UT(400,V)o R(4009V). ST(400,V)t 1S(400,V) LAMBDA(400*V). YT(400,V) EOUIVALENCE(DRgSTE) (UT.SYTX) (APRIMFLAMBDA), 1(V(2) N) VECTOR VALUES V2*1,O0 INTEGER NIgJtKgCH START READ AND PRINT DATA EXECUTE ZERO.(A(1,1)...A(N.N) B(1.l)-..B(N.N)) READ AND PRINT DATA THROUGH LOOPli FOR 1= 291.I.G.N THROUGH LOOP1 FOR J = 11.J.E.I A( IJ)=-2**B(I,J)*SQRT.(A( I)*A(JJ) ) A(JI): A(ItJ) LOOPI B(JI) = B(I,J) IND1=5. IND2=5. IND3=5. IND4=5. IND5=5* rND6=5. THROUGH LOOP1A, FOR 1=1,1, I.G.N*N LOOP1A D(I)=B(I) SCFACT = 1. IND1=EIGN.(D(1),N,1,UT(1),SCFACT) WHENEVER IND1*E%3. CONTINUE OR WHENEVER IND1.E.1. PRINT COMMENT $0 B MATRIX NOT ACCEPTED BY SUBROUTINE $ TRANSFER TO END OR WHENEVER IND1*.E2. PRINT COMMENT $0 CHARACTERISTIC VALUES OF B MATRIX SCALED BY$ PRINT RESULTS SCFACT TRANSFER TO END END OF CONDITIONAL THROUGH LOOP2, FOR I=1,1,I.G.N WHENEVER D(II),LE.O. PRINT COMMENT $0 B MATRIX IS NOT POSITIVE DEFINITE $ TRANSFER TO END OTHERWISE R(, I)=D(II).P.-.5 LOOP2 END OF CONDITIONAL THROUGH LOOP3, FOR I1,19.I.G.N THROUGH LOOP3, FOR J=ltl J.G.N WHENEVER I.E.J CONTINUE OTHERWISE R(I J)=0O LOOP3 END OF CONDITIONAL IND2=DPMAT.(N*ST(1)TUT(1)) WHENEVER IND2.E.O., TRANSFER TO END THROUGH LOOP 5, FOR I=1G1,I.G.N THROUGH LOOP 5, FOR J=l,11J*G.N StIJ)=ST(JgI) 109

110 LOOP5 APRIME(IJ)=ST(IJ) IND3=DPMAT.(NAPRIME(1)9A(1)) WHENEVER IND3.EO0., TRANSFER TO END IND4=DPMAT.(NtAPRIME(1),S(1)) WHENEVER IND4.E.0.. TRANSFER TO END THROUGH LOOP6, FOR I=291, I*G.N THROUGH LOOP6t FOR J=1l,1 J.E.I LOOP6 APRIME(IJ) = APRIME(J,!) SCFACT =1. INDSEIGN.(LAMBDA(1),N,1lYT(1),SCFACT) WHENEVER IND5.E.3o CONTINUE OR WHENEVER IND5SeE.1 PRINT COMMENT $0 APRIME MATRIX NOT ACCEPTED BY SUBROUTINE $ TRANSFER TO END OR WHENEVER IND5.E.2. PRINT COMMENT $0 CHARACTERISTIC VALUES SCALED BY $ PRINT RESULTS SCFACT END OF CONDITIONAL IND6=DPMAT.(NTYT(l) ST(t) THROUGH LOOP79 FOR I=1,1,I.G.N XSUMSQ = 0. THROUGH LOOP89 FOR Jul,1, J.G.N THROUGH LOOP8. FOR K=llK*G*N LOOP8 XSUMSQ=XSUMSQ+X(IJ)*X(IK)*B(JK) ROOT = XSUMSQ.P..5 THROUGH LOOP7. FOR J =1t1l J*G.N LOOP7 X(IJ)=X(I,J)/ROOT PRINT COMMENT $0 CHARACTERISTIC VALUES $ THROUGH LOOP8A, FORI=1*tI.G.NN LOOP8A PRINT RESULTS LAMBDA(1,I) PRINT COMMENT $0 THE ROWS OF THE FOLLOWING MATRIX ARE THE NOR 1MALIZED CHARACTERISTIC VECTORS $ PRINT RESULTS X(l,1).*.X(NN) DIMENSION MA(5),MB(2)MMC(3)tMD(2),ME(3),MZ(2)tMH(3). 1HOVA(5)tHOVB(2)gHOVC(3)tHOVD(2),HOVE(3),HOVZ(2),HOVH(3), 2CHA(5) CHB(2) CHC(3) CHDt2)tCHF(3) CHZ(2) CHH(3) WHENEVER CH.E.1 THROUGH LOA 9 FOR J=l11,J.G.5 MA(J)=S(Jt1)*S(J,1)+S( (2)$(J2)+S(Jt3)*S(Jt3) HOVA(J)=S(Jl)*S(Jt4)*B(194)+S(Jl)*S(J,5)*B(1,5)+S(J,2)*S( lJ,4)*B(2t4)+S(Je2)*S(Jt5)*B(2t5)+S(J,3)*S<(J4)*B(t34) CHA(J)=MA(J)+HOVA(J) LOA PRINT RESULTS MA(J),HOVA(J)sCHA(J) OR WHENEVER CH*E.2 THROUGH LOR, FOR J=ll1JeGe2 MB(J)=S(J'l)*S(J,1) HOVB(J)=S(Jl,)*S(J,2)*B(1,2) CHB(J)=MB(J)+HOVB(J) LOB PRINT RESULTS MB(J).HOVB(J)iCHB(J) OR WHENEVER CH.E.3 THROUGH LOC,FOR J=191,J.G*3 MC(J)=S(J,1)*S(J,1) HOVC(J)-S(Jl)*S(Jt2)*B(1,2)+S(J1l)*S(J,3)*R(1l3) CHC(J)=MC(J)+HOVC(J) LOC PRINT RESULTS MC(J)*HOVC(J)tCHC(J) OR WHENEVER CH.E.4 THROUGH LOD * FOR J=lplJ.*G.2 MD(J)-S(Jl1)*S(J,1)

11 HOVD(J)US(J,1)*S(J,2)8B(1,2) CHD(J)=MD(J)+HOVD(J) LOD PRINT RESULTS MD(J)tHOVD(J),CHD(J) OR WHENEVER CH.E.5 THROUGH LOE,FOR J=11J*G*3 ME(J)=S(J<1)*S(J1) HOVE(J)=*SJ1~)*S{J,2)*B{1,2)+S(Jt1)*S(J{3)~*B<13) CHF(J)=ME(J)+HOVE(J) LOE PRINT RESULTS ME(J).HOVE(J).CHE(J) OR WHENEVER CH.E.6 THROUGH LOZ, FOR J=11,J.G.22 MZ(J)=S(J,1)*S(J,1) HOVZ(J)=S(J,1)*S(J,2)*B(1,2) CHZ(J)=MZ(J)+HOVZ(J) LOZ PRINT RESULTS MZ(J) HOVZ(J) CHZ(J) OR WHENEVER CH.E.7 THROUGH LOH,FOR J=1,1,J.G.3 MH(J)=S(J,1)*S(J,1) HOVH(J)tS(Jsl)*S(J2)*B( 1,2)+S(Jl)*SS(J~3)*(1.S3) CHH(J)=MH(J)+HOVH(J) LOH PRINT RESULTS MH(J),HOVH(J),CHH(J) END OF CONDITIONAL WHENEVER SCFACT.E.l. CONTINUE OTHERWISE PRINT COMMENT SO ERROR MATRIX NOT COMPUTEDS TRANSFER TO END END OF CONDITIONAL END PRINT COMMENT $0 INDICATOR VALUES $ PRINT RESULTS IND1 IND2, IND3, IND4. IND5TIND6 TRANSFER TO START END OF PROGRAM SDATA N=5,CH1 * A ( l 1) =-81080* A( 2 2 ) =-096700e ~A (33 ) =-096700,A(44) =-10520'0*. A(5~5)=-154800., B( 1l)=l.B ~B(22)l. ~B( 3,3)=l.,B(41)=.493677,-*148237,-.052744'1., B(5l)=.416020,.190671B<(~,5~)1.. * N=t2CH=2 * A(1,1)=-96700.*A(222)=-105200o, B(1 1)=1.,B(321)=.251278,1.. * N=3tCH=3 * A( 11) -086000~A(2,2) -79400. A( 33) -79400. B(1,1)=1.,B(2,1)=.089535,1.,B(,1 )-.099466t0.,*1. * N*2~CH=4 * A( 11)=-086000.,A( 22) -79400.o B(1 11*B21)1 B(2s o110280,1. * N=3 CH=5 * A{(11)=-55800.,A(2~2)=-163230. A(3,3)=-79400., B(.1 )=l,,B(2,1)=.612076,1. B(3,1)a*3228160*.1.* * N=2 CH=6 * A( 1 1 =-62520. A<22) -105200. B(11)=11.,B(2*,1)=504168,1!. * Nu3,CH=7 * A(1~1)=-55800.AA(2Z2)=-112800. A(3.3)=-79400,t B( 11)sl,B(2,1)=.409328,1.,B(t3, )=.227028 0. 1.. *

APPENDIX K 5+ 5+ Mo AND W IN TiO2 5+ 5+Recently the ESR spectra of Mo and W in TiO2 were observed. The following values for the % and A tensor components were found experimentally A A A Ct 5 Ax Ay z X___ 2%_ %_ Y (in 10-4cm-1) 5+ TiOo Mo 1.8155 1.7923 1.9167 24.66 30.80 65.73 TiO2~W 1.4731 1.4463 1.5945 40.51 63.34 92.01 Following the calculation in Chapter VII-1, the ground state wave functions and the crystal field energy levels listed below were found: Ground State w.f. Energy Levels TiO2Mo5+ -.324 2i > +.9451X y> A (y ) ((%i)(43.000) - E("X _"(e) (12.800) (y )=^,J~) (.368) X (X^/^ - = ~ Tio2w5+ -.3971 t) +.92o01- Y> AE:(y)z( )(83.500) (X 2 )= (%. ) (3.930) (7:) = (lt) (o247) (xk y )= O 112

115 20. V. Heine, Group Theory in Quantum Mechanics, Pergamon Press (1960), p. 102. 21. V. Heine, Group Theory in Quantum Mechanics, Pergamon Press (1960), p. 119. 22. M. Wolfsberg and L. Helmheltz, J. Chem. Phys. 20, 837 (1952). 23. C. J. Ballhausen and H. B. Gray, Inorg. Chem. 1, 111 (1962). 24. C. J. Ballhausen, Ligand Field Theory, McGraw-Hill, New York (1962), 162. 25. L. Pauling, J. Am. Chenm Soc, 53, 1367 (1931). 26. R. S, Mulliken, J. Am, Chem, Soc. 72, 4493 (1950). 27. R. E. Watson, Iron Series Hartree-Fock Calculations Solid State and Molecular Theory Group, MIT Press (1959). 28. C. E. Moore, Atomic Energy Levels, NBS Circ. 467 (1949, 1952, 1958). 29. W. Moffitt, Repts. Prog. Phys. 17, 173 (1954). 30. H, A. Skinner and H. 0. Pritchard, Trans. Faraday Soc, 49, 1254 (1953) 31. J. C. Slater, Phys. Revo 98, 1039 (1955) 32. C. A. Coulson, Valence, Oxford University Press (1961). 33. R. S. Mulliken, J. Chimie Physique 46, 497 (1949). 34. R. S. Mulliken, J, Chemo Phys, 23, 1833 (1955) o 35, M, H Lo Pryce, Proc. Phys. Soc. (London) A63, 25 (1950). 36. C. P. Slichter, Principles of Magnetic Resonance, Harper and Row, New York (1963) 37. Ro Lacroix et G. Emch, Covalence et resonance paramagnetique, Helvetica Physica Acta 35, 592 (1962) 38. L. L. Lohr, Jr. and W. N. Lipscomb, J. Chemo Phys. 38, 1607 (1963). 39. J. D, Roberts, Notes on Molecular Orbital alculations, W. A. Benjamin, New York (1961). 40. D. K, Rei, SOV. Phys. Solid State 3, 1845 (1962).

116 41. K. W. H. Stevens, Proc. Roy. Soc. (London) A219, 542 (1953). 42. S. Karavelas, C. Kikuchi, and H. Watanabe, Bull. Am. Phys. Soc. 9, 403 (1964). 43. I. Chen, On the Theory of Superhyperfine Interaction in Iron Group Ion Complexes, PhD Thesis, University of Michigan, 1964. 44. T. Chang, Bull. Am. Phys. Soc. 9, 568 (1964).

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