THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Radiation-Solid-State Physics Laboratory Technical Report EVALUATION OF MATRIX ELEMENTS IN CRYSTALLINE FIELD THEORY Sow- Hsin.Chen ORA Project 04275 under contract with: AIR FORCE OFFICE OF SCIENTIFIC RESEARCH CONTRACT NOO AF 49(638)-987 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1962

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TABLE OF CONTENTS LIST OF TABLES INTRODUCTION PART I 1.1 Definition of Irreducible Tensor Operators 1.2 Tensor-Product and Scalar Product of Two Tensors 1.3 Double Tensors (Two-Sided Tensors) 1 4 Wigner-Eckart Theorem 1.5 Matrix Elements of Spin-Free Operators T (2).6 Matrix Elements of Tensor Product of Two Commuting Tensors 1.7 Matrix Elements of Scalar Product of Two Commuting Tensors 1.8 Matrix Elements o Salar Produ of Two Commuting Tensors 1.8 Matrix Elements of Double Tensors Page V 1 5 5 8 10 12 15 17 18 19 PART II 2.1 Evaluation of the Reduced Matrix Element (oSLITqPljcS ' L ) and (cSL IITk lc'SS 'L' ) 2.2 Recursion Formula 2.3 Matrix Elements of Electrostatic Interaction 2.4 Matrix Elements of Crystalline Field Potential 2.5 Matrix Element of the Spin-Orbit Interaction 2.6 Matrix Element of the Spin-Spin Interaction 2.7 Matrix Elements of Nuclear-Electron Magnetic Interaction 2.8 Matrix Elements of Nuclear-Electron Quadrupole Interaction 21 21 26 28 31 34 37 39 42 iii

TABLE OF CONTENTS (Concl) APPENDIX A: APPENDIX B: APPENDIX C: APPENDIX D: Rotation of Coordinate System and Rotation of Field Proof of the Wigner-Eckart Theorem Definitions and Properties of 3-j, 6-j, and 9-j Symbols n (K) n Numerical Tables for: (d aSLI|U )ld nW'S'L'), k = 2,4 n = 2,5 And (dnoSLIIVlklld no'S'L'), n = 2,4 k = 1,2 Page 47 52 57 63 75 REFERENCES iv

LIST OF TABLES Page TABLE D-l. (d2caLSISU (2)2) I2a'L'S') 63 TABLE D-2. (d 2cLS lU(4) (2) jd 2c'L'S ') 64 TABLE D-5'. (dcSSLIJu (2)I(2)idSa''L' ) 65 TABLE D-4. (d5cSL IIu4(2)Id.5a' S' L' ) 66 TABLE D-4'. (daWSLjU4(2)HId5a'S'L' ) 67 TABLE D-5. (d2aSL IIVp Id2a'S'L1) p = 1,2 68 TABLE D-6. (d5aSLIIv ld5a'S'LT) p = 1,2 69 V

INTRODUCTION In the interpretation of optical and microwave absorption spectra of crystalline solids, the so-called crystalline field theory, which is based on the ionic approximation, has been proved to be a remarkably successful one. The essential problem is to diagonalize the free ion Hamiltonian plus the perturbation due to ligand field. Two equivalent approaches are in use, namely, the strong-field scheme and the weak-field scheme. In the former approach the full use of the symmetry of ligand field from the beginning leads to simplified expressions for the matrix elements (see Sugano and Tanabe). In the latter approach the advantages are that one can make use of the results of the corresponding free-ion calculations and, as a result of the spherical symmetry, one can get a general formula for the reduced matrix element. In this report, we use the weak-field scheme for the evaluation of matrix elements in L-S coupling scheme. The interest is mainly on the iron-group (3d-group) elements so that Racah's results in the free atom case are still largely applicable. Racah's method combines the tensor algebra, which is the extension of vector algebra in Condon and Shortley, and the method of constructing the wave function for equivalent electrons using the vector-coupling formula and fractional-parentage coefficients. We shall find that the application of this powerful technique will not only give a systematic way of evaluating matrix elements of all terms that appeared in our Hamiltonian, but will also greatly simplify the computations of seemingly very complex problems of evaluating matrix elements between many electron wave functions. References to the use of this technique to atomic and nuclear physics are 1

scattered in various literatures and monographs. A variety of notations has been used, so that considerable care is needed in applying the indicated formulas. For this reason, we shall present a summary of relevant formulas in a manner convenient for application to solid-state physics. Also, the reference to tabulations of requisite numerical constants will be given. In Part I, we first introduce the concept of irreducible tensors and some of its algebra. The extended Wigner-Eckart theorem which is applicable to irreducible tensors of any rank is then stated. Using the Wigner-Eckart theorem,* we then evaluate the most general form of matrix elements involving spin-free (space-free) tensors, tensor product of two commuting tensors, scalar product of two commuting tensors, and double tensors between eigenfunctions in the L-S coupling scheme in terms of various recoupling-coefficients (3-j, 6-j and 9-j symbols) of angular momenta and certain types of reduced matrices. In Part II, then, we try to reduce each term of the Hamiltonian to these standard forms. We shall find that if we fix our attention on only a single configuration, evaluation of all the matrix elements can be reduced to the calculations of two types of reduced matrices, i.e., (aoSL|U( kI'S'L') and (oSLUVqPjCa'S'L'). For each of them, we then give formulas in terms of the fractional-parentage coefficients and Racah coefficients so that our problem of evaluating matrix elements is completely solved. The type of terms we consider in the Hamiltonian are:.electrostatic interation between equivalent electrons 2. crystalline field potential *And the vector-coupling formula. 2

3. spin-orbit coupling energy 4. spin-spin interaction energy 5. hyperfine interaction 6. quadrupole interaction. As supplements to the text we give: Appendix A: Rotation of coordinate system and rotation of field; Appendix B: Proof of Wigner-Eckart thorem; Appendix C: Definitions and properties of 3-j, 6-j and 9-j symbols; Appendix D: Numerical tables for: (dnaSLIU(k) lldnI'S'L' ), k = 2,4, n = 2,5 and (dnoSLLklldna'S'L'), n = 2,4, k = 1,2 which should be enough for evaluation of matrix elements of the six kinds of interaction considered above. 3

PART I 1.1 DEFINITION OF IRREDUCIBLE TENSOR OPERATORS A set of 2k+l operators T (k integer, q = -k, -k+l..., k-l, k) which transform irreducibly according to the k-th irreducible representation of the rotation group, under rotations of the frame of reference, is called an irreducible tensor of rank k. A rotation R(aBy) where (apy) are the Euler angles corresponds to a rotation matrix R(ay7) such that the components of a vector in unprimed (original) coordinate system S and the primed (rotated) coordinate system S' are related by Ir' R(agy) I - (1.1) This rotation R(apy) gives rise to an unitary operator OR by which the set of components of an irreducible tensor is transformed according to k k OR T R1 = T, Dq,q (R) (1.2) q'=-k and also by which a wave function IaSLJM > is rotated into (see Appendix A) (j) where DmIm (R) is defined by Dm'm (R) = < jmT OR I jm > (1.4) and |jm > is the simultaneous eigenfunction of total angular momentum J2 and Z-component of it Jz (Edmonds 4.l.1).* It is well known that the operator OR can be written in terms of the total See References*. *See References. 5

angular momentum operator as OR (e) = exp (i9Je) where ~ is the axis of rotation and 89 is the angle For an infinitesimal rotation '8 about axis I, OR (5e) - 1 + i5eJg OR (se ) 1 - seJ5 (k) Dqtq(50) = < kq' l + i5eJ j|kq > = qq + therefore Eq. (1.2) becomes k (1 + i eJ) Tq (1 - i~~J) = X Tkq, q q —k (1.5) of rotation (Edmonds, 4.1.9). ise < kq' IJ| kq >,q + i5e < kq' | Ikq > q~~~~~ or [J, T Tk] = q T, < kq'J kq > (1.6) q which is valid for any arbitrary 5e. From Eq. (1.6) it easily follows the following three commutation relations of Tk with respect to the angular momentum J, [J T] = [(k q q + Tk (1 7) [J+, ] [(k q)(k + + 1) T kq q 1 [Jo, = [Jo, Tq] = q Tq (1.8) where J+ = Jx + iJy; Jo = Jz Equations (1.7) and (1.8 ) are the alternative way of defining an irreducible tensor which is useful when one prefers an algebraic way of proving theorems about Tk (see Racah II). Examples of irreducible tensors: A. For k = q = 0, Doo(R) =1 therefore Eq. (1.2) gives OR To OR - T 6

which shows that To B. For k = 1, q = -1, 0 [J+, T1] [J+, To ] [J+, T,] is invariant under 1 Equ. (1.7) and = 0 [J_, T] = FT1 [J_, T ] \= 2T [J_, T_1] rotations and hence is a scalar. (1.8) give = \I2To [Jo, TI] = T' (1 0/-2To 1 = 'FT-1 [Jo, To] = 0 1 0 ~[J,, T1,] = -T.9) I I U -- - - -j - Let T1 = - (T + i T) Tl = (Tx - i T) T = TZ Then Eq. (1.9) leads to the following set of commutator relations, [Jx Tx] = [Jy, Ty] = 0 [Jz, T] = 0 [Jx, Ty] = iTz [Jy, Tz = iT [Jz Tx] = iTy (1.10) [Jx, Tz] = -iTy [Jy, T] - -iTz [Jz Ty] -iTx It is immediately seen that (Tx, Ty, Tz) defined this way is just the T type of vector defined in TAS.* Since J, L, and S are all T type vectors with respect to J, the following type of combinations of components of J, L, and S L - = J7^(Lx, ~iLy) = L+ L LZ (1.11) are examples of first rank irreducible tensors. C. In general, let us define C im()= (242 /2 1 ) Y m(r) (1.12) = (-1) 2 _(_-_m _ Iml im~ - -1 2 1+ I I), Pi (cos e)e and remember L4: C^m(e,4) = [( + m)(Q + m + 1)]/2 c, (e') Lz C m(e,4) = m C^m(e, ), *See References. 7

regarding then C(m(r) as an operator and noting that both L+ and Lz are first order linear differential operators, we obtain immediately [L+, C m('r)] = [(I m)( +~ m +1)]/2 C, m+~(r) [Lo CIm(r)] = mCIm(r) which shows that CAm(r) (m = -,,..., +~) is an ~-th rank irreducible tensor operator with respect to orbital angular momentum L, which are sometimes called "spherical tensors " For example, C11 = - - - (X + iy) |2 r Clo = - z (1.13) r 11 C11 = (x - iy) f2 r and r = (x,y,z) is an T type vector with respect to L. 1.2 TENSOR-PRODUCT AND SCALAR PRODUCT OF TWO TENSORS There is a general method of constructing irreducible tensors of higher (lower) ranks from two irreducible tensors analogous to the vector-coupling method. Let the two tensors be T -(Al) and Tk (A2) where Al and A2 indicate that the two tensors do not necessarily act on the same part of the system. Then p A K T. (Al) X T 2(KA2T = T '(Al) T 2(Al) T 2(A2) < kqk2q2 KQ > (1.14) = Tl(A) Tk2 (A2) < klqlk2Q-q JKQ > L Tq1 Q_-qj2 ql where < klqlk2q2 IKQ > is the vector-coupling coefficient and vanishes unless Iki - k21 K k ki + k2, (this condition will be denoted as A(klk2K) hereafter), and Q = ql + q2. Thus from two tensors of rank kl and k2, we can build up 8

irreducible tensors of rank ranging from Ikl - k21 to kl + k2. In a particular case, when kl = k2 = k, it is possible to have K = 0 = Q corresponding to contraction of tensors, i.e. FTk(Al) x Tk(A2) = TTk(Al) Tk (A2)< kqkq| 00 > Jo q-q (-l)k(2k + 1)/2 Z (-l)qT (Al)T (A2) ksince < kqk|o00> = (-1)k- (2k +1l)-/2 Let us define the scalar product of tensors Tq(Al) and Tk(A2) as Tq q k k kk Tk T (Al) ' Tk(A2) = (-1) (2k +1)1/2 Jo (A1) x Tk(A2T = (l)qT( Al) (2) (1.15) Examples of tensor product and scalar product: A. Consider the spin-orbit interaction energy XL S. From Eq. (1.11) 1 i Lx = = (L_1 - L+1); L = = (L_1 + L+1); Lz = Lo similarly for Sx, Sy, and Sz. We have then XS-L = \(-S1 L_1 -S S Li + SO Lo) (1.16) -= x (-l)q Sq L_ q which is the scalar product of Sp and Lv. B. Consider the nuclear-electron dipole-dipole interaction 3(_S - r)a Hne = -aI - 2 - r2 where I is a rank one tensor; therefore the expression in the bracket must be a rank one tensor too. We can construct it from a rank one 9

tensor S4 and rank two tensor Cv2)(r), i.e. (2) X Q A Sq CQ_(r) < iq 2 Q - lQ > q - 3(S.r)} Q by evaluating the Z-component (Q = O) of both sides, one can determine the constant A to be l0. Therefore Hne = -aZ (-1) I X-_ (1-17) = -af (-1)' I SqC (2q(r) < lq2-4-qll v > C. Another important example of the scalar product is given by the addition theorem of spherical hanmonics, which says Pj(cos 012) =_____22 P~(cos el2) = 22+ X (-1) Y( e$,) Ym(0e22) m=-2 in terms of Cem, we have P2(cos 012) = j (-1) C m(9l) Cmi(92()2) (1.18) m=-2 which is just the scalar product of Clm(1) and Cm(2) 1.3 DOUBLE TENSORS (TWO-SIDED TENSORS) (See Wigner, p. 273) When an operator in the Hamiltonian is made up of a sum of one-electron operators, each of which is the product of operators acting on spin part and orbital part of the wave function respectively, it is convenient to introduce the concept of double tensoro The definition of a double tensor operator has an intimate connection with the separability of the unitary operator OR into the spin part and the space part. In Pauli's theory of electron spin, the one 10

electron wave function is assumed to be a product of a space part and a spin part. Consequently, it is possible to split OR into product of two unitary and mutually commuting operators BR and QR acting on orbital part and spin part of the wave function respectively (Wigner, section 20), namely OR PRQR = QRPR (1.19) and PR |ISMLML > -= ISMLM > DM ML(R) (1-20) Q CSMsLML > = j IOSMGLM > D MS(R) (1.21) PR ML> ML(R) (S) gls~L1L> Zasm, > DM~t (R) (1.21) MS (See Appendix A). With this understanding, we can define a double tensor TqP which is of degree q with regard to QR and of degree p with regard to PR and irreducible with regard to each of them, i.e. QR T e QR = D v' (R) (1.22) v' pR Tci R- = V' qpl (P) (1.2) P Z p D () (R).23) PR V P1 - (!.23) I' With respect to OR, the Tqp is not irreducible but transforms according to the (q) (p) direct product of Dq) and D( OR T RP OR T V'~,' D (R) D (R) (1.24) V'1' qp However, one can form a linear combination of T using vector-coupling coefficient to get an irreducible tensor TQ analogous to the tensor product mentioned in Section 1.2 (Wigner, p. 284), namely. TQ = TVqp <" qvplIKQ >(1.25) v,iM. 11

It is evident that the simplest example of TqP is just the product Tq(S)T,(). A more familiar one is the spin-orbit coupling energy of n equivalent electrons: n n:s s(i ) = K (-l)Sv (i) (i) v where T = SV(i) v(i) = yt ^(i) * i 1 i=l ~ 1.4 WIGNER-ECKART THEOEEM (See Appendix B) The first simplification in calculation of matrix elements of tensor operators comes from application of the well-known theorem (Edmonds, p. 75) which states that if: (1) state JcJM > rotates irredicibly according to D( (2) state Ja'J'M' > rotates irredicibly according to D( O k (k) (3) Tq rotates irreducibly according to D q k then the matrix element of Tq between the two states can be separated into product of two factors: a vector-coupling coefficient which specifies the angular dependence (M, q and M') of the matrix element, and a reduced matrix which depends k on the magnitude of J, J', k and the physical nature of the operator Tq, namely *It is easy to see that when a one-electron operator tk(i) is an irreducible tensor of rank k, the sum of such operators over any finite number of electrons is also an irredicuble tensor of the same rank. The similar thing holds for one electron double tensors. 12

, k, ((JIITkIli ' J' ) < cJM T | c'J'M' > = < J'M'kq|JM > (2 + )1 (127) -= (-D^1)J4 J ) (w1 IT J) (1.28) <-~/J-M ( MI (see Appendix B) where M k J'1 < JIM'kq JM > -J+M M q Ml (2J + l)l/ (-) is the symmetrized V-C coefficient or the 3-j symbol of Wigner. The following points about the theorem should be noted: A. ca denotes all the necessary quantum number besides J and M to specify completely the state. It can be, for example, the principal quantum number, L, S, parity and seniority number. B. The selection rules are built in the V-C coefficient, i.e., the matrix element is zero unless A(JkJ') and M = q + MT and the parities satisfy the relation JTi TTk TTf = 1. C. The reduced (or double bar) matrix element is not a matrix element in the real: sense. It is essentially defined by Eq. (1.27) or Eq. (1.28); the symbol in it merely indicates the dependence of its value on them. k D. When Tk does not act on the spin (or space) part of the wave function, q k k that is, when [QR, T [QR, Tq] = O, we have k k —i k- OR TqOR1 = PR QR Tq Q PR = PR = T D( (1.29) that is, we have wave functions in the L representation that is, we have wave functions in the S, L2, S, Lz representation which rotate irreducibly with respect to PR and QR separately, = > (L)(R) (1.30) PRI CSMLML > = I lSMSML > D (R) (10) M'L 13

RM L> = MML, (s) QR lCMSTML > = I CQSMsML > M Ms (R) (13.51) MS and we have the W-E theorem for Tk(2) which act on space part only, < oSMsLML I Tk(2) I t'S'ML'M' > q L~~M = (-1)L-ML (2) SL') Ss MM = (-i),q (aSLIITk(2) II}C SL' } FSS' 6MSMs '~~~~'' q mi9 (1.532) (notice that the reduced matric element is MS and M6 independent) and similarly for Tk(l) we have q < CSMSLML I Tl)IcstM6LIML > q L = (-1)iS (. 4q M4 (oSL II Tk(l)lca'StL) bLL,1MLML (1.53) E. Equation (1.27) shows that for two tensors of the same rank k, their matrix elements between states of same cJ are proportional:* < amLL IT, (2) IaLML > < aLMLIUq(2) |aILM > (aL lTk (2) IcL) (aL ||IUk(2) ||aL) (for nonvanishing denominators) the special case is when k = 1, q = I(t = 0, 1, -1). Tl - rCi((r ) Ul = Lp kt 1 then < LML IrC(!(r) IaIMI > =,(L|r(l a)t < <LML I aLML > *Note: It is necessary that (aLjIIL'|aL) -+ 0, but (alL|IIIaL') = 0 for L + L'. 14

From Eq. (1.11) and (1.13) this becomes < aLM I * ( iy) > IIL) < <O L ML + (L ~ iLy) I|LM > (~ x +y IIM~L > = (o I ILIJL) /I I (aLJ{rC, {OL) < o MLI z laLML > = a(aL{lcL) < oLML Lz aLML > Thus < aLMLI x I mL M = L < aiLMILx |lmL[ > < LM2Ll y | cLML > = p< a;mL\LyyamL- > < ALMLI z I aLML > = < aLMLILZIaLML > (1.34) where p = (c|LI}rCl{cLL) which is constant for a fixed 2s+1 L term. Equation (1.34) (aL|IL||aL) is sometimes called the operator equivalence by the spin resonance worker, which says that in calculating the matrix element of x, y, z and their powers between states of a fixed 2+ L term (usually the ground terms of the paramagnetic ion) one can replace them by the corresponding powers of the angular momentum operators, provided that one pays proper attention to the non-commuting property of angular momentum operators. (See Bleaney and Stevens). k 1.5 MATRIX ELEMENTS OF SPIN-FREE OPERATORS Tq(2) If [Tk(2), QR] [Tk(2), QR1] 0 q q and and T(2)OR-1 = PRTk(2)PR-1 = kD (k)(R) = - = q'q 1 'q A. In S2 L, S, Lz representation. Since [S2, Tk(2)] = = [Sz, T (2)] there are no non-diagonal matrix q q elements with respect to S2 and Sz. So by Eq. (1.32) the W-E theorem < cSMLML ITk( 2) t' S 'MSL'M > q S L~'M = )SS',MSMs,()-1 LL ( (kLC)SLITk I'S'L') ( (1.32) 15

The fact that the reduced matrix element is independent of MS is shown in Appendix B. B. In S2, L2 J2, J representation. < aSLJM | Tk(2) | 'S'L'J'M' > = SSi < aSLJM T(2) a' S'L'J'M' > q since OR kISLJM > = IcSLJM' > DM,M(R) M' By the W-E theorem Eq. (1.28) < cSLJM I Tk(2) | 'tS'L'J'M' > /TJkJ~ j,\. ~(1 j55) = S JS' M) (OSLJI|Tk(2)Ia S' L J )(-1)J-M In (Edmonds, 7.1.8), it is shown that the reduced matrix in Eq. (1.35) is related to that of Eq. (1.32) by (CSLJlITk(2 )Ia'SL'J' ) = /(2J+l)(2Jt+l) W (JSkL';LTJ')(cGSLITk(2)IIl'SL') (1.36) = (-1)J+S+L'+k (2J+l)(2J +l) iJ, J. k (LIIT (2)||(aSL' (1 37) J L S where L,L ST k is the 6-j symbol of Wigner which is related to the Lwe J kj Racah coefficient by t,L = (-1)J+L+J+L' W(JLJ'L'; Sk) (1.38) (See Appendix C). Equations (13.2), (1.35), and (13.7) show that the evaluation of matrix elements of a spin-free operator in both S2, L2, Sz, LZ and S2, L2, J2, Jz representations are reduced to the evaluation of single reduced matrix element (sL |ITk (2) |lJ' SL '). 16

is 1.6 Similarly, when Tk(l) is space-free, the analogous equation to Equ. (1.37) q (QSLJITk(l) c'S'LJ' ) = ( J'+S+L)+k JJ- ' S + ( |k(^. 1 ^ JL' ()J+sV+L+k(2J+l)(2Jt+l)I4(cSLITk(l) I,'S'L) (1.39) E.~~~,,,,~~~~,,,,.,,~~~, (~9 MATRIX ELEMENTS OF TENSOR PRODUCT OF TWO COMMUTING TENSORS k Tkl Tk2 < TQ - cl q q2 kl [T k, L] = q1 - klqlk2q2 K1% > (1.14) and 0 [T k S ] Tq2 - 0 kA T -1 ORTqlR klQ -1 QRTqlR ql Tk D(k)(R) (140 ql qlql ORTkOR-1 = q2 OR OILJM > = pTk2p -1 PRT Pk q2:z q2 k2 Tq 2 D (2) (R) <2Cl2 M' (J) JcOLJM'> DMM (R) In S L2 J2 Jz representation, the W-E theorem gives < aSLJM I T Ic"''S'L' J'M' > - (-1)J-M )(LJ||T( |Ila'SL'J') (1.27) * On the other hand (Edmonds, 7.1*5) shows that < OSLJM | TQ | C'S'L'J' > - J /M 2KJ I L (CSL |ITk1TK2IAS.L) \Q1) ~ M ' I I(2K+l)(2J+l)(2J'+l) fS'L'J' (GSLlITk1Tk2IIcStL ) - (-1) v~MQM' Lkike (1.41) where the last factor is the 9-j symbol of Wigner. (See Appendix C). of Eq, (1.27) and (1.41) shows that SLJ (aSLJ|Tkll'S'L'J') = I(2k+l)(2J+l)(2J'+l) S'L'J3 x (SLlTklTk2Ha'S'Llk2K x (CSL |ITklTk2 Iac 'S' L' ) Comparison (1.42) 17

where (cSLIITklTk2lla'S'L') = Z (oaSiTkllat,,S )(a"L ITkl2aL') OgT (1.43) 1.7 MATRIX ELEMENTS OF SCALAR PRODUCT OF TWO COMMUTING TENSORS In Eq. (1.27) of the last section, if we put K = 0 = Q, kl = k2 = k, < oSLMITOa'cS'L'J'M ' > = ( -1JM J J') (aSLJ ITOil S ' L J' ) [Ko M 0 M'J But from Eq. (1.15) o *O = (- 2k+ -1/2 V ( 1) Tk (1)T (2) qq q q and further [from Appendix C (C-7)] M O M'J M M'/ = (-1)SJMj(2J+1) /2 SJJ'SMM' (1.44) Therefore < cSLJM I q (-l)qTk(1)Tk (2) a'S'L'J'M ' > = ()-k(2k+) /2 (2J+1)-1/2 BJJSMM(SLJI|T~|Ta'S'L'J') Now from Eq. (1.42), (1.45) (aSLJIITOII'S 'L' J) = (CSLIITkTkIICaS'L') x rSLJ I (2J+1) S'L'J) LkkO j r^ - whe re SLJ *i S'L'JS = lUkO J.J+L+S '+k r ( -1) _ SLJ J J —,lL'S 'kf \/ (2J+l)(2k+l) L J (1.46) [Appendix C (Eq. C-19)] Therefore (QSLJ IT~O la'S 'L'J) = (-1)J+L+S'+k S/LJ fT (2J+1) /2 (2k+l) / 2LsU1 'kfS x (aSLIITkTk Ca' SL') (1.47) 18

Substituting Eq. (1.47) into Eq. (1.45), we get finally: < aSLJMITk(1) * T (2) 'S'L'J'M' > = J MM ( -)J+L+S) ' L (CSLI(1) Tk(2)II S'LT) (1.48) = JMM(-1)J+L+SJ SL(-) ' J (cS ITk(l1) IaS ')(a"LITk(2) Ia'L') Equation (1.48) is the matrix element of two commuting operators in the S2 L2, J2 J representation. It reduces to the same type of reduced matrix element appeared in (1.43). The matrix is diagonal in J2 and Jz because it is a tensor of rank zero, scalar. 1.8 MATRIX ELEMENTS OF DOUBLE TENSORS A. In S2, L, Sz Lz representation. By the definition of a double tensor operator in Section 1.3, together with the W-E theorem in the form as in Section 1.4, D, we easily see that < OSMSLML ITI P 'S ML'ML > = (. 1)LML (ML (cSMsL llTqla 'S L' ) -( lS-Ms q S sq = (- vSLMLI TqPl( S'S'L'I) Hence < QSMSML I Tq I 'S'L'M > S-Ms+L-ML q SI p LT B I F (-D1)^^ y^ t i^. L( S -M P I) (SLIlT9PllSILt), (1.49) ~ MS ML B. In S2, L2, J, J representation. As we have mentioned in Section 1,3, we can construct an irreducible K tensor TQ by (1.25) 19

T = TqP < qvpp | KQ > (1.25) Therefore by W-E theorem < aSLJM TQ | c'S'L'J'M' > = ( JM Q M( ) (cSLJIITK kS ILJt analogous to Eq. (1.42), (see Trees, 1951) (aSLJIITKII' S'L'J' ) FSLJ = S/(2K+1)(2J+1)(2J'+l) 1S'L'J't(aSLIITqPl||c'StL), (1.50) lqpk Here we have a reduced matrix element (aSLIITq9lla'S'L') instead of (oaSLIIT kl()Tk2(2)Ia'S'L') as in Eq. (1.42). In fact the latter is the special case of the former, as was mentioned in the last paragraph of Section 1.3. Also analogous to Eq. (1.48), we have a< oSLJM| (-1) TPP | a'S'L'J'M' > = (-1)L.p (SL|TP|cS ' L)5jjMM (1.51) Thus in all cases we have shown that the evaluation of matrices reduces to the evaluation of two types of reduced matrix elements, namely (oLSLIITkI'S'L') and (GSLIITqPl|l'S'L'). 20

PART II 2.1 EVALUATION OF THE REDUCED MATRIX ELEMENT (aSLIITq la'S'L' ) and (cSLlITkll'S 'L' ) In Part I, we have reduced the problem of evaluation of matrix elements of: (1) spin-free (or space-free) tensor operators; (2) tensor product and scalar product of two commuting tensors; and (3) double tensors between wave functions of equivalent electrons in both LSJM and LSMLMS representation to essentially the problem of evaluating two types of reduced matrix elements written above. In this section, our task is to reduce them further into one-electron integrals using Racah's method of constructing wave functions for equivalent electrons. It is shown in Racah's paper (Racah III) that the vector-coupling formula does not lead to antisymmetric wave functions for equivalent electrons. However, suitable linear combinations of vector-coupled eigenfunctions would produce desired antisymmetric eigenfunctions. Racah called the coefficients of linear combination "coefficients of fractional parentage." Suppose we know an antisymmetric eigenfunction for the 2n-1 configuration: Ien-1 alS1MS1L1ML1 >. We want to get an eigenfunction for configuration 2n: iQncSMSLML > by adding one electron ~n. Therefore use the vector-coupling formula: Inl (aislSlL) nPSL >z In-lazS Ms4LMLM > ISnmsn > l nmin > x MS lmsn MLjm^n < SlMSisnmsn ISMs > < L1MLjnmnl IML >(2.l) However Eq. (2.1) is not antisymmetric with respect to the interchange of nth electron with the rest of n-l electrons. In order to get an antisymmetric 21

eigenfunction including the nth electron, a further linear combination of Eq. (2.1) using coefficient of fractional parentage has to be used, namely: InUOSMSLML > = IQnl(aSjL1)naSL >< ~n'-(aSIL1)~SL na S L >. (2.2) alS1L1 The fractional parentage coefficient < n- l(alSlL)IoCSL| ~naSL > in which the parent term is indicated by bracket, is tabulated in Racah III for pn and dn configurations. Now using Eq. (2.1) and (2.2), let us calculate the matrix element I = < ~nASMSLMLI TP I na'S'M 'ML > (2.3) in which Tnqp is made up of one electron operator: inv which Tn~ T q t qP(i) i=l Since electrons are equivalent, the matrix element of every term in Eq. is equal, i.e. n I = X < CXsfnMsL >Itqp(i)Inol'S'MgL'M{ > i=l = n < naQMsLML tqPL(n) P 'S'M' L > e we p r t nt Here we purposely retain the nth term in the sum. Now substituting Eq. (2.2) for the wave functions, we have (2.4) (2.4) I = n < ncSL jn-l(alSlL)) SL > alS1L1 a2S2L2 < Qn-l(alS1Ll)R nCtSL Itq(n) |n-l(a2S2L2)naS'S 'L' > x < In-1(aS2L2)ica'S'L' inxa'SL' > Again using Eq. (2.1) and notice < 2n- lSiMs Ln-cS2MS2 > = lac25SlS25MSjMS2 (2.5) 22

< Jn-lalLlMLI IBn 2LMLM > ~zzL'2> 5= salCSCLLL2LLML2 (2.6) we obtain I = n %< naSL In-(alSLl)1aSL > < In- (caiSiLi)a'S'L' lna'S'L' > clS 1L1 x y < SMS ISiMslSnimsn >< LML ILMLklnmln >< PSnmsn'nmn| t? (n) P snsnm nm > MSIML1 ftsnlsn X11 n,, ^ 0 md_. n I I M/T 5 / TMr O m t IT IM i /A r\ 7\ A < i0 11 lSlnml ' n I 1 - S " 1' ' Ll'ni1 n I ' I vl' - \co ( From now on we can drop the subscript n for the n-th electron and write (following Slater, II, p. 156) a( ncaSL; ciSiLi) = < nocSL jn-'l(alSjLl ) IoL >. (2,8) Further, for the one electron integral, we can use the W-E theorem in the form of Eq. (1.49), i.e., < psme > ltP Ssmg > = s-"m5 +2-ma, vmrs/ '2p L (-1),sq _ q, kmsvms)jmI1~L \ (psjlltqplljsl2) mf = < smsqvlsms > < Jmeppjlxm s s,,,/ "au~"' > (pBsltqPJjlps) / (2s+l) (21)+l ) Therefore Eq. (2.7) becomes I = n a(lnsL;aSlLl)a(na'S'L,';ctlSiLi) ( ll alS 1L1 V (2s+l)( 2+l) < S1Msmsni SMs > < sm'qv Isms> < S Ms sm I S 'M' > MS 1l4L1 isms im'm < LIMLlQn ILML > < im'p|l|mf > < L1MLiM. IL'ML > (2.9) The last summation in Eq. (2.9) can be done using the formula for product of three V-C coefficients [Appendix C(Eq. C.13)]. 23

m< jlmljm2 Iji2m12 > < jiam2jsms3 Ij m > < jm2js3m3 lj23m23 > m2 n12 m23 = J2jl2+l)(2j23+l) W;(jlj2js3;j12jl3) < jlmlj23m-mi jm > (2.10) and get < S1Ms smS SMS > < smqqvlsms > < S1MslsmS I S'M > x Ms 1msmS ML1MQMQ < LMLm lLML > < 1mKplmM > < L1MLiM' IL'ML > = ()-sS+l)3'+1 T2s+1)(2S'+T) W(SsS's;Slq) < S'MsqvlSMs > x (.1)p+++L'+L1 J(22+1)(2L'+1) W(L2L'B;Llp) < L'M LPiLML > (2.11) Substituting back to Eq. (2.9), we finally obtain I = n 2S'+. )(2L'+]) 1a(1naSL;alSlLl)a(Bna'S'L';alSlLl)(Ps1,|tqP|| s. ) alS1L1 (_l~q+s+S'+SZW(SsS+s;Slq) < S'MsqvISMs > (-_)P++L' +LW(LL,';Llp) < L'M'pI|LML > * (2.12) While from Eq. (1.49) we have another expression for I, i.e. I - < S'MsqvISMs > < L'MptIILML > (L (1'49) Hence the comparison of Eq. (2.12) and (1.49) gives the reduced matrix element (cSLITqPlla'S 'L') as Zy n ~q+=+S'+S1 p+2+L'+L1 x a(2SL;calSlLl)a( a'S 'L';oalSlL1)(-l) 2 (-1) alSlLl W(S2S ';S q)W(LBL.'L QiLp) (2.13) Since the special case of Tp when q = 0, p = k is n Tn = t (i) (2.14) i=l 24

We easily get the reduced matrix element for this spin-free (or space-free) operator by the same procedure: k (SLIITn(2 ) la'S 'L') = ss n (2L+l)(2L'+l) x (-1 )k++L+L a ( enoSL;ciSiL ) a( nc 'S'L;caiSiLi) olS1Li (2.15) W(LL';;Llk) (B3Iltk(2) 11p) and (aSL Tk(1 )a' S 'L) = 5LL n (S+)(2S' +) k+l+' +S x (-1) 2 a( nasL;alSLl)a ( nnc 'S Ll;alSlL ) 1S 1L1 1 k W(S St;Skl) (P tk(l) II) (2.16) Equations (2.13), (2.15) and (2.16) formally contain the final results we obtain for the task we set forth in this section. The one-electron reduced matrix elements in them can be evaluated by an ordinary method. It depends, of course, on the specific nature of the operator. For the convenience of numerical tabulation, Racah (II,III) defined the socalled unit double tensor operator V and the unit tensor operator U(). As can be seen from Eq. (2.13) and (2.15), if we divide through both sides by the respective one-electron reduced matrices, we obtain: T qP q~pi1 Tqp _ q+P+-+ +S'+L' (nSLII|| n P IIa na'S'L' ) = n (2S+1)(2S'+)(2L+1)(2L'+)(-1) 2 xt (-1 ) 1 la(=naSL;alSlLl)a( naS'S' L';lSlL)) cx(SS(L x W(SS ';Slq)W(LIL'; Llp) (2.13) 25

(k) Tn) (2) n. 1tk(2)11I) i (galltk(2) (2.15' SS n(2L+)(2L+l) (-)+' x;<-l) Za(n0L; alSLz)a(In aS.L';azSLz)W(LL' 1;Llk) (2.15') Notice that the right hand sides of the formulas do not contain the specific nature of operators TqP and T(k)(2) but only depend on their rank and the conn figuration I Therefore one can tabulate the value of them for various n, q, p and k thus define: ( TP ^ ^ *^~v~ ~(2 13a) (B Die tqP |,5l~ ) and (k) Tn (2) (k)2 u (2) (2.15a) (~11tk(2) I1u) We list some values for the one-electron reduced matrices below for the latter references (Edmonds, p. 76): (03pI (p110) = (+ )(+1)(2.15b) (Ifs2L-) = | (2ol6b). (P CIIc I()I) (-1) (22+1) (o (2.15c) 0 2.4 RECURSION FORMULA So far our consideration has been confined to operators of the form n n i=l i=l interaction the operator of which contains, coordinate of two electrons such as 26

the spin-spin interaction or the electrostatic interaction, then evaluation of the general formulas for the reduced matrix element becomes very complicated. In this case, it is perhaps more convenient to use the recursion formula by which we derive all the matrix elements of dn configurations starting from the knowledge of d2 configuration. Our tensor operator is of the form: n Tqp= tqP(ij) (2.17) n i> j I = < nc aSMIML TqP Inl>'S'ML'M > S L Since all the electrons are equivalent, each term in Eq. (2,17) contributes the equal amount to the matrix element. If we now remove one electron, the value of the matrix element will become (n-2'2)/itimesthe original one (since there is 'n(n-l) terms in Eq. (2.17), therefore 2 I = n noSMsLML ITqp 1,nS LM n-2 L n-1 S L We substitute Eq. (2.2) for the wave function on both sides and get n n I n= n- a(Qn oSL;alSlLl)a( t'S'L't;a2S2L2) ajq nn-2 1L1~ < en-Y(QlSlLlz)nsL TP IT nl (a2S2L2)ynS'L' > c2 s2L2 (2.18) Then use Eq. (2.1) to get the explicit dependence of wave functions on the n-th (removed) electron to evaluate the matrix element in Eq. (2.18), and remember Eq. (2.11), and Eqo (2.13), we get 27

(POaSL||T pII,,'S'L') -. = (_-L+S —2+p n + [(2S+l)(2L+l)(2S'+l)(2L'+l)]/ x n-2 ( -1) a()nSL'TalSlLl)a((na tS 'L';2S2L2) x aolS1L1 CL2S2L2 < n -olSLl llTqP I nl2S2L2 > W(SlSS2S';2 q)W(LlLL2L1';p) 2.3 MATRIX ELEMENTS OF ELECTROSTATIC INTERACTION The operator under consideration is: He s = )ri- (2.20) i>j It is evident that QR He.s. QR-1 He.s. PR He.s. PR- = He.s. OR He.s. OR = He.s. (2.21) Therefore both in SLJM and SMSLML representations He os is diagonal. However, if the state is not completely characterized by the angular momentum quantum numbers only, it can have a non-diagonal element between other quantum numbers (like a). Since < dnSLJMIHes Idno'SLJM > = < SMsLML JM > < dn MSLML He. dnaSMLMU > < SMsI J|M> MSML MtM? (2.22) Consider < d aSMSLMLJHe.s. In d ' SMSLM? > Equation (2.21) shows that He.s is scalar with respect to QR and PR respectively. Therefore we have by W-E theorem 28

< dnosMSIML He. s dno' SM[LM> > (-)S- MS IS (dQSLI-Hes Idn&'SLML) 0 M S x ( d\naS-jL Hesl Sdvnov ' SL) ) 0. = (2S+l)-/2 6MsMg (dnC~IhllHe' JIdn'SM~) and i E < d(-1Hes L-ML L > L(dnM, llHe.S. I SML) (2L+l) 25MLML (dnMs L IIHe.se l ) or < dnSMsLMLIHe.s. Idn'SMLM >= [(2S+1)(2L+1) ]- /2 MSM~6MLML (2.23) x (dnasLJIHe.. Ildna'SL) Substituting Eq. (2.23) into Eq. (2.22) < danSLJMIe.s. Idna'SLJM. > = SMsML JM> < SMSIML JM > [(2S+l)(2L+l) ]- /2(dncSL He.Ina'SL) MSML = [(2S+l)(2L+l)]1 (d nLSLIHe.s. Idnl SL) (2.24) Equations (2.23) and (2.24) show that the term values in both LSJM and LMLSMS representations are exactly the same. This is just the familiar result that the term values are characterized only by S and L values and a which, in the case of dn configuration, is just the "seniority number" v (see Racad III). It is quite easy to seqthat the operator He.s is a particular case of the operator considered in Section 2.2. Therefore, the value of matrix elements for dn configurations in general can be obtained from the recursion formula (2.18). In this special case, the same procedure as used in deriving (2.18) yields;:[. 29

< dn SMsLMLIHe.s. dnaSMSLML > = < dncSLJMIHe.. Idno'SLJM > = [(2S+l)(2L+l) ]/2 (dncSL I|Hn s dna SL) = -2 a(dn OL;alS lLl)a(dno'SL; a;SlLl) < dn-lalSLL IHe. s. Id iS1L1 > ~,(2.25) SitLl Since the < d aSL - J d2ocSL > can be calculated by an ordinary method rl2 and the result is well known (Racah III): TABLE I L' aSL a| P v tD | F 2 3. 2 oCSL b 3 0 1S A+14B+7C 3 -2P'. A+7B 2D A-3B+2C 3 * F A-8B 2G A+4B+2C Table I, together with Eq. (2.25) can be used to obtain all the matrix elements of dn configurations. Constants A, B, and C in the table, called Racah's parameters, are related to the Slater integral by A = Fo -49F4 = F - F 9 1 B = F2 - 5F4 = 1 (9F2 - 5F4) (2.26) 5 C = 35F4 = 63F4 It should be noted that in Table I the coefficients of A in every matrix element are the same. Therefore, in discussing the relative term values (as in 30

the case of optical absorption spectra) only two parameters, B and C, need be considered. Further, the recursion formula (2.25) shows also that the relative term values in all dn configurations can be characterized by two parameters, B and C. alone. 2.4 MATRIX ELEMENTS OF CRYSTALLINE FIELD POTENTIAL The crystalline field potential can be expressed as n - VC.F. = c kqr Ykq( i) (2.27) i=l k For the d-electron in the crystalline field of one of the 32-point groups, k < 4 and actually k can take two values, 2 and 4 only. Therefore, our basic matrix element is n < dn~SLJM E Ckqr Ykq(91 ) d 'S'L'JM > and i=l n < d SMSLM Ckqr Yk(eti) Id ''ML'ML > (2.28) i=l If we notice k Ykq t (2) (acts on space part of wave function only) Then n kq( i = Ti ) = ti (2) (2.29) kq kq (2) t. (2 i=l i=l We have shown in Part I that both matrix elements in (2.28) can be reduced to the problem of evaluating one-reduced matrix element. (dln T )^(k) (dn SL IT(k)(2) ITdn' SI'L' ) and we further showed that this reduced matrix element is given by Eq. (2.15), i.e. 31

(dnSLITk(2)lldno'S'L').= SS, n (2L+l)(2L'+l) x n.llntl1 sS o ( -1S ) L l) a ( L '; lL l1S mL, W(LQL' B;Lk)(p ||t (2) I1i) (2.15) The summation over olS1L1 is extended to all common parent terms for terms aSL and a'S'L' Now our problem is to evaluate the one-electron matrix element ( itk (2) 11Ii ) k(2) q = CkqrkYk (, Itk llB ) = (P ICkqrk l(IIYkqllI) (23.0) But (lYklW) = (-l)J 42il 0k 0\ (2i+l) to O ~0 (2.31) [see (2.15c) and (1.12)] (PI|Ckqrllp) = Ckq < r >3d We have (p lltkll ) k 2k+ (2+)( 0 k = Ckq >r3d(-1)^ - 4 (l ) oo (2o32) For example, for k = 4, i = 2 (~211t(4) 11~2) - C40 < r4 > 3d whe re Dq = - 14 = C40 < r4 >3d = / 7 Dq (2.55) is the cubic crystalline field parameter. We have from (2o15') (dnoSL!U~ )(2)ldnoa'S'L') - = 8SS n (2L+l)(2L'+l),a(dnouSL;calSlLl)a(dnca'S'L',aClSlL) x OlS1L1 (-1) +l W(L2L'2;LL4) (2.34) 32

n 4)(2) = ui (2) (2.34') i=l ((4)(2) and U(4)(2) = t (2). This is called the "one-electron unit tensor i fia2 it (4;i( 2 )li 2) (4) operator" by Racah (Racah III), since (P211Ui (2)1||2) = 1. For the actual calculation of the matrix elements crystalline field potential, it is convenient to tabulate Eq. (2.34) for k = 2, 4 and n = 2, 3, 4, 5. We have done this for n = 2 5, k = 2, 4.* The table for n = 2, 5 and k = 2 is taken from Slater II. As an actual application of this method in the crystalline field calculations and the usage of the table for (dnoSL U( (2) IdnS 'L' ) in Appendix D, n let us calculate, for example, the matrix element: I = < d54G4 lIVc.F. d54F4F4 > with 5 VCF. C40r4 [Y40(i) + X (Y44(i) + Y44(i)] i=l T4+ - (T4 + T4) -o 0 14 We have (Griffith, Appendix) 5d.G4 F = > jd5 3 L 4, ML 4><dd5, 3 Id54, >= {id S = -, MS MsL = 4, 4ML = 44 > 1(44 - ~4) Id54F441 > = IdS = -, MS L = 3, ML = 0 > = <3o 2' Therefore;*See Append. - *See Appendix Dr 33

< d54G4 lVC.F. d54F44l > = (44 - 44 IVC.F. IF3o) ( I44ITO I30) = (4 IT4o o30) = ( i44IT I30) ( 44 IT4 I /30) = 0 since M # q + M' so the V-C coefficient vanishes in the W-E theorem. Thus I= \14 [(44 IT 3o) - (44 IT4 1 30)] 5 3)(d54||4|d ) 4 - (4 j )(d54GIT4ld54F) 28 4 4 4 4 But T4 = (2llt41I2)U( ) = 3470 Dq U )(2) by Eq. (2.33) (da54G| U( (2)Ild54F) = F - from Appendix D. and 4 3 ( 4 =- 14 (from Rotenberg). Hence, I7'1011 Hence, I / —\ 14 I 28=f7O 2x K 7 l/l 14) = 2t Dq 2.5 MATRIX ELEMENT OF THE SPIN-ORBIT INTERACTION For the d configuration the Hs.o. is given by n Hs.. o = X3d s(i)._(i) i=l 1 = d (- (i)(i) i p=-1 1 = dJ-. 1. (2.35) |-j=-1 as in Eq. (1.26). Therefore, from Eq. (1.51) [dnoSLJM >, where T = s_.(ii)(i) we have in the representation 34

I -- < d SLJMIHs.o. Idn'S'L'J'M' > I = < d naSLJMI X3d (-1) T' IJdna'S'L'J'M' > = JJ ' MM(-l) S (dncSL13dTn|dnaSL). (2'36) We see that HS.o. is diagonal with respect to J2 and Jz. The J dependence L+S J, Esp ecially, of the matrix element is given by the factor (-1) '. Especially, for the diagonal element (L' = L, S' = S, J1 = J, M' = M), we have L J S+L+J J(J+) - S(S+) - L(L+l) (2.37) {L JS 1j 2[S(S+1)(2S+1)L(L+l)(2L+l) ]1/2 in which the factor J(J+l) - S(S+1) - L(L+l) is just the J dependence which gives risel to- the Lande interval.rule. The reduced matrix element (aSLl3sdTLIIc 'S'L') is given by Eq. (.13): (dnOSLlX3dT1 Ildn's' L' ) = n (2S+1)(2S'+l)(2L+l)(2L'+l) (-1 ) x ( -l)Sl+la((dncSL;xlSlL)a((dn 'S'L';IlSlLl) x (2.13) olS lL1 1 1 allh~.t 1 x W(S- S3;S11)W(L2L'2;Lil)(P1 g11|3dt|P 1 ) 2 2 2 2 We again notice that the factor besides (51-.e|3dtf11l- ) is independent 2P I Ill3dt -2110-) is independent of the nature of the double tensor T'll Therefore, we call it 4 (dnSLIV.llldnxa'S'L' ) as in Eq. (2.13a), i.e., (dnSLlIIx3dT1Idna'S'L') = (dncSLIIVlllIldn'S'tL) 2J x (2.38) 2 2 I1n 1n 2 5 The matrix (d aSLIIV"lld W'S'L') is tabulated for d and d in Appendix D. We have to evaluate 35

( 2 1 2ll )3a 11i ) j 2 2 since t"= S& ( 2 IIp ) 2 2 = (1|SI il)(2 ||I)(PI\3d IP) 2 2 But since ( jIJlij) = j(j+l)(2j+l) (2.15b) we have (212 1X3dt lI2I ) 2 2 = %3d 3 (+1)(21+1) = _3d 45 ( dsL3dT lxaladns' L' ) = d ' (dnSL IIV" I dn' S'L' ) Hence, for dn configurations (2.39) < dnaSLJMIHS.O. IdnT'S'L'J'M' > +0 8 ( 1VL+S'+J rS L i n I ) = X3d TO5 FJJjt)M~t(-l) [ L S1i (dnaSL I hISVL) i ~ l i i J (2.40) In some cases, one is interested in the matrix element in the dnOSMSLML > representation. Using Eq. (1.49) we obtain I = < dnCSMSLML 3di (-l)4TIldW'STMsLM'M > =F (1)< SLML = 3d (-1)i < dncSMsLMLT lTdntS 'ML' M' > I- 3d ( )(- S-MS+L-ML S 1 S't(L VM L' (dansLll a' d'S'L') ML d= X3d. (dnoSLvIIllIdnlts sLI )(-1) Sms (-L) S 1 S '.MSIS ML 1:: L ' (2.40') using Eq. (2.39) for (dnUSLITIlldna 'S'L'). Both (2.40) and (2.40') show that spin-orbit interaction depends on one parameter k3d only. 36

2.6 MATRIX ELEMENT OF THE SPIN-SPIN INTERACTION HSJ. -a2rj [3( i j (j rij) - (i s,) r ] (2.41) where a2 = e/mc, can be converted into a standard form using tensor product analogous to example given in Eq. (1.17) ij ~s. 1 ~ T(2) (2) ~S. -3 ( -1) {X, x R (2.42) where _ 2 R = arij r Thus HS.S. = u(2)(i) T2ij) i >j n (-1)g t22 (ij) i >j which is the scalar product of two second rank tensors each of which contains two indices ij. This is the form we discussed in Section 2.2 in which we derived the recursion formula. First from Eq. (1.51), we have < d nSLJMJHS S Id 'S'L'J'M' > = Sj,8 MM' )1^ '(- + S d SLTd (ij)ldna ') * (1.51) The J dependence is again contained in the 6-j symbol. For the diagonal element, we have for the J dependence ( L+ l S L J (2S-2)!(2L-2)L( - 4S+ )L weeK J L S 2 (+3) )!(2L+5); where K = J(J+1) - L(L+l) - S(S+1). 37

The reduced matrix element in Eq. (1.51) can be evaluated by the recursion formula (2o19): ~~~~~~~1 ~1/2 (dnfSLIIT 22idnaSL) (-L+S- n [ (2S+l)(2S+l)(2L+l)(2Lt+l)] x n-2 (-1) L22a(dnoaSL;alS1Ll)a(d at'SL';a2S2L2)(dn lclSlLlaTn1|dn-z2S2L2) x x2S2L2 W(S1SS2S';22)W(L1LL2L';22) (2.19) 2 The matrix elements of d configuration was calculated by Marvin. He obtained for the non-vanishing elements in d the following: 3F,3F = - 6Mo + 228 M2 3P, 3P = 14 Mo + 168 M2 1/2 3F2, P2 = (-) (24 Mo - 512 M2) whe re 00 00 Mo = M~(ab) -= / r -~ R (a)R2(b)drzdr2 00 00r2 M2 = M (ab) = 44 5 R (a)R2(b)drldr2 4,4, r> 0 0 Starting from these values, (deoSLIIT2 (ij)I|d a'S'L') can be found. Then n the recursion formula (2,19) is used to get reduced matrix elements for all d configurations. This work was done by Trees (Trees, 1951). He found that nonvanishing matrix elements exist only between states with the same seniority number, and the values of the (dnaSLIITn2 lddnc'S'L') are fully defined if = v, S, L are given, i.e. (dnvSLIIT2 Idnv'S'L') = (d vSLlT22 IdvvS 'L') For the calculations of 210 elements for all ten configurations in dn one needs only to evaluate 48 (d vSLIIT22ljd vS'L') due to above relation. These 48 elements n 38

are tabulated in Trees' paper (1951). 2.7 MATRIX ELEMENTS OF NUCLEAR-ELECTRON MAGNETIC INTERACTION (Trees, 1953) The nuclear-electron magnetic interactions contain three terms: HIj H H.?U' i.e. HI S, S H i e I ' n n jafI+.S.=ai +3 -aI - ri(ri "S i); Hh.f. = HI + HI.S. a= n I i - i S - i=L i=l n' Hh.f. = HIS = a SiS., (n' = no. of u.paired S-electrons); i = 0 i=l where a = R c2a -( < r I as = R Ra2a2 (e i (o) 12g 3~~~~~~p/ As we have seen in Section 1.2, B, si - 3i(ri si) = N [SixC2) ] / i ' (1) = X. —. which is a scalar product of two first rank tensors. The matrix element of it in representation IdnoeJ,cII, FMF > is given by Eq. (1.48). < d c'eJqOcI:FMFII * idl eJ1aIIIFMp > i I 1 1 = SFF. SMp -| Jj(cI IIIIlaI) (dneJ Vi ldJ?) -= FF 'MFMF (-1)F+I+1)(2I+ i (d eJ|Vidnc J'). (2.45) U J 39

Consider first the diagonal element (J = J). The F dependence of the first order energy is contained in the factor (_F-)+ I = F(F+1) - J(J+) - I(I+1) [Appendix C L J 1 2[I(I+1)(2I+1)J(J+1)(2J+1l)]/ (Eq. C.15)] For the evaluation of the reduced matrix (dceJil VjillIdn 'J') we have the foli lowing three cases: A. HI e = I *~a = I Vi i i XVi = ai i = a (llIeP)u (2) i i = a2 i(i+l)(22+l) U( (2) = a2\0 U(1)(2) Therefore by Eq. (1.37) (dn el Vi IldnJ') = a '3 (d SLJIU(1)(2) idnaS 'LJ' ) i J+S+1l +I LSnp ( = a \~3 (-1)J+ +L'+(2J+l)(2J'+l)LJilSt(dnaSLU( )(2),Idn'S'L') K The reduced matrix element of the unit operator U (2) can be evaluated easily by observing that -1/2 [(+1)(2+1) ]/2 L (2 = 1,0,1) is a unit operator, for which -1/2 < dncoMsLML1 [(2+1))(22+1) ] Lod n's 'MSL'ML > = &5o'15SS1 'LLML [2(+1)(2i+l)] = (-1)L- LML (dncSLJIU() (2)lld ' ) S L (2.44)

Since (-1) L-ML L 1 LL 0 M]) ML { L(L+l)(2L+l) (Appendix Q-7.) ' (2.45) Substituting Eq. (2.45) into Eq. (2.44), we obtain (dnaSL Il( ) (2) Idn'c ' a ' ) 'L(L+1) (2L+1) - y)tSS I LL1 e(1+_1)(2H+l) - a' SS'?LL', L(L+l)(2L+l) = ' SSbLL ' L((30 (2.46) B. His = I aSi = I. V. i = aS S- = as(1 IIS 11)u(l)(1) 2 2 = a 3 U() (1) Therefore n, (dn eJll ViIlld te' J ) i = a 4(dnc(eJlU ( ) (IldnJ ) = a,dnoSLJiU ( )(l)lldna'S'L'J') 2S~Y by Eq. (1.39) J'+S+L+l S J Ln (1) n = a -)J+s+ -1) (2J+1) (2J+l) s, (dn, SL| ) ( ) ldn ' 'L ) (2.47) again, analogous to Eq. (2.46) we have (d a LlU( l ) () dn 'S 'L' ) S(S+1)(2S+1) = 50 'SSS'LL'1 1... 1( 1 + )(1+1) (2.48) Substituting Eq. (2.48) into Eq. (2.47), we have finally: (dnc eJI ZViIld nJ') i = 8c,'~SS'aLL 't aS (-1) [(2J+1)(2J'+1)(2S+l)(S+1)S] S (2.49 oal6SS'66LL S 1 2.4

Ci *Li i_ ri S -i)3 I iV V.S x Ci)] i i (dC l. C I J) i = - (a^d aSLJ|| [S. x C ) )|d UfS'L'J)?liw z(dnosL(i ' S1 -i I- a \(2+1)(2J+1)(2J'+1) 21' ((d)SL| iC a'S'L' by Eq. (1.42) S C(2) (c(2) ) zi i (ddcneJVi jv., J') i knowing (2 k (, we have (llc)(2J+) LJ)(dl l) 11C I -^(2a-1)(2~+3) =h- te for 2 S L J = 10 ~ ~(2J+l)(2J'+l) Si'L'IJ (dnaSL Vlldna'S'L') (2.50) N7 ' [.1 2 1J where the last factor is given by Eq. (2.13), and the numerical values for d2 and d are given in Appendix D. 2.8 MATRIX ELEMENTS OF NUCLEAR-ELECTRON QUADRUPOLE INTERACTION (Trees, 1953) The electrostatic interaction between a nucleus with z protons and n equivalent electrons is given by

n z HI,e =p i<l p=l eiep I-i - In this expansion, changes, I = 1 term the action. Thus HQ r = eep e p P (cos eip i,p,2 2 = 0 term, is the electrostatic interaction between point dipole interaction and i = 2 term the quadrupole inter= eiep P2 (cos eip) ip i ip1 where = eiep C( )(i~C((p)OC( ) by Eq (1.18) r ip ipi V() Q(2) v(2) = eiZ (2) Y''= C (eSi(i) i (2) eprC(2) P We are interested i.e. in the matrix element in the representation!|eSLJ,apI,FMF >, < ceJ,capI,FMFIHQ |eJ't,pI,F 'MF > = < eJ,capI,FMFIV() Q( l pI,F'M > = SFF'SMFMF(-l)1 I F (aeJIIV( )IaJ' )(apI Q(1) I IapI) (2.51) using Eq. (1.48) for the matrix element of scalar product of two commuting tensors. For the diagonal element (J = J'), the F dependence of the matrix element 43

is contained in the factor (l)F+I+J{J I F}j Jwhere where 3K(K-1) - 4J(J+l)I(I+l) 2 [(2J-1)J(J+1)(2J+1)(2J+3)(21-1)I(I+1)(2I+1)(2I+3) ]-2 K = J(J+1) + I(I+1) - F(F+1) The reduced matrix elements in Eq. (2.51) can be evaluated as follows: A. (a(eJIIV(2) lcJ') = e (dnaSLJ||2 ri ldQCSTi dJt) i = e < 1 > (oCSLJII = i ci Ila'S'LfJ') 1 = e <1 > (||C1(2) 1|)(adnaSLJIlU()(2)(2)ldna'S'LJ) r3 d = -e < - > (i2+1)( 2J'+ l) (2i-l1)(2i+3) LJ L SJ L'J'2 ur J (dnaSLilu(2) Idn' S 'L' ) 10 = _ - e 1 r3 d > (-J+S+ (2J+'l(2J'+l) by Eq. (1.37) {LJ ( d(SLiU (2) d 'SL ) (2.52) B. (apI IIQ(2) CpI) Since Q(2) 2 = e rp p 2 = e r / P P= t P., (2) (p C (epcp) C0 (ep~fr) (2) Qo0 - e Q - ) e_= 2 (3z2 rp) 2 p p P-i (2)> Qo between states IcpIMj > with MI = I: consider the matrix element of

< ApII |2)|apII > = < apII | =2 0 I p (3zp - rp) lapL > OG IQ(2) l I ) p (2.53) where /I 2 IX v4\OI I -I 2= \I I 0) I(2I-1) - N (2I+5)(2I+l) (I+1) (Appendix III-7' ') Define Q = < apII | j p (35z - rp)I II > Then Eq. (2,53) becomes I+ )(2I-1) (2I ( ) 2 = (2I+3)(2IEL)(I+l) ('IlQ Ila') Therefore (,p I IIQ (2)11 I I) eQ ~ (2I+3) (2I+1) (!+) 2 I1(2I-1) (2.54) Equations (2.51), (2.52), and (2.54) combine to give finally < dndeJ,I FMF IHQ dneJapI,F'M >M e~~~l~~~p ~ F 9 ^ FF ', F+I+J+J'+S+L'+1 e2Q = 8FFi,FMFOss(-1) ) 0 r 1 -1-< -r )7 r3 d I (2I-1) fJ I Fl L ST (dn SL (2) I nL x I J 2 J -I IIJUL2'2SLJ) IJ2 L J2 2.55) where the last factor is given in Eq. (2.15) and its numerical values tabulated in Appendix D. 45

APPENDIX A ROTATION OF COORDINATE SYSTEM AND ROTATION OF FIELDConsider a point P in space which is described by a coordinate (x,y,z) or z p r X/0 x or a vector r in a coordinate system S. If we perform a rotation R(cap) of the coordinate system, where (acy) denote the Euler angles, then in the new coordinate system S" the point P is described by a coordinate (x",y",z,) or a vector r" which is connected to the original unprimed one by a rotation matrix Rapy: r" = R,7r (A.1) where Rlpy is a product of three successive rotation matrices about (a) z-axis by y, (b) y-axis by A, and (c) z-axis by a Rza = Dzt()Dy:()Dz(7) Cos a sin a O s sin os y sin y 0 = I-sin a cos a 0 1 0 -sin y cos y ) (A2) \0 0 1/\in 0 cos 0 1/ It can be shown also that the same Rgby would be obtained if one rotates, instead of the coordinate system, the point P successively about (a) the z-axis by ca, (b) the y-axis by p,- and (c) the z-axis by 7 (Edmonds, 1.3), i.e. 47

Ra7 = Dz(y)Dy(D)Dz(a) A5) Therefore, we may consider r" as either the same vector op expressed in the new coordinate system S" or a new rotated vector OQ in the old coordinate system S, and in each case (x",y",z") and (x,y,z) are connected by Racy as in Eq. (Aol)o Next, consider a scalar field t(P) for which there is a definite value for each space point P. Suppose in coordinate system S the scalar field is expressed by a function *(r), then in the rotated coordinate system the field is in general expressed by a different function *'(r'). Suppose that r and r' correspond to the same point P in space, that is, they are connected by a rotation matrix Rdoy as in Eq. (Aol), i.e. r' = Rs r (Aol) Then by the definition of a scalar field, one must have 4'(r') = 4(r) (Ao4) Equation (Aol) together with Eq. (A.4), then, expresses the transformation law of a scalar field under coordinate rotations. However, we may consider Eqo (Aol) and (A.4) from a different point of view. The difference in functional form of 4' and 4 allow us to imagine that associated with the rotation of coordinate system R(cay) one has an operator PR which, upon acting on the function r, converts it into Tf, ioe. PRI = '.o Then Eq. (A.4) can be written as PR*(r') = tr(r) (Ao5) or PR$(R7rr): = r(r) or - PR*(r) = (R'1 r).(A,5) Equation (A.5) effectively defines the operational effect of PR on t(r), e.g.

replace r by R71 r. It is shown in Wigner, p. 106, that operator PR is unitary and linear, and further, for two successive rotations, R and S, one has PSR = PS PR (Ak6) Now, as we have mentioned earlier, Eq. (A.6) can be interpreted equivalently as rotating the field point P about the fixed axes of the coordinate system S in a reverse order. This interpretation, together with Eq. (Aa5), allows us to picture the process as "rotation of the field." In this picture, (A.1) states that R(6y7) we rotate the point P(x,y,z) to a point Q(x',y',z') in the same coordinate system S and Eq. (A.5) says that then the value of the rotated function PR* evaluated at the new point Q is equal to the value of original function V evaluated at old point P.:, Next, suppose we have a spinor field t(r)X (T = -1/2, 1/2). Then associated T with every rotation R(qPy) there is a unitary, linear operator OR such that the transformation law is given by: I1/2/1/2 0 R(r )1 = D TT (R)*r(R r)XT (A.7) T Careful inspection of Eq. (A.7) reveals that OR can be decomposed into a product of two operators PR and QR such that each of them operate on space part and spin part of the spinor field, respectively, e.g. OR = PRQR (A.8) PR*(r)X1 = 1 (A.9) -- TI gly T Wl-r^~~~~~~~~~~~~~~~~ - U-. DX 1/2 - (12 ) "/2 QR*(r )X D / (R)*(r)XTl - T T T - ' (A.10) Wigner (p. 223) showed that both PR and QR are linear, unitary and further, they commute, i.e. 49

PRQR ' QRPR A. This separation of OR into PR and QR is possible only when the wave function is separable into product of space part and spin part which is the basic assumption of the Pauli spin theoryo In atomic problems, one is interested in a spinor field of the type n 1/2 (r)X/ Rm - T the transformation properties of which is given by OR m (r)X' = PRVKe(r)QRXT2 - 7D n (R-t (1/2) 1/2 D( )(R) (r) DT (R)X<, 'm Tm' - 'E D ((R)D (R )R n (r )XT (A.12) - I ' m ' m'T' that is, the one-electron eigenfunctions transform irreducibly under PR and QR separately, but not irreducibly under OR, One can take a linear combination of these one-electron functions (with same n and i) to get an eigenfunction of L2, 2 Lz, S, SZ, (aSMSLML > which transforms irreducibly under PR and QR, respectively, i.~e. IR LL L PR icSMsL > = D( (R) SMILM > (A.14) M L ML S M' M'M3 M S SIt is easily seen that aSML > does nottransform irreducibly under It is easily seen that IaSMSLML > does not" transform irreducibly under O 50

(L) (S) but according to the product representation D (R) - D (R). However, one can form linear combinations of I|SMSLML > by the help of the V-C coefficient and obtain eigenfunctions of L2, S2 J2, Jz which rotate irreducibly under OR, i.e. IaSLJM > = IaSMLML > < SMsLMLIJM > MSML (A.15) OR |SLJM> = )M'M(R) iaSLJM' > -M k Lastly, for an operator T which acts on the field _T(r), by requiring the invariance of the sealar product under the coordinate rotations we have (VTk~) = (ORV, ORTq) k k = (OR ORTq OR ) But we know under rotations + R ORR; < + OR. k Therefore, under the rotation, the operator T must transform according to Tk OR -T q R qO R 51

APPENDIX B PROOF OF THE WIGNER-ECKART THEOREM The validity of this theorem depehds essentially only on the k properties of the wave function |aJM > and of the operator T. q Write J = ICJM > in the following. One assumes Z ' (J)M'M OR' RM',) MI rotational (B.1) and ORTq R = (R) (B.2) q ~~k OiJ k a'J' Since< <aJM T l'J'M' > T (M, T M, ) is just a number, it is invariant under rotation, i.e. -- k W'J' I - (IM, Tq ' ) C-J k Oa'J' (ORM, ORTq ) q M' c J k - t1 /tJ' (OR ORT 0R OR M = mM (R)D (R)DM < aJmlT k'Jm' > by (II-1) and (II-2) m,q 'm' The product of two representations can be decomposed into sum of irreducible representations by the V-C coefficient (Edmonds, 4.3.1), i.e. k+J' Dq,(R)D (R') <= kj kq J'm' IN q'+m' >D N mq+M' kqJ'M DqTgJR )DmIMI(R)=7 ><N~q+M'jkqj'M' (B.3) N=lk-J' I Thus 53

I = ) < kq'J'm' N,q'+m' > < N,q+M'kqJ'M' > N mq'm' k <aJm |Tq, ''m' m> q q (B. ) q'+m', rnM '. Integrate both sides of Eq. (B.4) with respect to all rotations R(ay7), i.e. /dIR a= / du sin PdP d'y (ca,,y Euler angles) We have IdR -= Z < kq'J'm' N,q'+m' > < lNq+MI' qJM' >< cJM|T, Ja'J'M' >, -j IdR:.-..'. > N mq'mt x WD (,M R (B.5) mM q'+m',q+M' But from the orthogonality properties of D( )s, we have J ( qJ+m'q+MdR M q'+m' 8Mq+M' -,N (B.6) and IdR = tI. (B.7) N m~qtm..q+mM,q+M 6j:ifm, q'+m' MM, q+M' ><J_, ' 'kqJM~~ q1 2J+1 q mt = Z < kq'J'm' J'' Jq'+m' >< | +'kJ'm' > < cJm' > q 'm' < J'M'kqlJM > V 1+ k 1 (2J+1)M' 2, (2J+l2 < J'm'kq 'Jq'+m' > < oJq '+m' T c'J'm' > q'm' <J'M'kqJM >- (2J+l)1/2 54

where (IJITkll'J) -(2J+l)/- X < J'm k'q' IJ,q'+m' > < aJq'+m' IT I'Ja m1 > (aJIITkIc J(2J+1)12 qqm ' q 'm ' is independent of M, M' and q. It may be remarked that this theorem holds whenever T rotates like Eq. q (B.2). Therefore, even if Tq is a one-electron operator and |aJM > is a manyelectron wave function, the theorem is also true. 55

APPENDIX C DEFINITIONS AND PROPERTIES OF 3-j, 6-j, and 9-j SYMBOLS 1. 1 3-J'S:YMLBOL a. Definition: j1 2 j) lm m2 m j l-j2-m (-1) -< > < j< imj2m2.1Jm > (C.1) b. Properties: (1) (m1 j2 ji) = o ml m2 m A(jlj2j) unless mln+m2+m = 0 j +j2+j = integer (C.2) (2) Even permutations of columns leave the numerical value unchanged, i.e. lml j2 J3\ mi m2 m3 m32 j3 l\ \m2 13 ml/ \m3 ml m2 (c.3) Odd permutation is equivalent to multiplication by (-l)jl+j2+j3 ijl j2 j33 ml m2 m3 jl+j2+j3 ( -1+3+33) (3 J2 Jil Im3 nm2 M j2 jl j3\ \m2 mi m3) J l j3 32 - \ml m3 m2 (3) Simultaneous change of signs of m's multiplies the ( -1 )+J2+j3 i.e. numerical value by (C.4) /l1 JS J32 zml m2 mI = (_l)Jl+j2+j3 (31 2 j31 \m1 ms i3/ (4) Orthogonality properties:,.1 (2j+l) (m jm 32 j ( 32 m t t I I2 M II n2 ~.\. 'I.. m ).Mipl B Mmma -.', (c.5) Z bJl j2 A, >m m2 mm m Im2 il j2 \ml m2 'm ) = 5(j12j) jj I' am' Inm 2j+l (C.6) where 5(jlj2j) = 1 if A(jlj2j) and zero otherwise. (5) Specialized formulas: t1i m2 O/ M l ms 0 ('-1) ) l(2j1 /2 5jlj2 5mlm2 57 (c.7)

/i J J% 1\4 j-M ~ J J l0) = ((-)J) 3M2 _ J(J+1) (C.y) (JJ2M J-M 3M2 2 \MM =0/ ( [(2J+3)(J+l)(2J+l)J(2J-1)]7/2 (J J32 33\) )-J 2 (jl+i2-j3)!(jl+j3- j2)! (j2+j3 ) 1/23(\ j jl_ vo / _ _ _ _ _ _ _ - _ _ _J__ _ _ _ _)-(__ _)_ (_3 )_ 0 0 (.) L (jl+j2+j3+1)!I( 2 2 2 if jl + j2 + j3 = J = even (c.8) 0 if ji + j2 + j3 = J = odd 2. 6-j SYMBOL a. Definition: The 6-j symbol is defined in terms of the recoupling coefficient of three angular momenta, j, j2, and j3 and is in turn expressible as a sum of products of four V-C coefficients, ie. r. 2 +j2 j121j2+j3 1/2 3132 2 = (-) l+2+3+(2j12+l)(2j23+l). <(jlj2)jl2,j3,JIjl,(j2j3)J23,J> j3 J 323j = l+j2+j32+J J ( -1/2 =- (-1) ++J[(2j 12+1)(2j23+1 ) ] X 7 < j1mlj2m2Ijl2ml+m2 ><$L2ml+m2j3M-ml-m2IJM > x mlm2 < J2m2j3M —m2-m j23sM-ml > < jlmlj23 M-ml JM > (C.9) or in terms of 3-j symbol, a more symmetrical expression: Jl J2 J 2j3 + 1 2 M3 M1M2M3: (C.10) mim2 MJa J3 il\ /J3 J. \. ja2 j3 M2 M3 mI \M3 M1 M2\ml m2m 3 58&

Relation to Racah coefficient: 6-j symbol is identical to the Racah coefficient within a phase factor: J1 J2 ji2 Jl+j2+J+j3 32 J12K= (- l)l +J+3 W(jlJj3;jl2j23) (C.ll) 33 J j23J b. Properties: (1) Symmetry properties: (a) 6-j symbol is invariant under any permutation of the columns~ (b) 6-j symbol is invariant against interchange of the upper and lower arguments in each of any two columns. For example: Jl Ji2 J23 r p J5 M6 j4 j5 6 4j4 j2 j3f (a) and (b) together consist of twenty-four operations which leave 6-j symbol invariant. &(c) 2 3 = O unless A(jlj2j3), A(jl2I3),A(28j2j3),A(2122j3) (2) Two important relations between 3-j.and 6-j symbols. (a) ( )j2+J2-ml-M1 l J2 J3 (J3 j2 J1 _ ( S. + 2 Jl x( 2 J3 (3 J2 J1 mi M2 M sAM3 m2 M1/ - 1lJ J2 Ji M m2m/ 3 M2 M21 j3 (C.12) (b), Jl+J2+J3+Ml+M2+M3 Jl: J2 J3\ J1 j2 J3 fJi J2 j3 MM2 3 1 m2 M3kMM2M3 2 (il j2 J3\fjl j2 j3l (C.15) -\ml m2m 3 131 J (C.13)J3J km1 j2 m3j ) 1 j2 J 59

(3) Orthogonality property. ' j' l J2 3 J3 J2 j (2j+l)(2j'+l) j3 j4 j i j4 = j'j" (C.14) J that is 4(2j+l)(2j'+l) 1 2 j forms a real orthogonal 3 j4 J matrix, with rows and columns rabelled by j and j' respectively. (4) Two particular values of the 6-j symbol. [J s 4 ( ) J+S+L j(J+l)- s(S+1) L(L+1) 2 [S(S+l)(2s+l)L(L+1)(2L'+1) ]1 (c.15) (_) J+S+L ( 3X(X-1) - 4S(S+l)L(L+l) L S 2 2[S(S+l)(2S-1)(2S+1)(2S+3)L(L+1)(2L-1)(2L+l)(2L+ ) ] where X = S(S+1)(+ L(L+1) - J(J+l) (C.16) 3. 9-j SYMBOL a. Definition: The 9-j- symbol is defined in terms of the recoupling coefficient of four angular momenta: jl i2 12 -1/ 31 j3 j j34 = [2 12+l)(2j34+l)(2J l1)(2J24+1)1 X,, 13i24i (C.17) < (jl 2)l2,(m o34)J34fJ (Jlj3)Jl,(J2J4)J24,f > or in terms of 6-j symbols. 21 j2 J2 J23 = (-1) (2K+1) Js Js j i J31 J32 J33j K ' J (C.18) or in terms of 3-j symbols, 60

jl j2 3J2f j3 j4 J34 LJ13J24J J /J13 J24 J) \M13 M24 M =1 (s mlm2m3m4 m12M34 j2 J12 /j3 j4 m2 M12 i 3 m4 J34 hi M34 \Ml J3 J13\ m3 M13) (C.19)..~~~ X J12 J34 J\ \M12 M34 M/ b. Properties: Look upon rJll j12 jl3l 21 j22 j23. LJ31 32 33J as a matrix. (1) Odd permutation of rows or columns produces sign change of (-1)ll+jl2+jl3s+j21+j22+j23+j31+j32+j33 (2) Even permutation or a transposition (with respect to both diagonals) leaves the symbol unchanged. (3) Special value of the 9-j symbol..a b el c de f f b+c+e+f (- -1) 2a fb ec {i21l) (2f +1 (c.20)

APPENDIX D NUMERICAL TABLES FOR: (dncSLiU( K)Id. S' L' ), AND (dnoaSLIVlklldna'S'L'), n = 2,4, k = 2,4 k = 1,2 n = 2,5 TABLE D-l* (d2LS IIU( (2) (lidZ'L'S ' ) 'S'L'F f5. 1 1 cS^ 2F2^. 2 22 O2S 2 5 5 - 32 Q soorP~ ~ ~ ~ ~ ~ ~ j5 12 2 2 7 35 5 1 12 5 3 2 G ~35 'i5 7 ' 1 2 OS O o 0,0 - \1 *Taken from Slater II, Appendix... 26. 63

TABLE D-2 (daLS I U (2) l &a 'L'S' ) ""'~.........3 "3 1 1 I1 2tS'L' l' l ' F 3p | D G S oSL^ - 2 2 2 0 3: F11 0 O O 2 5 55 1 2 0 0 0 " TABLE D-3* (dJas LI5U( )(2)i||d5 'S'L') '" StL' 6 4 4 G4 G p D F 56 s o o o o o 4 5 0 0 0 0 O 5 G o o o o - _3 7 p 0 0 0 0 35 45 555 35 O ^F O 0 0 3~~~~~~~~~~~~~~ 38 "35 *Taken from Slater II, Appendix 26 64

(dCSLU( ) (2) Ild5a'S'L' ) W'S 'L' 2 1 2 2 12 1 2 2 2 22 2 I H F F D I F 5 5 S Cia~~sLz~~ 5 j5 153~~~. 5 I 35 1 5, i3 5 I I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - 2 5 0 5 5 4 35 0 0 0 0 0 0 0 0 2 H 3 2 3G 5 4 35273 0 0 6 o - 5470 2 o - 5462 4 o 42 355~ *0 2 7- 21 7 0 0 4 - 0~ r^6 0 0 0 30 3 5 t~ 0 0 2 5G 6 _5T 9 -g 3550 \Jl 2 3 F 0 0 O 0 0 0 0 0 0 2 515 6 0 O 0 0 5 0 0 0 0 0 2 F 5 2 D 1 0 0 2 345142 55 55 2 o~~7 53 5 0 0 3 - 0 7..3~ 1 0 -1530 0 0 2 -0 - 45 2 3 D 5 D D 5 0 0 6 o55 O _ W5 16 ~O 0 72 70 O O 3, 5o '7 - 0 0 -o 7 4.~. '-70 35 -0 TO 0 4 0 V\ 0 7 2 3 2 S 5 0 0 0 0 0 0 0 0 0 35 0 5 4 7 7 0 0 0 0 0 0 0 *Taken from Slater II, Appendix 26.

TABLE D-4 (dE5aSL IU4(2) IdE5a'tS 'L' ) W-a'SL 6444 4 4G 00-g O - |4 4 5 7 4 P O O O O 5 7 4 D 10 0_ _ ~7J 0 0 0:~ ~ ~~~~~~~~~ ~Ul 66

TABLE D-4' (da5asLIU4 (2) Ilda'S 'L' ) 'SL' 2 22 2 1 2 2 2 2 2 12 2 SL, 5\ I 3 H G 5G F 3F D 5 3D 5D 5s cSL 5 3 i 3i ~i5 3 553 5 i 53 I 5 o 2- 1 iS 3 2 221 1 O -t 69ji~ 0 0 17 0 0 0 12 H 3 0 0 1 13 2 7 0 _ 7 - 34 6s 7 15 o - 2 357 0 0 0 0 139 - 14 0 2G 5 1 1 o 2 2 145 3 J21 O _ 11 C 2 7 1 137 - 14 0 0 o ^[T 0 3 5 7 2? 0\ 2 3 - 6 N91 0 0 1 ~q 7 O - \ 6 6 0 0 2F 5 61 14 3 6 7 14 3 0 -411 6 0 11 1 I 14 217 21. 7 0 2 1 O t 22 | 5'3 *21 0 0 5' o _3 0 0 0 D 3 2 13 38 7 0 0 3 7 2 1 41 0 14N 3 o -11 1 ' 5 3i 14 J7 10 0 t1 2 - 0 0 2 5 211 3 7 0 0 J102 21 0 0 0 2 3 0 0 0 1 11 2 7 0 3 2s7 0 0 0 0 0 25 5 O 0 22 21 0 O 0 0 0 0 0O

TABLE D-5* (d2LIIV'PlIldia2'S'L') p =1,2 V" V12 3 2 3 F 2 F 0 3/5 6/5 3 3 2 2 3 1 F +2 G 2 2 - 35-0/10 -53 42/14 3 1 2 F+ 2 D 4 lo05/35 -35 14/10 3 P.-* 3 P J30/5 p-3 D 2 2 ' 3 1 ' P + S 1 1 2. 2G - \"l-'o0 j15/5 0 -3r53/10 0 1 1 2G-2D 1 1 2 +2D 0 0 0 0 1 1 D + S 2 0 0 0 *Owing to the limited space, the following two tables are presented in a different form. Matrix components are zero for transitions omitted from the table. When there are two signs in front of the numerical value, the lower one refers to the component taken in the opposite order.** These two tables are taken from Slater II, Appendix 26. **The relation (dnaSLI|VPq||dn 'SL' ) = (1)L+SL (d n'S'L'1VPqldnaSL) holds for all the table. 68

TABLE D-6 (d5uSL IIV1P lld5a'S 'L ' ) p = 1,2 V" 0 V12 vF _ 7 6 4 5 5 6 4 S+ 3 P 58 ^P 4 4 G + G 5 5 4 4 5 G 3F.f 0 0 - \30/7 - T 0 4 4 5 G 5 D 4 2 5 G + 5 I 4 2 5G+ H 4 2 G G 3 G 0 0 - \|110/5 -4 \/7: j2730/35 0 ~ \l15/5 0 4 5 2 G > G 5. 0 F 4 2 G- F 3 Q 4 2 5 G - 5 F 4 2 5 G + 5 D' 4 4 F - F 5Ff33 4 4 3F -+ 5D 0 0 2\1l5/7 0 - 7/5 ~ 2 /3 0 69

TABLE D-6 (Cont) 4. 4 F+ p 3 3 4 2 3F 3H 4 2 3 F+3G 4 2 3 F 5G 4 2 F~ F 4 2 F F 3 5 4 2 3F5 4 2 F+ 3 4 2 35 5 4 2 F3 P 3 3 4 4 5 D + 5D 4 4 5 D+ 3P 4 2 5 D 5 4 2 5 D+ 3F if,. 2 y 0 0 +- 2~r/3 - V/5 0 0. /15 0 ~ 521/5 0 70 4fl5/15 V12 - 2J1/5 ~ 2\ 385/55 - 9355/35 o + 7/5 0 0 2\120/755 0 + llJ\l4/35 415/7 0 A. 4\5330/35 0

TABLE D-6 (Cont) _ 2..... -,, -. V ~~~~~~~~~~~~V2........ 4 2 D5 + D 5 3 ~ 2\ 3 0 4 2 5D D5 D ') 5 4 2 D+ P 5 3 4 2 D+ 5 S 5 0 20l-5i /15 0 - 6 \3/7 0 ~ 35J30555 4 4 3 3 3 0 4 3 2 P -+ F 3 0 + 4/5 4 2 P+ D 4 2 3 3 D - 443/15 0 - 2 4l05/15 0 - 2410/5 4 3 2 P 5D 0 4 3 2 P 5 0 + 2/5 4 3P 2 + 5 S 5 Jo05 /15 0 2 2 I + I 5 5 5 2 2 T -* H 5 '- 3 2 2 5I G 5 3 0 + \390/10 0 - \858/22 0 0 71

TABLE D-6 (Cont) V v12 2 2 5 I -* 5 G 2 2 I + G3 H 5 2 2 3 H 5 2 G 2 2 2 "52 2 2 H3 35 0 0 0 4 j30,00/385 - 42,002/70 T 677/35 ~ 2 0/75 0 0 - 2\3T85/35 2 2 3 H -5F 2 2 3G3 G 5 0 0 0 3y3/70 2 2 G+ 5 G -:\i65/10 2 2 G F 3 0 0 3 3 /70 0 2 3 2 5 2 5 F 35i/i170 2 G - 1 D 0 0 2 2 3G 3 D 15 0 - 62/7 2 2 G-* D 5 0 0 2 2 5 G+ 5 G 5G^55 2 2 5 G 3 F 0 25 25/154 ~ /0lo 0 72

TABLE D-6 (Contat) V" V12 ^ < G + F 5 5 0 i 1,155/70 G 2D G G D 5 5 0 0 2 G 2 5 5 D 0 - 2J46755 2 2F 3- 3 0 - 11/10 2 F 2F 3 5 I 4J2i/6 0 2 2 F -+ D 5 1 iT 2 f105/15 0 2 F 2 35 0 ~ 2210/35 2 F D 3 5 4 3so/15 0 2 F 5 3 2. F 3 0 - 4/2 5 - 3/2 2F 2 F 5 5 2 2 F - D 5 3 0 - ~ 2357 0 F -5 2D 5 0 ~ 2pfi2/7 2 2 p 5 3 2 D 2 D 1 1 D-+ D 1 1 0 0 0 2. 2 D - D 1 3 - 2/1i6 0 73

TABLE D-6 (Conci) VI-. V12 v v 2 2 D + P 3 ~ J05/15 0 2 2 D + D 3.3 0 2 2 -D+ D 5 2 2 D +3 P 3 3 2 2 D 5 S 5 D 5D 2 2 D5 D 5. 5 7513 - '6/14 04 - J45J /55 0 0 0 0 3g/14 2 2 D~ P 5 3 2 2 D + 5 S 53 3 2 2 3 p 3 F d30/30 0 0 ~ 2507115 0 - 2J210/75 - 19114/70 0 2 P + 3 2 S 5

REFERENCES Bleaney and Stevens, Rep. Prog. Phys., 16, 108 (1953). Condon and Shortley, The Theory of Atomic Spectra (Cambridge, 1935). Abbreviated TAS. Edmonds, A. R., Angular Momentum in Quantum Mechanics (Princeton University, 1957). Most of the notations and conventions used in the text follow those of this book. Griffith, J. S. The Theory of Transition-Metal Ions (Cambridge, 1961), Appendix 2, Table A19. Innes, F. R., Phys. Rev., 91, 31 (1953). Marvin, H. H., Phys. Rev.,,71 102 (1947) Racah, G., Phys. Rev., 62, 438 (1942), II. Racah, G., Phys. Rev., 63, 367 (1943), III. Rose, M. E., Elementary Theory of Angular Momentum (Wiley, 1957). Rotenberg, M., The 3- and 6-j Symbols (MIT Technology Press, 1959). Slater, J. C., Quantum Theory of Atomic Structure, vol. II (McGraw-Hill, 1960). Sugano and Tanabe, Journ. Phys. Soc. Japan,, 753 (1954). Trees, R. E., Phys. Rev., 82, 683 (1951). Trees, R. E., Phys. Rev., 9 308 (1953). Wigner, E. P., Group Theory (Princeton, 1959). REFERENCES FOR APPENDICES Numerical Tables for 3-j, 6-j, and 9-j Symbols Biedenharn, L. C., Tables of the Racah Coefficients, ORNL 1098 (1952). W(abcd:ef) for e ~ 3, f = 0, 1/2,...8 including half-integers. 75

Howell, K. M., Tables of 9-j Symbols, Research Report 59-2, University of Southhampton Mathematics Department (1959). Obi, S. Y., et al, "Tables of the Racah Coefficients W(abcd:ef)," Annals of the Tokyo Astronomical Observatory, Second Series, III, No. 3 (1953). This is a convenient source when one prefers to use the Racah coefficient instead of the 6-j symbol. Tabulations are only for integer arguments. Rotenberg, M., The 3-j and 6-j Symbols (MIT Technology Press, 1959). This is probably the most convenient source for the practical purposes. Smith, K., Table of Wigner 9-j Symbols for Integral and Half-Integral Values of the Parameters, ANL-5860 (1958). Numerical Values of (d.nSLjIU( lidn'S 'L') and (dnaSLIIVqPIdna'S'L') Racah, G., "Theory of Complex Spectra III," Phys. Rev. 63, 367 (1943). For n = 3,4,5, k = 2, q = 1, p = 1 Slater, J. C., Quantum Theory of Atomic Structure, vol. II (1960). Appendix 26: for the value n = 2,3,4,5, k = 2, q = 1, p = 1,2. Numerical Values of Fractional Parentage Coefficients Racah, G.) "Theory of Complex Spectra III," Phys. Rev. 63, 367 (1943). (dnaSLddn-i(('S'L't)dSL):for n = 3,4,5. Slater, J. C., Quantum Theory of Atomic Structure, vol. II (1960). Appendix 27: a(dnaSL = a'S'L') for n = 3,4,5. Note: a(dncaSL;a'S'L') = 1. 76

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