* Appendix A: Spherical Tensor Formalism The geometrical aspects of the quantum-mechanical properties of molecular systems are described in terms of quantities familiar from angular momentum theory and the reader is referred to the text books existing on this subject. 1- 3 In this appendix we collect together the quantities and relations most fre­ quently used in calculations concerning the properties of solid hydrogen and other systems of interacting molecules. A.1 Spherical Harmonics The spherical harmonics Y,m(O,<jJ) are defined as the angular parts of the simultaneous eigenfunctions of the square and the :: component of the orbital angular momentum of a single particle, measured in units h (A.1) where m = I, I - 1, ... , -I and I = 0, 1,2, .... These functions are mu­ tually orthogonal and normalized on the unit sphere, (1m I I'm') == SOh S: Y;';" Y,'m,sine de d<jJ = ()ll'()mm' (A.2) The Y,m are determined uniquely by eqs. (A.1)-(A.2) up to arbitrary phase factors which we fix by choosing the so-called Condon and Shortley phases defined by the conditions that the nonvanishing matrix elements of the operators L, ± iLy and the values of the Y,o at 0 = °are real and positive (Lx ± iL)Y,m = [(I ± m + 1)(1 =+= m)]1'2y,.m±1 (A.3) and 21 + 1)1.12 Y,o(O,<jJ) = ( ~ (AA) 277 278 Appendix A • Spherical Tensor Formalism F or this choice of phases Y'tr,(8,c/J) = (-lrr;m(8,c/J) (A.5) where m == -m. Many expressions are simplified by using the Racah spherical harmonics defined as (A.6) which satisfy (A.7) where PI is the Legendre polynomial of degree I. For I = 0, ... , 4 and m > 0 these functions are given in Table A.1, and for m > 0 can be obtained from eg. (A.5). Hence explicit minus signs in the expressions for the elm and r;m occur only for positive odd values of m. The transformation law of the spherical harmonics under rotations of the coordinate frame is (A.S) m where 8,c/J and 8',c/J' are the polar angles of a given direction (unit vector) in the old and the new frame, respectively, and ~, /3, " are the Euler angles of the rotation carrying the old into the new frame. This rotation is effected by Table A.1 The Racah spherical harmonics up to I = 4. Coo = 1 C IO = cosO C 20 = t(3cos28 - 1) C 30 = lcos8(icos18 - 1) C40 = i(¥cos4 0 - lOcos28 + 1) C ll = -rtll/2sin!ie'4> C 21 = - (~)1!2sin8cosOe'4> C 31 = -i,/3sin8(5cos28 - Ije'4> C41 = -i,JSsin8cos8(1cos28 - l)e,4> C22 = t(~)1'2sin28e2i,p C 31 = HY/ 2sin18cos8e2',p C 41 = i(W 2 sin28(7cos28 - l)e2i 4> C 33 = -i.j5sin38e 3,,p 3 C43 = -iJSsin38cos!ie ',p C 44 = ~"isin 40e4,,p A.1 • Spherical Harmonics 279 first rotating the old frame x,y,.: into X 1,Y1,':1 = .: by a rotation through an angle oc around the z axis, then through an angle [3 around the )' 1 axis to produce a frame XhV2 = )'1' ':2 = .:' and finally through an angle ~' around the z' axis to produce the new frame x',y',.:'. The same transformation is obtained by rotations around the original axes, first through ~' around .:, then through [3 around y, and finally through :x around.: again. The rotatioll operator in the Hilbert space of a quantum-mechanical system with total angular momentum J. corresponding to the rotation R = (:x.[3.~') is and the rotatioll matrices are defined as D!",,(R) = (jmiD(R)ijll> = e-im~d~n([3)e-in;' (A.10) where d;m,(/3) = (jmie-i/lJ'ijll> (A.11) The rotation matrices are unitary (A.12) LD~n(R)*D!nll,(R) = b'III' m and satisfy the symmetry relations D~n(R)* = (-l)m-nD~n(R) = D~m(R-1) (A.13) where R - 1 = (-~' + n, [3, -:x + n) is the rotation inverse to R, and the orthonormality relations (" (2" (2,,"". 8n 2 Jo Jo Jo D:"n(:x[3~')* D:"·II·(:x[3·,')sin[3 d[3 d:x d~' = 2j + 1 bjj.bmm ·b,I/J' (A.14) For integer j, j = I, and Il = 0 the rotation matrices reduce to spherical harmonics (A.15) and eq. (A.8) then reduces to the spherical harmollics additioll theorem, (A.16) III where 8 is the angle between the directions specified by 8b ¢1 and 82'¢2' The matrices (A.11) have the symmetry properties d!"n([3) = (-l)m-lId~m([3) = dkm([3) (A.17) and forj = 1,2 are given in Table A.2. 280 Appendix A • Spherical Tensor Formalism Table A.2 The reduced rotation matrices d~" for j = 1,2. dl[ = cos21fJ di o = -(WI2sinfJ dbo = cosfJ dlr = sin21fJ d~2 = cos41fJ dL = 10 + cosfJ)(2cosfJ - 1) dL = -1sinfJ(1 + cosfJ) dio = _(~f2sinfJcosfJ d~o = (~)[ 2sin2fJ dh = 10 - cosfJ)(2cosfJ + 1) dh = -1sinfJ(1 - cosfJ) d~2 = sin41fJ A.2 Clebsch-Gordan Coefficients The vector-addition or Clebsch-Gordan coeffiCients are the coefficients C(i! ;2i~;m,m,m~) = <;, ;,m,m, 1;1;' ;m> (A.18) in the unitary transformation Ijd2jm) = I Ijd2mlm2)Ud2mlm2 Ulj2jm) (A.19) corresponding to the addition, J = J 1 + J 2, of the angular momenta J 1 and J 2' The unitarity implies the relations I C(jljzj;m 1m2m )C(jljzj';m1m 2m') = (jjj,(jmm' (A.20) I C(jljzj;mlmzm)C(jdzj;m~m2m) = (jm[m\(jm2m2 jm and the relation J = J I + J 2 implies C(jljzj3;m 1m Zm 3) = 0, unless m3 = m1 + m2, and l'l(jlj2j3) (A.21) where the triangle condition l'l(jlj2h) means that j3 is one of the numbers (A.22) The phases are chosen in such a way that all the coefficients (A.18) are real and (A.23) where C(jlj2j3;m 1mZ ) == C(jdzj3;m 1,mZ,ml + mz) (A.24) We note the further properties C(jjOj3;m I 0) = (jith (A.25) Sec. A.2 • Clebsch-Gordan Coefficients 281 and C(jjj2j3;OO) = O. unless 2J == jl + j2 + h is even. (A.26) For J integer, i.e.,jl + j2 + j3 even, we have J . J! C(jlhi3;OO) = (-1) - J3 (J _ JJ!-(J---j-z)-!(-J---j-'3-)! x i(2i3 + 1)(j1 + j2 - j3)!(jl - j2 + h)!( -j~+ j2 + j3)!jl 2 L (2J+l)! (A.27) where 2J = jl + jz + j3' The symmetry properties of the Clebsch-Gordan coefficients are most easily expressed in terms of the Wigner 3-j symbol defined by which is invariant under cyclic permutations ofthe columns, and is multiplied by a phase factor under noncyclic ones = = (_ l)a + b + c (a b c) (b c a) (h a '/c) . etc. (A.29) 'X f3~' f3"/ 'X f3 'X and (~ ~ ~) = (_l)a+b+C(a b c) (A.30) 'X f3 7 'X f3 7 gIvmg C(jlj2h;11l1111 2111 3) = (_I)h + h-hC(jlj2j3;ifllifl2ifz 3) 2' 1)1/2 = (_I)h+m2 ( J.3 + C('hhh .. ;11121113-111 1 ) 211+ 1 2' + 1)12 = (_ l)h +",2 ( -'1-=.3'---_ C('"hhh ;1113- 111 2ml-) 211+ 1 = (_I)h -m, (-,21-=>_+_11)1/2 C(j3jlj2;111 3ifz I 111 2) 212 + (A.31) The numerical values of some often occurring Clebsch-Gordan coefficients can be obtained from Table A.3 and eg. (A.31). More extensive tables are available.4 . 5 282 Appendix A • Spherical Tensor Formalism Table A.3 Numerical values of frequently occurring Clebsch-Gordan coefficients; C = C(/t1zl;mtmzm). 1,/ 2 1 1n 11112 m C 1,lzl m}m2m C 1 1 1 000 0 223 1 1 2 0 1 1 1 1 0 1 (W'z 223 2 1 3 (WIZ 1 12 000 (W z 224 000 3(is)12 1 1 2 1 10 (i)'z 224 iTO 2(is)1/2 1 1 2 101 (l:)' Z 224 210 (10)12 1 12 1 1 2 224 10 1 (W z 224 2 11 (i4)' 2 212 000 0 224 202 (1:,:)' 2 212 o 1 1 _(1)12 224 1 1 2 2(W z 2 I 2 101 (i)12 224 2 I 3 (WO z 212 202 (W 2 224 224 1 2 I 2 1 1 2 -(W 2 3 I 3 000 0 2 I 3 000 (W 2 3 1 3 01 1 -(W 2 2 1 3 110 (!)IZ J ! J ! 0 ! ~!~}1.'2 2 I 3 o I I (W Z 3 1 3 1 1 2 z 2 I 3 101 2(is)1 Z -1(W 3 I 3 202 Ct)t2 2 1 3 2T 1 (is)12 3 1 3 2 1 3 I 2 1 3 I 12 (W 2 -2 3 1 3 303 z 2 1 3 202 (t)1 2 1(W 2 I 3 2 I 3 3 1 4 000 2(W Z 3 I 4 1 10 (1:,:)1 Z 222 000 -(W z 3 1 4 101 1(1;)IZ 222 101 _(i4)12 3 1 4 I I (~)I 2 222 I 12 -(W 2 o 3 1 4 2T 1 1(W Z 222 202 (W 2 3 I 4 202 (W 2 223 000 0 3 1 4 1 1 2 1(1;)12 2 223 1 I 1 (W 3 I 4 312 1(W 2 (io)12 223 220 3 1 4 303 (W 2 /2 223 1 0 1 (W 314 2 1 3 1(W 2 (1o)1!2 223 2 I I 3 I 4 3 14 I 223 202 (W/2 The Clebsch-Gordan coefficients also occur in the decomposition rule for the rotation matrices, D!,:l n l(R)D!;2n2(R) = L C(j Ij2j;m 1 m2f1)C(j d2j;n 1 n2v)D{,,(R) jl1\' L C(j d2j;ml m2)C(j Ij 2j ;nln2)D~1 +mpi +1I2(R) (A.32) Sec.