THE UNIVE R S I T Y OF M I C H IGA. N COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report MOLECULAR ORBITAL CALCULATION OF VANADIUM IN THE TUNGSTEN SITE OF CaW04 S. Karavelas C. Kikuchi ORA Project 06029 under contract with: U S. ARMY MATERIEL COMMAND HARRY DIAMOND LABORATORIES CONTRACT NO. DA-49-186-AMC-80(X) WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR September 1965

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES v'ii ABSTRACT ix CHAPTER Io THEORETICAL BACKGROUND 1 Solution of the One-Electron Problem 3 Simplification of the Secular Determinant 4 II. SEMIEMPIRICAL MOLECULAR ORBITAL CALCULATION OF VANADIUM IN THE TUNGSTEN SITE OF CaWO4 6 Selection of the Atomic Orbital 8 Group Overlap Integrals 13 Oxygen 13 Vanadium 13 Resonance (or Exchange) Integrals 14 Coulomb Integrals 14 a) The Case of 3d VSIP 16 b) Cases of 4s and 4p VSIP's 18 Results of the Charge-Self-Consistent Calculation 19 Highest Occupied Orbital 23 III. EPR PROPERTIES OF THE COMPLEX [V04] 24 Comments 25 Charge-Selfconsistent Calculation with the Excited F2 Level 28 IV. CONCLUSIONS AND REMARKS 33 REFERENCES 37 iii

LIST OF TABLES Tab le Page I. Character Table of S4 Point Symmetry Group 6 II. Combination of AO Transforming According to the I R.'s of the S4 Group 12 III. Valence-State Ionization Potentials for Vanadium in Kcm-l 17 IV. Coefficients of the Interpolation Trinomal Aq2+Bq+C for the Various Vanadium Configurations 18 Vo Eigenvalues and Eigenfunctions of the Complex 20 VI. Distribution of the Metal Electronic Charge Among the AO 22 VII. Distribution of the Metal Electronic Charge Among the AO 29 VIII. Eigenvalues and Eigenfunctions of the Complex [V04] 52 or

LIST OF FIGURES Figure Page 1. Coordinate systems of vanadium and oxygen ions. 7 2. Part of the unit cell showing the coordination of an oxygen atom. 10 35 Charge self-consistent-MO calculation of vanadium in the Tungsten site of CaWO4. 21 4. Coupling of the F2 I.R. by the angular momentum operators. 27 45. MO calculation of the complex [V04] 5 31 vii

ABSTRACT Semiempirical MO calculations of vanadium in the tungsten site of CaWO4 crystals are presented. The two cases of [V04]3- and [V04]4- are considered. The true symmetry S4 of the tungsten site is used in the construction of the ]LCAO and ligand hybrid orbitals are constructed to accomodate the crystal structure. Small positive Ag shifts are predicted and possible accidental isotropy of the g tensor. ix

CHAPTER I THEORETICAL BACKGROUND The Scrodinger equation for a collection of k nuclei and N electrons in the Born-Oppenheimer approximation is; H(Y(rlr2.. rN) = EY(rlr2...rN) (I-l) where the Hamiltonian is given in (1-2) N r k 25 N Z ^ 2 2 21 e H - i { Vi vj- i- +Z (I-2) 2m rij i r,i i=!Lj=l _ J. i<i' In addition to the Borm-Oppenheimer approximation which neglects the kinetic energy of the nuclei, we have neglected terms depending on the electron spin, the nuclear spin and the nuclear quadrupole momento Even so the solution of Eqo (1-1) is impossible and only approximate numerical solutions can be obtained in the simple caseso The difficulty arises from the last term in Eq. (I-2) which expresses the electron-electron interaction. Without this term the Hamiltonian in Eq. (I-2) would become a sum of one-electron operators and the solutions of Eq. (I-l) would be in the form of a product of oneelectron functions (or orbitals;) i.e., t(rlr2.. rN) = 1(rl)12(r2)..O N(rN) (I-3) In the Self-consistent Field (SCF) approximation we keep the electron-electron interaction term but assume that the solution of Eq. (I-l) can be written in the form of a product of orbitals as in Eq. (I-n. Using this expression for 1

the solution and applying the variational principle to minimize the energy in (I-4) E = IHL (I-4) <E I E> we get the following N Hartree equations~ k N 11 2 7 e2 () 2 -- v -Vi+ < -—,(2)1 ~,,i(2)> ~i(l) = Ii-() 2m rij i L ~ j=ll 2- i'. =1 i=l,2,i3..N (I-5) We interpret Eq. (I-5) as expressing the motion of the electron that occupies the i-th orbital, under the influence of the stationary nuclei (second term) and the fixed effective (average) field of the rest of the electrons which occupy the other orbitals (third term). The method used for solving the equations in (I-5) is by successive iterations. However this is still a formidable problem and therefore additional assumptions are introduced. In the valence electron approximation we divide the electrons into two groups. The core electrons and the valence electrons, The core electrons are those found in closed atomic shells and are considered to interact with the rest of the electrons (the valence electrons) only through the screening of the corresponding nuclei6 Therefore Eq. (I-5) simplifies into the following: r k Nr 2 2 Zje + e< () ( Ep( j=l rb e =2 2

where Z is the effective nuclear charge of the j-th nucleus and Nr is the J number of the valence electrons. Of course the only simplification that occurs in Eq. (I-6) with respect to Eq. (I-5) is in the number of the equations, The third term in Eq. (I-6) (as well as in Eq. (I-5)) varies with the index ~J because it expresses the potential energy of an electron in the t-th orbital due to the rest of the electrons which occupy the other orbitals and all orbitals are not the same. Nevertheless as an additional assumption we consider quite frequently that this variation is not great so that all valence electrons are considered to move under the same effective fieldo Therefore we write one effective Hamiltonian for all valence electrons k He ff ~ V2-Z- el + v() (I-7) 2m r Ij=l where V(r ) is the average value of the third term in (I-6) taken over all values of p. Thus the original many-electron problem has been reduced into a oneelectron problem, namely the following. Hefft( ) E= (r) (-8) SOLUTION OF THE ONE-ELECTRON PROBLEM In order to solve Eq. (I-8) usually we approximate the molecular orbital <,(r) by a linear combination of atomic orbitals (LCA.O) centered at the different nuclei; ie, we write, i~e., we wr

n Vt () ( CiP9i -(I9) 1=1 Then by using the variational method we minimize the expression (I-10) E = <*f I Heff |> ( 10) where *, is expressed by Eq. (1-9). This procedure gives the following equations: > ci(Hij-ESij) = 0 j=l,2...n (I-l) i=l For a nontrivial solution we must have det|Hij-ESijl = 0 i,j=l,2o..n (I-12) where by definition Hii - <i lHeff i> = the Coulomb Integral Hi.j <epilHeffl Pj> = the Resonance Integral Si <cpil cp> = the Overlap Integral The selection of the proper atomic orbital cpi to be used in Eq. (I-9) rests on intuition and experience but the best coefficients ci's are determined rigorously by the variational method, SIMPLIFICATION OF THE SECULAR DETERMINANT The secular determinant (12) is usually quite cumbersome to solve. Its rank i equal to the number n of the AO used in the expansion (I-9) which is often twenty, thirty or forty.S In order to factor out the secular determinant 4

it is convenient to use Group Theory to form combinations of the originally selected AO so that these combinations transform according to the irreducible representations (I.R.) of the group G of the symmetry transformations of the Hamiltonian Heffv Then according to group theory the only nonzero matrix elements Hij and Sij occur between functions belonging to the same irreducible representation since the Hamiltonian operator and the identity operator transform according to the identity I.Ro of the group G (see for example Heine "Group Theory"). The overlap integrals Sij are called group overlap integrals when combinations of AO transforming according to the various I.R. are used. They are calculated by using SCF atomic orbitals like the ones given by Freeman and Watson or if the latter are not available Slater-type functions may be tried, For the Coulomb and resonance integrals either we try to calculate them directly from first principles or we approximate them by using experimental data concerning the ionization potential of free atoms and molecules. The latter case is referred to as a semiempirical calculation. 5

CHAPTER II SEMIEMPIRICAL MOLECULAR ORBITAL CALCULATION OF VANADIUM IN THE TUNGSTEN SITE OF CaW04 In this chapter a semiempirical molecular orbital calculation of vanadium in the tungsten site of CaW04 is tried. The true symmetry of the tungsten site is S4. However, if one restricts himself to the four neighboring oxygen atoms, then the symmetry is higher, namely D2d. The coordinate systems have been chosen according to Fig. 1. Notice that the coordinate systems on the oxygen atoms are left-handed. This facilitates the calculation of the group overlap integrals. The character table of the point group S4 is given in Table I. The atomic orbitals s, p, d are classified according to the I.R as well as the angular momentum A A A operators Lx, Ly, Lz. In the lower part of Table I the reduction of some direct products of I.R. is given. TABLE I CHARACTER TABLE OF S4 POINT SYMMETRY GROUP S4 E C2 S4 S4 _ A Fr 1 1 1 1 s; Z2; X2+y2; Lz F2 1 1 -1 -1 z; x2-y2; xy E 2 -2 0 0 x, y; xz, yz; Lx, Ly r x 2 1 -1 1 -1 r2 I xE 2 0 -2 0 =E r2x E 2 0 -2 0 =E E;x E 4 0 4 0 = 2F + 22 6

X4 \4 /I \ X2 7 3//\ o Fig. 1. Coordinate systems of vanadium and oxygen ions.

SELECTION OF THE ATOMIC ORBITAL Now the problem of which AO are to be used in the LCAO approximation is considered. The electronic configurations of the vanadium and oxygen atoms are as follows: 2 2 V: Is22s22p63s23p63 d 4s = [Ar]3d3 4s 0: ls22s22p4 = [He]2s22p4 Since the vanadium 3d, 4s, and 4p AO's are energetically close as well as the oxygen 2s and 2p ones, it is reasonable to consider them as candidates in the expansion (II-1) I,(r) = cicpi (II-1) i=l Of course the greater the number n of the AO's used in (II-1) the better. However for practical reasons we try to limit this number as much as possible. Therefore we shall try to solve the problem of vanadium in the tungsten site using only the nine vanadium AO 5d, 4s, 4p and the sixteen AO 2s, 2p of the four neighboring oxygen atoms; i.eo, we shall consider only the complex [V04]. Thus the molecular orbitals 4r(r) may be approximated by the (11-2) 25 p(r) C X ~ccpi (II-2) i=l However, the approximation (II-2) is deficient in one important aspect, namely it does not take into consideration the surroundings of the complex [VthO4]t Thus one would be inclined to use the point symmetry group D2d instead of the S4J Furthermore the approximation (II-2) would result in the following 8

a oxygen hybrid bonding orbitals sinQi(2s)i + cos@i(2p)i i=1,2,5,4 whose hybridization angle @ would be about 25~~5~, and the corresponding nonbonding orbitals cosoi(2s )-sinQi(2p)i directed away from vanadium. What we intend to do, instead of the above program, is to incorporate the influence of the rest of the crystal in the construction of oxygen hybrid orbitals which will be combined afterwards according to the S4 point group. Each of the oxygen atoms in the complex [V04] is at a distance of 1.78 0 0 A from the vanadium atom and at 2.44 A and 2.48 A from two neighbor calcium atoms (see Figo 2)o The vanadium atom and the two calcium atoms are the closest metal atoms and lie almost on a plane with the oxygen atom (oxygen atom about.25 A off the plane defined by the vanadium and the calcium atoms). 0 Approximately perpendicular to this plane and at a distance of 2.91 A lies a tungsten atom. We make the assumption that the hybrid orbitals, which are constructed pointing towards the four metal atom, will result in the most stable situationo Since it is convenient to have these hybrid orbitals orthogonal to each other, we construct first the three orthogonal hybrid orbitals whch are directed towards the vanadium atom and the two calcium atom as follows: 9

Co I3 0 V ~ C - a2 b Fig. 2. Part of the unit cell showing the coordination of an oxygen atom. 10

cpv -.76473 (2s)+ 64435(2p), cca2 = -38671(2s)+o92220(2p)2 (l-5) ca3 = " 49268(2s)+.87020(2p)3 Then we construct the orbital h which is orthogonal to the three orbitals in Eqso (II-3) as follows: h = 15133 (2s)+ 98848(2p)wl (1I-4) The oxygen orbitals (2p)v, (2p)2, (2p)3 point towards the corresponding metal atomso When checked the orbital h points quite close towards the direction of the Wl-siteo To be more specific the direction angles of the h orbital are 125~, 535, 56~ and those of the Wl-site, 116~, 510, 50~, The two directions form an angle of about 10~. We take the orbitals in (II-3) and (II-4) as the proper ones to be used in (II-1)o However, two of them, namely the ca2 and ca3 overlap very little with the vanadium orbitals and will be considered as nonbonding orbitalso Thus referring to Figo 1 the following orbitals (II-5) are to be usedo (a) the nine vanadium orbitals 3d, 4s, 3p (b) the four a orbitals ai - o76473 (2s) +.64435(2p) i i=- 2,3,4 (c) the four h-orbitals, which are almost it-orbitals, h = 5133 (2s )+()o.52582(2px)i 81949(2py)i o 17957(2pz)i 1 for i=1,4 -1 Cfor i-2,3 (I15 ) 11

Using the Character Table I for the S4 group and the transformation properties of the functions in (II-5) (see the Appendix) the combinations of orbitals which transform according to the I.R. of the S4 group are constructed. The results are summarized in Table II. We observe that the symmetry arguments alone are not sufficient to determine the coefficients in the doubly degenerate irreducible representation E. This is because the S4 group is mathematically an abelian group with nondegenerate irreducible representations. The double degeneracy of the E I.R. which is shown in Table I is due to time reversal symmetry and not to the S4 point symmetry group. TABLE II COMBINATION OF AO TRANSFORMING ACCORDING TO THE I.R.' S OF THE S4 GROUP I. R. Metal o-Ligand 7r-Ligand z2s I +o2.+ 03 +04 hl + h2 + h3 + h4 ~F, z2; 4s 0 H+I ______ F2 x2-y2;xy;z I 0 2-03 +4 H=hI -h2 -h3+h4 2 2 H2 2 (sin a(xz) + cos a(yz)) (sin l13 + cos P3') (sinYH3 + cosYH3) E cos a(xz) - sin a(yz) cos 1P3 - sin 1)33 cosYH3 - sinY H3' (sinf(y) + cos a'(x) cos a'(y) - sin a'(x) (1) 3- o -o 4. (3) H3 h - h4 (2) *3'-,, (4) H3,'-h 12

GROUP OVERLAP INTEGRALS The group overlap integrals are calculated according to the standard techniques; see for example Reference 1 Chapter 7, Appendix I. The University of Michigan IBM 7090 computer was used. The following SCF radial functions were used. Oxygen R(2s) = 5459cp2(1.8O)+4 e4839cp2(2.8o) (II-6) R(2p) =.6804cp2(155)+. 4038p2(3.45) Vanadium R(3d) =.5243cp5(18289)+.4989cp3 (3 6102) +. 1153 1p3 (6.8020 )+.0055p3 ( 12.4322) R(4s) = R(4p) 0o677cp3( 9 3 319)-. 09713(5.1562) -. 0246cp3 (35o78)+. o441cp4((38742) +o36077cp4( 8764)+. 6090cp4(l 1462) +. 1487p4 (.7800 (II-7) where ( \ - (2 ) 2n 1 n-1 -e r Pn') r e (2n)r The following nonzero overlap integrals were found: Gr(z2, pl).02022; Gr2(x2 y H2) = -.15657 Gp1(z2,Hl).09518; Gr2(xy,cp2) =.25564 Grl(4scpm) =.67470; Gr2(xy,H2) = -.05509 Gr1(4s,Hi) =.04082; Gr2(CzH2) =.l8355 (Ii-8)

cGE(xz,cp5) = GE(xz,qp) = GE(yz,cp3) = -GE(yz,cp ) = 16792 GE(Y,)3) = GE(y,cpq) = GE(x,cp5) = -GE(x,cp) =.37166 GE(xz,H5) = -GE(yz,H) = -.02774 GE(xz,H) = GE(Yz,H3) =.07510 GE(Y,H5) = -GE(x,H) =.14880 G HE(y,H) = GE(x,H3) = -.19477 (I1-8) RESONANCE (or Exchange) INTEGRALS In the semiempirical calculations we approximate the Coulomb integrals by experimental data and the resonance integrals by the following relation (II-9) Hij = FGj HiiHjj (-9) where F is a constant between 1.67 and 2.00. The Gij is the group overlap integral between the i-th and j-th orbitals. In the following, F will be taken equal to 2.00 according to common practice. COULOMB INTEGRALS The results of any semiempirical calculation depend crucially on the Coulomb integrals Hii used in the Secular Equation (II-10) iHi-jESijI = - (II-o0) On the other hand th.e Coulomb integrals H.i, which are approximated as 14

valence-state ionization potentials (VSIP), depend again crucially on the charges and the electronic configurations of the atoms. Therefore it seems that one is faced with an ambiguous problem. The way out of this ambiguity is to use the trial and error method in connection with some monitoring device. The most commonly used monitoring device is the "charge self-consistency"; i.e., (a) we assume a certain electronic charge and configuration (b) we estimate the corresponding VSIP (c) the secular equation is solved (d) the charge and configurations is calculated from the solution and (e) we stop when agreement between assumed and calculated values is reached. Any overlap charge is devided in half between metal and ligand orbitals according to Mulliken's suggestion. To start the calculation we assume a charge of minus three for the com2plex [V04] which makes it isoelectronic with the complex [W04] with 32 electrons. We expect most of this charge to be located in the periphery of the complex leaving the vanadium atom almost neutral. Thus each oxygen atom will have a charge in the vicinity of -o75. From previous calculations the following VSIP of the a and h oxygen orbitals are deduced: VSIP(Oxygen a) -77000 cm-1 (11-11) VSIP (Oxygen h) -49500 cm1 These VSIP are held fixed in the following trial and error calculation. In general form the calculation runs as follows: Assumed vanadium configuration: d s p Assumed.vanadium charge: q = 5-d-s-p Assumed electronic charge: n = 5-q = d+s+p 15

a) The Case of 3d VSIP The assumed configuration 3dd4ss4pP is split into components corresponding to the configurations 3dn, 3dn14s and 3dn-14p as follows: Conf.3d 4sS4p = (l-s-p)Conf3dn+sConf3dn 14s + pConf3dn-14p Then the VSIP of the 3d electron is given by the following expression 3dVSIP = (l-s-p)(3dVSIP of 3dn)+s(3dVSIP of 3d n1s) +p(3dVSIP of 3dn-14p) (II-12) Generally n is not an integer. However, available estimations for the various VSIP's exist only for integer values of n. In Table III the values given3 for vanadium by A. Viste and H. Basch are listed. The strong dependence of the VSIP's on the vanadium charge and configuration is obvious from the Table III. In order to be able to use formula (II-12) in the cases where the electronic charge n is not an integer we interpolate the VSIP's assuming at most a quadratic dependence on the charge q. Therefore for every configuration we have an expression of the form 2 VSIP(q) = Aq +BqC (II-13) Table IV gives the calculated values of A, B, C for the corresponding transitions of Table III. Whenever the VSIP's for only two integer values n of a certain configuration are known, the coefficient A of the quadratic dependence is taken from the preceding configuration. Also, whenever the VSIP of a single integer value is known then both the coefficients A and B are 16

\ID 00 Kch -t I 0~ 01\: f' T ffC\ O I ro 8 Xro I r ad Fr h r-A H | C | D s_ I O X I 4H c'i)~ 0- C,-S H' \oL L — 0 ) Z \-,D -- 0 1; ~. - ~' r-]:- C- c - J — i Cr- U I F1 S + C\l -.-\ 4 KJ p c ) ro t t C\ t + - Q KC\ p ) 9 U) n g-1'H +~Z1~2t _1II - -z cII H H rOi c-~ l-C\CC N K l\ -t Cii rorrro ro rord ro ro H E-] 0 KC\ K' C K-C\ N i\ K 0 pq 0 0 OC.4- (Di C; ^ C0\ (\ 8' O7 s,.) ~ e ~,, Q f 8 n. n Ct U 0 b 9o i > 0 8 - rO i U) ^^ - -f Pi 0 - t <r I^O s^g ^-^n~ ~ ^ ~ ( r )O d CoO C E-4- C ICil + P- 4 I\ + 9 > IW~T + + VC- -0 +0) Pi - po ono no rorKr O r TO rO O 0 C r d ro ro \ K 0 \o t 1 C\j 8 C t C ^,H V^_- _._,- cn )a ci 0 0 U)' -i cii 3Hg c P -P 4- p U) 0 ri r H S c) c) c)..0h 0 <r p D 00 FLJ 6S n m ( 1 tM nc? -f' ^ ^ OJ *-~ *T CVJ ^h -^5 ^~ie O C;^ 5-^ - L^^s^ -t ( I^ ^t K^ ^ 8-D1I $-TI ^ I ^ T!^ Nj T r a, K a,^ I a, 6d -- 8 - -- _ ^ _ ^ _ _ ~ _ _, ~ ~. -- - - gd ~~-e a ~~~n ~~a o &~~~~~~ I p-~~~~~~~~~~~~~~C (Dq

taken from the preceding configuration. It is obvious that the obtained curves in the latter two cases are less reliable but that is the best we can do for the present time. TABLE IV COEFFICIENTS OF THE INTERPOLATION TRINOMAL Aq +Bq+C FOR THE VARIOUS VANADIUM CONFIGURATIONS ^vsm T Starting vsIP A B C Configuration 3d 3d5 -15.8 -68.O -31.4 3d 3d 4s -14.0 -87.0 -51.4 3d d4 4p -14.0 -87.5 -61.4 4s 3d 4s - 8.6 -54.1 -51.0 4s 3d3 4s2 - 8.6 -62.9 -60.4 4s 3d3 4s4p - 8.6 -57.5 -70.6 4p 3d 4p - 7,5 -45.4 -27.7 4p 3d3 4p2 - 7.5 -50.8 -36.8 4p 3d3 4s4p - 7-5 -50.8 -36.4 b) Cases of 4s and 4p VSIP's / These cases are similar to the previous one of the 3d VSIP with the difference that when the assumed configuration 3dd4ss4pP is decomposed to its components the results depend on what electronic orbital we have in mind. For example for the VSIP of a 4s electron the assumed configuration should be decomposed in contributions from the configurations 3d4 4s, 3d3 4s and 18

and 3d3 4s4p. Thus the obtained result is Conf.3dd4s 4pP (2-s-p)Conf5d 4s+(s-l)Conf3d 4s +pConf 3d5 4s4p (11i-14) Similarly when the VSIP of the 4p electron is involved the configurations that should be considered are the 3d4 4p, d 4p2 and 3d 4s4p Thus we have instead Conf3dd4ss4pP = (2-s-p)Conf3d 4p+(p-l)Conf3d3 4p2+sConf3d3 4s4p (II 15) RESULTS OF THE CHARGE-SELFCONSISTENT CALCULATION By trial and error the following vanadium configuration has been found to give charge self-consistent results: 3 8281 54394.5744 (3d) (4s) (4p) (11-16) Assuming this configuration the VSIP's for the vanadium atomic orbitals are found as follows. 3d VSIP = -[-o0138(42.5)+.4394(65.5)+.5744(75.5)] = -71.6 Kcm4s VSIP = -[..9862(59.8)-.56o6(70o6)+.5744(79.9)] = -65-3 Kcm4p VSIP = -[.9862(35.1)-.4256(45.0)+.4394(44.6)] = -35.0 Kcm. ( 1-17) We have taken already as VSIPTs for the c and h oxygen hybrid orbitals -77. 0 Kcm and -49.5 Kcm correspondingly. Approximating the Coulomb integrals Hii with the above VSIP's four secular equations are solved, one for each of the irreducible representations of the S point group~ The results are given in the following Table V and Figo 3. 19

1 ip oEr-] r-4 1 co Lnr- C\OJ O r-C co o O (D 01 \ I o O C\ C\O 0 *CO 0 e ~ P=n'IO w EcOLN OKO 3M NO O OcO 0 K^ ro 00 C\JOa 00 +SX + * 9k + + I t- rl -~r-0Oj 0OT 0\-ZOO WI 0N C 0 C\ Nr CC\ > \ C\ e o ( t O %1!,c'O 0 CN o r- i' 0 * o * P++ e t- NL + \O ++ * I ONC _ t C N ON I r-] C\J 0 r-l + -N Kn - 9- [N o ON9- c1 9-CU - r C z \9 C00 ~ \ 9-r L[NA OJ 0 N -0 O0 O, t- - I o - LP O cO F ~> Jr CSO- C\J + OJ CO + C\NJ \0 + -_t JO O e9- CO O CJ r1 o\ Cl ri- CJ * r-4 9- C10 9- * 9- \O 91X X. + I o CO0 r-I \0 + I 00 + C00 0 t — 00 IX C01 X M r- + > X N\ 0J Ln J- 0 \O X O LN r- N - WIt X Mr- 00 C 00 O * -t - O n \ID t- ON O \ C OON \I D ~O - r-4 + C\* OJ O * -zrQ *-r C)0D 0 * V- ON N I O O + r-l O + O + 0 U + * O 00 N 0 * r- N O N O r- I + ]J NI + c I\J + n + 4 g9 r-4+l r-l Ln >^ + 01 + 0 + D- D - ON 0 9 O e- o\ O r- rJ 0\ 0 rE- - N-C O — t 1 0N C(O r- - 9- 0 r 0 91 rl > -*+ \D O O Nt 4 O\ N * N ON\j ( I + 0 0 CO O L + r-l r- + CO) + rO ~ < * X 0 O > O* * r O ~ O rI I I N X X + l O. X o V2 + Nt N *n 0, I n ON O 0 ti^1 Cn 1 NU CI 00 r-A 1 ON N! -2- + O 01 0t — r-] \ -z- \-c -OCJ U2 0 o 00 - o H [On C\i t- n N CN\ C01 \ - n [n \O 0 O con LOn n > EE-l ^D K\ [ 1n Ln ON * t-0 0 0 0 0 \ o N CJ1 01 ( - j - 0 N CCK, OI 0ON Ln^ 00 9 ^ * b o ~ X o O 0 C r- 0 -t 5 r- 1+ I+ + + + + +9 + 0 o N I N CU I + N C\1 I N N I 9 + N 8 + Z 0 C\J N 01N NX N C01 a NX 1 c N X. N N RiX C 0 \ CQ -g N X CCt N O >- > N X >15 L\n \I n L Cl CO O C H ~ L[NK ONj ON0 - CcO t — CO O t — L O L a\ r-] Xr O 0 C( r-4 ) l(n r CN \0 1 k ) N\ \O O r 0 CU\ 9 j II I 0 * I 9 1 + * 9 z H rrK-4 CC\J C\J r 0 C0\ r-4 L 0 (\J K N -1 -1- ON r O ON I\r- [N ON ON 0 0 j &O - O r- 0- 0O -I- L0 C[O O U O 0 1& \O 00 OO [O- 1O 0 0 t- 01 01 -z -- - [N t — - 00 00 ^ + + I I 9 9 9 i et I 9 20

VANADIUM OXYGEN ORBITALS ORBITALS +77 +27- I\ i' // r \1 -30- / / \ \\ 4p. \-'-40 z / _60- / \E the Tungsten h (a)o C -70 - 3d r, / 21 \E / -90 f- \ 21

Using the eigenfunctions of Table V and the overlap integrals in (II-8) the output configuration is calculated and matches exactly the assumed configuration 3.8281.4394 5744 3d 4s 4p To be more specific the following distribution of electronic charge is found: TABLE VI DISTRIBUTION OF THE METAL ELECTRONIC CHARGE AMONG THE AO I.R Orbital Electronic charge 2 2 rl 3z r2.9490 4s 4394 r2 x2_ 2.8849 22'2 x -y.8849 xy.4299 z o0008 E xz.7822 yz.7822 y.2868 x.2868 The total electronic charge is 4.8419 corresponding to a net vanadium charge of q = +,1581 (TI-18) On the right hand side of Fig, 3 in addition to the C and h oxygen hybrid 22

orbitals the oxygen nonbonding orbitals which are directed towards the neighboring calcium atoms are shown. Their energies are calculated from the same values of the oxygen 2s and 2p VSIP which were used in calculating the C and h orbitals, namely 2pVSIP = -48.4 Kcm 1 and 2sVSIP = -97.3 Kcm-l HIGHEST OCCUPIED ORBITAL The assumed charge on the complex VT04 and -3 corresponding to a total number of 32 electron. We start filling the levels in Fig. 3 from the bottom going up until the number of electrons is exhausted. The highest filled level is the h(Ca2). However, the levels h(Ca2) and h(Ca3) are considered as nonbonding and therefore do not enter in the interactions of the vanadium center. For the latter the highest filled level is the doubly degenerate level E at -62.5 Kcm. 23

CHAPTER III EPR PROPERTIES OF THE COMPLEX [V04] According to the results obtained in the previous chapter no EPR signal is expected from the complex [V04] because all electrons are paired. This is in agreement with the experimental fact that the crystals of CaWO4:V do not show any EPR signal. However, after x-irradiation many EPR lines are observed bearing the signature of the vanadium hyperfine interaction. In particular an intense set of eight lines is observed which shows an isotropic g value of 2.0245 and a hyperfine separation of 19 to 21 gauss. Both of these properties are a little unusual for vanadium and the possibility of identifying the complex [V04] as the source of this EPR signal need to be considered. Obviously the x-irradiation pVrduces unpair spins giving the EPR signal. We distinguish the following possibilities: a) An electron is knocked out of the vanadium center by the x-rays. This leaves a hole in the highest occupied level. When examined the EPR properties of this hole are found far from being the isotropic tensor. b) An electron is excited from the level E to a higher level. This gives spin zero or one contrary to the observed effective spin of 1/2. c) An electron is added to the vanadium center. In this case the calculated g values are found as follows when the eigenvalues and eigenfunctions of Table V are used: 24

gl1 2.010 (III-1) gl = 2.003 COMMENTS Only in the last case do we have some resemblance with the observed set of the eight strong lines; namely (a) the calculated g values are greater than 2 (b) the extra -electron in the F2 level spends less than 42% of its time in the vanadium neighborhood in accordance, perhaps, with the observed small hyperfine interaction. However, the numerical results are fair because they show a pronounced anisotropy which is compatible with the symmetry of the tungsten site but not in agreement with the experimental values. In addition the calculated values for the Ag's are small. Probable inadequacies of calculation may be due to the following reasons: a) The complex [V04] is not in vaccum but in the CaW04 crystal, Therefore the VSIP may be altered by a type of Madelung potential. b) With the extra electron in the F2 orbital the charge of the complex has been altered and consequently the VSIP's have changed too. The alteration may be due either to the addition of a completely new electron to the complex bringing its charge up to -4 or to the transfer of one electron from the lone pairs of the nonbonding orbitals to the F2 molecular orbital. The latter transfer leaves the overall charge of the complex unchanged but redistributes the electronic cloud so that an excited state results; In any case due to the necessity of charge neutrality the extra 25

negative charge of the vanadium center has to be compensated by a positive charge in the neighborhoods We consider the case (b) in which extra negative charge is added to vanadium. The usual formula7 for nondegenerate ground state is used; ie., gij = 2(aij-ij) (II-2) where Aij = <~olinXn||o> (III-3) En - E n0O However an extra difficulty arises from the fact that the ground state |0> and the excited states In>, which are Slater determinants, consist of wave functions having both metal and ligand parts. Thus in one of the matrix elements, for example the <O|tiln> only ligand-ligand and metalmetal contributions need to be taken thinking of it as being the matrix element due to the spin-orbit coupling (There is a l/r3 dependence in the spin-orbit coupling constant). In this case the other matrix element <n|fIjIO> has to be due to the Zeeman interaction and the contributions from metal-ligand and ligand-metal should be taken in addition to the others. For more details see the example References 5 and 8. In Fig. 4 the coupling of the r2 I.Ro is shown by the singular momentum operators. As for the -l one-electron spin-orbit coupling constant the value of -150cm is usedo From the spectroscopic data the corresponding values of VO and V+ are -158cm 1 and -136cm-. Actually for the parts coming from the ligand contribution a higher constant should be used. However, by using 180cmz1 for the ligand-ligand evaluations of the spin-orbit matrix elements no 26

r, LZ r2 -- E E --- L -- E r,,, I..... Fig. 4. Coupling of the F2 I.R. by the angular momentum operators. 27

effect of importance is observed. CHA.RGE-SELFCONSISTENT CALCUIATION WITH THE EXCITED F2 LEVEL OCCUPIED BY AN EXTRA ELECTRON 4The calculation for the complex [V04] follows the same steps as that of [V04]3- We start by fixing the values of oxygen (2p) VSIP " -16 Kcmnand (2s)VSIP; -70 Kcm-1. The higher VSIP are due now to the approximate-1 charge that each of the oxygen atoms is suppose to have. Various vanadium configurations are tried. Self-consistency is obtained for the following configuration 4o6718.4967 0o102 35d 4s 4 (III-4) The total electronic charge is n = 5.1788 corresponding to a net vanadium charge of q = -.1788. The VSIP corresponding to this configuration are as follows: Ed =-[.4390(19.7)+.4968(3563)+.0102(46 2)] = -28.22 Es = -[1.4930(41.6)-.5032(49.4)+.0102(60.6) = -37.87 (III-5) Ep = -[l.4930(19.8)-.9898(2709) +4965(27.5)1] -15.61 The above results are obtained in this case by extrapolating the quadratic fit of Table IV in the region of negative vanadium charge and are less reliable than before. Specially it was felt that the (2p)VSIP value should be increased a little. Summarizing, the following VSIP values were used, which gave the above mentioned configuration (III-4). (4p)VSIP = -5.93 Kcm (h)VSIP = -16.2 Kcm-1 (3d)VSIP - 2853 Kcm1 (III-6) (4s)VSIP = -37o7 Kcm-1 (oC)VSIP = -38.1 Kcmnl 28

Except for the increase in the (4p)VSIP of the order of 1 ev. the calculation is rigorously charge self-consistent. The distribution of the electronic charge is as follows: TABLE VII DISTRIBUTION OF THE METAL ELECTRONIC CHARGE AMONG THE AO I R Orbital Electronic charge 3z2-r2.9780 4s.4968 F2 |x2-y2.9493 xy.9574 z E xz.89355 yz.8935 y.0053 x.0053 The calculated g values are: gll = 2.0700 (III-?) gl = 2,0608 We observe that the g tensor is almost isotropic but the absolute values are greater this time than the observed one. At this point we observe by mere arithmetic that if all VSIP are made deeper by a factor of 2.528 the following values are obtained for the g tensor. 29

gl = 2.0268 and gl = 2.0231 These values are just above and below the observed g value of 2.0245. The above mention reduction involves energies of the order of 50 Kcmr for the vanadium 3d and 4s VSIP's. A rough calculation shows that a positive unit charge located at the site of a neighboring calcium atom lowers the potential energy of an -l electron at the tungsten site by about 30 Kcm Similarly a positive unit 0 charge at a distance of 1.784 A which is the vanadium-oxygen distance creates a lowering of about 65 Kcm l From these magnitudes we observe that perhaps the final answer to the problem lies in the way the extra negative charge which is located at the vanadium site is compensated by the crystal. The results are summarized in Fig. 5 and Table VIII. 50

VANADIUM LIGAND ORBITALS ORBITALS +401 r,+ 30 +lo 0 2 E -20 _ _\ -30 ~-40~ \\ -1 0 -/ V \ \ / r \\\ -60 \ \-0 E 0 Fig. 5. M calculation of the complex [v /! 1 /s E 51~~~~~~~~~~ C)

TABLE VIII EIGENVALUES AND EIGENFUNCTIONS OF THE COMPLEX [V04]Energy cm-1 I.R. Eigenfunctions +41218 rl -. 485z2-1.2432s+1.2390p1+. 0869H1 (104199) +5167 E -. 2869xz+. 0014yz-1.2200y+. o698x+. 4268c3+.4698cp+. 2080H3-. 2150H3 (13063) +. 0014xz+. 2869yz+. 0698y+1. 2200x-. 4698q3 +. 4268c3 +. 2150H3 +. 2080H~ -5170 r -.0738(x 2-y)-.0378xy+1.0123z+.0178c2-. 2895H2 (-135071) -14515 r2.3081(x2-y2)+.2286xy+.1044z-. 1270p2+. 9595H2 (-36693) -15493 E.2980xz+.o896yz-. 000y+.0127x-. 1406c3 -. 676p+. 0984H-. 9720H' (-39167) -.o896xz+.2980yz-.0127y-.0300x-. 0676cp3+.1406cp-. 9720H3-.0984H3 -15739 rF -. 200lz2+.0337s-.0522cpl+.9966H1 (-39787) -20928 r2.0823 (x2-y2)-.9266xy-.0029z+.6188cp2+. 177612 (-52907) -22380 E -.5669xz+. 6598yz+. 0783y-.0953+-. 0400cp3+.5373cp +. 203 2H3-. 0801H1 (-56576).6598xz+.5669yz-. 0953y-.0783x-. 5373p3-. 0400p +. 0801H3 +. 2032H -28490 rl.9837z2-.0244s+o0042pl+.l 079H1 (-72023) -28734 r2.9586(x2-y2)+.0134xy+.0126z-. 013362-. 1719H2 (-72639) -41138 r.00003 (x2-y2)+. 4014xy+. 0002z+.8189p2_-.0016H2 (-103996) -41595 E.3532xz-.1614yz-.. 438y+.0658x+.3193cp3+.8440cp.+.0286H-. 0147H1 (-105151).1614xz+.3532yz-.0658y-. 1438x+.8440cp3-.3143c3-. 0147H3-.0286H~ -53148 r *.0057z2+.5432s+.5496Tcp-.0024H1 (-134358) Note: The numbers in parenthesis correspond to the case where all the VSIP have been divided by the factor 2.528. 32

CHAPTER IV CONCLUSIONS AND REMARKS The results obtained in the Chapters III and IV point decidedly in identifying the complex [V04] as the source of the set of the eight intense lineso During the trial and error method all calculations were giving positive Ag shifts with values varying from.002 to.080. The complex [V04] seems to satisfy better the charge self-consistency and the isotropy of the g tensor although not both of them simultaneously,. However, one may find many reasons for changing the extrapolated VSIP's, 4as we did, in the case of the [V04], in order to accomodate both the isotropy of tensor and the charge self-consistency. It is felt that further calculations at this point will not help much. On the contrary more experimental results are needed. For example a serious objection that one might have for the scheme of [VO04] is the two extra negative charges of this complex as compared to [W042] Even in the milder case of [V04]3, charge compensation is required which, if the sample is pure CaWO4 with vanadium only as an impurity, must come from vanadium itself. Thus one can picture the complex [V04] adjacent to a which occupies a calcium site. Then a very appealing scheme is the followingBy x-irradiation a lone pair occupying the nonbonding orbitals h(Ca2) or h(Ca3) is broken upo One of the electrons is transfered to the r2 MO giving the above mentioned EPR signal and the other electron is attracted 33

by the adjecent vanadium reducing it to V2+ Therefore the total charge of the complex is actually decreased to [V04] and the adjacent vanadium has an odd number of electrons which should produce an EPR signal characteristic of ionic vanadium. Furthermore this signal should be anisotropic having some relation with the broken lone pair. Referring to Fig. 3 it seems that the h(Ca2) lone pairs are more likely to be affected by the x-irradiation since they lie higher in energy than the h(Ca3) ones. On the other hand the h(Ca3) are closer to the calcium sites that presumably the vanadium impurity occupies. Perhaps it is significant that such anisotropic spectra characteristic of ionic vanadium have been reported. A further experimental study of these spectra, specially their angular properties with respect to orbitals h(Ca2) and h(Ca3) seems to be highly desirable. Finally in Chapter III we saw that a reduction of the estimated VSIP by a factor 2.528 would bring the g values in the right position. This reduction amounts in using the following VSIP's. 4pVSIP = -15.0 Kcm-1 VSIP = -41.0 Kcm-1 4sVSIP = -950 Kcm 1 VSIP -96.2 Kcm-1 (IV-l) 3dVSIP = -71.6 Kcm-1 We notice that according to the above model in which a lone pair is split we can accomplish two things. First bring an electron to the orbital which gives the EPR signal of the strong lines and second reduce the overall charge of the complex to -2 so that the VSIP are lowered as required 34

to produce the observable g values. In fact both -the a VSIP and 4sVSIP in (IV-1) are lower than in the case of the IV04]3 The 4s and 3d interchange order probably because the rF orbital adds electronic charge to the 3rd orbitals x2-y2 and xy. Similarly charge is added to the h and 4p orbitals accounting perhaps for the relative increase in these VSIPts3 55

APPENDIX TRANSFORMATION OF AO BY THE S4 GROUP (Coordinates and Indices as in Fig. 1) s4 E C2 S4 S4 a1 c o'4 02`3 02 a2 ~3 a4 1l 03 cr3 02 l 04 04 c4 1 053 2 xl 1 x4 -x2 "3 x2 X2 3 -x4 -X X3 x3 x2 -xi x4 x4 x4 x1 X3 X2 Y Yl Y4 Y2 Y3 Y2 Y2 Y3 Y4 Y Y3 Y3 Y2 Y1 Y4 Y4 4 Y Y3 y2 x x -x y -y y -y -x x z z z -Z - I Y~~5

REFERENCES lo C. J. Ballhausen, Ligand Field Theory, McGraw-Hill, New York (1962). 2. C. Jo Ballhausen and H. B. Gray, Inorg. Chem. 1, 111 (1962), 35 C. Jo Ballhausen and Ho B. Gray, Molecular Orbital theory, W. A. Benjamin New York (1964). 4, Ro S. Mulliken, Jo Chem. Phys. 23, 1833 (1955). 5. So Karavelas and C. Kikuchi, University of Michigan Technical Report 04381-7-T. 6. N. Mahootian and C. Kikuchi, Bull. Am, Phys. Soc. Series II, 10 No. 5 (1965). 7. M. H. L. Pryce, Proc. Phys. Soc. (London) A63, 25 (1950). 8. R. Lacroix, Helvetica Physica Acta 35, 592 (1962). 37